Subgroups — C₂²⋊PSU(3,𝔽₂) #64247

Order 288.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#157	C₂	1	1	yes	maximal, characteristic
2	C₂×PSU(3,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊#157	C₂	1	2	yes	maximal
4	#3777	C₂	1	2	yes	maximal, complemented
5	#3777	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₃⋊D₃	C₂²	1	1	yes	characteristic
7	C₂×#157	—	2	2	no	—
8	C₂×#157	C₂²	1	2	yes	—
9	C₂×#157	C₂²	1	1	yes	characteristic
10	C₂×#157	C₂²	1	2	yes	—
11	C₂×#157	C₂²	1	1	yes	characteristic
12	PSU(3,𝔽₂)	—	2	2	no	—
13	C₂²×C₃²	Q₈	1	1	yes	characteristic, complemented
14	C₂×C₃⋊D₃	—	2	1	no	—
15	C₂×C₃⋊D₃	—	2	1	no	—
16	C₂×C₃⋊D₃	Q₈	1	1	yes	characteristic, complemented
17	C₂×C₃⋊D₃	C₂³	1	1	yes	characteristic
18	#157	—	2	2	no	—
19	#157	—	2	1	no	—
20	#157	—	2	2	no	—
21	#157	D₄	1	2	yes	—
22	C₂²⋊Q₈	—	9	1	no	maximal, Sylow
23	C₂²×D₃	—	12	1	no	—
24	C₂×C₃²	—	2	1	no	—
25	C₂×C₃²	C₂×Q₈	1	1	yes	characteristic
26	C₃⋊D₃	—	2	1	no	—
27	C₃⋊D₃	C₂×D₄	1	1	yes	characteristic
28	C₃⋊D₃	#41	1	1	yes	characteristic
29	C₂×Q₈	—	9	1	no	—
30	C₂²×C₄	—	9	1	no	—
31	C₄⋊C₄	—	9	1	no	—
32	C₄⋊C₄	—	9	2	no	—
33	C₂²⋊C₄	—	9	2	no	—
34	C₂²×C₃	—	4	1	no	—
35	C₂×D₃	—	12	1	no	—
36	C₂×D₃	—	12	1	no	—
37	C₂×D₃	—	12	2	no	—
38	C₂×D₃	—	12	2	no	—
39	C₃²	C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
40	C₂³	—	9	1	no	—
41	Q₈	—	18	2	no	—
42	C₂×C₄	—	9	1	no	—
43	C₂×C₄	—	9	2	no	—
44	C₂×C₄	—	9	2	no	—
45	C₂×C₄	—	9	1	no	—
46	C₂×C₄	—	18	2	no	—
47	C₂×C₃	—	4	2	no	—
48	C₂×C₃	—	4	1	no	—
49	D₃	—	12	1	no	—
50	D₃	—	12	1	no	—
51	D₃	—	12	2	no	—
52	C₂²	—	9	1	no	—
53	C₂²	—	9	1	no	—
54	C₂²	—	18	1	no	—
55	C₂²	—	18	1	no	—
56	C₂²	PSU(3,𝔽₂)	1	1	yes	characteristic, complemented
57	C₄	—	9	2	no	—
58	C₄	—	18	1	no	—
59	C₄	—	18	2	no	—
60	C₄	—	18	2	no	—
61	C₃	—	4	1	no	—
62	C₂	—	2	1	no	—
63	C₂	—	9	1	no	—
64	C₂	—	9	1	no	—
65	C₂	—	18	1	no	—
66	C₂	C₂×PSU(3,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×PSU(3,𝔽₂)
commutator, cosocle	C₂×C₃⋊D₃	C₂³
Fitting	C₂²×C₃²	Q₈
socle	C₂×C₃²	C₂×Q₈
prime radical	C₂²×C₃⋊D₃	C₂²
hypercenter	C₂²	PSU(3,𝔽₂)
hypocenter	C₃²	C₂²⋊Q₈
LCS	G ▷ C₂×C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₂²

Complements

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

N	H ≅ G/N	type
2. C₂×PSU(3,𝔽₂)	C₂	semidirect
4. #3777	C₂	semidirect
5. #3777	C₂	semidirect
13. C₂²×C₃²	Q₈	semidirect
16. C₂×C₃⋊D₃	Q₈	semidirect
39. C₃²	C₂²⋊Q₈	semidirect
56. C₂²	PSU(3,𝔽₂)	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#157	2
C₂×PSU(3,𝔽₂)	2
C₂²⋊#157	2
#3777	2
C₂²⋊Q₈	9

Subgroups — C22⋊PSU(3,𝔽2) #64247

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²⋊PSU(3,𝔽₂) #64247