Subgroups — #11723985

Order 864.

952 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
2	C₂×#6308	C₂	1	1	yes	maximal, characteristic
3	#88857	C₂	1	4	yes	maximal
4	C₄×C₃⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic
5	#64153	—	3	13	no	maximal
6	C₃³×Q₈	—	2	2	no	—
7	C₂×C₃³×C₄	C₂²	1	2	yes	—
8	C₂×C₃³×C₄	C₂²	1	1	yes	characteristic
9	C₂×C₃⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
10	C₂×C₃⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
11	#6308	—	2	2	no	—
12	C₂×C₃²×Q₈	D₃	1	13	yes	—
13	C₂×#634	—	3	13	no	—
14	#3750	—	3	52	no	—
15	C₄×C₃⋊Dic₃	—	3	13	no	—
16	C₂²×C₃³	C₂³	1	1	yes	characteristic
17	C₃³×C₄	—	2	2	no	—
18	C₃³×C₄	D₄	1	2	yes	—
19	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
20	C₃⋊(C₃⋊Dic₃)	Q₈	1	4	yes	complemented
21	#944	—	9	13	no	—
22	C₃²×Q₈	—	2	26	no	—
23	C₂×C₃²×C₄	C₂×D₃	1	26	yes	—
24	C₂×C₃²×C₄	C₂×D₃	1	13	yes	—
25	C₂×C₃⋊Dic₃	—	3	26	no	—
26	C₂×C₃⋊Dic₃	—	3	26	no	—
27	#634	—	6	26	no	—
28	C₂×C₃³	C₂×D₄	1	1	yes	characteristic
29	C₂×C₃³	C₂×Q₈	1	2	yes	—
30	C₂×C₃×Q₈	C₃⋊D₃	1	13	yes	—
31	C₂×Dic₆	—	9	13	no	—
32	Dic₃⋊C₄	—	9	52	no	—
33	C₄×Dic₃	—	9	13	no	—
34	C₂²×C₃²	C₂²×D₃	1	13	yes	—
35	C₃²×C₄	—	2	26	no	—
36	C₃²×C₄	#67	1	26	yes	—
37	C₃⋊Dic₃	—	3	52	no	—
38	C₃⋊Dic₃	—	6	26	no	—
39	C₄⋊Q₈	—	27	1	no	Sylow
40	C₃³	C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
41	C₃×Q₈	—	2	26	no	—
42	C₂×C₃×C₄	C₂×C₃⋊D₃	1	26	yes	—
43	C₂×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
44	C₂×Dic₃	—	9	26	no	—
45	C₂×Dic₃	—	9	26	no	—
46	Dic₆	—	18	26	no	—
47	C₂×C₃²	D₃×Q₈	1	26	yes	—
48	C₂×C₃²	C₂×#67	1	13	yes	—
49	C₂×Q₈	—	27	1	no	—
50	C₂×Q₈	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
51	C₄⋊C₄	—	27	4	no	—
52	C₄²	—	27	1	no	—
53	C₂²×C₃	C₂²×C₃⋊D₃	1	13	yes	—
54	C₃×C₄	—	2	26	no	—
55	C₃×C₄	#638	1	26	yes	—
56	Dic₃	—	9	52	no	—
57	Dic₃	—	18	26	no	—
58	C₃²	#944	1	13	yes	complemented
59	Q₈	—	2	2	no	—
60	Q₈	—	54	2	no	—
61	C₂×C₄	—	27	2	no	—
62	C₂×C₄	—	27	2	no	—
63	C₂×C₄	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
64	C₂×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
65	C₂×C₃	Q₈×C₃⋊D₃	1	26	yes	—
66	C₂×C₃	C₂×#638	1	13	yes	—
67	C₂²	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
68	C₄	—	2	2	no	—
69	C₄	—	27	4	no	—
70	C₄	—	54	2	no	—
71	C₄	#6312	1	2	yes	—
72	C₃	#64153	1	13	yes	complemented
73	C₂	Q₈×C₃⋊(C₃⋊D₃)	1	2	yes	—
74	C₂	C₂×#6312	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×C₃⋊(C₃⋊D₃)
commutator, socle, prime radical, cosocle	C₂²×C₃³	C₂³
Fitting	C₂×C₃³×Q₈	C₂
hypercenter	C₂×Q₈	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	C₄⋊Q₈
LCS	G ▷ C₂²×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
20. C₃⋊(C₃⋊Dic₃)	Q₈	semidirect
40. C₃³	C₄⋊Q₈	semidirect
58. C₃²	#944	semidirect
72. C₃	#64153	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
C₂×#6308	2
#88857	2
C₄×C₃⋊(C₃⋊Dic₃)	2
#64153	3