Editing List of small groups (section)

== List of small non-abelian groups==
The numbers of non-abelian groups, by order, are counted by {{OEIS|id=A060689}}. However, many orders have no non-abelian groups. The orders for which a non-abelian group exists are
: 6, 8, 10, 12, 14, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 34, 36, 38, 39, 40, 42, 44, 46, 48, 50, ... {{OEIS|id=A060652}}

{| class="wikitable"
|+ List of all nonabelian groups up to order 31
|-
! Order
! Id.{{efn|name=id}}
! G<sub>''o''</sub><sup>''i''</sup>
! Group
! Non-trivial proper subgroups<ref name= Dockchitser >{{cite web |last=Dockchitser |first=Tim |title=Group Names |url=https://people.maths.bris.ac.uk/~matyd/GroupNames/ |access-date=23 May 2023}}</ref>
! [[Cycle graph (algebra)|Cycle <br />graph]]
! Properties
|-
! 6
! 7
! G<sub>6</sub><sup>1</sup>
| D<sub>6</sub> = S<sub>3</sub> = Z<sub>3</sub> ⋊ Z<sub>2</sub>
| Z<sub>3</sub>, Z<sub>2</sub> (3) 
| [[Image:GroupDiagramMiniD6.svg|40px]]
| [[Dihedral group]], [[Dihedral group of order 6|Dih<sub>3</sub>]], the smallest non-abelian group, symmetric group, smallest [[Frobenius group]].
|-
! rowspan="2" | 8
! 12
! G<sub>8</sub><sup>3</sup>
| D<sub>8</sub>
| Z<sub>4</sub>, Z<sub>2</sub><sup>2</sup> (2), Z<sub>2</sub> (5)
| [[Image:GroupDiagramMiniD8.svg|40px]]
| Dihedral group, [[dihedral group of order 8|Dih<sub>4</sub>]]. [[Extraspecial group]]. [[Nilpotent group|Nilpotent]]. 
|-
! 13
! G<sub>8</sub><sup>4</sup>
| Q<sub>8</sub>
| Z<sub>4</sub> (3), Z<sub>2</sub>
| [[Image:GroupDiagramMiniQ8.svg|40px]]
| [[Quaternion group]], [[Hamiltonian group]] (all subgroups are [[normal subgroup|normal]] without the group being abelian). The smallest group ''G'' demonstrating that for a normal subgroup ''H'' the [[quotient group]] ''G''/''H'' need not be isomorphic to a subgroup of ''G''.  [[Extraspecial group]]. [[Dicyclic group|Dic<sub>2</sub>]],<ref name="ChenTang2020">{{cite journal|last1=Chen|first1=Jing|last2=Tang|first2=Lang|title=The Commuting Graphs on Dicyclic Groups|journal=Algebra Colloquium|volume=27|issue=4|year=2020|pages=799–806|issn=1005-3867|doi=10.1142/S1005386720000668|s2cid=228827501}}</ref> [[Binary dihedral group]] <2,2,2>.<ref name=anglebracket>{{cite book |last = Coxeter |first = H. S. M. |title = Generators and relations for discrete groups |publisher = Springer |location = Berlin |year = 1957 |isbn = 978-3-662-23654-3 |doi=10.1007/978-3-662-25739-5 |quote=<l,m,n>: R<sup>l</sup>=S<sup>m</sup>=T<sup>n</sup>=RST}}:</ref> Nilpotent.
|-
! 10
! 17
! G<sub>10</sub><sup>1</sup>
| D<sub>10</sub>
| Z<sub>5</sub>, Z<sub>2</sub> (5)
| [[Image:GroupDiagramMiniD10.svg|40px]]
| Dihedral group, Dih<sub>5</sub>, Frobenius group.
