Subgroups — #64243

Order 288.

64 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#157	C₂	1	2	yes	maximal
2	C₄×#157	C₂	1	1	yes	maximal, characteristic
3	#3777	C₂	1	4	yes	maximal
4	C₂×#157	C₂²	1	2	yes	—
5	C₂×#157	C₂²	1	4	yes	—
6	C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
7	C₂×C₃⋊D₃	C₂³	1	1	yes	characteristic
8	#157	—	2	2	no	—
9	#157	—	2	4	no	—
10	C₃²×C₄	Q₈	1	1	yes	characteristic
11	C₃⋊Dic₃	Q₈	1	1	yes	characteristic
12	#125	—	9	1	no	maximal, Sylow
13	C₄×D₃	—	12	1	no	—
14	C₂×C₃²	C₂×Q₈	1	1	yes	characteristic
15	C₃⋊D₃	#41	1	2	yes	—
16	C₄⋊C₄	—	9	4	no	—
17	C₄⋊C₄	—	9	2	no	—
18	C₄²	—	9	1	no	—
19	C₂×D₃	—	12	1	no	—
20	C₃×C₄	—	4	1	no	—
21	Dic₃	—	12	1	no	—
22	C₃²	#125	1	1	yes	characteristic, complemented, Sylow
23	C₂×C₄	—	9	1	no	—
24	C₂×C₄	—	9	2	no	—
25	C₂×C₄	—	9	4	no	—
26	C₂×C₃	—	4	1	no	—
27	D₃	—	12	2	no	—
28	C₂²	—	9	1	no	—
29	C₄	—	9	1	no	—
30	C₄	—	18	2	no	—
31	C₄	—	18	4	no	—
32	C₄	PSU(3,𝔽₂)	1	1	yes	characteristic
33	C₃	—	4	1	no	—
34	C₂	—	9	2	no	—
35	C₂	C₂×PSU(3,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×PSU(3,𝔽₂)
commutator, prime radical, cosocle	C₂×C₃⋊D₃	C₂³
Fitting	C₃²×C₄	Q₈
socle	C₂×C₃²	C₂×Q₈
hypercenter	C₄	PSU(3,𝔽₂)
hypocenter	C₃²	#125
LCS	G ▷ C₂×C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₄

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

N	H ≅ G/N	type
22. C₃²	#125	semidirect

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