Subgroups — #11758104

Order 972.

498 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, Frattini	C₃	C₃³×A₄
commutator, socle, cosocle	C₂²×C₃	C₃⁴
Fitting	C₂²×C₃×3₋¹⁺²	C₃
prime radical	C₃×C₂²⋊H(3,𝔽₃)	C₃
hypercenter	C₃×3₋¹⁺²	A₄
hypocenter	C₂²	3₋¹⁺⁴
LCS	G ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₃ ◁ C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×3₋¹⁺²	C₃	semidirect
3. C₃×#1094	C₃	semidirect
4. #66440	C₃	semidirect
5. A₄×3₋¹⁺²	C₃	semidirect
6. #66436	C₃	semidirect
12. C₂²×3₋¹⁺²	C₃²	semidirect
13. C₂²×C₃×C₉	C₃²	semidirect
17. #1094	C₃²	semidirect
18. #1094	C₃²	semidirect
19. C₃×C₂²⋊C₉	C₃²	semidirect
20. C₉⋊A₄	C₃²	semidirect
21. C₉×A₄	C₃²	semidirect
26. C₃×3₋¹⁺²	A₄	semidirect
47. 3₋¹⁺²	C₃×A₄	semidirect
52. C₃×C₉	C₃×A₄	semidirect
75. 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×C₃×3₋¹⁺²	3
C₃×C₂²⋊H(3,𝔽₃)	3
C₃×#1094	3
#66440	3
A₄×3₋¹⁺²	3
#66436	3
3₋¹⁺⁴	4