Subgroups — #11724471

Order 864.

460 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#41⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃⋊(C₃⋊GL(2,𝔽₃))	C₂	1	2	yes	maximal, complemented
3	#64267	—	3	12	no	maximal
4	#64158	—	3	1	no	maximal
5	C₄⋊(C₃⋊(C₃⋊D₃))	—	4	1	no	maximal
6	C₃²×SL(2,𝔽₃)	C₂²	1	1	yes	characteristic, complemented
7	C₃²×#41	D₃	1	1	yes	characteristic, complemented
8	C₂×C₄⋊(C₃⋊D₃)	—	3	1	no	—
9	C₃×#41⋊C₃	D₃	1	12	yes	complemented
10	C₃⋊GL(2,𝔽₃)	—	3	24	no	—
11	#3755	—	3	2	no	—
12	#3753	—	3	2	no	—
13	#3748	—	3	1	no	—
14	C₂×C₃⋊(C₃⋊D₃)	—	4	2	no	—
15	C₃³×C₄	—	4	1	no	—
16	#986	—	9	9	no	—
17	#949	—	9	4	no	—
18	C₂²×C₃⋊D₃	—	6	1	no	—
19	C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
20	C₃²×D₄	—	3	1	no	—
21	C₂×C₃²×C₄	—	3	1	no	—
22	C₄⋊(C₃⋊D₃)	—	3	2	no	—
23	C₄⋊(C₃⋊D₃)	—	6	1	no	—
24	C₄⋊(C₃⋊D₃)	—	12	12	no	—
25	C₃×SL(2,𝔽₃)	C₂×D₃	1	12	yes	complemented
26	C₃⋊(C₃⋊C₈)	—	3	2	no	—
27	C₂×C₃³	—	4	1	no	—
28	C₃⋊(C₃⋊D₃)	—	8	2	no	—
29	C₃×#41	C₃⋊D₃	1	4	yes	complemented
30	C₂×D₁₂	—	9	4	no	—
31	#41⋊C₃	C₃⋊D₃	1	9	yes	complemented
32	GL(2,𝔽₃)	—	9	18	no	—
33	#215	—	9	8	no	—
34	#213	—	9	8	no	—
35	#208	—	9	4	no	—
36	C₂²×C₃²	—	3	1	no	—
37	C₂×C₃⋊D₃	—	6	2	no	—
38	C₂×C₃⋊D₃	—	12	24	no	—
39	C₂×C₃⋊D₃	—	12	2	no	—
40	C₃²×C₄	—	3	1	no	—
41	C₃²×C₄	—	4	12	no	—
42	C₃²×C₄	S₄	1	1	yes	characteristic
43	AGL(1,ℤ/8)	—	27	1	no	Sylow
44	C₃³	—	4	1	no	Sylow
45	C₂²×D₃	—	18	4	no	—
46	C₃×Q₈	C₂×C₃⋊D₃	1	4	yes	—
47	C₃×D₄	—	3	4	no	—
48	C₂×C₃×C₄	—	3	4	no	—
49	D₁₂	—	9	8	no	—
50	D₁₂	—	18	4	no	—
51	D₁₂	—	36	9	no	—
52	SL(2,𝔽₃)	C₂×C₃⋊D₃	1	9	yes	complemented
53	C₃⋊C₈	—	9	8	no	—
54	C₂×C₃²	—	4	12	no	—
55	C₂×C₃²	—	6	1	no	—
56	C₂×C₃²	C₂×S₄	1	1	yes	characteristic
57	C₃⋊D₃	—	12	2	no	—
58	C₃⋊D₃	—	24	24	no	—
59	#41	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
60	C₂×D₄	—	27	1	no	—
61	SD₁₆	—	27	2	no	—
62	D₈	—	27	2	no	—
63	M₁₆	—	27	1	no	—
64	C₂²×C₃	—	3	4	no	—
65	C₂×D₃	—	18	8	no	—
66	C₂×D₃	—	36	18	no	—
67	C₂×D₃	—	36	8	no	—
68	C₃×C₄	—	3	4	no	—
69	C₃×C₄	—	4	9	no	—
70	C₃×C₄	C₃⋊S₄	1	4	yes	—
71	C₃²	—	4	12	no	—
72	C₃²	#986	1	1	yes	characteristic, complemented
73	C₂³	—	54	1	no	—
74	Q₈	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
75	D₄	—	3	1	no	—
76	D₄	—	27	2	no	—
77	D₄	—	54	1	no	—
78	C₂×C₄	—	3	1	no	—
79	C₈	—	27	2	no	—
80	C₂×C₃	—	4	9	no	—
81	C₂×C₃	—	6	4	no	—
82	C₂×C₃	C₂×C₃⋊S₄	1	4	yes	—
83	D₃	—	36	8	no	—
84	D₃	—	72	18	no	—
85	C₂²	—	3	1	no	—
86	C₂²	—	54	2	no	—
87	C₂²	—	108	2	no	—
88	C₄	—	3	1	no	—
89	C₄	C₃⋊(C₃⋊S₄)	1	1	yes	characteristic
90	C₃	—	4	9	no	—
91	C₃	#64267	1	4	yes	complemented
92	C₂	—	6	1	no	—
93	C₂	—	108	2	no	—
94	C₂	C₂×C₃⋊(C₃⋊S₄)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂×C₃⋊(C₃⋊S₄)
commutator, cosocle, hypocenter	C₃²×SL(2,𝔽₃)	C₂²
Fitting	C₃²×#41	D₃
socle	C₂×C₃²	C₂×S₄
hypercenter	C₄	C₃⋊(C₃⋊S₄)
LCS	G ▷ C₃²×SL(2,𝔽₃)
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
1. C₃²×#41⋊C₃	C₂	semidirect
2. C₃⋊(C₃⋊GL(2,𝔽₃))	C₂	semidirect
6. C₃²×SL(2,𝔽₃)	C₂²	semidirect
7. C₃²×#41	D₃	semidirect
9. C₃×#41⋊C₃	D₃	semidirect
25. C₃×SL(2,𝔽₃)	C₂×D₃	semidirect
29. C₃×#41	C₃⋊D₃	semidirect
31. #41⋊C₃	C₃⋊D₃	semidirect
52. SL(2,𝔽₃)	C₂×C₃⋊D₃	semidirect
59. #41	C₃⋊(C₃⋊D₃)	semidirect
72. C₃²	#986	semidirect
91. C₃	#64267	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#41⋊C₃	2
C₃⋊(C₃⋊GL(2,𝔽₃))	2
#64267	3
#64158	3
C₄⋊(C₃⋊(C₃⋊D₃))	4