Subgroups — #11757450

Order 972.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×C₉	C₃	1	1	yes	maximal, characteristic
2	C₃×#1094	C₃	1	1	yes	maximal, characteristic
3	C₃²⋊(C₂²⋊C₉)	C₃	1	1	yes	maximal, characteristic
4	C₃²⋊(C₂²⋊C₉)	C₃	1	1	yes	maximal, characteristic
5	3₋¹⁺²⋊C₉	—	4	1	no	maximal, Sylow
6	C₂×C₃²×C₉	—	3	1	no	—
7	C₂²×C₃³	C₃²	1	1	yes	characteristic
8	C₂²×C₃×C₉	—	3	1	no	—
9	C₂²×C₃×C₉	—	3	3	no	—
10	#1094	—	3	2	no	—
11	#1094	C₉	1	3	yes	complemented
12	C₃×C₂²⋊C₉	—	3	1	no	—
13	C₃×C₂²⋊C₉	—	3	1	no	—
14	C₃×C₂²⋊C₉	—	3	1	no	—
15	C₃×3₋¹⁺²	—	4	1	no	—
16	C₃²×C₉	A₄	1	1	yes	characteristic
17	C₃²⋊C₉	—	4	1	no	—
18	C₃²⋊C₉	—	4	1	no	—
19	C₂×C₃³	—	3	1	no	—
20	C₂×C₃×C₉	—	3	9	no	—
21	C₂×C₃×C₉	—	3	3	no	—
22	C₂²×C₃²	—	3	1	no	—
23	C₂²×C₃²	—	3	1	no	—
24	C₂²×C₃²	—	3	1	no	—
25	C₂²×C₃²	C₃×C₉	1	1	yes	characteristic
26	C₂²×C₃²	H(3,𝔽₃)	1	1	yes	characteristic
27	C₂²×C₃²	3₋¹⁺²	1	2	yes	—
28	C₂²×C₉	—	3	3	no	—
29	C₂²⋊C₉	—	3	3	no	—
30	C₂²⋊C₉	—	9	1	no	—
31	C₂²⋊C₉	—	9	1	no	—
32	C₃³	C₃×A₄	1	1	yes	characteristic
33	3₋¹⁺²	—	4	3	no	—
34	3₋¹⁺²	—	12	2	no	—
35	C₃×C₉	—	3	1	no	—
36	C₃×C₉	—	3	3	no	—
37	C₃×C₉	—	12	1	no	—
38	C₃×C₉	—	12	1	no	—
39	C₃×C₉	—	12	1	no	—
40	C₂×C₃²	—	3	3	no	—
41	C₂×C₃²	—	3	3	no	—
42	C₂×C₃²	—	3	3	no	—
43	C₂×C₃²	—	3	1	no	—
44	C₂×C₃²	—	3	2	no	—
45	C₂×C₃²	—	3	1	no	—
46	C₂×C₉	—	3	9	no	—
47	C₂²×C₃	—	3	1	no	—
48	C₂²×C₃	—	3	2	no	—
49	C₂²×C₃	C₃²⋊C₉	1	1	yes	characteristic
50	C₂²×C₃	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
51	C₂²×C₃	#731	1	1	yes	characteristic
52	C₂²×C₃	#733	1	1	yes	characteristic
53	C₃²	—	3	1	no	—
54	C₃²	—	3	1	no	—
55	C₃²	—	3	1	no	—
56	C₃²	C₉×A₄	1	1	yes	characteristic
57	C₃²	C₉⋊A₄	1	2	yes	—
58	C₃²	C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
59	C₉	—	3	3	no	—
60	C₉	—	12	3	no	—
61	C₉	—	36	1	no	—
62	C₉	—	36	1	no	—
63	C₂×C₃	—	3	3	no	—
64	C₂×C₃	—	3	6	no	—
65	C₂×C₃	—	3	1	no	—
66	C₂×C₃	—	3	1	no	—
67	C₂×C₃	—	3	1	no	—
68	C₂×C₃	—	3	1	no	—
69	C₂²	3₋¹⁺²⋊C₉	1	1	yes	characteristic, complemented, Sylow
70	C₃	—	3	1	no	—
71	C₃	—	3	2	no	—
72	C₃	#66357	1	1	yes	characteristic
73	C₃	#66359	1	1	yes	characteristic
74	C₃	(C₃×A₄)⋊C₉	1	1	yes	characteristic
75	C₃	#66368	1	1	yes	characteristic
76	C₂	—	3	1	no	—

	subgroup	G/H
center	C₃²	C₂²⋊H(3,𝔽₃)
commutator	C₂²×C₃²	C₃×C₉
Frattini	C₃³	C₃×A₄
Fitting	C₂²×C₃²×C₉	C₃
socle	C₂²×C₃²	H(3,𝔽₃)
prime radical, cosocle	C₂²×C₃³	C₃²
hypercenter	C₃²×C₉	A₄
hypocenter	C₂²	3₋¹⁺²⋊C₉
LCS	G ▷ C₂²×C₃² ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃²×C₉

Complements

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

N	H ≅ G/N	type
11. #1094	C₉	semidirect
69. C₂²	3₋¹⁺²⋊C₉	semidirect

Maximal subgroups

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

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