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.

ℤ₂²: 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