|-
! rowspan="3" | 12
! 20
! G<sub>12</sub><sup>1</sup>
| style="white-space:nowrap;" | Q<sub>12</sub> = Z<sub>3</sub> ⋊ Z<sub>4</sub>
| Z<sub>2</sub>, Z<sub>3</sub>, Z<sub>4</sub> (3), Z<sub>6</sub>
| [[Image:GroupDiagramMiniX12.svg|40px]]
| [[Dicyclic group]] Dic<sub>3</sub>, Binary dihedral group, <3,2,2>.<ref name=anglebracket/>
|-
! 22
! G<sub>12</sub><sup>3</sup>
| A<sub>4</sub> = K<sub>4</sub> ⋊ Z<sub>3</sub> = (Z<sub>2</sub> × Z<sub>2</sub>) ⋊ Z<sub>3</sub>
| Z<sub>2</sub><sup>2</sup>, Z<sub>3</sub> (4), Z<sub>2</sub> (3)
| [[Image:GroupDiagramMiniA4.svg|40px]]
| [[Alternating group]]. No subgroups of order 6, although 6 divides its order. Smallest Frobenius group that is not a dihedral group.<br />Chiral [[tetrahedral symmetry]] (T).
|-
! 23
! G<sub>12</sub><sup>4</sup>
| D<sub>12</sub> = D<sub>6</sub> × Z<sub>2</sub>
| Z<sub>6</sub>, D<sub>6</sub> (2), Z<sub>2</sub><sup>2</sup> (3), Z<sub>3</sub>, Z<sub>2</sub> (7)
| [[Image:GroupDiagramMiniD12.svg|40px]]
| Dihedral group, Dih<sub>6</sub>, product.
|-
! 14
! 26
! G<sub>14</sub><sup>1</sup>
| D<sub>14</sub>
| Z<sub>7</sub>, Z<sub>2</sub> (7)
| [[Image:GroupDiagramMiniD14.svg|40px]]
| Dihedral group, Dih<sub>7</sub>, Frobenius group.
|-
! rowspan="9" | 16<ref>{{cite journal|journal=Am. Math. Mon.|first1=Marcel|last1=Wild|url=http://math.sun.ac.za/~wild/Marcel%20Wild%20-%20Home%20Page_files/Groups16AMM.pdf|title=The Groups of Order Sixteen Made Easy|doi=10.1080/00029890.2005.11920164|year=2005|pages=20–31|volume=112|number=1|jstor=30037381|s2cid=15362871 |archive-url=https://web.archive.org/web/20060923012610/http://math.sun.ac.za/~wild/Marcel%20Wild%20-%20Home%20Page_files/Groups16AMM.pdf |archive-date=2006-09-23 }}</ref>
! 31
! G<sub>16</sub><sup>3</sup>
| K<sub>4</sub> ⋊ Z<sub>4</sub>
| Z<sub>2</sub><sup>3</sup>, Z<sub>4</sub> × Z<sub>2</sub> (2), Z<sub>4</sub> (4), Z<sub>2</sub><sup>2</sup> (7), Z<sub>2</sub> (7)
| [[Image:GroupDiagramMiniG44.svg|40px]]
| Has the same number of elements of every [[order (group theory)|order]] as the Pauli group. Nilpotent. 
|-
! 32
! G<sub>16</sub><sup>4</sup>
| Z<sub>4</sub> ⋊ Z<sub>4</sub>
| Z<sub>4</sub> × Z<sub>2</sub> (3), Z<sub>4</sub> (6), Z<sub>2</sub><sup>2</sup>, Z<sub>2</sub> (3)
| [[Image:GroupDiagramMinix3.svg|40px]]
| The squares of elements do not form a subgroup. Has the same number of elements of every order as Q<sub>8</sub> × Z<sub>2</sub>. Nilpotent.
|-
! 34
! G<sub>16</sub><sup>6</sup>
| Z<sub>8</sub> ⋊ Z<sub>2</sub>
| Z<sub>8</sub> (2), Z<sub>4</sub> × Z<sub>2</sub>, Z<sub>4</sub> (2), Z<sub>2</sub><sup>2</sup>, Z<sub>2</sub> (3)
| [[File:GroupDiagramMOD16.svg|40px]]
| Sometimes called the [[Iwasawa group|modular group]] of order 16, though this is misleading as abelian groups and Q<sub>8</sub> × Z<sub>2</sub> are also modular. Nilpotent.
