Subgroups — #3774

Order 144.

38 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#625	C₂	1	2	yes	maximal, complemented
2	#622	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×D₃	—	2	2	no	—
4	C₃⋊Dic₃	C₂²	1	1	yes	characteristic
5	#67	—	6	2	no	—
6	C₂×C₃²	D₄	1	1	yes	characteristic
7	C₃×D₃	—	4	2	no	—
8	D₈	—	9	1	no	maximal, Sylow
9	C₂²×C₃	—	6	2	no	—
10	C₂×D₃	—	2	2	no	—
11	Dic₃	—	6	2	no	—
12	C₃²	D₈	1	1	yes	characteristic, complemented, Sylow
13	D₄	—	9	2	no	—
14	C₈	—	9	1	no	—
15	C₂×C₃	—	2	2	no	—
16	C₂×C₃	—	12	2	no	—
17	D₃	—	4	2	no	—
18	C₂²	—	6	2	no	—
19	C₄	—	9	1	no	—
20	C₃	—	2	2	no	—
21	C₂	—	12	2	no	—
22	C₂	GO⁺(4,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	GO⁺(4,𝔽₂)
commutator, cosocle	C₃⋊Dic₃	C₂²
Fitting, socle	C₂×C₃²	D₄
hypocenter	C₃²	D₈
LCS	G ▷ C₃⋊Dic₃ ▷ C₂×C₃² ▷ 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.