Subgroups — #3773

Order 144.

42 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#157	C₂	1	1	yes	maximal, characteristic
2	#624	C₂	1	2	yes	maximal
3	C₂×C₃⋊D₃	C₂²	1	1	yes	characteristic
4	#157	C₄	1	2	yes	complemented
5	C₃×Dic₃	—	2	2	no	—
6	C₄×D₃	—	6	2	no	—
7	C₂×C₃²	D₄	1	1	yes	characteristic
8	C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
9	C₃⋊D₃	Q₈	1	1	yes	characteristic
10	C₄⋊C₄	—	9	1	no	maximal, Sylow
11	C₂×D₃	—	6	2	no	—
12	C₃×C₄	—	6	2	no	—
13	Dic₃	—	2	2	no	—
14	C₃²	C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
15	C₂×C₄	—	9	2	no	—
16	C₂×C₄	—	9	1	no	—
17	C₂×C₃	—	2	2	no	—
18	D₃	—	6	2	no	—
19	D₃	—	6	2	no	—
20	C₂²	—	9	1	no	—
21	C₄	—	6	2	no	—
22	C₄	—	9	2	no	—
23	C₃	—	2	2	no	—
24	C₂	—	9	1	no	—
25	C₂	—	9	1	no	—
26	C₂	GO⁺(4,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, hypercenter	C₂	GO⁺(4,𝔽₂)
commutator	C₃⋊D₃	C₂×C₄
Fitting, socle	C₂×C₃²	D₄
prime radical, cosocle	C₂×C₃⋊D₃	C₂²
hypocenter	C₃²	C₄⋊C₄
LCS	G ▷ C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₂

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

N	H ≅ G/N	type
4. #157	C₄	semidirect
14. C₃²	C₄⋊C₄	semidirect

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