|-
! 35
! G<sub>16</sub><sup>7</sup>
| D<sub>16</sub>
| Z<sub>8</sub>, D<sub>8</sub> (2), Z<sub>2</sub><sup>2</sup> (4), Z<sub>4</sub>, Z<sub>2</sub> (9)
| [[Image:GroupDiagramMiniD16.svg|40px]]
| Dihedral group, Dih<sub>8</sub>. Nilpotent.
|-
! 36
! G<sub>16</sub><sup>8</sup>
| QD<sub>16</sub>
| Z<sub>8</sub>, Q<sub>8</sub>, D<sub>8</sub>, Z<sub>4</sub> (3), Z<sub>2</sub><sup>2</sup> (2), Z<sub>2</sub> (5)
| [[Image:GroupDiagramMiniQH16.svg|40px]]
| The order 16 [[quasidihedral group]]. Nilpotent.
|-
! 37
! G<sub>16</sub><sup>9</sup>
| Q<sub>16</sub>
| Z<sub>8</sub>, Q<sub>8</sub> (2), Z<sub>4</sub> (5), Z<sub>2</sub>
| [[Image:GroupDiagramMiniQ16.svg|40px]]
| [[Generalized quaternion group]], Dicyclic group Dic<sub>4</sub>, binary dihedral group, <4,2,2>.<ref name=anglebracket/> Nilpotent.
|-
! 39
! G<sub>16</sub><sup>11</sup>
| D<sub>8</sub> × Z<sub>2</sub>
| D<sub>8</sub> (4), {{nowrap|Z<sub>4</sub> × Z<sub>2</sub>}}, Z<sub>2</sub><sup>3</sup> (2), Z<sub>2</sub><sup>2</sup> (13), Z<sub>4</sub> (2), Z<sub>2</sub> (11)
| [[Image:GroupDiagramMiniC2D8.svg|40px]]
| Product. Nilpotent.
|-
! 40
! G<sub>16</sub><sup>12</sup>
| Q<sub>8</sub> × Z<sub>2</sub>
| Q<sub>8</sub> (4), Z<sub>4</sub> × Z<sub>2</sub> (3), Z<sub>4</sub> (6), Z<sub>2</sub><sup>2</sup>, Z<sub>2</sub> (3)
| [[Image:GroupDiagramMiniC2Q8.svg|40px]]
| [[Hamiltonian group]], product. Nilpotent.
|-
! 41
! G<sub>16</sub><sup>13</sup>
| (Z<sub>4</sub> × Z<sub>2</sub>) ⋊ Z<sub>2</sub>
| Q<sub>8</sub>, D<sub>8</sub> (3), Z<sub>4</sub> × Z<sub>2</sub> (3), Z<sub>4</sub> (4), Z<sub>2</sub><sup>2</sup> (3), Z<sub>2</sub> (7)
| [[Image:GroupDiagramMiniC2x2C4.svg|40px]]
| The [[Pauli group]] generated by the [[Pauli matrix|Pauli matrices]]. Nilpotent.
|-
! rowspan="3" | 18
! 44
! G<sub>18</sub><sup>1</sup>
| D<sub>18</sub>
| Z<sub>9</sub>, D<sub>6</sub> (3), Z<sub>3</sub>, Z<sub>2</sub> (9)
| [[File:GroupDiagramMiniD18.png|40px]] || Dihedral group, Dih<sub>9</sub>, Frobenius group.
|-
! 46
! G<sub>18</sub><sup>3</sup>
| Z<sub>3</sub> ⋊ Z<sub>6</sub> = D<sub>6</sub> × Z<sub>3</sub> = S<sub>3</sub> × Z<sub>3</sub>
| Z<sub>3</sub><sup>2</sup>, D<sub>6</sub>, Z<sub>6</sub> (3), Z<sub>3</sub> (4), Z<sub>2</sub> (3)
| [[File:GroupDiagramMiniC3D6.png|40px]] || Product.
