Subgroups — #11758106

Order 972.

493 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#737	C₃	1	1	yes	maximal, characteristic, complemented
2	#66440	C₃	1	16	yes	maximal, complemented
3	A₄×3₋¹⁺²	C₃	1	8	yes	maximal, complemented
4	A₄×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
5	#66437	C₃	1	2	yes	maximal, complemented
6	#66436	C₃	1	8	yes	maximal, complemented
7	C₃×C₉⋊A₄	C₃	1	4	yes	maximal, complemented
8	3₋¹⁺⁴	—	4	1	no	maximal, Sylow
9	C₂×#737	—	3	1	no	—
10	C₃²×A₄	C₃²	1	4	yes	—
11	C₂²×3₋¹⁺²	C₃²	1	8	yes	complemented
12	C₂²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
13	C₂²×C₃×C₉	C₃²	1	4	yes	complemented
14	C₂²⋊H(3,𝔽₃)	C₃²	1	8	yes	—
15	#1094	C₃²	1	16	yes	complemented
16	C₃×C₂²⋊C₉	C₃²	1	8	yes	complemented
17	C₉⋊A₄	C₃²	1	1	yes	characteristic
18	C₉⋊A₄	C₃²	1	8	yes	—
19	C₉⋊A₄	C₃²	1	8	yes	complemented
20	C₉⋊A₄	C₃²	1	16	yes	complemented
21	C₉⋊A₄	C₃²	1	24	yes	complemented
22	C₉×A₄	C₃²	1	24	yes	complemented
23	#737	—	4	16	no	—
24	#737	—	4	2	no	—
25	#737	—	4	8	no	—
26	#737	A₄	1	1	yes	characteristic, complemented
27	C₃×3₋¹⁺²	—	4	8	no	—
28	C₃×3₋¹⁺²	—	4	4	no	—
29	C₃×H(3,𝔽₃)	—	4	1	no	—
30	C₂×3₋¹⁺²	—	3	8	no	—
31	C₂×H(3,𝔽₃)	—	3	1	no	—
32	C₂×C₃×C₉	—	3	4	no	—
33	C₂²×C₃²	C₃³	1	4	yes	—
34	C₃×A₄	—	9	4	no	—
35	C₃×A₄	C₃³	1	8	yes	—
36	C₃×A₄	C₃³	1	1	yes	characteristic
37	C₂²×C₉	C₃³	1	8	yes	—
38	C₂²×C₉	C₃³	1	1	yes	characteristic
39	C₂²⋊C₉	C₃³	1	16	yes	—
40	C₂²⋊C₉	C₃³	1	2	yes	—
41	C₃³	—	4	4	no	—
42	3₋¹⁺²	—	4	16	no	—
43	3₋¹⁺²	—	4	8	no	—
44	3₋¹⁺²	—	4	1	no	—
45	3₋¹⁺²	—	4	16	no	—
46	3₋¹⁺²	—	4	24	no	—
47	3₋¹⁺²	—	4	8	no	—
48	3₋¹⁺²	C₃×A₄	1	8	yes	complemented
49	H(3,𝔽₃)	—	4	8	no	—
50	H(3,𝔽₃)	C₃×A₄	1	1	yes	characteristic
51	C₃×C₉	—	4	8	no	—
52	C₃×C₉	—	4	24	no	—
53	C₃×C₉	C₃×A₄	1	4	yes	complemented
54	C₂×C₃²	—	3	4	no	—
55	C₂×C₉	—	3	1	no	—
56	C₂×C₉	—	3	8	no	—
57	C₂²×C₃	—	3	4	no	—
58	C₂²×C₃	C₃⁴	1	1	yes	characteristic
59	A₄	—	3	1	no	—
60	A₄	—	3	8	no	—
61	C₃²	—	4	8	no	—
62	C₃²	—	4	1	no	—
63	C₃²	—	36	4	no	—
64	C₃²	C₃²×A₄	1	4	yes	—
65	C₉	—	4	2	no	—
66	C₉	—	4	16	no	—
67	C₉	C₃²×A₄	1	8	yes	—
68	C₉	C₃²×A₄	1	1	yes	characteristic
69	C₂×C₃	—	3	1	no	—
70	C₂×C₃	—	9	4	no	—
71	C₂²	3₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
72	C₃	—	3	4	no	—
73	C₃	—	12	1	no	—
74	C₃	—	12	8	no	—
75	C₃	C₃³×A₄	1	1	yes	characteristic
76	C₂	—	3	1	no	—

	subgroup	G/H
center, Frattini	C₃	C₃³×A₄
commutator, socle, cosocle	C₂²×C₃	C₃⁴
Fitting	C₂²×#737	C₃
prime radical	A₄×H(3,𝔽₃)	C₃
hypercenter	#737	A₄
hypocenter	C₂²	3₋¹⁺⁴
LCS	G ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₃ ◁ #737

Complements

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

N	H ≅ G/N	type
1. C₂²×#737	C₃	semidirect
2. #66440	C₃	semidirect
3. A₄×3₋¹⁺²	C₃	semidirect
5. #66437	C₃	semidirect
6. #66436	C₃	semidirect
7. C₃×C₉⋊A₄	C₃	semidirect
11. C₂²×3₋¹⁺²	C₃²	semidirect
13. C₂²×C₃×C₉	C₃²	semidirect
15. #1094	C₃²	semidirect
16. C₃×C₂²⋊C₉	C₃²	semidirect
19. C₉⋊A₄	C₃²	semidirect
20. C₉⋊A₄	C₃²	semidirect
21. C₉⋊A₄	C₃²	semidirect
22. C₉×A₄	C₃²	semidirect
26. #737	A₄	semidirect
48. 3₋¹⁺²	C₃×A₄	semidirect
53. C₃×C₉	C₃×A₄	semidirect
71. C₂²	3₋¹⁺⁴	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#737	3
#66440	3
A₄×3₋¹⁺²	3
A₄×H(3,𝔽₃)	3
#66437	3
#66436	3
C₃×C₉⋊A₄	3
3₋¹⁺⁴	4