Subgroups — #11721713

Order 864.

300 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄²×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	#88536	C₂	1	3	yes	maximal
3	#88534	C₂	1	3	yes	maximal, complemented
4	#64085	—	3	1	no	maximal
5	#63438	—	3	3	no	maximal
6	C₂²×C₃⋊D₉	C₂²	1	1	yes	characteristic
7	C₂×C₃×C₄×C₉	C₂²	1	3	yes	—
8	C₂×C₃⋊Dic₉	C₂²	1	3	yes	—
9	C₃²×C₄²	D₃	1	1	yes	characteristic
10	#3752	—	3	3	no	—
11	#3750	—	3	3	no	—
12	C₄²×C₉	D₃	1	3	yes	complemented
13	(C₂×D₉)⋊C₄	—	3	9	no	—
14	Dic₉⋊C₄	—	3	9	no	—
15	C₂²×C₃×C₉	C₂³	1	1	yes	characteristic
16	C₂×C₃⋊D₉	—	2	3	no	—
17	C₃×C₄×C₉	—	2	3	no	—
18	C₃⋊Dic₉	—	2	3	no	—
19	#876	—	9	3	no	—
20	#876	—	9	1	no	—
21	C₂²×C₃⋊D₃	—	3	1	no	—
22	C₂×C₃²×C₄	C₂×D₃	1	3	yes	—
23	C₂×C₃⋊Dic₃	—	3	3	no	—
24	C₂²×D₉	—	3	3	no	—
25	C₂×C₄×C₉	C₂×D₃	1	9	yes	—
26	C₂×Dic₉	—	3	9	no	—
27	C₂×C₃×C₉	#41	1	3	yes	—
28	C₃⋊D₉	—	4	1	no	—
29	C₃×C₄²	D₉	1	3	yes	complemented
30	C₃×C₄²	C₃⋊D₃	1	1	yes	characteristic
31	(C₂×D₃)⋊C₄	—	9	9	no	—
32	(C₂×D₃)⋊C₄	—	9	3	no	—
33	Dic₃⋊C₄	—	9	9	no	—
34	Dic₃⋊C₄	—	9	3	no	—
35	C₂²×C₃²	C₂²×D₃	1	1	yes	characteristic
36	C₂×C₃⋊D₃	—	6	3	no	—
37	C₃²×C₄	—	2	3	no	—
38	C₃⋊Dic₃	—	6	3	no	—
39	C₂²×C₉	C₂²×D₃	1	3	yes	—
40	C₂×D₉	—	6	9	no	—
41	C₄×C₉	—	2	9	no	—
42	Dic₉	—	6	9	no	—
43	#126	—	27	1	no	Sylow
44	C₃×C₉	#126	1	1	yes	characteristic, complemented, Sylow
45	C₂²×D₃	—	9	3	no	—
46	C₂²×D₃	—	9	1	no	—
47	C₂×C₃×C₄	C₂×D₉	1	9	yes	—
48	C₂×C₃×C₄	C₂×C₃⋊D₃	1	3	yes	—
49	C₂×Dic₃	—	9	9	no	—
50	C₂×Dic₃	—	9	3	no	—
51	C₂×C₃²	#235	1	3	yes	—
52	C₃⋊D₃	—	12	1	no	—
53	C₂×C₉	#235	1	9	yes	—
54	D₉	—	12	3	no	—
55	C₄⋊C₄	—	27	3	no	—
56	C₂²⋊C₄	—	27	3	no	—
57	C₄²	C₃⋊D₉	1	1	yes	characteristic, complemented
58	C₂²×C₃	C₂²×D₉	1	3	yes	—
59	C₂²×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
60	C₂×D₃	—	18	9	no	—
61	C₂×D₃	—	18	3	no	—
62	C₃×C₄	—	2	9	no	—
63	C₃×C₄	—	2	3	no	—
64	Dic₃	—	18	9	no	—
65	Dic₃	—	18	3	no	—
66	C₃²	#876	1	1	yes	characteristic
67	C₉	#876	1	3	yes	complemented
68	C₂³	—	27	1	no	—
69	C₂×C₄	—	27	3	no	—
70	C₂×C₄	C₂×C₃⋊D₉	1	3	yes	—
71	C₂×C₃	#3697	1	9	yes	—
72	C₂×C₃	#3828	1	3	yes	—
73	D₃	—	36	1	no	—
74	D₃	—	36	3	no	—
75	C₂²	—	54	3	no	—
76	C₂²	C₂²×C₃⋊D₉	1	1	yes	characteristic
77	C₄	—	2	3	no	—
78	C₄	—	54	3	no	—
79	C₃	#63438	1	3	yes	complemented
80	C₃	#64085	1	1	yes	characteristic
81	C₂	—	108	1	no	—
82	C₂	#88736	1	3	yes	—

	subgroup	G/H
center	C₂²	C₂²×C₃⋊D₉
commutator, cosocle	C₂²×C₃×C₉	C₂³
Frattini	C₂²×C₃	C₂²×C₃⋊D₃
Fitting	C₃×C₄²×C₉	C₂
socle	C₂²×C₃²	C₂²×D₃
prime radical	C₂²×C₃⋊D₉	C₂²
hypercenter	C₄²	C₃⋊D₉
hypocenter	C₃×C₉	#126
LCS	G ▷ C₂²×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
1. C₃×C₄²×C₉	C₂	semidirect
3. #88534	C₂	semidirect
12. C₄²×C₉	D₃	semidirect
29. C₃×C₄²	D₉	semidirect
44. C₃×C₉	#126	semidirect
57. C₄²	C₃⋊D₉	semidirect
67. C₉	#876	semidirect
79. C₃	#63438	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₄²×C₉	2
#88536	2
#88534	2
#64085	3
#63438	3