|-
! 47
! G<sub>18</sub><sup>4</sup>
| (Z<sub>3</sub> × Z<sub>3</sub>) ⋊ Z<sub>2</sub>
| Z<sub>3</sub><sup>2</sup>, D<sub>6</sub> (12), Z<sub>3</sub> (4), Z<sub>2</sub> (9)
| [[File:GroupDiagramMiniG18-4.png|40px]] || Frobenius group.
|-
! rowspan="3" | 20
! 50
! G<sub>20</sub><sup>1</sup>
| Q<sub>20</sub>
| Z<sub>10</sub>, Z<sub>5</sub>, Z<sub>4</sub> (5), Z<sub>2</sub>
| [[File:GroupDiagramMiniQ20.png|40px]] || Dicyclic group Dic<sub>5</sub>, Binary dihedral group, <5,2,2>.<ref name=anglebracket/>
|-
! 52
! G<sub>20</sub><sup>3</sup>
| Z<sub>5</sub> ⋊ Z<sub>4</sub>
| D<sub>10</sub>, Z<sub>5</sub>, Z<sub>4</sub> (5), Z<sub>2</sub> (5)
| [[File:GroupDiagramMiniC5semiprodC4.png|40px]] || Frobenius group.
|-
! 53
! G<sub>20</sub><sup>4</sup>
| D<sub>20</sub> = D<sub>10</sub> × Z<sub>2</sub>
| Z<sub>10</sub>, D<sub>10</sub> (2), Z<sub>5</sub>, Z<sub>2</sub><sup>2</sup> (5), Z<sub>2</sub> (11)
| [[File:GroupDiagramMiniD20.png|40px]] || Dihedral group, Dih<sub>10</sub>, product.
|-
! 21
! 55
! G<sub>21</sub><sup>1</sup>
| Z<sub>7</sub> ⋊ Z<sub>3</sub> || Z<sub>7</sub>, Z<sub>3</sub> (7) || [[File:Frob21 cycle graph.svg|40px]] || Smallest non-abelian group of [[parity (mathematics)|odd]] order. Frobenius group.
|-
! 22
! 57
! G<sub>22</sub><sup>1</sup>
| D<sub>22</sub>
| Z<sub>11</sub>, Z<sub>2</sub> (11)
| 
| Dihedral group Dih<sub>11</sub>, Frobenius group.
|-
! rowspan="12" | 24
! 60
! G<sub>24</sub><sup>1</sup>
| Z<sub>3</sub> ⋊ Z<sub>8</sub>
| Z<sub>12</sub>, Z<sub>8</sub> (3), Z<sub>6</sub>, Z<sub>4</sub>, Z<sub>3</sub>, Z<sub>2</sub>
| [[File:Cycle graph Z3xiZ8.svg|40px]] || Central extension of ''S''<sub>3</sub>.
|-
! 62
! G<sub>24</sub><sup>3</sup>
| [[Special linear group|SL]](2,3) = Q<sub>8</sub> ⋊ Z<sub>3</sub>
| Q<sub>8</sub>, Z<sub>6</sub> (4), Z<sub>4</sub> (3), Z<sub>3</sub> (4), Z<sub>2</sub>
| [[File:SL(2,3); Cycle graph.svg|40px]] || [[Binary tetrahedral group]], [[Binary tetrahedral group|2T]] = <3,3,2>.<ref name=anglebracket/>
|-
! 63
! G<sub>24</sub><sup>4</sup>
| Q<sub>24</sub> = Z<sub>3</sub> ⋊ Q<sub>8</sub>
| Z<sub>12</sub>, Q<sub>12</sub> (2), Q<sub>8</sub> (3), Z<sub>6</sub>, Z<sub>4</sub> (7), Z<sub>3</sub>, Z<sub>2</sub>
| [[File:GroupDiagramMiniQ24.png|40px]] || Dicyclic group Dic<sub>6</sub>, Binary dihedral, <6,2,2>.<ref name=anglebracket/>
|-
! 64
! G<sub>24</sub><sup>5</sup>
| D<sub>6</sub> × Z<sub>4</sub> = S<sub>3</sub> × Z<sub>4</sub>
| Z<sub>12</sub>, D<sub>12</sub>, Q<sub>12</sub>, Z<sub>4</sub> × Z<sub>2</sub> (3), Z<sub>6</sub>, D<sub>6</sub> (2), Z<sub>4</sub> (4), Z<sub>2</sub><sup>2</sup> (3), Z<sub>3</sub>, Z<sub>2</sub> (7)
| || Product.
