Subgroups — #11724682

Order 864.

2044 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃³×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#89080	C₂	1	4	yes	maximal, complemented
3	Q₈×C₃⋊(C₃⋊D₃)	C₂	1	4	yes	maximal, complemented
4	#89076	C₂	1	6	yes	maximal, complemented
5	#64364	—	3	13	no	maximal
6	C₃³×Q₈	C₂²	1	4	yes	—
7	C₂×C₃³×C₄	C₂²	1	3	yes	—
8	#6312	C₂²	1	4	yes	—
9	C₄⋊(C₃⋊(C₃⋊D₃))	C₂²	1	6	yes	—
10	C₄×C₃⋊(C₃⋊D₃)	C₂²	1	12	yes	—
11	#6308	C₂²	1	6	yes	—
12	C₂×C₃²×Q₈	D₃	1	13	yes	complemented
13	#3832	—	3	52	no	—
14	Q₈×C₃⋊D₃	—	3	52	no	—
15	#3828	—	3	78	no	—
16	C₂²×C₃³	C₂³	1	1	yes	characteristic
17	C₂×C₃⋊(C₃⋊D₃)	C₂³	1	4	yes	—
18	C₃³×C₄	C₂³	1	6	yes	—
19	C₃⋊(C₃⋊Dic₃)	C₂³	1	4	yes	—
20	#1007	—	9	13	no	—
21	C₃²×Q₈	C₂×D₃	1	52	yes	—
22	C₂×C₃²×C₄	C₂×D₃	1	39	yes	—
23	#638	—	3	52	no	—
24	C₄⋊(C₃⋊D₃)	—	3	78	no	—
25	C₄×C₃⋊D₃	—	3	156	no	—
26	#634	—	3	78	no	—
27	C₂×C₃³	—	2	1	no	—
28	C₂×C₃³	C₂⁴	1	1	yes	characteristic
29	C₃⋊(C₃⋊D₃)	—	2	4	no	—
30	C₂×C₃×Q₈	C₃⋊D₃	1	13	yes	complemented
31	#239	—	9	52	no	—
32	D₃×Q₈	—	9	52	no	—
33	#235	—	9	78	no	—
34	C₂²×C₃²	C₂²×D₃	1	13	yes	—
35	C₂×C₃⋊D₃	—	3	52	no	—
36	C₃²×C₄	C₂²×D₃	1	78	yes	—
37	C₃⋊Dic₃	—	3	52	no	—
38	2₋¹⁺⁴	—	27	1	no	Sylow
39	C₃³	2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
40	C₃×Q₈	C₂×C₃⋊D₃	1	52	yes	—
41	C₂×C₃×C₄	C₂×C₃⋊D₃	1	39	yes	—
42	#67	—	9	52	no	—
43	D₁₂	—	9	78	no	—
44	C₄×D₃	—	9	156	no	—
45	Dic₆	—	9	78	no	—
46	C₂×C₃²	—	2	13	no	—
47	C₂×C₃²	C₂³×D₃	1	13	yes	—
48	C₃⋊D₃	—	6	52	no	—
49	#41	—	27	4	no	—
50	#41	—	27	6	no	—
51	C₂×Q₈	—	27	4	no	—
52	C₂×Q₈	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
53	C₂²×C₃	C₂²×C₃⋊D₃	1	13	yes	—
54	C₂×D₃	—	9	52	no	—
55	C₃×C₄	C₂²×C₃⋊D₃	1	78	yes	—
56	Dic₃	—	9	52	no	—
57	C₃²	#1007	1	13	yes	complemented
58	Q₈	—	27	6	no	—
59	Q₈	C₂×C₃⋊(C₃⋊D₃)	1	4	yes	—
60	D₄	—	27	4	no	—
61	D₄	—	27	6	no	—
62	C₂×C₄	—	27	12	no	—
63	C₂×C₄	C₂×C₃⋊(C₃⋊D₃)	1	3	yes	—
64	C₂×C₃	—	2	13	no	—
65	C₂×C₃	C₂³×C₃⋊D₃	1	13	yes	—
66	D₃	—	18	52	no	—
67	C₂²	—	27	4	no	—
68	C₂²	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
69	C₄	—	27	4	no	—
70	C₄	C₂²×C₃⋊(C₃⋊D₃)	1	6	yes	—
71	C₃	#64364	1	13	yes	complemented
72	C₂	—	2	1	no	—
73	C₂	—	54	4	no	—
74	C₂	C₂³×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×C₃⋊(C₃⋊D₃)
commutator, socle, cosocle	C₂×C₃³	C₂⁴
Fitting	C₂×C₃³×Q₈	C₂
hypercenter	C₂×Q₈	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	2₋¹⁺⁴
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃³×Q₈	C₂	semidirect
2. #89080	C₂	semidirect
3. Q₈×C₃⋊(C₃⋊D₃)	C₂	semidirect
4. #89076	C₂	semidirect
12. C₂×C₃²×Q₈	D₃	semidirect
30. C₂×C₃×Q₈	C₃⋊D₃	semidirect
39. C₃³	2₋¹⁺⁴	semidirect
52. C₂×Q₈	C₃⋊(C₃⋊D₃)	semidirect
57. C₃²	#1007	semidirect
71. C₃	#64364	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃³×Q₈	2
#89080	2
Q₈×C₃⋊(C₃⋊D₃)	2
#89076	2
#64364	3