Subgroups — C₄×PSU(3,𝔽₂)⋊C₃ #11724087

Order 864.

82 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×PSU(3,𝔽₂)⋊C₃	C₂	1	1	yes	maximal, characteristic
2	C₄×PSU(3,𝔽₂)	C₃	1	1	yes	maximal, characteristic, complemented
3	PSU(3,𝔽₂)⋊C₃	C₄	1	1	yes	characteristic, direct product factor
4	C₄×(C₃⋊D₃)⋊C₃	—	4	1	no	maximal
5	C₂×PSU(3,𝔽₂)	C₂×C₃	1	1	yes	characteristic
6	C₄×#157	—	3	1	no	—
7	#3777	—	3	1	no	—
8	C₂×(C₃⋊D₃)⋊C₃	—	4	1	no	—
9	C₄×H(3,𝔽₃)	—	4	1	no	—
10	(C₃⋊Dic₃)⋊C₃	—	4	1	no	—
11	C₄×SL(2,𝔽₃)	—	9	1	no	maximal
12	C₂×#157	—	3	1	no	—
13	C₂×#157	—	3	1	no	—
14	PSU(3,𝔽₂)	—	3	1	no	—
15	PSU(3,𝔽₂)	C₃×C₄	1	1	yes	characteristic, complemented
16	C₄×C₃⋊D₃	A₄	1	1	yes	characteristic
17	C₃×C₄×D₃	—	12	1	no	—
18	C₂×H(3,𝔽₃)	—	4	1	no	—
19	(C₃⋊D₃)⋊C₃	—	4	1	no	—
20	(C₃⋊D₃)⋊C₃	—	4	1	no	—
21	C₂×SL(2,𝔽₃)	—	9	1	no	—
22	C₂×C₃⋊D₃	C₂×A₄	1	1	yes	characteristic
23	C₂×C₃×D₃	—	12	1	no	—
24	#157	—	3	1	no	—
25	#157	—	3	1	no	—
26	#157	—	6	1	no	—
27	C₃²×C₄	—	4	1	no	—
28	C₃²×C₄	—	8	1	no	—
29	C₃²×C₄	SL(2,𝔽₃)	1	1	yes	characteristic, complemented
30	C₃⋊Dic₃	SL(2,𝔽₃)	1	1	yes	characteristic, complemented
31	C₃×Dic₃	—	12	1	no	—
32	C₄×Q₈	—	9	1	no	Sylow
33	H(3,𝔽₃)	—	4	1	no	Sylow
34	C₂×C₃×C₄	—	36	1	no	—
35	C₄×D₃	—	12	1	no	—
36	SL(2,𝔽₃)	—	9	1	no	—
37	C₂×C₃²	—	4	1	no	—
38	C₂×C₃²	—	8	1	no	—
39	C₂×C₃²	C₂×SL(2,𝔽₃)	1	1	yes	characteristic
40	C₃⋊D₃	C₄×A₄	1	1	yes	characteristic
41	C₃⋊D₃	#41⋊C₃	1	1	yes	characteristic
42	C₃×D₃	—	12	1	no	—
43	C₃×D₃	—	12	1	no	—
44	C₂×Q₈	—	9	1	no	—
45	C₄⋊C₄	—	27	1	no	—
46	C₄²	—	27	1	no	—
47	C₂²×C₃	—	36	1	no	—
48	C₂×D₃	—	12	1	no	—
49	C₃×C₄	—	4	1	no	—
50	C₃×C₄	—	12	1	no	—
51	C₃×C₄	—	24	1	no	—
52	C₃×C₄	—	36	1	no	—
53	Dic₃	—	12	1	no	—
54	C₃²	—	4	1	no	—
55	C₃²	—	8	1	no	—
56	C₃²	C₄×SL(2,𝔽₃)	1	1	yes	characteristic, complemented
57	Q₈	—	9	1	no	—
58	Q₈	—	27	1	no	—
59	C₂×C₄	—	9	1	no	—
60	C₂×C₄	—	27	1	no	—
61	C₂×C₄	—	27	1	no	—
62	C₂×C₃	—	4	1	no	—
63	C₂×C₃	—	12	1	no	—
64	C₂×C₃	—	24	1	no	—
65	C₂×C₃	—	36	1	no	—
66	C₂×C₃	—	36	1	no	—
67	D₃	—	12	1	no	—
68	D₃	—	12	1	no	—
69	C₂²	—	9	1	no	—
70	C₄	—	9	1	no	—
71	C₄	—	27	1	no	—
72	C₄	—	27	1	no	—
73	C₄	—	54	1	no	—
74	C₄	PSU(3,𝔽₂)⋊C₃	1	1	yes	characteristic, direct product factor
75	C₃	—	4	1	no	—
76	C₃	—	12	1	no	—
77	C₃	—	24	1	no	—
78	C₂	—	9	1	no	—
79	C₂	—	9	1	no	—
80	C₂	C₂×PSU(3,𝔽₂)⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₄	PSU(3,𝔽₂)⋊C₃
commutator, hypocenter	PSU(3,𝔽₂)	C₃×C₄
Frattini	C₂	C₂×PSU(3,𝔽₂)⋊C₃
Fitting	C₃²×C₄	SL(2,𝔽₃)
socle	C₂×C₃²	C₂×SL(2,𝔽₃)
prime radical	C₂×PSU(3,𝔽₂)⋊C₃	C₂
cosocle	C₂×PSU(3,𝔽₂)	C₂×C₃
LCS	G ▷ PSU(3,𝔽₂)
UCS	C₁ ◁ C₄

Complements

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

N	H ≅ G/N	type
2. C₄×PSU(3,𝔽₂)	C₃	semidirect
3. PSU(3,𝔽₂)⋊C₃	C₄	direct (N × H)
15. PSU(3,𝔽₂)	C₃×C₄	semidirect
29. C₃²×C₄	SL(2,𝔽₃)	semidirect
30. C₃⋊Dic₃	SL(2,𝔽₃)	semidirect
56. C₃²	C₄×SL(2,𝔽₃)	semidirect
74. C₄	PSU(3,𝔽₂)⋊C₃	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×PSU(3,𝔽₂)⋊C₃	2
C₄×PSU(3,𝔽₂)	3
C₄×(C₃⋊D₃)⋊C₃	4
C₄×SL(2,𝔽₃)	9

Subgroups — C4×PSU(3,𝔽2)⋊C3 #11724087

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×PSU(3,𝔽₂)⋊C₃ #11724087