Subgroups — #11723096

Order 864.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#88989	C₂	1	2	yes	maximal
2	C₃×C₈×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic
3	#63771	—	3	1	no	maximal
4	C₃×C₄×C₃⋊D₃	C₂²	1	1	yes	characteristic
5	C₃⋊#622	—	2	2	no	—
6	C₃³×C₈	C₄	1	1	yes	characteristic
7	C₃×C₃⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
8	#3788	—	3	2	no	—
9	C₈×C₃⋊D₃	D₃	1	1	yes	characteristic
10	C₃×D₃×C₈	—	6	2	no	—
11	C₂×C₃×C₃⋊D₃	Q₈	1	1	yes	characteristic
12	C₃³×C₄	C₂×C₄	1	1	yes	characteristic
13	C₃×C₃⋊Dic₃	D₄	1	1	yes	characteristic
14	#819	—	9	1	no	maximal
15	C₄×C₃⋊D₃	C₂×D₃	1	1	yes	characteristic
16	C₃×C₄×D₃	—	6	2	no	—
17	#622	—	6	2	no	—
18	C₃²×C₈	—	2	2	no	—
19	C₃²×C₈	—	4	2	no	—
20	C₃²×C₈	Dic₃	1	1	yes	characteristic
21	C₃⋊(C₃⋊C₈)	Dic₃	1	1	yes	characteristic
22	C₃×C₃⋊C₈	—	6	2	no	—
23	C₂×C₃³	C₄⋊C₄	1	1	yes	characteristic
24	C₃×C₃⋊D₃	—	2	1	no	—
25	C₂×C₃×C₈	—	9	1	no	—
26	#208	—	9	2	no	—
27	D₃×C₈	—	6	2	no	—
28	C₂×C₃⋊D₃	Dic₆	1	1	yes	characteristic
29	C₂×C₃×D₃	—	6	2	no	—
30	C₃²×C₄	—	2	2	no	—
31	C₃²×C₄	—	4	2	no	—
32	C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
33	C₃⋊Dic₃	D₁₂	1	1	yes	characteristic
34	C₃×Dic₃	—	6	2	no	—
35	#108	—	27	1	no	Sylow
36	C₃³	#108	1	1	yes	characteristic, complemented, Sylow
37	C₂×C₃×C₄	—	9	1	no	—
38	C₄×D₃	—	6	2	no	—
39	C₃×C₈	—	2	2	no	—
40	C₃×C₈	—	4	2	no	—
41	C₃×C₈	—	9	1	no	—
42	C₃×C₈	#157	1	1	yes	characteristic
43	C₃⋊C₈	—	6	2	no	—
44	C₃⋊C₈	—	18	2	no	—
45	C₂×C₃²	—	2	2	no	—
46	C₂×C₃²	—	4	2	no	—
47	C₂×C₃²	C₄⋊Dic₃	1	1	yes	characteristic
48	C₃⋊D₃	—	2	1	no	—
49	C₃×D₃	—	6	4	no	—
50	M₁₆	—	27	2	no	—
51	C₂×C₈	—	9	1	no	—
52	C₂²×C₃	—	9	1	no	—
53	C₂×D₃	—	6	2	no	—
54	C₃×C₄	—	2	2	no	—
55	C₃×C₄	—	4	2	no	—
56	C₃×C₄	—	9	1	no	—
57	C₃×C₄	C₂×#157	1	1	yes	characteristic
58	Dic₃	—	6	2	no	—
59	C₃²	—	2	2	no	—
60	C₃²	—	4	2	no	—
61	C₃²	#819	1	1	yes	characteristic, complemented
62	C₂×C₄	—	9	1	no	—
63	C₈	—	9	1	no	—
64	C₈	—	54	2	no	—
65	C₈	C₃⋊#157	1	1	yes	characteristic
66	C₂×C₃	—	2	2	no	—
67	C₂×C₃	—	4	2	no	—
68	C₂×C₃	—	18	1	no	—
69	C₂×C₃	C₄⋊#157	1	1	yes	characteristic
70	D₃	—	6	4	no	—
71	C₂²	—	9	1	no	—
72	C₄	—	9	1	no	—
73	C₄	C₂×C₃⋊#157	1	1	yes	characteristic
74	C₃	—	2	2	no	—
75	C₃	—	4	2	no	—
76	C₃	#63771	1	1	yes	characteristic, complemented
77	C₂	—	18	1	no	—
78	C₂	C₄⋊(C₃⋊#157)	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₄⋊(C₃⋊#157)
commutator	C₃³×C₄	C₂×C₄
Frattini	C₄	C₂×C₃⋊#157
Fitting	C₃³×C₈	C₄
socle	C₂×C₃³	C₄⋊C₄
prime radical	C₂×C₃×C₃⋊D₃	Q₈
cosocle	C₃×C₄×C₃⋊D₃	C₂²
hypercenter	C₈	C₃⋊#157
hypocenter	C₃³	#108
LCS	G ▷ C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈

Complements

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

N	H ≅ G/N	type
36. C₃³	#108	semidirect
61. C₃²	#819	semidirect
76. C₃	#63771	semidirect

Maximal subgroups

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

subgroup	index
#88989	2
C₃×C₈×C₃⋊D₃	2
#63771	3
#819	9