Subgroups — C₄⋊PSU(3,𝔽₂) #64245

Order 288.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×PSU(3,𝔽₂)	C₂	1	2	yes	maximal
2	C₄⋊#157	C₂	1	2	yes	maximal
3	C₄×#157	C₂	1	1	yes	maximal, characteristic
4	#3777	C₂	1	2	yes	maximal
5	C₂×#157	C₂²	1	4	yes	—
6	C₂×#157	C₂²	1	1	yes	characteristic
7	C₂×#157	C₂²	1	1	yes	characteristic
8	PSU(3,𝔽₂)	—	2	4	no	—
9	C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
10	C₂×C₃⋊D₃	C₂³	1	1	yes	characteristic
11	#157	—	2	4	no	—
12	#157	D₄	1	2	yes	—
13	#157	Q₈	1	2	yes	complemented
14	C₃²×C₄	Q₈	1	1	yes	characteristic, complemented
15	C₃⋊Dic₃	Q₈	1	1	yes	characteristic, complemented
16	C₄⋊Q₈	—	9	1	no	maximal, Sylow
17	C₄×D₃	—	12	1	no	—
18	C₂×C₃²	C₂×Q₈	1	1	yes	characteristic
19	C₃⋊D₃	C₂×D₄	1	1	yes	characteristic
20	C₃⋊D₃	C₂×Q₈	1	1	yes	characteristic
21	C₂×Q₈	—	9	2	no	—
22	C₄⋊C₄	—	9	2	no	—
23	C₄⋊C₄	—	9	2	no	—
24	C₄²	—	9	1	no	—
25	C₂×D₃	—	12	1	no	—
26	C₃×C₄	—	4	1	no	—
27	Dic₃	—	12	1	no	—
28	C₃²	C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
29	Q₈	—	18	4	no	—
30	C₂×C₄	—	9	1	no	—
31	C₂×C₄	—	9	1	no	—
32	C₂×C₄	—	9	1	no	—
33	C₂×C₄	—	9	4	no	—
34	C₂×C₃	—	4	1	no	—
35	D₃	—	12	1	no	—
36	D₃	—	12	1	no	—
37	C₂²	—	9	1	no	—
38	C₄	—	9	2	no	—
39	C₄	—	9	2	no	—
40	C₄	—	9	1	no	—
41	C₄	—	18	4	no	—
42	C₄	PSU(3,𝔽₂)	1	1	yes	characteristic, complemented
43	C₃	—	4	1	no	—
44	C₂	—	9	1	no	—
45	C₂	—	9	1	no	—
46	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	C₄⋊Q₈
LCS	G ▷ C₂×C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₄

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.