Subgroups — #11723988

Order 864.

1064 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	C₂×C₄⋊(C₃⋊(C₃⋊D₃))	C₂	1	1	yes	maximal, characteristic
3	#88859	C₂	1	4	yes	maximal, complemented
4	C₄×C₃⋊(C₃⋊Dic₃)	C₂	1	1	yes	maximal, characteristic, complemented
5	#64156	—	3	13	no	maximal
6	C₂²×C₃⋊(C₃⋊D₃)	C₂²	1	2	yes	—
7	C₃³×Q₈	—	2	2	no	—
8	C₂×C₃³×C₄	C₂²	1	2	yes	—
9	C₂×C₃³×C₄	C₂²	1	1	yes	characteristic
10	C₂×C₃⋊(C₃⋊Dic₃)	C₂²	1	2	yes	—
11	C₄⋊(C₃⋊(C₃⋊D₃))	—	2	2	no	—
12	C₂×C₃²×Q₈	D₃	1	13	yes	complemented
13	C₂×C₄⋊(C₃⋊D₃)	—	3	13	no	—
14	#3752	—	3	52	no	—
15	C₄×C₃⋊Dic₃	—	3	13	no	—
16	C₂²×C₃³	C₂³	1	1	yes	characteristic
17	C₂×C₃⋊(C₃⋊D₃)	—	2	2	no	—
18	C₂×C₃⋊(C₃⋊D₃)	—	2	4	no	—
19	C₃³×C₄	—	2	2	no	—
20	C₃³×C₄	D₄	1	2	yes	—
21	C₃⋊(C₃⋊Dic₃)	—	2	2	no	—
22	#947	—	9	13	no	—
23	C₂²×C₃⋊D₃	—	3	26	no	—
24	C₃²×Q₈	—	2	26	no	—
25	C₂×C₃²×C₄	C₂×D₃	1	26	yes	—
26	C₂×C₃²×C₄	C₂×D₃	1	13	yes	—
27	C₂×C₃⋊Dic₃	—	3	26	no	—
28	C₄⋊(C₃⋊D₃)	—	6	26	no	—
29	C₂×C₃³	C₂×D₄	1	1	yes	characteristic
30	C₂×C₃³	#41	1	2	yes	—
31	C₃⋊(C₃⋊D₃)	—	4	2	no	—
32	C₂×C₃×Q₈	C₃⋊D₃	1	13	yes	complemented
33	C₂×D₁₂	—	9	13	no	—
34	(C₂×D₃)⋊C₄	—	9	52	no	—
35	C₄×Dic₃	—	9	13	no	—
36	C₂²×C₃²	C₂²×D₃	1	13	yes	—
37	C₂×C₃⋊D₃	—	6	52	no	—
38	C₂×C₃⋊D₃	—	6	26	no	—
39	C₃²×C₄	—	2	26	no	—
40	C₃²×C₄	#67	1	26	yes	—
41	C₃⋊Dic₃	—	6	26	no	—
42	#124	—	27	1	no	Sylow
43	C₃³	#124	1	1	yes	characteristic, complemented, Sylow
44	C₂²×D₃	—	9	26	no	—
45	C₃×Q₈	—	2	26	no	—
46	C₂×C₃×C₄	C₂×C₃⋊D₃	1	26	yes	—
47	C₂×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
48	C₂×Dic₃	—	9	26	no	—
49	D₁₂	—	18	26	no	—
50	C₂×C₃²	#239	1	26	yes	—
51	C₂×C₃²	C₂×#67	1	13	yes	—
52	C₃⋊D₃	—	12	26	no	—
53	C₂×Q₈	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
54	C₂×D₄	—	27	1	no	—
55	C₂²⋊C₄	—	27	4	no	—
56	C₄²	—	27	1	no	—
57	C₂²×C₃	C₂²×C₃⋊D₃	1	13	yes	—
58	C₂×D₃	—	18	52	no	—
59	C₂×D₃	—	18	26	no	—
60	C₃×C₄	—	2	26	no	—
61	C₃×C₄	#638	1	26	yes	—
62	Dic₃	—	18	26	no	—
63	C₃²	#947	1	13	yes	complemented
64	C₂³	—	27	2	no	—
65	Q₈	—	2	2	no	—
66	D₄	—	54	2	no	—
67	C₂×C₄	—	27	2	no	—
68	C₂×C₄	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
69	C₂×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
70	C₂×C₃	#3832	1	26	yes	—
71	C₂×C₃	C₂×#638	1	13	yes	—
72	D₃	—	36	26	no	—
73	C₂²	—	54	4	no	—
74	C₂²	—	54	2	no	—
75	C₂²	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
76	C₄	—	2	2	no	—
77	C₄	—	54	2	no	—
78	C₄	#6312	1	2	yes	—
79	C₃	#64156	1	13	yes	complemented
80	C₂	—	108	2	no	—
81	C₂	#89080	1	2	yes	—
82	C₂	C₂×#6312	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×C₃⋊(C₃⋊D₃)
commutator, socle, cosocle	C₂²×C₃³	C₂³
Fitting	C₂×C₃³×Q₈	C₂
prime radical	C₂×C₄⋊(C₃⋊(C₃⋊D₃))	C₂
hypercenter	C₂×Q₈	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#124
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
1. C₂×C₃³×Q₈	C₂	semidirect
3. #88859	C₂	semidirect
4. C₄×C₃⋊(C₃⋊Dic₃)	C₂	semidirect
12. C₂×C₃²×Q₈	D₃	semidirect
32. C₂×C₃×Q₈	C₃⋊D₃	semidirect
43. C₃³	#124	semidirect
53. C₂×Q₈	C₃⋊(C₃⋊D₃)	semidirect
63. C₃²	#947	semidirect
79. C₃	#64156	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×C₄⋊(C₃⋊(C₃⋊D₃))	2
#88859	2
C₄×C₃⋊(C₃⋊Dic₃)	2
#64156	3