Subgroups — #11757836

Order 972.

204 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, complemented
2	C₃×C₂²⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×#1094	C₃	1	2	yes	maximal, complemented
4	#66368	C₃	1	9	yes	maximal, complemented
5	#6909	—	4	1	no	maximal, Sylow
6	C₂×C₃×H(3,𝔽₃)	—	3	1	no	—
7	C₂²×C₃³	C₃²	1	1	yes	characteristic
8	C₂²×C₃³	C₃²	1	3	yes	complemented
9	C₃²×A₄	—	3	1	no	—
10	C₂²×H(3,𝔽₃)	—	3	2	no	—
11	C₂²×H(3,𝔽₃)	—	3	1	no	—
12	C₂²⋊H(3,𝔽₃)	—	3	2	no	—
13	C₂²⋊H(3,𝔽₃)	C₃²	1	3	yes	—
14	#1094	—	3	2	no	—
15	#1094	—	3	2	no	—
16	#1094	C₃²	1	6	yes	complemented
17	C₃×C₂²⋊C₉	—	3	2	no	—
18	C₃×3₋¹⁺²	—	4	2	no	—
19	C₃×H(3,𝔽₃)	—	4	1	no	—
20	C₃×H(3,𝔽₃)	A₄	1	1	yes	characteristic, complemented
21	AΣL(1,𝔽₂₇)	—	4	9	no	—
22	C₂×C₃³	—	3	1	no	—
23	C₂×C₃³	—	3	3	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	C₃×A₄	—	3	3	no	—
33	C₃×A₄	—	9	3	no	—
34	C₂²⋊C₉	—	3	6	no	—
35	C₃³	—	12	1	no	—
36	C₃³	C₃×A₄	1	1	yes	characteristic
37	C₃³	C₃×A₄	1	3	yes	complemented
38	3₋¹⁺²	—	4	6	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)	—	3	1	no	—
43	H(3,𝔽₃)	—	4	3	no	—
44	H(3,𝔽₃)	—	12	2	no	—
45	C₃×C₉	—	12	2	no	—
46	C₂×C₃²	—	3	1	no	—
47	C₂×C₃²	—	3	9	no	—
48	C₂×C₃²	—	3	2	no	—
49	C₂×C₃²	—	3	1	no	—
50	C₂×C₃²	—	9	3	no	—
51	C₂×C₃²	—	9	9	no	—
52	C₂²×C₃	—	3	1	no	—
53	C₂²×C₃	—	3	1	no	—
54	C₂²×C₃	—	3	2	no	—
55	C₂²×C₃	—	9	3	no	—
56	C₂²×C₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
57	A₄	—	9	3	no	—
58	C₃²	—	3	3	no	—
59	C₃²	—	9	1	no	—
60	C₃²	—	9	3	no	—
61	C₃²	—	12	3	no	—
62	C₃²	—	36	3	no	—
63	C₃²	C₂²⋊H(3,𝔽₃)	1	2	yes	—
64	C₃²	C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
65	C₃²	C₃²×A₄	1	1	yes	characteristic
66	C₉	—	12	6	no	—
67	C₂×C₃	—	3	3	no	—
68	C₂×C₃	—	3	1	no	—
69	C₂×C₃	—	9	1	no	—
70	C₂×C₃	—	9	2	no	—
71	C₂×C₃	—	9	9	no	—
72	C₂²	#6909	1	1	yes	characteristic, complemented, Sylow
73	C₃	—	3	1	no	—
74	C₃	—	3	1	no	—
75	C₃	—	3	2	no	—
76	C₃	—	9	3	no	—
77	C₃	—	36	3	no	—
78	C₃	C₃×C₂²⋊H(3,𝔽₃)	1	1	yes	characteristic
79	C₂	—	3	1	no	—

	subgroup	G/H
center	C₃	C₃×C₂²⋊H(3,𝔽₃)
commutator, cosocle	C₂²×C₃²	C₃³
Frattini	C₃²	C₃²×A₄
Fitting	C₂²×C₃×H(3,𝔽₃)	C₃
socle	C₂²×C₃	C₃×H(3,𝔽₃)
hypercenter	C₃×H(3,𝔽₃)	A₄
hypocenter	C₂²	#6909
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
1. C₂²×C₃×H(3,𝔽₃)	C₃	semidirect
2. C₃×C₂²⋊H(3,𝔽₃)	C₃	semidirect
3. C₃×#1094	C₃	semidirect
4. #66368	C₃	semidirect
8. C₂²×C₃³	C₃²	semidirect
16. #1094	C₃²	semidirect
20. C₃×H(3,𝔽₃)	A₄	semidirect
37. C₃³	C₃×A₄	semidirect
72. C₂²	#6909	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×C₂²⋊H(3,𝔽₃)	3
C₃×#1094	3
#66368	3
#6909	4