Subgroups — #11724517

Order 864.

228 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3832	C₂	1	1	yes	maximal, characteristic
2	#88989	C₂	1	3	yes	maximal, complemented
3	#88988	C₂	1	3	yes	maximal, complemented
4	#64289	—	3	1	no	maximal
5	C₃³×Q₈	C₄	1	1	yes	characteristic
6	C₃×C₄⋊(C₃⋊D₃)	C₄	1	3	yes	—
7	C₃×C₄×C₃⋊D₃	C₂²	1	3	yes	—
8	C₃⋊#622	C₂²	1	1	yes	characteristic
9	C₃⋊#622	C₂²	1	3	yes	—
10	#3832	D₃	1	1	yes	characteristic
11	C₃×#239	—	6	2	no	—
12	#3788	—	3	3	no	—
13	#3787	—	3	3	no	—
14	C₂×C₃×C₃⋊D₃	C₂×C₄	1	3	yes	—
15	C₃³×C₄	C₂×C₄	1	3	yes	—
16	C₃×C₃⋊Dic₃	C₂³	1	1	yes	characteristic
17	#948	—	9	1	no	maximal
18	C₃²×Q₈	—	2	2	no	—
19	C₃²×Q₈	—	4	2	no	—
20	C₃²×Q₈	Dic₃	1	1	yes	characteristic
21	C₄⋊(C₃⋊D₃)	Dic₃	1	3	yes	—
22	C₄×C₃⋊D₃	C₂×D₃	1	3	yes	—
23	C₃×D₁₂	—	6	6	no	—
24	C₃×C₄×D₃	—	6	6	no	—
25	#622	—	3	1	no	—
26	#622	—	3	3	no	—
27	C₂×C₃³	C₂²×C₄	1	1	yes	characteristic
28	C₃×C₃⋊D₃	—	2	3	no	—
29	C₃×#41	—	9	1	no	—
30	#239	—	6	2	no	—
31	#208	—	9	3	no	—
32	C₂×C₃⋊C₈	—	9	3	no	—
33	C₂×C₃⋊D₃	C₂×Dic₃	1	3	yes	—
34	C₂×C₃×D₃	—	6	6	no	—
35	C₃²×C₄	—	2	6	no	—
36	C₃²×C₄	—	4	6	no	—
37	C₃²×C₄	C₂×Dic₃	1	3	yes	—
38	C₃⋊Dic₃	C₂²×D₃	1	1	yes	characteristic
39	C₃×Dic₃	—	6	2	no	—
40	#131	—	27	1	no	Sylow
41	C₃³	#131	1	1	yes	characteristic, complemented, Sylow
42	C₃×Q₈	—	2	2	no	—
43	C₃×Q₈	—	4	2	no	—
44	C₃×Q₈	#157	1	1	yes	characteristic
45	C₃×D₄	—	9	3	no	—
46	C₂×C₃×C₄	—	9	3	no	—
47	D₁₂	—	6	6	no	—
48	C₄×D₃	—	6	6	no	—
49	C₃⋊C₈	—	9	1	no	—
50	C₃⋊C₈	—	9	3	no	—
51	C₂×C₃²	—	2	2	no	—
52	C₂×C₃²	—	4	2	no	—
53	C₂×C₃²	C₂²×Dic₃	1	1	yes	characteristic
54	C₃⋊D₃	—	2	3	no	—
55	C₃×D₃	—	12	6	no	—
56	#41	—	9	1	no	—
57	M₁₆	—	27	3	no	—
58	C₂×C₈	—	27	3	no	—
59	C₂²×C₃	—	9	3	no	—
60	C₂×D₃	—	6	6	no	—
61	C₃×C₄	—	2	6	no	—
62	C₃×C₄	—	4	6	no	—
63	C₃×C₄	—	9	1	no	—
64	C₃×C₄	C₂×#157	1	3	yes	—
65	Dic₃	—	6	2	no	—
66	C₃²	—	2	2	no	—
67	C₃²	—	4	2	no	—
68	C₃²	#948	1	1	yes	characteristic, complemented
69	Q₈	C₃⋊#157	1	1	yes	characteristic
70	D₄	—	9	3	no	—
71	C₂×C₄	—	9	3	no	—
72	C₈	—	27	1	no	—
73	C₈	—	27	3	no	—
74	C₂×C₃	—	2	2	no	—
75	C₂×C₃	—	4	2	no	—
76	C₂×C₃	—	18	3	no	—
77	C₂×C₃	C₂²×#157	1	1	yes	characteristic
78	D₃	—	12	6	no	—
79	C₂²	—	9	3	no	—
80	C₄	—	9	1	no	—
81	C₄	C₂×C₃⋊#157	1	3	yes	—
82	C₃	—	2	2	no	—
83	C₃	—	4	2	no	—
84	C₃	#64289	1	1	yes	characteristic, complemented
85	C₂	—	18	3	no	—
86	C₂	C₂²×C₃⋊#157	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×C₃⋊#157
commutator, socle	C₂×C₃³	C₂²×C₄
Fitting	C₃³×Q₈	C₄
prime radical	C₃×#3832	C₂
cosocle	C₃×C₃⋊Dic₃	C₂³
hypercenter	Q₈	C₃⋊#157
hypocenter	C₃³	#131
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂ ◁ Q₈

Complements

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

N	H ≅ G/N	type
2. #88989	C₂	semidirect
3. #88988	C₂	semidirect
41. C₃³	#131	semidirect
68. C₃²	#948	semidirect
84. C₃	#64289	semidirect

Maximal subgroups

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

subgroup	index
C₃×#3832	2
#88989	2
#88988	2
#64289	3
#948	9