|-
! 65
! G<sub>24</sub><sup>6</sup>
| D<sub>24</sub>
| Z<sub>12</sub>, D<sub>12</sub> (2), D<sub>8</sub> (3), Z<sub>6</sub>, D<sub>6</sub> (4), Z<sub>4</sub>, Z<sub>2</sub><sup>2</sup> (6), Z<sub>3</sub>, Z<sub>2</sub> (13)
| || Dihedral group, Dih<sub>12</sub>.
|-
! 66
! G<sub>24</sub><sup>7</sup>
| Q<sub>12</sub> × Z<sub>2</sub> = Z<sub>2</sub> × (Z<sub>3</sub> ⋊ Z<sub>4</sub>)
| Z<sub>6</sub> × Z<sub>2</sub>, Q<sub>12</sub> (2), Z<sub>4</sub> × Z<sub>2</sub> (3), Z<sub>6</sub> (3), Z<sub>4</sub> (6), Z<sub>2</sub><sup>2</sup>, Z<sub>3</sub>, Z<sub>2</sub> (3)
| || Product.
|-
! 67
! G<sub>24</sub><sup>8</sup>
| (Z<sub>6</sub> × Z<sub>2</sub>) ⋊ Z<sub>2</sub> = Z<sub>3</sub> ⋊ D<sub>8</sub>
| Z<sub>6</sub> × Z<sub>2</sub>, D<sub>12</sub>, Q<sub>12</sub>, D<sub>8</sub> (3), Z<sub>6</sub> (3), D<sub>6</sub> (2), Z<sub>4</sub> (3), Z<sub>2</sub><sup>2</sup> (4), Z<sub>3</sub>, Z<sub>2</sub> (9)
| || Double cover of dihedral group.
|-
! 69
! G<sub>24</sub><sup>10</sup>
| D<sub>8</sub> × Z<sub>3</sub>
| Z<sub>12</sub>, Z<sub>6</sub> × Z<sub>2</sub> (2), D<sub>8</sub>, Z<sub>6</sub> (5), Z<sub>4</sub>, Z<sub>2</sub><sup>2</sup> (2), Z<sub>3</sub>, Z<sub>2</sub> (5)
| || Product. Nilpotent.
|-
! 70
! G<sub>24</sub><sup>11</sup>
| Q<sub>8</sub> × Z<sub>3</sub>
| Z<sub>12</sub> (3), Q<sub>8</sub>, Z<sub>6</sub>, Z<sub>4</sub> (3), Z<sub>3</sub>, Z<sub>2</sub>
| || Product. Nilpotent.
|-
! 71
! G<sub>24</sub><sup>12</sup>
| S<sub>4</sub>
| A<sub>4</sub>, D<sub>8</sub> (3), D<sub>6</sub> (4), Z<sub>4</sub> (3), Z<sub>2</sub><sup>2</sup> (4), Z<sub>3</sub> (4), Z<sub>2</sub> (9)<ref>{{Cite web|url=https://groupprops.subwiki.org/wiki/Subgroup_structure_of_symmetric_group:S4|title = Subgroup structure of symmetric group:S4 - Groupprops}}</ref>
| [[File:Symmetric group 4; cycle graph.svg|40px]] || Symmetric group. Has no normal [[Sylow subgroup]]s. Chiral [[octahedral symmetry]] (O), Achiral [[tetrahedral symmetry]] (T<sub>d</sub>).
