Subgroups — #11757461

Order 972.

392 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃⁴	C₃	1	1	yes	maximal, characteristic
2	C₃²⋊(C₂²⋊C₉)	C₃	1	3	yes	maximal, complemented
3	#6866	—	4	1	no	maximal, Sylow
4	C₂×C₃⁴	—	3	1	no	—
5	C₂²×C₃³	—	3	9	no	—
6	C₂²×C₃³	—	3	1	no	—
7	C₂²×C₃³	—	3	2	no	—
8	C₂²×C₃³	C₉	1	3	yes	complemented
9	C₂²×C₃³	C₃²	1	1	yes	characteristic
10	C₃×C₂²⋊C₉	—	3	3	no	—
11	C₃⁴	A₄	1	1	yes	characteristic
12	C₃²⋊C₉	—	4	3	no	—
13	C₂×C₃³	—	3	27	no	—
14	C₂×C₃³	—	3	6	no	—
15	C₂×C₃³	—	3	3	no	—
16	C₂×C₃³	—	3	3	no	—
17	C₂×C₃³	—	3	1	no	—
18	C₂²×C₃²	—	3	18	no	—
19	C₂²×C₃²	—	3	3	no	—
20	C₂²×C₃²	—	3	9	no	—
21	C₂²×C₃²	—	3	3	no	—
22	C₂²×C₃²	—	3	9	no	—
23	C₂²×C₃²	C₃×C₉	1	1	yes	characteristic
24	C₂²×C₃²	H(3,𝔽₃)	1	1	yes	characteristic
25	C₂²×C₃²	3₋¹⁺²	1	2	yes	—
26	C₂²⋊C₉	—	9	3	no	—
27	C₃³	—	3	9	no	—
28	C₃³	—	3	2	no	—
29	C₃³	—	3	1	no	—
30	C₃³	C₂²⋊C₉	1	3	yes	complemented
31	C₃³	C₃×A₄	1	1	yes	characteristic
32	C₃×C₉	—	12	3	no	—
33	C₂×C₃²	—	3	27	no	—
34	C₂×C₃²	—	3	54	no	—
35	C₂×C₃²	—	3	9	no	—
36	C₂×C₃²	—	3	27	no	—
37	C₂×C₃²	—	3	9	no	—
38	C₂×C₃²	—	3	1	no	—
39	C₂×C₃²	—	3	2	no	—
40	C₂×C₃²	—	3	1	no	—
41	C₂²×C₃	—	3	9	no	—
42	C₂²×C₃	—	3	1	no	—
43	C₂²×C₃	—	3	2	no	—
44	C₂²×C₃	C₃²⋊C₉	1	1	yes	characteristic
45	C₂²×C₃	AΣL(1,𝔽₂₇)	1	3	yes	—
46	C₃²	—	3	9	no	—
47	C₃²	—	3	18	no	—
48	C₃²	—	3	3	no	—
49	C₃²	—	3	9	no	—
50	C₃²	—	3	3	no	—
51	C₃²	C₃×C₂²⋊C₉	1	1	yes	characteristic
52	C₃²	#1094	1	2	yes	—
53	C₃²	C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
54	C₉	—	36	3	no	—
55	C₂×C₃	—	3	27	no	—
56	C₂×C₃	—	3	3	no	—
57	C₂×C₃	—	3	6	no	—
58	C₂×C₃	—	3	1	no	—
59	C₂×C₃	—	3	3	no	—
60	C₂²	#6866	1	1	yes	characteristic, complemented, Sylow
61	C₃	—	3	9	no	—
62	C₃	—	3	1	no	—
63	C₃	—	3	2	no	—
64	C₃	C₃²⋊(C₂²⋊C₉)	1	1	yes	characteristic
65	C₃	#66368	1	3	yes	—
66	C₂	—	3	1	no	—

	subgroup	G/H
center	C₃²	C₂²⋊H(3,𝔽₃)
commutator	C₂²×C₃²	C₃×C₉
Frattini	C₃³	C₃×A₄
Fitting, prime radical	C₂²×C₃⁴	C₃
socle	C₂²×C₃²	H(3,𝔽₃)
cosocle	C₂²×C₃³	C₃²
hypercenter	C₃⁴	A₄
hypocenter	C₂²	#6866
LCS	G ▷ C₂²×C₃² ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₃² ◁ C₃³ ◁ C₃⁴

Complements

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

N	H ≅ G/N	type
2. C₃²⋊(C₂²⋊C₉)	C₃	semidirect
8. C₂²×C₃³	C₉	semidirect
30. C₃³	C₂²⋊C₉	semidirect
60. C₂²	#6866	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃⁴	3
C₃²⋊(C₂²⋊C₉)	3
#6866	4