Subgroups — #11720534

Order 864.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₈×C₉	C₂	1	1	yes	maximal, characteristic
2	C₃⋊(C₉⋊C₁₆)	C₂	1	2	yes	maximal, complemented
3	#63639	—	3	1	no	maximal
4	#63371	—	3	3	no	maximal
5	C₂×C₃×C₄×C₉	C₄	1	1	yes	characteristic
6	C₃×C₈×C₉	C₄	1	1	yes	characteristic
7	C₃×C₈×C₉	C₂²	1	1	yes	characteristic
8	C₂×C₃²×C₈	D₃	1	1	yes	characteristic
9	C₃⋊(C₃⋊C₁₆)	—	3	2	no	—
10	C₂×C₈×C₉	D₃	1	3	yes	—
11	C₉⋊C₁₆	—	3	6	no	—
12	C₂²×C₃×C₉	C₈	1	1	yes	characteristic
13	C₃×C₄×C₉	C₈	1	1	yes	characteristic
14	C₃×C₄×C₉	C₂×C₄	1	1	yes	characteristic
15	#812	—	9	3	no	—
16	#812	—	9	1	no	—
17	C₂×C₃²×C₄	Dic₃	1	1	yes	characteristic
18	C₃²×C₈	Dic₃	1	1	yes	characteristic
19	C₃²×C₈	C₂×D₃	1	1	yes	characteristic
20	C₂×C₄×C₉	Dic₃	1	3	yes	—
21	C₈×C₉	Dic₃	1	3	yes	—
22	C₈×C₉	C₂×D₃	1	3	yes	—
23	C₂×C₃×C₉	—	2	1	no	—
24	C₂×C₃×C₉	C₂×C₈	1	1	yes	characteristic
25	C₂×C₃×C₈	D₉	1	3	yes	—
26	C₂×C₃×C₈	C₃⋊D₃	1	1	yes	characteristic
27	C₃⋊C₁₆	—	9	6	no	—
28	C₃⋊C₁₆	—	9	2	no	—
29	C₂²×C₃²	C₃⋊C₈	1	1	yes	characteristic
30	C₃²×C₄	C₃⋊C₈	1	1	yes	characteristic
31	C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
32	C₂²×C₉	C₃⋊C₈	1	3	yes	—
33	C₄×C₉	C₃⋊C₈	1	3	yes	—
34	C₄×C₉	C₂×Dic₃	1	3	yes	—
35	M₃₂	—	27	1	no	Sylow
36	C₃×C₉	M₃₂	1	1	yes	characteristic, complemented, Sylow
37	C₂×C₃×C₄	Dic₉	1	3	yes	—
38	C₂×C₃×C₄	C₃⋊Dic₃	1	1	yes	characteristic
39	C₃×C₈	Dic₉	1	3	yes	—
40	C₃×C₈	C₂×D₉	1	3	yes	—
41	C₃×C₈	C₃⋊Dic₃	1	1	yes	characteristic
42	C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
43	C₂×C₃²	—	2	1	no	—
44	C₂×C₃²	C₂×C₃⋊C₈	1	1	yes	characteristic
45	C₂×C₉	—	2	3	no	—
46	C₂×C₉	C₂×C₃⋊C₈	1	3	yes	—
47	C₂×C₈	C₃⋊D₉	1	1	yes	characteristic
48	C₁₆	—	27	2	no	—
49	C₂²×C₃	C₉⋊C₈	1	3	yes	—
50	C₂²×C₃	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
51	C₃×C₄	C₉⋊C₈	1	3	yes	—
52	C₃×C₄	C₂×Dic₉	1	3	yes	—
53	C₃×C₄	C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
54	C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
55	C₃²	#812	1	1	yes	characteristic
56	C₉	#812	1	3	yes	complemented
57	C₂×C₄	C₃⋊Dic₉	1	1	yes	characteristic
58	C₈	C₃⋊Dic₉	1	1	yes	characteristic
59	C₈	C₂×C₃⋊D₉	1	1	yes	characteristic
60	C₂×C₃	—	2	3	no	—
61	C₂×C₃	—	2	1	no	—
62	C₂×C₃	C₂×C₉⋊C₈	1	3	yes	—
63	C₂×C₃	C₂×C₃⋊(C₃⋊C₈)	1	1	yes	characteristic
64	C₂²	C₃⋊(C₉⋊C₈)	1	1	yes	characteristic
65	C₄	C₃⋊(C₉⋊C₈)	1	1	yes	characteristic
66	C₄	C₂×C₃⋊Dic₉	1	1	yes	characteristic
67	C₃	#63371	1	3	yes	complemented
68	C₃	#63639	1	1	yes	characteristic
69	C₂	—	2	1	no	—
70	C₂	C₂×C₃⋊(C₉⋊C₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂×C₃⋊D₉
commutator	C₂×C₃×C₉	C₂×C₈
Frattini	C₃×C₈	C₂×C₃⋊D₃
Fitting	C₂×C₃×C₈×C₉	C₂
socle	C₂×C₃²	C₂×C₃⋊C₈
prime radical	C₂²×C₃²	C₃⋊C₈
cosocle	C₃×C₈×C₉	C₂²
hypercenter	C₂×C₈	C₃⋊D₉
hypocenter	C₃×C₉	M₃₂
LCS	G ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₈ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
2. C₃⋊(C₉⋊C₁₆)	C₂	semidirect
36. C₃×C₉	M₃₂	semidirect
56. C₉	#812	semidirect
67. C₃	#63371	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×C₈×C₉	2
C₃⋊(C₉⋊C₁₆)	2
#63639	3
#63371	3