Subgroups — #11757838

Order 972.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
2	C₃×#1094	C₃	1	2	yes	maximal, complemented
3	C₃×#1094	C₃	1	1	yes	maximal, characteristic, complemented
4	#66361	C₃	1	9	yes	maximal, complemented
5	#6910	—	4	1	no	maximal, Sylow
6	C₂×C₃×H(3,𝔽₃)	—	3	1	no	—
7	C₂²×C₃³	—	3	1	no	—
8	C₂²×C₃³	C₃²	1	1	yes	characteristic
9	C₂²×H(3,𝔽₃)	—	3	2	no	—
10	C₂²×H(3,𝔽₃)	C₃²	1	3	yes	—
11	#1094	—	3	2	no	—
12	#1094	—	3	2	no	—
13	#1094	—	3	2	no	—
14	#1094	—	3	1	no	—
15	#1094	C₃²	1	6	yes	complemented
16	C₃×C₂²⋊C₉	—	3	2	no	—
17	C₃×C₂²⋊C₉	C₃²	1	3	yes	complemented
18	C₃×3₋¹⁺²	—	4	1	no	—
19	C₃×3₋¹⁺²	—	4	2	no	—
20	C₃×H(3,𝔽₃)	A₄	1	1	yes	characteristic
21	#731	—	4	9	no	—
22	C₂×C₃³	—	3	3	no	—
23	C₂×C₃³	—	3	1	no	—
24	C₂×H(3,𝔽₃)	—	3	6	no	—
25	C₂×H(3,𝔽₃)	—	3	3	no	—
26	C₂²×C₃²	—	3	3	no	—
27	C₂²×C₃²	—	9	1	no	—
28	C₂²×C₃²	—	9	3	no	—
29	C₂²×C₃²	H(3,𝔽₃)	1	2	yes	—
30	C₂²×C₃²	H(3,𝔽₃)	1	1	yes	characteristic
31	C₂²×C₃²	C₃³	1	1	yes	characteristic
32	C₂²⋊C₉	—	3	6	no	—
33	C₂²⋊C₉	—	3	3	no	—
34	C₃³	—	3	1	no	—
35	C₃³	C₃×A₄	1	1	yes	characteristic
36	3₋¹⁺²	—	4	6	no	—
37	3₋¹⁺²	—	12	2	no	—
38	3₋¹⁺²	—	12	1	no	—
39	3₋¹⁺²	—	12	2	no	—
40	3₋¹⁺²	—	12	2	no	—
41	H(3,𝔽₃)	—	3	2	no	—
42	H(3,𝔽₃)	C₃×A₄	1	3	yes	—
43	C₃×C₉	—	4	3	no	—
44	C₃×C₉	—	12	2	no	—
45	C₂×C₃²	—	3	1	no	—
46	C₂×C₃²	—	3	9	no	—
47	C₂×C₃²	—	3	2	no	—
48	C₂×C₃²	—	3	1	no	—
49	C₂×C₃²	—	9	3	no	—
50	C₂×C₃²	—	9	9	no	—
51	C₂²×C₃	—	3	1	no	—
52	C₂²×C₃	—	3	2	no	—
53	C₂²×C₃	—	3	1	no	—
54	C₂²×C₃	—	9	3	no	—
55	C₂²×C₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
56	C₃²	—	3	3	no	—
57	C₃²	—	9	1	no	—
58	C₃²	—	9	3	no	—
59	C₃²	C₂²⋊H(3,𝔽₃)	1	2	yes	—
60	C₃²	C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
61	C₃²	C₃²×A₄	1	1	yes	characteristic
62	C₉	—	12	3	no	—
63	C₉	—	12	6	no	—
64	C₂×C₃	—	3	3	no	—
65	C₂×C₃	—	3	1	no	—
66	C₂×C₃	—	9	2	no	—
67	C₂×C₃	—	9	1	no	—
68	C₂×C₃	—	9	9	no	—
69	C₂²	#6910	1	1	yes	characteristic, complemented, Sylow
70	C₃	—	3	1	no	—
71	C₃	—	3	2	no	—
72	C₃	—	3	1	no	—
73	C₃	—	9	3	no	—
74	C₃	C₃×C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
75	C₂	—	3	1	no	—

	subgroup	G/H
center	C₃	C₃×C₂²⋊H(3,𝔽₃)
commutator, cosocle	C₂²×C₃²	C₃³
Frattini	C₃²	C₃²×A₄
Fitting, prime radical	C₂²×C₃×H(3,𝔽₃)	C₃
socle	C₂²×C₃	C₃×H(3,𝔽₃)
hypercenter	C₃×H(3,𝔽₃)	A₄
hypocenter	C₂²	#6910
LCS	G ▷ C₂²×C₃² ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₃ ◁ C₃³ ◁ C₃×H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
2. C₃×#1094	C₃	semidirect
3. C₃×#1094	C₃	semidirect
4. #66361	C₃	semidirect
15. #1094	C₃²	semidirect
17. C₃×C₂²⋊C₉	C₃²	semidirect
69. C₂²	#6910	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×H(3,𝔽₃)	3
C₃×#1094	3
#66361	3
#6910	4