Subgroups — #11757809

Order 972.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	A₄×H(3,𝔽₃)	C₃	1	3	yes	maximal, complemented
2	C₃×C₉⋊A₄	C₃	1	1	yes	maximal, characteristic, complemented
3	C₂²×#732	C₃	1	1	yes	maximal, characteristic, complemented
4	#66362	C₃	1	6	yes	maximal, complemented
5	#66358	C₃	1	2	yes	maximal, complemented
6	#6911	—	4	1	no	maximal, Sylow
7	C₂×#732	—	3	1	no	—
8	C₃²×A₄	—	3	3	no	—
9	C₃²×A₄	C₃²	1	1	yes	characteristic
10	C₂²×H(3,𝔽₃)	C₃²	1	3	yes	complemented
11	C₂²×C₃×C₉	C₃²	1	1	yes	characteristic, complemented
12	C₂²⋊H(3,𝔽₃)	—	3	6	no	—
13	C₂²⋊H(3,𝔽₃)	C₃²	1	6	yes	complemented
14	C₃×C₂²⋊C₉	C₃²	1	2	yes	complemented
15	C₉⋊A₄	—	3	3	no	—
16	C₃×3₋¹⁺²	—	4	1	no	—
17	C₃×H(3,𝔽₃)	—	4	3	no	—
18	#732	—	4	6	no	—
19	#732	—	4	2	no	—
20	#732	A₄	1	1	yes	characteristic, complemented
21	C₂×H(3,𝔽₃)	—	3	3	no	—
22	C₂×C₃×C₉	—	3	1	no	—
23	C₂²×C₃²	—	3	3	no	—
24	C₂²×C₃²	C₃³	1	1	yes	characteristic
25	C₃×A₄	—	3	6	no	—
26	C₃×A₄	—	9	1	no	—
27	C₃×A₄	—	9	9	no	—
28	C₃×A₄	H(3,𝔽₃)	1	3	yes	—
29	C₂²×C₉	—	3	1	no	—
30	C₂²⋊C₉	—	3	2	no	—
31	C₃³	—	4	1	no	—
32	C₃³	—	12	3	no	—
33	3₋¹⁺²	—	12	3	no	—
34	H(3,𝔽₃)	—	4	6	no	—
35	H(3,𝔽₃)	—	12	6	no	—
36	H(3,𝔽₃)	C₃×A₄	1	3	yes	complemented
37	C₃×C₉	—	4	2	no	—
38	C₃×C₉	C₃×A₄	1	1	yes	characteristic, complemented
39	C₂×C₃²	—	3	1	no	—
40	C₂×C₃²	—	9	3	no	—
41	C₂×C₉	—	9	1	no	—
42	C₂²×C₃	—	3	1	no	—
43	C₂²×C₃	—	9	3	no	—
44	C₂²×C₃	C₃×H(3,𝔽₃)	1	1	yes	characteristic
45	A₄	—	3	3	no	—
46	A₄	—	9	6	no	—
47	C₃²	—	3	3	no	—
48	C₃²	—	4	3	no	—
49	C₃²	—	12	6	no	—
50	C₃²	—	36	1	no	—
51	C₃²	—	36	9	no	—
52	C₃²	C₃²×A₄	1	1	yes	characteristic
53	C₉	—	3	1	no	—
54	C₉	—	12	2	no	—
55	C₂×C₃	—	3	1	no	—
56	C₂×C₃	—	9	1	no	—
57	C₂×C₃	—	27	3	no	—
58	C₂²	#6911	1	1	yes	characteristic, complemented, Sylow
59	C₃	—	3	1	no	—
60	C₃	—	9	3	no	—
61	C₃	—	12	3	no	—
62	C₃	—	36	6	no	—
63	C₃	A₄×H(3,𝔽₃)	1	1	yes	characteristic
64	C₂	—	3	1	no	—

	subgroup	G/H
center	C₃	A₄×H(3,𝔽₃)
commutator, cosocle	C₂²×C₃²	C₃³
Frattini	C₃²	C₃²×A₄
Fitting	C₂²×#732	C₃
socle	C₂²×C₃	C₃×H(3,𝔽₃)
hypercenter	#732	A₄
hypocenter	C₂²	#6911
LCS	G ▷ C₂²×C₃² ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₃ ◁ C₃² ◁ #732

Complements

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

N	H ≅ G/N	type
1. A₄×H(3,𝔽₃)	C₃	semidirect
2. C₃×C₉⋊A₄	C₃	semidirect
3. C₂²×#732	C₃	semidirect
4. #66362	C₃	semidirect
5. #66358	C₃	semidirect
10. C₂²×H(3,𝔽₃)	C₃²	semidirect
11. C₂²×C₃×C₉	C₃²	semidirect
13. C₂²⋊H(3,𝔽₃)	C₃²	semidirect
14. C₃×C₂²⋊C₉	C₃²	semidirect
20. #732	A₄	semidirect
36. H(3,𝔽₃)	C₃×A₄	semidirect
38. C₃×C₉	C₃×A₄	semidirect
58. C₂²	#6911	semidirect

Maximal subgroups

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

subgroup	index
A₄×H(3,𝔽₃)	3
C₃×C₉⋊A₄	3
C₂²×#732	3
#66362	3
#66358	3
#6911	4