|-
! 72
! G<sub>24</sub><sup>13</sup>
| A<sub>4</sub> × Z<sub>2</sub>
| A<sub>4</sub>, Z<sub>2</sub><sup>3</sup>, Z<sub>6</sub> (4), Z<sub>2</sub><sup>2</sup> (7), Z<sub>3</sub> (4), Z<sub>2</sub> (7)
| [[File:GroupDiagramMiniA4xC2.png|40px]] || Product. [[Pyritohedral symmetry]] (T<sub>h</sub>).
|-
! 73
! G<sub>24</sub><sup>14</sup>
| D<sub>12</sub> × Z<sub>2</sub>
| Z<sub>6</sub> × Z<sub>2</sub>, D<sub>12</sub> (6), Z<sub>2</sub><sup>3</sup> (3), Z<sub>6</sub> (3), D<sub>6</sub> (4), Z<sub>2</sub><sup>2</sup> (19), Z<sub>3</sub>, Z<sub>2</sub> (15)
| || Product.
|-
! 26
! 77
! G<sub>26</sub><sup>1</sup>
| D<sub>26</sub>
| Z<sub>13</sub>, Z<sub>2</sub> (13)
| || Dihedral group, Dih<sub>13</sub>, Frobenius group.
|-
! rowspan=2|27
! 81
! G<sub>27</sub><sup>3</sup>
| Z<sub>3</sub><sup>2</sup> ⋊ Z<sub>3</sub>
| Z<sub>3</sub><sup>2</sup> (4), Z<sub>3</sub> (13)
| || All non-trivial elements have order 3. [[Extraspecial group]]. Nilpotent.
|-
! 82
! G<sub>27</sub><sup>4</sup>
| Z<sub>9</sub> ⋊ Z<sub>3</sub>
| Z<sub>9</sub> (3), Z<sub>3</sub><sup>2</sup>, Z<sub>3</sub> (4)
| || [[Extraspecial group]]. Nilpotent.
|-
! rowspan=2|28
! 84
! G<sub>28</sub><sup>1</sup>
| Z<sub>7</sub> ⋊ Z<sub>4</sub>
| Z<sub>14</sub>, Z<sub>7</sub>, Z<sub>4</sub> (7), Z<sub>2</sub>
| || Dicyclic group Dic<sub>7</sub>, Binary dihedral group, <7,2,2>.<ref name=anglebracket/>
|-
! 86
! G<sub>28</sub><sup>3</sup>
| D<sub>28</sub> = D<sub>14</sub> × Z<sub>2</sub>
| Z<sub>14</sub>, D<sub>14</sub> (2), Z<sub>7</sub>, Z<sub>2</sub><sup>2</sup> (7), Z<sub>2</sub> (9)
| || Dihedral group, Dih<sub>14</sub>, product.
|-
! rowspan=3|30
! 89
! G<sub>30</sub><sup>1</sup>
| D<sub>6</sub> × Z<sub>5</sub>
| Z<sub>15</sub>, Z<sub>10</sub> (3), D<sub>6</sub>, Z<sub>5</sub>, Z<sub>3</sub>, Z<sub>2</sub> (3)
| || Product.
|-
! 90
! G<sub>30</sub><sup>2</sup>
| D<sub>10</sub> × Z<sub>3</sub>
| Z<sub>15</sub>, D<sub>10</sub>, Z<sub>6</sub> (5), Z<sub>5</sub>, Z<sub>3</sub>, Z<sub>2</sub> (5)
| || Product.
|-
! 91
! G<sub>30</sub><sup>3</sup>
| D<sub>30</sub>
| Z<sub>15</sub>, D<sub>10</sub> (3), D<sub>6</sub> (5), Z<sub>5</sub>, Z<sub>3</sub>, Z<sub>2</sub> (15)
| || Dihedral group, Dih<sub>15</sub>, Frobenius group.
|}