# Table of Small Finite Groups

## Order 1: 1 Group

 Abelian Cyclic Simple ℤ1 = S1 = A2 Y Y Y

## Order 2: 1 Group

 Abelian Cyclic Simple ℤ2 = S2 Y Y Y

## Order 3: 1 Group

 Abelian Cyclic Simple ℤ3 = A3 Y Y Y

## Order 4: 2 Groups

 Abelian Cyclic Simple ℤ4 Y Y ℤ22 = K4 Y

## Order 5: 1 Group

 Abelian Cyclic Simple ℤ5 Y Y Y

## Order 6: 2 Groups

 Abelian Cyclic Simple ℤ6 = ℤ3 × ℤ2 Y Y Dih3 = S3

## Order 7: 1 Group

 Abelian Cyclic Simple ℤ7 Y Y Y

## Order 8: 5 Groups

 Abelian Cyclic Simple ℤ8 Y Y ℤ4 × ℤ2 Y ℤ23 Y Dih4 Dic2 = Q8

## Order 9: 2 Groups

 Abelian Cyclic Simple ℤ9 Y Y ℤ3 × ℤ3 Y

## Order 10: 2 Groups

 Abelian Cyclic Simple ℤ10 = ℤ5 × ℤ2 Y Y Dih5

## Order 11: 1 Group

 Abelian Cyclic Simple ℤ11 Y Y Y

## Order 12: 5 Groups

 Abelian Cyclic Simple ℤ12 = ℤ4 × ℤ3 Y Y ℤ6 × ℤ2 = ℤ3 × ℤ22 Y Dic3 A4 Dih6 = Dih3 × ℤ2

## Order 13: 1 Group

 Abelian Cyclic Simple ℤ13 Y Y Y

## Order 14: 2 Groups

 Abelian Cyclic Simple ℤ14 = ℤ7 × ℤ2 Y Y Dih7

## Order 15: 1 Group

 Abelian Cyclic Simple ℤ15 = ℤ5 × ℤ3 Y Y

## Order 16: 14 Groups

 Abelian Cyclic Simple ℤ16 Y Y ℤ8 × ℤ2 Y ℤ42 Y ℤ4 × ℤ22 Y ℤ24 Y Dih8 Dic4 Dih4 × ℤ2 Dic2 × ℤ2 K4 ⋊ ℤ4 ℤ4 ⋊ ℤ4 ℤ8 ⋊ ℤ2 (ℤ4 × ℤ2) ⋊ ℤ2 QD16

## Order 17: 1 Group

 Abelian Cyclic Simple ℤ17 Y Y Y

## Order 18: 5 Groups

 Abelian Cyclic Simple ℤ18 = ℤ9 × ℤ2 Y Y ℤ6 × ℤ3 = ℤ32 × ℤ2 Y Dih9 S3 × ℤ3 (ℤ3 × ℤ3) ⋊ ℤ2

## Order 19: 1 Group

 Abelian Cyclic Simple ℤ19 Y Y Y

## Order 20: 5 Groups

 Abelian Cyclic Simple ℤ20 = ℤ5 × ℤ4 Y Y ℤ10 × ℤ2 = ℤ5 × ℤ3 × ℤ2 Y Dih10 = Dih5 × ℤ2 Dic5 ℤ5 ⋊ ℤ4

## Order 21: 2 Groups

 Abelian Cyclic Simple ℤ20 = ℤ7 × ℤ3 Y Y ℤ7 ⋊ ℤ3

## Order 22: 2 Groups

 Abelian Cyclic Simple ℤ22 = ℤ11 × ℤ2 Y Y Dih11

## Order 23: 1 Group

 Abelian Cyclic Simple ℤ23 Y Y Y

## Order 24: 14 Groups

 Abelian Cyclic Simple ℤ24 = ℤ8 × ℤ3 Y Y ℤ12 × ℤ2 = ℤ6 × ℤ4 Y ℤ6 × ℤ22 = ℤ3 × ℤ23 Y Dih12 Dih6 × ℤ2 Dih4 × ℤ3 S4 S3 × ℤ4 A4 × ℤ2 Dic6 Dic3 × ℤ2 Dic2 × ℤ3 ℤ3 ⋊ ℤ8 ℤ3 ⋊ Dih4 SL(2,3) = 2T

## Order 25: 2 Groups

 Abelian Cyclic Simple ℤ25 Y Y ℤ52 Y

## Order 26: 2 Groups

 Abelian Cyclic Simple ℤ26 = ℤ13 × ℤ2 Y Y Dih13

## Order 27: 5 Groups

 Abelian Cyclic Simple ℤ27 Y Y ℤ9 × ℤ3 Y ℤ33 Y ℤ32 ⋊ ℤ3 ℤ9 ⋊ ℤ3

## Order 28: 3 Groups

 Abelian Cyclic Simple ℤ28 = ℤ7 × ℤ4 Y Y ℤ14 × ℤ2 = ℤ7 × ℤ22 Y Dih14 Z7 ⋊ Z4

## Order 29: 1 Group

 Abelian Cyclic Simple ℤ29 Y Y Y

## Order 30: 4 Groups

 Abelian Cyclic Simple ℤ30 = ℤ15 × ℤ2 = ℤ10 × ℤ3 = ℤ6 × ℤ5 Y Y Dih15 Dih5 × ℤ3 S3 × ℤ5

## Order 31: 1 Group

(Order p)

 Abelian Cyclic Simple ℤ31 Y Y Y

## Order 32: 51 Groups

I’m not going to list these all here, but seven of them are abelian.

(All cycle graphs stolen from Wikipedia)

# Notable Groups

A5: Smallest simple group that isn’t cyclic. Order 60.

22: Smallest non-cyclic group.

Dih3: Smallest non-abelian group.

A4: First example where n divides |G| but there is no element of order n. (n = 6)

Z7 ⋊ Z3: Smallest non-abelian group of odd size.

Dih3: Smallest group with a normal subgroup that isn’t isomorphic to one of its subgroups.

# General Patterns

1 group of order p
2 groups of order p2
5 groups of order p3
15 groups of order p4, for p > 2
2 groups of order pq for q-1 divisible by p
1 group of order pq for q-1 not divisible by p