Subgroups — C₃×(C₃×A₄)⋊C₉ #11757827

Order 972.

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

	subgroup	G/H
center	C₃³	C₃×A₄
commutator	C₂²×C₃	C₃²×C₉
Frattini	C₃²	C₃²×A₄
Fitting	C₂²×C₃²×C₉	C₃
socle	C₂²×C₃³	C₃²
prime radical	C₃³×A₄	C₃
cosocle	C₂²×C₃²	C₃³
hypercenter	C₃²×C₉	A₄
hypocenter	C₂²	C₃×C₃²⋊C₉
LCS	G ▷ C₂²×C₃ ▷ C₂²
UCS	C₁ ◁ C₃³ ◁ C₃²×C₉

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃²×C₉	C₃	semidirect
3. C₃²×C₂²⋊C₉	C₃	semidirect
4. (C₃×A₄)⋊C₉	C₃	direct (N × H)
10. C₃²×A₄	C₉	semidirect
13. C₂²×C₃×C₉	C₃²	semidirect
15. C₃×C₂²⋊C₉	C₃²	semidirect
18. C₃²×C₉	A₄	semidirect
30. C₃×A₄	C₃×C₉	semidirect
41. C₃×C₉	C₃×A₄	semidirect
47. C₂²×C₃	C₃²⋊C₉	semidirect
66. C₂²	C₃×C₃²⋊C₉	semidirect
68. C₃	(C₃×A₄)⋊C₉	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×C₃²×C₉	3
C₃²×C₂²⋊C₉	3
(C₃×A₄)⋊C₉	3
C₃×C₃²⋊C₉	4

Subgroups — C3×(C3×A4)⋊C9 #11757827

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×(C₃×A₄)⋊C₉ #11757827