Subgroups — C₃×GU(3,𝔽₂) #14519052

Order 1944.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×SU(3,𝔽₂)	C₃	1	1	yes	maximal, characteristic, complemented
2	GU(3,𝔽₂)	C₃	1	3	yes	maximal, direct product factor
3	C₃×#4174	—	4	1	no	maximal
4	C₃×#1088	—	3	1	no	—
5	C₃×AΣL(1,𝔽₂₇)	—	4	1	no	Sylow
6	C₃²×SL(2,𝔽₃)	—	9	1	no	maximal
7	SU(3,𝔽₂)	C₃²	1	1	yes	characteristic, complemented
8	C₃³×D₃	—	12	1	no	—
9	C₃×#272	A₄	1	1	yes	characteristic
10	#4174	—	4	3	no	—
11	#1088	—	3	1	no	—
12	C₃⁴	—	4	1	no	—
13	C₃×3₋¹⁺²	—	8	1	no	—
14	C₃×H(3,𝔽₃)	SL(2,𝔽₃)	1	1	yes	characteristic, complemented
15	AΣL(1,𝔽₂₇)	—	4	3	no	—
16	AΣL(1,𝔽₂₇)	—	8	3	no	—
17	C₃²×Q₈	—	9	1	no	—
18	C₃×SL(2,𝔽₃)	—	9	3	no	—
19	C₃×SL(2,𝔽₃)	—	9	9	no	—
20	C₂×C₃³	—	36	1	no	—
21	C₃²×D₃	—	12	1	no	—
22	C₃²×D₃	—	12	3	no	—
23	C₃²×D₃	—	12	9	no	—
24	#272	C₃×A₄	1	1	yes	characteristic
25	C₃²×C₄	—	27	1	no	—
26	C₃³	—	4	1	no	—
27	C₃³	—	4	3	no	—
28	C₃³	—	12	1	no	—
29	C₃³	—	12	9	no	—
30	C₃³	—	24	1	no	—
31	3₋¹⁺²	—	8	3	no	—
32	3₋¹⁺²	—	24	2	no	—
33	H(3,𝔽₃)	—	8	1	no	—
34	H(3,𝔽₃)	C₃×SL(2,𝔽₃)	1	1	yes	characteristic, complemented
35	C₃×C₉	—	24	1	no	—
36	C₃×Q₈	—	9	1	no	—
37	C₃×Q₈	—	9	3	no	—
38	SL(2,𝔽₃)	—	9	9	no	—
39	C₂×C₃²	—	9	1	no	—
40	C₂×C₃²	—	36	3	no	—
41	C₂×C₃²	—	36	9	no	—
42	C₃×D₃	—	12	1	no	—
43	C₃×D₃	—	12	3	no	—
44	C₃×D₃	—	12	9	no	—
45	C₃×C₄	—	27	1	no	—
46	C₃×C₄	—	27	3	no	—
47	C₃²	—	4	1	no	—
48	C₃²	—	8	1	no	—
49	C₃²	—	12	9	no	—
50	C₃²	—	12	3	no	—
51	C₃²	—	12	9	no	—
52	C₃²	—	12	3	no	—
53	C₃²	—	24	9	no	—
54	C₃²	PSU(3,𝔽₂)⋊C₃	1	1	yes	characteristic
55	C₉	—	24	3	no	—
56	Q₈	—	9	1	no	Sylow
57	C₂×C₃	—	9	1	no	—
58	C₂×C₃	—	9	3	no	—
59	C₂×C₃	—	36	9	no	—
60	D₃	—	12	1	no	—
61	C₄	—	27	1	no	—
62	C₃	—	12	1	no	—
63	C₃	—	12	9	no	—
64	C₃	—	24	1	no	—
65	C₃	GU(3,𝔽₂)	1	3	yes	direct product factor
66	C₃	C₃×PSU(3,𝔽₂)⋊C₃	1	1	yes	characteristic
67	C₂	—	9	1	no	—

	subgroup	G/H
center, socle, hypercenter	C₃²	PSU(3,𝔽₂)⋊C₃
commutator, cosocle, hypocenter	SU(3,𝔽₂)	C₃²
Frattini	C₃	C₃×PSU(3,𝔽₂)⋊C₃
Fitting	C₃×H(3,𝔽₃)	SL(2,𝔽₃)
LCS	G ▷ SU(3,𝔽₂)
UCS	C₁ ◁ C₃²

Complements

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

N	H ≅ G/N	type
1. C₃×SU(3,𝔽₂)	C₃	semidirect
2. GU(3,𝔽₂)	C₃	direct (N × H)
7. SU(3,𝔽₂)	C₃²	semidirect
14. C₃×H(3,𝔽₃)	SL(2,𝔽₃)	semidirect
34. H(3,𝔽₃)	C₃×SL(2,𝔽₃)	semidirect
65. C₃	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
C₃×SU(3,𝔽₂)	3
GU(3,𝔽₂)	3
C₃×#4174	4
C₃²×SL(2,𝔽₃)	9

Subgroups — C3×GU(3,𝔽2) #14519052

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×GU(3,𝔽₂) #14519052