Subgroups — #11757844

Order 972.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
2	C₃×#1094	C₃	1	1	yes	maximal, characteristic
3	C₃²⋊(C₂²⋊C₉)	C₃	1	2	yes	maximal
4	C₉⋊(C₂²⋊C₉)	C₃	1	2	yes	maximal, complemented
5	(C₂²⋊C₉)⋊C₉	C₃	1	4	yes	maximal, complemented
6	(C₂²⋊C₉)⋊C₉	C₃	1	2	yes	maximal, complemented
7	C₉×C₂²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
8	#6896	—	4	1	no	maximal, Sylow
9	C₂×C₃×3₋¹⁺²	—	3	1	no	—
10	C₂²×C₃³	C₃²	1	1	yes	characteristic
11	C₂²×3₋¹⁺²	—	3	3	no	—
12	C₂²×C₃×C₉	C₃²	1	2	yes	—
13	C₂²×C₃×C₉	C₃²	1	1	yes	characteristic
14	#1094	—	3	3	no	—
15	C₃×C₂²⋊C₉	C₃²	1	2	yes	—
16	C₃×C₂²⋊C₉	C₃²	1	4	yes	—
17	C₃×C₂²⋊C₉	C₃²	1	2	yes	—
18	C₃×C₂²⋊C₉	C₃²	1	1	yes	characteristic
19	C₃×3₋¹⁺²	—	4	1	no	—
20	C₃×3₋¹⁺²	A₄	1	1	yes	characteristic
21	C₉⋊C₉	—	4	2	no	—
22	C₉⋊C₉	—	4	4	no	—
23	C₉⋊C₉	—	4	2	no	—
24	C₃²⋊C₉	—	4	2	no	—
25	C₉²	—	4	1	no	—
26	C₂×C₃³	—	3	1	no	—
27	C₂×3₋¹⁺²	—	3	9	no	—
28	C₂×C₃×C₉	—	3	2	no	—
29	C₂×C₃×C₉	—	3	1	no	—
30	C₂²×C₃²	—	3	1	no	—
31	C₂²×C₃²	—	3	1	no	—
32	C₂²×C₃²	—	3	2	no	—
33	C₂²×C₃²	C₃³	1	1	yes	characteristic
34	C₂²×C₉	—	3	2	no	—
35	C₂²×C₉	3₋¹⁺²	1	3	yes	complemented
36	C₂²⋊C₉	—	3	2	no	—
37	C₂²⋊C₉	—	3	2	no	—
38	C₂²⋊C₉	—	3	4	no	—
39	C₂²⋊C₉	3₋¹⁺²	1	3	yes	complemented
40	C₃³	C₃×A₄	1	1	yes	characteristic
41	3₋¹⁺²	—	3	3	no	—
42	3₋¹⁺²	—	12	3	no	—
43	C₃×C₉	—	4	1	no	—
44	C₃×C₉	—	4	2	no	—
45	C₃×C₉	—	4	4	no	—
46	C₃×C₉	—	4	2	no	—
47	C₃×C₉	C₃×A₄	1	2	yes	—
48	C₃×C₉	C₃×A₄	1	1	yes	characteristic
49	C₂×C₃²	—	3	1	no	—
50	C₂×C₃²	—	3	3	no	—
51	C₂×C₃²	—	9	1	no	—
52	C₂×C₃²	—	9	2	no	—
53	C₂×C₉	—	3	6	no	—
54	C₂×C₉	—	3	3	no	—
55	C₂²×C₃	—	9	1	no	—
56	C₂²×C₃	C₃×3₋¹⁺²	1	1	yes	characteristic
57	C₂²×C₃	C₃×3₋¹⁺²	1	1	yes	characteristic
58	C₂²×C₃	#737	1	2	yes	—
59	C₃²	—	3	1	no	—
60	C₃²	—	3	2	no	—
61	C₃²	—	3	1	no	—
62	C₃²	C₃²×A₄	1	1	yes	characteristic
63	C₉	—	3	2	no	—
64	C₉	—	4	3	no	—
65	C₉	—	12	2	no	—
66	C₉	—	12	2	no	—
67	C₉	—	12	4	no	—
68	C₉	#1094	1	3	yes	complemented
69	C₂×C₃	—	3	1	no	—
70	C₂×C₃	—	3	2	no	—
71	C₂×C₃	—	3	1	no	—
72	C₂×C₃	—	9	3	no	—
73	C₂²	#6896	1	1	yes	characteristic, complemented, Sylow
74	C₃	—	9	1	no	—
75	C₃	A₄×3₋¹⁺²	1	1	yes	characteristic
76	C₃	#66440	1	2	yes	—
77	C₃	C₃×#1094	1	1	yes	characteristic
78	C₂	—	3	1	no	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×A₄
commutator, socle, cosocle	C₂²×C₃²	C₃³
Fitting	C₂²×C₃×3₋¹⁺²	C₃
prime radical	C₂²×C₃³	C₃²
hypercenter	C₃×3₋¹⁺²	A₄
hypocenter	C₂²	#6896
LCS	G ▷ C₂²×C₃² ▷ C₂²
UCS	C₁ ◁ C₃² ◁ C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
4. C₉⋊(C₂²⋊C₉)	C₃	semidirect
5. (C₂²⋊C₉)⋊C₉	C₃	semidirect
6. (C₂²⋊C₉)⋊C₉	C₃	semidirect
7. C₉×C₂²⋊C₉	C₃	semidirect
35. C₂²×C₉	3₋¹⁺²	semidirect
39. C₂²⋊C₉	3₋¹⁺²	semidirect
68. C₉	#1094	semidirect
73. C₂²	#6896	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×3₋¹⁺²	3
C₃×#1094	3
C₃²⋊(C₂²⋊C₉)	3
C₉⋊(C₂²⋊C₉)	3
(C₂²⋊C₉)⋊C₉	3
C₉×C₂²⋊C₉	3
#6896	4