Subgroups — #11757832

Order 972.

222 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	3	yes	maximal, complemented
3	C₃²×C₂²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃²⋊(C₂²⋊C₉)	C₃	1	6	yes	maximal, complemented
5	C₃²⋊(C₂²⋊C₉)	C₃	1	2	yes	maximal, complemented
6	C₉⋊H(3,𝔽₃)	—	4	1	no	maximal, Sylow
7	C₂×C₃×H(3,𝔽₃)	—	3	1	no	—
8	C₂²×C₃³	C₃²	1	3	yes	—
9	C₂²×C₃³	C₃²	1	1	yes	characteristic
10	C₂²×H(3,𝔽₃)	—	3	3	no	—
11	#1094	—	3	9	no	—
12	C₃×C₂²⋊C₉	—	3	3	no	—
13	C₃×C₂²⋊C₉	C₃²	1	2	yes	—
14	C₃×C₂²⋊C₉	C₃²	1	6	yes	—
15	C₃×C₂²⋊C₉	C₃²	1	1	yes	characteristic
16	C₃×3₋¹⁺²	—	4	3	no	—
17	C₃×H(3,𝔽₃)	A₄	1	1	yes	characteristic
18	C₃²×C₉	—	4	1	no	—
19	C₃²⋊C₉	—	4	6	no	—
20	C₃²⋊C₉	—	4	2	no	—
21	C₂×C₃³	—	3	3	no	—
22	C₂×C₃³	—	3	1	no	—
23	C₂×H(3,𝔽₃)	—	3	9	no	—
24	C₂²×C₃²	—	3	1	no	—
25	C₂²×C₃²	—	3	2	no	—
26	C₂²×C₃²	—	3	3	no	—
27	C₂²×C₃²	—	3	3	no	—
28	C₂²×C₃²	—	3	6	no	—
29	C₂²×C₃²	3₋¹⁺²	1	3	yes	complemented
30	C₂²×C₃²	C₃³	1	1	yes	characteristic
31	C₂²⋊C₉	—	3	2	no	—
32	C₂²⋊C₉	—	3	6	no	—
33	C₂²⋊C₉	H(3,𝔽₃)	1	3	yes	complemented
34	C₃³	C₃×A₄	1	3	yes	—
35	C₃³	C₃×A₄	1	1	yes	characteristic
36	3₋¹⁺²	—	12	9	no	—
37	H(3,𝔽₃)	—	3	3	no	—
38	C₃×C₉	—	4	1	no	—
39	C₃×C₉	—	4	2	no	—
40	C₃×C₉	—	4	6	no	—
41	C₃×C₉	—	12	3	no	—
42	C₂×C₃²	—	3	1	no	—
43	C₂×C₃²	—	3	9	no	—
44	C₂×C₃²	—	3	3	no	—
45	C₂×C₃²	—	9	2	no	—
46	C₂×C₃²	—	9	6	no	—
47	C₂×C₃²	—	9	1	no	—
48	C₂×C₃²	—	9	3	no	—
49	C₂²×C₃	—	3	3	no	—
50	C₂²×C₃	—	9	3	no	—
51	C₂²×C₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
52	C₂²×C₃	C₃×3₋¹⁺²	1	1	yes	characteristic
53	C₂²×C₃	#737	1	2	yes	—
54	C₃²	—	3	2	no	—
55	C₃²	—	3	1	no	—
56	C₃²	—	3	6	no	—
57	C₃²	—	3	3	no	—
58	C₃²	—	3	3	no	—
59	C₃²	#1094	1	3	yes	complemented
60	C₃²	C₃²×A₄	1	1	yes	characteristic
61	C₉	—	4	3	no	—
62	C₉	—	12	2	no	—
63	C₉	—	12	6	no	—
64	C₂×C₃	—	3	2	no	—
65	C₂×C₃	—	3	1	no	—
66	C₂×C₃	—	3	1	no	—
67	C₂×C₃	—	9	3	no	—
68	C₂×C₃	—	9	9	no	—
69	C₂²	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
70	C₃	—	3	3	no	—
71	C₃	—	9	3	no	—
72	C₃	#66437	1	2	yes	—
73	C₃	A₄×H(3,𝔽₃)	1	1	yes	characteristic
74	C₃	C₃×#1094	1	1	yes	characteristic
75	C₂	—	3	1	no	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×A₄
commutator, socle, cosocle	C₂²×C₃²	C₃³
Fitting, prime radical	C₂²×C₃×H(3,𝔽₃)	C₃
hypercenter	C₃×H(3,𝔽₃)	A₄
hypocenter	C₂²	C₉⋊H(3,𝔽₃)
LCS	G ▷ C₂²×C₃² ▷ C₂²
UCS	C₁ ◁ C₃² ◁ 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. C₃²×C₂²⋊C₉	C₃	semidirect
4. C₃²⋊(C₂²⋊C₉)	C₃	semidirect
5. C₃²⋊(C₂²⋊C₉)	C₃	semidirect
29. C₂²×C₃²	3₋¹⁺²	semidirect
33. C₂²⋊C₉	H(3,𝔽₃)	semidirect
59. C₃²	#1094	semidirect
69. C₂²	C₉⋊H(3,𝔽₃)	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
C₃²×C₂²⋊C₉	3
C₃²⋊(C₂²⋊C₉)	3
C₉⋊H(3,𝔽₃)	4