Subgroups — #11723945

Order 864.

868 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#89076	C₂	1	2	yes	maximal, complemented
2	C₂×C₃³×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#88853	C₂	1	1	yes	maximal, characteristic, complemented
4	#88852	C₂	1	1	yes	maximal, characteristic, complemented
5	#88851	C₂	1	2	yes	maximal, complemented
6	#64113	—	3	13	no	maximal
7	C₂×C₃³×C₄	C₂²	1	1	yes	characteristic
8	#6312	—	2	2	no	—
9	C₄⋊(C₃⋊(C₃⋊D₃))	C₂²	1	2	yes	—
10	C₄×C₃⋊(C₃⋊D₃)	—	2	2	no	—
11	#6308	C₂²	1	2	yes	—
12	C₃³×C₈	C₂²	1	1	yes	characteristic
13	C₃³×C₈	C₂²	1	1	yes	characteristic
14	#3828	—	3	26	no	—
15	C₂×C₃²×C₈	D₃	1	13	yes	complemented
16	#3746	—	3	13	no	—
17	#3745	—	3	13	no	—
18	#3744	—	3	26	no	—
19	C₂²×C₃³	D₄	1	1	yes	characteristic
20	C₂×C₃⋊(C₃⋊D₃)	—	2	2	no	—
21	C₃³×C₄	D₄	1	1	yes	characteristic
22	C₃³×C₄	C₂³	1	1	yes	characteristic
23	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
24	#904	—	9	13	no	—
25	C₂×C₃²×C₄	C₂×D₃	1	13	yes	—
26	#638	—	6	26	no	—
27	C₄⋊(C₃⋊D₃)	—	3	26	no	—
28	C₄×C₃⋊D₃	—	6	26	no	—
29	#634	—	3	26	no	—
30	C₃²×C₈	C₂×D₃	1	13	yes	—
31	C₃²×C₈	C₂×D₃	1	13	yes	—
32	C₂×C₃³	—	2	1	no	—
33	C₂×C₃³	C₂×D₄	1	1	yes	characteristic
34	C₃⋊(C₃⋊D₃)	—	4	2	no	—
35	#235	—	9	26	no	—
36	C₂×C₃×C₈	C₃⋊D₃	1	13	yes	complemented
37	Dic₁₂	—	9	13	no	—
38	D₂₄	—	9	13	no	—
39	#204	—	9	26	no	—
40	C₂²×C₃²	D₁₂	1	13	yes	—
41	C₂×C₃⋊D₃	—	6	26	no	—
42	C₃²×C₄	D₁₂	1	13	yes	—
43	C₃²×C₄	C₂²×D₃	1	13	yes	—
44	C₃⋊Dic₃	—	6	26	no	—
45	#135	—	27	1	no	Sylow
46	C₃³	#135	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
48	#67	—	18	26	no	—
49	D₁₂	—	9	26	no	—
50	C₄×D₃	—	18	26	no	—
51	Dic₆	—	9	26	no	—
52	C₃×C₈	C₂×C₃⋊D₃	1	13	yes	—
53	C₃×C₈	C₂×C₃⋊D₃	1	13	yes	—
54	C₂×C₃²	—	2	13	no	—
55	C₂×C₃²	C₂×D₁₂	1	13	yes	—
56	C₃⋊D₃	—	12	26	no	—
57	#41	—	27	2	no	—
58	Q₁₆	—	27	1	no	—
59	SD₁₆	—	27	2	no	—
60	D₈	—	27	1	no	—
61	C₂×C₈	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
62	C₂²×C₃	C₄⋊(C₃⋊D₃)	1	13	yes	—
63	C₂×D₃	—	18	26	no	—
64	C₃×C₄	C₄⋊(C₃⋊D₃)	1	13	yes	—
65	C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
66	Dic₃	—	18	26	no	—
67	C₃²	#904	1	13	yes	complemented
68	Q₈	—	27	2	no	—
69	D₄	—	27	2	no	—
70	D₄	—	54	2	no	—
71	C₂×C₄	—	54	2	no	—
72	C₂×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
73	C₈	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
74	C₈	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
75	C₂×C₃	—	2	13	no	—
76	C₂×C₃	C₂×C₄⋊(C₃⋊D₃)	1	13	yes	—
77	D₃	—	36	26	no	—
78	C₂²	—	54	2	no	—
79	C₂²	C₄⋊(C₃⋊(C₃⋊D₃))	1	1	yes	characteristic
80	C₄	—	54	2	no	—
81	C₄	C₄⋊(C₃⋊(C₃⋊D₃))	1	1	yes	characteristic
82	C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
83	C₃	#64113	1	13	yes	complemented
84	C₂	—	2	1	no	—
85	C₂	—	108	2	no	—
86	C₂	C₂×C₄⋊(C₃⋊(C₃⋊D₃))	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄⋊(C₃⋊(C₃⋊D₃))
commutator, cosocle	C₃³×C₄	C₂³
Frattini	C₄	C₂²×C₃⋊(C₃⋊D₃)
Fitting	C₂×C₃³×C₈	C₂
socle	C₂×C₃³	C₂×D₄
hypercenter	C₂×C₈	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#135
LCS	G ▷ C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
1. #89076	C₂	semidirect
2. C₂×C₃³×C₈	C₂	semidirect
3. #88853	C₂	semidirect
4. #88852	C₂	semidirect
5. #88851	C₂	semidirect
15. C₂×C₃²×C₈	D₃	semidirect
36. C₂×C₃×C₈	C₃⋊D₃	semidirect
46. C₃³	#135	semidirect
61. C₂×C₈	C₃⋊(C₃⋊D₃)	semidirect
67. C₃²	#904	semidirect
83. C₃	#64113	semidirect

Maximal subgroups

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

subgroup	index
#89076	2
C₂×C₃³×C₈	2
#88853	2
#88852	2
#88851	2
#64113	3