Subgroups — C₂×C₃⋊PSU(3,𝔽₂) #89111

Order 432.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃⋊#157	C₂	1	2	yes	maximal
2	C₂×C₃×#157	C₂	1	1	yes	maximal, characteristic
3	C₃⋊PSU(3,𝔽₂)	C₂	1	4	yes	maximal, direct product factor
4	C₂×PSU(3,𝔽₂)	—	3	1	no	maximal
5	C₂×C₃×C₃⋊D₃	C₂²	1	1	yes	characteristic
6	C₃⋊#157	C₂²	1	4	yes	—
7	C₃×#157	C₂²	1	2	yes	—
8	C₂×#157	—	3	2	no	—
9	C₂×#157	D₃	1	1	yes	characteristic
10	PSU(3,𝔽₂)	—	3	4	no	—
11	C₂×C₃³	Q₈	1	1	yes	characteristic, complemented
12	C₃×C₃⋊D₃	Q₈	1	1	yes	characteristic, complemented
13	C₃×C₃⋊D₃	C₂³	1	1	yes	characteristic
14	C₂×Dic₆	—	9	1	no	maximal
15	C₂×C₃⋊D₃	C₂×D₃	1	1	yes	characteristic
16	C₂×C₃×D₃	—	12	1	no	—
17	#157	—	3	4	no	—
18	#157	C₂×D₃	1	2	yes	—
19	C₃³	C₂×Q₈	1	1	yes	characteristic, complemented, Sylow
20	C₂×C₃×C₄	—	9	1	no	—
21	C₂×Dic₃	—	9	2	no	—
22	Dic₆	—	9	4	no	—
23	C₂×C₃²	—	4	1	no	—
24	C₂×C₃²	—	8	1	no	—
25	C₂×C₃²	Dic₆	1	1	yes	characteristic, complemented
26	C₃⋊D₃	Dic₆	1	1	yes	characteristic, complemented
27	C₃⋊D₃	C₂²×D₃	1	1	yes	characteristic
28	C₃×D₃	—	12	1	no	—
29	C₃×D₃	—	12	1	no	—
30	C₂×Q₈	—	27	1	no	Sylow
31	C₂²×C₃	—	9	1	no	—
32	C₂×D₃	—	12	1	no	—
33	C₃×C₄	—	9	2	no	—
34	Dic₃	—	9	4	no	—
35	C₃²	—	4	1	no	—
36	C₃²	—	8	1	no	—
37	C₃²	C₂×Dic₆	1	1	yes	characteristic, complemented
38	Q₈	—	27	4	no	—
39	C₂×C₄	—	9	1	no	—
40	C₂×C₄	—	27	2	no	—
41	C₂×C₃	—	4	1	no	—
42	C₂×C₃	—	8	1	no	—
43	C₂×C₃	—	9	1	no	—
44	C₂×C₃	—	9	1	no	—
45	C₂×C₃	PSU(3,𝔽₂)	1	1	yes	characteristic, complemented
46	D₃	—	12	1	no	—
47	D₃	—	12	1	no	—
48	C₂²	—	9	1	no	—
49	C₄	—	9	2	no	—
50	C₄	—	27	4	no	—
51	C₃	—	4	1	no	—
52	C₃	—	8	1	no	—
53	C₃	C₂×PSU(3,𝔽₂)	1	1	yes	characteristic, complemented
54	C₂	—	9	1	no	—
55	C₂	—	9	1	no	—
56	C₂	C₃⋊PSU(3,𝔽₂)	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	C₃⋊PSU(3,𝔽₂)
commutator, cosocle	C₃×C₃⋊D₃	C₂³
Frattini	C₁	G
Fitting, socle	C₂×C₃³	Q₈
prime radical	C₂×C₃×C₃⋊D₃	C₂²
hypocenter	C₃³	C₂×Q₈
LCS	G ▷ C₃×C₃⋊D₃ ▷ C₃³
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
3. C₃⋊PSU(3,𝔽₂)	C₂	direct (N × H)
11. C₂×C₃³	Q₈	semidirect
12. C₃×C₃⋊D₃	Q₈	semidirect
19. C₃³	C₂×Q₈	semidirect
25. C₂×C₃²	Dic₆	semidirect
26. C₃⋊D₃	Dic₆	semidirect
37. C₃²	C₂×Dic₆	semidirect
45. C₂×C₃	PSU(3,𝔽₂)	semidirect
53. C₃	C₂×PSU(3,𝔽₂)	semidirect
56. C₂	C₃⋊PSU(3,𝔽₂)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃⋊#157	2
C₂×C₃×#157	2
C₃⋊PSU(3,𝔽₂)	2
C₂×PSU(3,𝔽₂)	3
C₂×Dic₆	9

Subgroups — C2×C3⋊PSU(3,𝔽2) #89111

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃⋊PSU(3,𝔽₂) #89111