Subgroups — C₂×GU(3,𝔽₂) #13049177

Order 1296.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	GU(3,𝔽₂)	C₂	1	1	yes	maximal, characteristic, direct product factor
2	C₂×SU(3,𝔽₂)	C₃	1	1	yes	maximal, characteristic, complemented
3	C₂×#4174	—	4	1	no	maximal
4	C₂×#1088	—	3	1	no	—
5	SU(3,𝔽₂)	—	3	1	no	—
6	SU(3,𝔽₂)	C₂×C₃	1	1	yes	characteristic, complemented
7	C₂×AΣL(1,𝔽₂₇)	—	4	1	no	—
8	#4174	—	4	1	no	—
9	#4174	—	4	1	no	—
10	C₂×C₃×SL(2,𝔽₃)	—	9	1	no	maximal
11	C₂×C₃²×D₃	—	12	1	no	—
12	C₂×#272	A₄	1	1	yes	characteristic
13	#1088	—	3	1	no	—
14	#1088	—	3	1	no	—
15	AΣL(1,𝔽₂₇)	—	4	1	no	Sylow
16	C₃×SL(2,𝔽₃)	—	9	1	no	—
17	C₂×C₃³	—	4	1	no	—
18	C₃²×D₃	—	12	1	no	—
19	C₃²×D₃	—	12	1	no	—
20	C₂×3₋¹⁺²	—	8	1	no	—
21	C₂×H(3,𝔽₃)	SL(2,𝔽₃)	1	1	yes	characteristic, complemented
22	#272	SL(2,𝔽₃)	1	1	yes	characteristic, complemented
23	#272	C₂×A₄	1	1	yes	characteristic
24	C₂×C₃×Q₈	—	9	1	no	—
25	C₂×SL(2,𝔽₃)	—	9	3	no	—
26	C₂²×C₃²	—	36	1	no	—
27	C₂×C₃×D₃	—	12	1	no	—
28	C₂×C₃×D₃	—	12	3	no	—
29	C₃³	—	4	1	no	—
30	3₋¹⁺²	—	8	1	no	—
31	H(3,𝔽₃)	C₂×SL(2,𝔽₃)	1	1	yes	characteristic, complemented
32	C₃×Q₈	—	9	1	no	—
33	C₃×Q₈	—	27	1	no	—
34	C₂×C₃×C₄	—	27	1	no	—
35	SL(2,𝔽₃)	—	9	3	no	—
36	C₂×C₃²	—	4	1	no	—
37	C₂×C₃²	—	12	1	no	—
38	C₂×C₃²	—	12	3	no	—
39	C₂×C₃²	—	36	1	no	—
40	C₂×C₃²	—	36	1	no	—
41	C₃×D₃	—	12	1	no	—
42	C₃×D₃	—	12	1	no	—
43	C₃×D₃	—	12	3	no	—
44	C₃×D₃	—	12	3	no	—
45	C₂×C₉	—	24	1	no	—
46	C₂×Q₈	—	9	1	no	Sylow
47	C₂²×C₃	—	9	1	no	—
48	C₂²×C₃	—	36	3	no	—
49	C₂×D₃	—	12	1	no	—
50	C₃×C₄	—	27	1	no	—
51	C₃×C₄	—	27	1	no	—
52	C₃²	—	4	1	no	—
53	C₃²	—	12	3	no	—
54	C₃²	—	12	1	no	—
55	C₉	—	24	1	no	—
56	Q₈	—	9	1	no	—
57	Q₈	—	27	1	no	—
58	C₂×C₄	—	27	1	no	—
59	C₂×C₃	—	9	1	no	—
60	C₂×C₃	—	9	1	no	—
61	C₂×C₃	—	12	1	no	—
62	C₂×C₃	—	12	3	no	—
63	C₂×C₃	—	36	3	no	—
64	C₂×C₃	—	36	3	no	—
65	C₂×C₃	PSU(3,𝔽₂)⋊C₃	1	1	yes	characteristic
66	D₃	—	12	1	no	—
67	D₃	—	12	1	no	—
68	C₂²	—	9	1	no	—
69	C₄	—	27	1	no	—
70	C₄	—	27	1	no	—
71	C₃	—	12	1	no	—
72	C₃	—	12	3	no	—
73	C₃	C₂×PSU(3,𝔽₂)⋊C₃	1	1	yes	characteristic
74	C₂	—	9	1	no	—
75	C₂	—	9	1	no	—
76	C₂	GU(3,𝔽₂)	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, socle, hypercenter	C₂×C₃	PSU(3,𝔽₂)⋊C₃
commutator, cosocle, hypocenter	SU(3,𝔽₂)	C₂×C₃
Frattini	C₃	C₂×PSU(3,𝔽₂)⋊C₃
Fitting	C₂×H(3,𝔽₃)	SL(2,𝔽₃)
LCS	G ▷ SU(3,𝔽₂)
UCS	C₁ ◁ C₂×C₃

Complements

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

N	H ≅ G/N	type
1. GU(3,𝔽₂)	C₂	direct (N × H)
2. C₂×SU(3,𝔽₂)	C₃	semidirect
6. SU(3,𝔽₂)	C₂×C₃	semidirect
21. C₂×H(3,𝔽₃)	SL(2,𝔽₃)	semidirect
22. #272	SL(2,𝔽₃)	semidirect
31. H(3,𝔽₃)	C₂×SL(2,𝔽₃)	semidirect
76. C₂	GU(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
GU(3,𝔽₂)	2
C₂×SU(3,𝔽₂)	3
C₂×#4174	4
C₂×C₃×SL(2,𝔽₃)	9

Subgroups — C2×GU(3,𝔽2) #13049177

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×GU(3,𝔽₂) #13049177