Subgroups — C₃×PSU(3,𝔽₂)⋊C₄ #11724096

Order 864.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×PSU(3,𝔽₂)	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×AGL(1,𝔽₉)	C₂	1	1	yes	maximal, characteristic
3	C₃×#3773	C₂	1	1	yes	maximal, characteristic
4	PSU(3,𝔽₂)⋊C₄	C₃	1	1	yes	maximal, characteristic, direct product factor
5	C₂×C₃×#157	—	2	1	no	—
6	C₂×C₃×#157	C₂²	1	1	yes	characteristic
7	C₃×PSU(3,𝔽₂)	—	2	1	no	—
8	C₃×PSU(3,𝔽₂)	C₄	1	2	yes	complemented
9	C₃×AGL(1,𝔽₉)	—	2	1	no	—
10	C₃×#624	—	2	1	no	—
11	C₂×PSU(3,𝔽₂)	C₂×C₃	1	1	yes	characteristic
12	C₂×AGL(1,𝔽₉)	C₂×C₃	1	1	yes	characteristic
13	#3773	C₂×C₃	1	1	yes	characteristic
14	C₂×C₃×C₃⋊D₃	D₄	1	1	yes	characteristic
15	C₃×#157	—	2	2	no	—
16	C₃×#157	C₂×C₄	1	1	yes	characteristic
17	C₃×#157	D₄	1	1	yes	characteristic
18	C₃²×Dic₃	—	4	1	no	—
19	C₃×#103	—	9	1	no	maximal
20	C₂×#157	—	2	1	no	—
21	C₂×#157	C₂²×C₃	1	1	yes	characteristic
22	PSU(3,𝔽₂)	—	2	1	no	—
23	PSU(3,𝔽₂)	C₃×C₄	1	2	yes	complemented
24	AGL(1,𝔽₉)	—	2	1	no	—
25	C₃×C₄×D₃	—	12	1	no	—
26	#624	—	2	1	no	—
27	C₂×C₃³	SD₁₆	1	1	yes	characteristic
28	C₃×C₃⋊D₃	C₂²⋊C₄	1	1	yes	characteristic
29	C₃×C₃⋊D₃	Q₁₆	1	1	yes	characteristic
30	C₂×C₃×Q₈	—	9	1	no	—
31	C₂×C₃×C₈	—	9	1	no	—
32	C₃×C₄⋊C₄	—	9	1	no	—
33	C₂×C₃⋊D₃	C₃×D₄	1	1	yes	characteristic
34	C₂×C₃×D₃	—	12	1	no	—
35	#157	—	2	2	no	—
36	#157	C₂×C₃×C₄	1	1	yes	characteristic
37	#157	C₃×D₄	1	1	yes	characteristic
38	C₃²×C₄	—	12	1	no	—
39	C₃×Dic₃	—	4	1	no	—
40	C₃×Dic₃	—	4	1	no	—
41	C₃×Dic₃	—	8	1	no	—
42	#103	—	9	1	no	Sylow
43	C₃³	#103	1	1	yes	characteristic, complemented, Sylow
44	C₃×Q₈	—	9	2	no	—
45	C₃×Q₈	—	18	1	no	—
46	C₂×C₃×C₄	—	9	1	no	—
47	C₂×C₃×C₄	—	18	1	no	—
48	C₂×C₃×C₄	—	18	1	no	—
49	C₄×D₃	—	12	1	no	—
50	C₃×C₈	—	18	1	no	—
51	C₂×C₃²	—	4	1	no	—
52	C₂×C₃²	—	8	1	no	—
53	C₂×C₃²	C₃×SD₁₆	1	1	yes	characteristic
54	C₃⋊D₃	C₃×C₂²⋊C₄	1	1	yes	characteristic
55	C₃⋊D₃	C₃×Q₁₆	1	1	yes	characteristic
56	C₃×D₃	—	12	1	no	—
57	C₃×D₃	—	12	1	no	—
58	C₂×Q₈	—	9	1	no	—
59	C₂×C₈	—	9	1	no	—
60	C₄⋊C₄	—	9	1	no	—
61	C₂²×C₃	—	9	1	no	—
62	C₂×D₃	—	12	1	no	—
63	C₃×C₄	—	9	1	no	—
64	C₃×C₄	—	9	1	no	—
65	C₃×C₄	—	12	1	no	—
66	C₃×C₄	—	12	1	no	—
67	C₃×C₄	—	18	2	no	—
68	C₃×C₄	—	24	1	no	—
69	Dic₃	—	4	1	no	—
70	C₃²	—	4	1	no	—
71	C₃²	—	8	1	no	—
72	C₃²	C₃×#103	1	1	yes	characteristic, complemented
73	Q₈	—	9	2	no	—
74	Q₈	—	18	1	no	—
75	C₂×C₄	—	9	1	no	—
76	C₂×C₄	—	18	1	no	—
77	C₂×C₄	—	18	1	no	—
78	C₈	—	18	1	no	—
79	C₂×C₃	—	4	1	no	—
80	C₂×C₃	—	8	1	no	—
81	C₂×C₃	—	9	1	no	—
82	C₂×C₃	—	9	1	no	—
83	C₂×C₃	AΓL(1,𝔽₉)	1	1	yes	characteristic
84	D₃	—	12	1	no	—
85	D₃	—	12	1	no	—
86	C₂²	—	9	1	no	—
87	C₄	—	9	1	no	—
88	C₄	—	9	1	no	—
89	C₄	—	12	1	no	—
90	C₄	—	18	2	no	—
91	C₃	—	4	1	no	—
92	C₃	—	8	1	no	—
93	C₃	PSU(3,𝔽₂)⋊C₄	1	1	yes	characteristic, direct product factor
94	C₂	—	9	1	no	—
95	C₂	—	9	1	no	—
96	C₂	C₃×AΓL(1,𝔽₉)	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃	AΓL(1,𝔽₉)
commutator	#157	C₂×C₃×C₄
Frattini	C₂	C₃×AΓL(1,𝔽₉)
Fitting, socle	C₂×C₃³	SD₁₆
prime radical	C₂×C₃×C₃⋊D₃	D₄
cosocle	C₂×#157	C₂²×C₃
hypocenter	C₃²	C₃×#103
LCS	G ▷ #157 ▷ C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₂×C₃

Complements

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

N	H ≅ G/N	type
4. PSU(3,𝔽₂)⋊C₄	C₃	direct (N × H)
8. C₃×PSU(3,𝔽₂)	C₄	semidirect
23. PSU(3,𝔽₂)	C₃×C₄	semidirect
43. C₃³	#103	semidirect
72. C₃²	C₃×#103	semidirect
93. C₃	PSU(3,𝔽₂)⋊C₄	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×PSU(3,𝔽₂)	2
C₂×C₃×AGL(1,𝔽₉)	2
C₃×#3773	2
PSU(3,𝔽₂)⋊C₄	3
C₃×#103	9

Subgroups — C3×PSU(3,𝔽2)⋊C4 #11724096

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×PSU(3,𝔽₂)⋊C₄ #11724096