Subgroups — #11723948

Order 864.

868 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#89076	C₂	1	1	yes	maximal, characteristic
2	C₃³×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂	1	1	yes	maximal, characteristic, complemented
4	#88850	C₂	1	2	yes	maximal, complemented
5	C₈×C₃⋊(C₃⋊D₃)	C₂	1	2	yes	maximal, complemented
6	#64116	—	3	13	no	maximal
7	C₂×C₃³×C₄	C₂²	1	1	yes	characteristic
8	#6312	C₄	1	2	yes	—
9	C₄⋊(C₃⋊(C₃⋊D₃))	C₄	1	1	yes	characteristic
10	C₄×C₃⋊(C₃⋊D₃)	C₂²	1	2	yes	—
11	#6308	C₄	1	1	yes	characteristic
12	C₃³×C₈	C₂²	1	2	yes	—
13	C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
14	C₃⋊(C₃⋊(C₃⋊C₈))	C₂²	1	1	yes	characteristic
15	#3828	—	3	13	no	—
16	C₃²×M₁₆	D₃	1	13	yes	complemented
17	C₂×C₃⋊(C₃⋊C₈)	—	3	13	no	—
18	#3743	—	3	26	no	—
19	C₈×C₃⋊D₃	—	3	26	no	—
20	C₂²×C₃³	C₂×C₄	1	1	yes	characteristic
21	C₂×C₃⋊(C₃⋊D₃)	C₂×C₄	1	2	yes	—
22	C₃³×C₄	C₂×C₄	1	1	yes	characteristic
23	C₃³×C₄	C₂³	1	1	yes	characteristic
24	C₃⋊(C₃⋊Dic₃)	C₂×C₄	1	2	yes	—
25	#907	—	9	13	no	—
26	C₂×C₃²×C₄	C₂×D₃	1	13	yes	—
27	#638	—	3	26	no	—
28	C₄⋊(C₃⋊D₃)	—	3	13	no	—
29	C₄×C₃⋊D₃	—	3	26	no	—
30	#634	—	3	13	no	—
31	C₃²×C₈	C₂×D₃	1	26	yes	—
32	C₃⋊(C₃⋊C₈)	—	3	13	no	—
33	C₃⋊(C₃⋊C₈)	—	3	13	no	—
34	C₂×C₃³	—	2	1	no	—
35	C₂×C₃³	C₂²×C₄	1	1	yes	characteristic
36	C₃⋊(C₃⋊D₃)	—	2	2	no	—
37	#235	—	9	13	no	—
38	C₃×M₁₆	C₃⋊D₃	1	13	yes	complemented
39	C₂×C₃⋊C₈	—	9	13	no	—
40	ΓL(1,𝔽₂₅)	—	9	26	no	—
41	D₃×C₈	—	9	26	no	—
42	C₂²×C₃²	C₄×D₃	1	13	yes	—
43	C₂×C₃⋊D₃	—	3	26	no	—
44	C₃²×C₄	C₄×D₃	1	13	yes	—
45	C₃²×C₄	C₂²×D₃	1	13	yes	—
46	C₃⋊Dic₃	—	3	26	no	—
47	#131	—	27	1	no	Sylow
48	C₃³	#131	1	1	yes	characteristic, complemented, Sylow
49	C₂×C₃×C₄	C₂×C₃⋊D₃	1	13	yes	—
50	#67	—	9	26	no	—
51	D₁₂	—	9	13	no	—
52	C₄×D₃	—	9	26	no	—
53	Dic₆	—	9	13	no	—
54	C₃×C₈	C₂×C₃⋊D₃	1	26	yes	—
55	C₃⋊C₈	—	9	13	no	—
56	C₃⋊C₈	—	9	13	no	—
57	C₂×C₃²	—	2	13	no	—
58	C₂×C₃²	C₂×C₄×D₃	1	13	yes	—
59	C₃⋊D₃	—	6	26	no	—
60	#41	—	27	1	no	—
61	M₁₆	—	27	2	no	—
62	M₁₆	C₃⋊(C₃⋊D₃)	1	1	yes	characteristic, complemented
63	C₂×C₈	—	27	1	no	—
64	C₂×C₈	—	27	2	no	—
65	C₂²×C₃	C₄×C₃⋊D₃	1	13	yes	—
66	C₂×D₃	—	9	26	no	—
67	C₃×C₄	C₄×C₃⋊D₃	1	13	yes	—
68	C₃×C₄	C₂²×C₃⋊D₃	1	13	yes	—
69	Dic₃	—	9	26	no	—
70	C₃²	#907	1	13	yes	complemented
71	Q₈	—	27	1	no	—
72	D₄	—	27	2	no	—
73	D₄	—	27	1	no	—
74	C₂×C₄	—	27	2	no	—
75	C₂×C₄	C₂×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
76	C₈	—	27	1	no	—
77	C₈	—	27	1	no	—
78	C₈	C₂×C₃⋊(C₃⋊D₃)	1	2	yes	—
79	C₂×C₃	—	2	13	no	—
80	C₂×C₃	C₂×C₄×C₃⋊D₃	1	13	yes	—
81	D₃	—	18	26	no	—
82	C₂²	—	27	2	no	—
83	C₂²	C₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
84	C₄	—	27	2	no	—
85	C₄	C₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
86	C₄	C₂²×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic
87	C₃	#64116	1	13	yes	complemented
88	C₂	—	2	1	no	—
89	C₂	—	54	2	no	—
90	C₂	C₂×C₄×C₃⋊(C₃⋊D₃)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂²×C₃⋊(C₃⋊D₃)
commutator, socle	C₂×C₃³	C₂²×C₄
Fitting	C₃³×M₁₆	C₂
prime radical	#89076	C₂
cosocle	C₃³×C₄	C₂³
hypercenter	M₁₆	C₃⋊(C₃⋊D₃)
hypocenter	C₃³	#131
LCS	G ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₄ ◁ M₁₆

Complements

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

N	H ≅ G/N	type
2. C₃³×M₁₆	C₂	semidirect
3. C₂×C₃⋊(C₃⋊(C₃⋊C₈))	C₂	semidirect
4. #88850	C₂	semidirect
5. C₈×C₃⋊(C₃⋊D₃)	C₂	semidirect
16. C₃²×M₁₆	D₃	semidirect
38. C₃×M₁₆	C₃⋊D₃	semidirect
48. C₃³	#131	semidirect
62. M₁₆	C₃⋊(C₃⋊D₃)	semidirect
70. C₃²	#907	semidirect
87. C₃	#64116	semidirect

Maximal subgroups

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

subgroup	index
#89076	2
C₃³×M₁₆	2
C₂×C₃⋊(C₃⋊(C₃⋊C₈))	2
#88850	2
C₈×C₃⋊(C₃⋊D₃)	2
#64116	3