Subgroups — #11723953

Order 864.

868 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#89078	C₂	1	2	yes	maximal, complemented
2	C₃³×D₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#88861	C₂	1	2	yes	maximal, complemented
4	#88853	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×C₃⋊(C₃⋊D₃)	C₂	1	1	yes	maximal, characteristic, complemented
6	#64121	—	3	13	no	maximal
7	C₃³×D₄	C₂²	1	2	yes	—
8	#6312	—	2	2	no	—
9	C₂×C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
10	C₄×C₃⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic
11	#6308	C₂²	1	2	yes	—
12	C₃³×C₈	C₂²	1	1	yes	characteristic
13	C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
14	#3830	—	3	26	no	—
15	C₃²×D₈	D₃	1	13	yes	complemented
16	#3754	—	3	26	no	—
17	#3746	—	3	13	no	—
18	C₈×C₃⋊D₃	—	3	13	no	—
19	C₂²×C₃³	—	2	2	no	—
20	C₂×C₃⋊(C₃⋊D₃)	D₄	1	1	yes	characteristic
21	C₃³×C₄	C₂³	1	1	yes	characteristic
22	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
23	C₃⋊(C₃⋊Dic₃)	D₄	1	1	yes	characteristic
24	#912	—	9	13	no	—
25	C₃²×D₄	C₂×D₃	1	26	yes	—
26	#638	—	6	26	no	—
27	C₂×C₃⋊Dic₃	—	6	26	no	—
28	C₄×C₃⋊D₃	—	3	13	no	—
29	#634	—	3	26	no	—
30	C₃²×C₈	C₂×D₃	1	13	yes	—
31	C₃⋊(C₃⋊C₈)	—	3	13	no	—
32	C₂×C₃³	—	4	2	no	—
33	C₂×C₃³	C₂×D₄	1	1	yes	characteristic
34	C₃⋊(C₃⋊D₃)	—	2	1	no	—
35	#237	—	9	26	no	—
36	C₃×D₈	C₃⋊D₃	1	13	yes	complemented
37	#214	—	9	26	no	—
38	Dic₁₂	—	9	13	no	—
39	D₃×C₈	—	9	13	no	—
40	C₂²×C₃²	—	2	26	no	—
41	C₂×C₃⋊D₃	—	3	13	no	—
42	C₃²×C₄	C₂²×D₃	1	13	yes	—
43	C₃⋊Dic₃	—	3	13	no	—
44	C₃⋊Dic₃	—	6	26	no	—
45	#135	—	27	1	no	Sylow
46	C₃³	#135	1	1	yes	characteristic, complemented, Sylow
47	C₃×D₄	C₂×C₃⋊D₃	1	26	yes	—
48	#67	—	18	26	no	—
49	C₂×Dic₃	—	18	26	no	—
50	C₄×D₃	—	9	13	no	—
51	Dic₆	—	9	26	no	—
52	C₃×C₈	C₂×C₃⋊D₃	1	13	yes	—
53	C₃⋊C₈	—	9	13	no	—
54	C₂×C₃²	—	4	26	no	—
55	C₂×C₃²	D₃×D₄	1	13	yes	—
56	C₃⋊D₃	—	6	13	no	—
57	#41	—	27	2	no	—
58	Q₁₆	—	27	1	no	—
59	SD₁₆	—	27	2	no	—
60	D₈	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
61	C₂×C₈	—	27	1	no	—
62	C₂²×C₃	—	2	26	no	—
63	C₂×D₃	—	9	13	no	—
64	C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
65	Dic₃	—	9	13	no	—
66	Dic₃	—	18	26	no	—
67	C₃²	#912	1	13	yes	complemented
68	Q₈	—	27	2	no	—
69	D₄	—	54	2	no	—
70	D₄	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
71	C₂×C₄	—	27	1	no	—
72	C₂×C₄	—	54	2	no	—
73	C₈	—	27	1	no	—
74	C₈	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
75	C₂×C₃	—	4	26	no	—
76	C₂×C₃	D₄×C₃⋊D₃	1	13	yes	—
77	D₃	—	18	13	no	—
78	C₂²	—	2	2	no	—
79	C₂²	—	27	1	no	—
80	C₄	—	27	1	no	—
81	C₄	—	54	2	no	—
82	C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
83	C₃	#64121	1	13	yes	complemented
84	C₂	—	4	2	no	—
85	C₂	—	54	1	no	—
86	C₂	D₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×C₃⋊(C₃⋊D₃)
commutator, cosocle	C₃³×C₄	C₂³
Frattini	C₄	C₂²×C₃⋊(C₃⋊D₃)
Fitting	C₃³×D₈	C₂
socle	C₂×C₃³	C₂×D₄
hypercenter	D₈	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#135
LCS	G ▷ C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. #89078	C₂	semidirect
2. C₃³×D₈	C₂	semidirect
3. #88861	C₂	semidirect
4. #88853	C₂	semidirect
5. C₈×C₃⋊(C₃⋊D₃)	C₂	semidirect
15. C₃²×D₈	D₃	semidirect
36. C₃×D₈	C₃⋊D₃	semidirect
46. C₃³	#135	semidirect
60. D₈	C₃⋊(C₃⋊D₃)	semidirect
67. C₃²	#912	semidirect
83. C₃	#64121	semidirect

Maximal subgroups

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

subgroup	index
#89078	2
C₃³×D₈	2
#88861	2
#88853	2
C₈×C₃⋊(C₃⋊D₃)	2
#64121	3