Subgroups — A₄×#737 #11758105

Order 972.

513 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	C₃×#737	—	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	48	yes	complemented
18	C₉×A₄	C₃²	1	1	yes	characteristic
19	C₉×A₄	C₃²	1	8	yes	—
20	C₉×A₄	C₃²	1	8	yes	complemented
21	C₉×A₄	C₃²	1	16	yes	complemented
22	#737	—	4	16	no	—
23	#737	—	4	2	no	—
24	#737	—	4	8	no	—
25	#737	A₄	1	1	yes	characteristic, direct product factor
26	C₃×3₋¹⁺²	—	4	8	no	—
27	C₃×H(3,𝔽₃)	—	4	1	no	—
28	C₃²×C₉	—	4	4	no	—
29	C₂×3₋¹⁺²	—	3	8	no	—
30	C₂×H(3,𝔽₃)	—	3	1	no	—
31	C₂×C₃×C₉	—	3	4	no	—
32	C₂²×C₃²	C₃³	1	4	yes	—
33	C₃×A₄	—	3	12	no	—
34	C₃×A₄	C₃³	1	8	yes	—
35	C₃×A₄	C₃³	1	1	yes	characteristic
36	C₂²×C₉	C₃³	1	1	yes	characteristic
37	C₂²×C₉	C₃³	1	8	yes	—
38	C₂²⋊C₉	C₃³	1	2	yes	—
39	C₂²⋊C₉	C₃³	1	16	yes	—
40	C₃³	—	4	4	no	—
41	3₋¹⁺²	—	4	16	no	—
42	3₋¹⁺²	—	4	48	no	—
43	3₋¹⁺²	C₃×A₄	1	8	yes	complemented
44	H(3,𝔽₃)	—	4	8	no	—
45	H(3,𝔽₃)	C₃×A₄	1	1	yes	characteristic
46	C₃×C₉	—	4	8	no	—
47	C₃×C₉	—	4	8	no	—
48	C₃×C₉	—	4	16	no	—
49	C₃×C₉	—	4	1	no	—
50	C₃×C₉	—	4	8	no	—
51	C₃×C₉	C₃×A₄	1	4	yes	complemented
52	C₂×C₃²	—	3	4	no	—
53	C₂×C₉	—	3	8	no	—
54	C₂×C₉	—	3	1	no	—
55	C₂²×C₃	—	3	4	no	—
56	C₂²×C₃	C₃⁴	1	1	yes	characteristic
57	A₄	—	3	8	no	—
58	A₄	#737	1	3	yes	direct product factor
59	C₃²	—	4	1	no	—
60	C₃²	—	4	8	no	—
61	C₃²	—	12	12	no	—
62	C₃²	C₃²×A₄	1	4	yes	—
63	C₉	—	4	2	no	—
64	C₉	—	4	16	no	—
65	C₉	C₃²×A₄	1	8	yes	—
66	C₉	C₃²×A₄	1	1	yes	characteristic
67	C₂×C₃	—	3	1	no	—
68	C₂×C₃	—	9	4	no	—
69	C₂²	C₃×#737	1	1	yes	characteristic, complemented, Sylow
70	C₃	—	3	4	no	—
71	C₃	—	4	3	no	—
72	C₃	—	12	8	no	—
73	C₃	C₃³×A₄	1	1	yes	characteristic
74	C₂	—	3	1	no	—

	subgroup	G/H
center	C₉	C₃²×A₄
commutator, socle, cosocle	C₂²×C₃	C₃⁴
Frattini	C₃	C₃³×A₄
Fitting	C₂²×#737	C₃
prime radical	A₄×H(3,𝔽₃)	C₃
hypercenter	#737	A₄
hypocenter	C₂²	C₃×#737
LCS	G ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₉ ◁ #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
17. C₉⋊A₄	C₃²	semidirect
20. C₉×A₄	C₃²	semidirect
21. C₉×A₄	C₃²	semidirect
25. #737	A₄	direct (N × H)
43. 3₋¹⁺²	C₃×A₄	semidirect
51. C₃×C₉	C₃×A₄	semidirect
58. A₄	#737	direct (N × H)
69. C₂²	C₃×#737	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
C₃×#737	4

Subgroups — A4×#737 #11758105

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — A₄×#737 #11758105