Subgroups — C₃×A₄×3₋¹⁺² #11758102

Order 972.

614 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₃²	C₃²×A₄
commutator, cosocle	C₂²×C₃	C₃⁴
Frattini	C₃	C₃³×A₄
Fitting	C₂²×C₃×3₋¹⁺²	C₃
socle	C₂²×C₃²	C₃³
prime radical	C₃³×A₄	C₃
hypercenter	C₃×3₋¹⁺²	A₄
hypocenter	C₂²	C₃²×3₋¹⁺²
LCS	G ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₃² ◁ C₃×3₋¹⁺²

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×3₋¹⁺²	C₃	semidirect
3. C₃×#1094	C₃	semidirect
4. A₄×3₋¹⁺²	C₃	direct (N × H)
5. C₃×C₉⋊A₄	C₃	semidirect
6. C₃×C₉×A₄	C₃	semidirect
14. C₂²×3₋¹⁺²	C₃²	semidirect
15. C₂²×C₃×C₉	C₃²	semidirect
16. #1094	C₃²	semidirect
17. C₃×C₂²⋊C₉	C₃²	semidirect
18. C₉⋊A₄	C₃²	semidirect
19. C₉×A₄	C₃²	semidirect
24. C₃×3₋¹⁺²	A₄	direct (N × H)
34. C₃×A₄	3₋¹⁺²	direct (N × H)
37. C₂²×C₉	C₃³	semidirect
38. C₂²⋊C₉	C₃³	semidirect
46. 3₋¹⁺²	C₃×A₄	direct (N × H)
49. C₃×C₉	C₃×A₄	semidirect
55. C₂²×C₃	C₃×3₋¹⁺²	semidirect
58. A₄	C₃×3₋¹⁺²	direct (N × H)
68. C₉	C₃²×A₄	semidirect
72. C₂²	C₃²×3₋¹⁺²	semidirect
76. C₃	A₄×3₋¹⁺²	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃³×A₄	3
C₂²×C₃×3₋¹⁺²	3
C₃×#1094	3
A₄×3₋¹⁺²	3
C₃×C₉⋊A₄	3
C₃×C₉×A₄	3
C₃²×3₋¹⁺²	4

Subgroups — C3×A4×3−1+2 #11758102

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×A₄×3₋¹⁺² #11758102