Subgroups — #624

Order 72.

37 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
2	C₃×Dic₃	C₂	1	2	yes	maximal, complemented
3	C₄×D₃	—	3	2	no	maximal
4	C₂×C₃²	C₂²	1	1	yes	characteristic
5	C₃⋊D₃	C₄	1	2	yes	complemented
6	C₂×D₃	—	3	2	no	—
7	C₂×D₃	—	6	1	no	—
8	C₃×C₄	—	3	2	no	—
9	Dic₃	D₃	1	2	yes	complemented
10	C₃²	C₂×C₄	1	1	yes	characteristic, complemented, Sylow
11	C₂×C₄	—	9	1	no	Sylow
12	C₂×C₃	—	2	1	no	—
13	C₂×C₃	C₂×D₃	1	2	yes	—
14	D₃	—	3	4	no	—
15	D₃	—	6	2	no	—
16	C₂²	—	9	1	no	—
17	C₄	—	3	2	no	—
18	C₃	—	2	1	no	—
19	C₃	C₄×D₃	1	2	yes	complemented
20	C₂	—	9	2	no	—
21	C₂	D₃²	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₃²
commutator, hypocenter	C₃²	C₂×C₄
Fitting, socle, cosocle	C₂×C₃²	C₂²
prime radical	C₂×C₃⋊D₃	C₂
LCS	G ▷ C₃²
UCS	C₁ ◁ C₂

Splittings G = N ⋊ H with N normal; H ≅ G/N.

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.