Subgroups — C₉×A₄² #13049231

Order 1296.

275 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×A₄²	C₃	1	1	yes	maximal, characteristic
2	C₉×C₂²⋊A₄	C₃	1	2	yes	maximal, complemented
3	C₂²×C₉×A₄	C₃	1	2	yes	maximal, complemented
4	(C₂²⋊A₄)⋊C₉	C₃	1	4	yes	maximal, complemented
5	A₄×C₂²⋊C₉	C₃	1	4	yes	maximal, complemented
6	C₃×C₉×A₄	—	4	2	no	maximal
7	C₂×C₉×A₄	—	3	2	no	—
8	C₃×C₂²⋊A₄	C₃²	1	2	yes	—
9	C₂²×C₃×A₄	C₃²	1	2	yes	—
10	A₄²	C₉	1	9	yes	direct product factor
11	C₂⁴×C₉	C₃²	1	1	yes	characteristic, complemented
12	C₂²⋊(C₂²⋊C₉)	C₃²	1	4	yes	complemented
13	C₂²×C₂²⋊C₉	C₃²	1	4	yes	complemented
14	C₃²×A₄	—	4	2	no	—
15	C₂²×C₃×C₉	—	4	2	no	—
16	C₃×C₂²⋊C₉	—	4	4	no	—
17	C₉×A₄	—	4	4	no	—
18	C₉×A₄	—	4	4	no	—
19	C₉×A₄	—	4	8	no	—
20	C₉×A₄	—	12	2	no	—
21	C₉×A₄	A₄	1	2	yes	direct product factor
22	C₃²×C₉	—	16	1	no	Sylow
23	C₂×C₃×A₄	—	3	2	no	—
24	C₂³×C₉	—	3	2	no	—
25	C₂³×C₉	—	9	1	no	—
26	C₂×C₂²⋊C₉	—	3	4	no	—
27	C₂×C₃×C₉	—	12	2	no	—
28	C₂⁴×C₃	C₃³	1	1	yes	characteristic
29	C₂²⋊A₄	C₃×C₉	1	6	yes	complemented
30	C₂²×A₄	C₃×C₉	1	6	yes	complemented
31	C₂²×C₃²	—	4	2	no	—
32	C₃×A₄	—	4	4	no	—
33	C₃×A₄	—	4	18	no	—
34	C₃×A₄	—	12	2	no	—
35	C₃×A₄	C₃×A₄	1	2	yes	—
36	C₂²×C₉	—	3	2	no	—
37	C₂²×C₉	—	4	4	no	—
38	C₂²×C₉	—	9	1	no	—
39	C₂²×C₉	—	9	2	no	—
40	C₂²×C₉	C₃×A₄	1	2	yes	complemented
41	C₂²⋊C₉	—	4	8	no	—
42	C₂²⋊C₉	—	12	4	no	—
43	C₂²⋊C₉	C₃×A₄	1	4	yes	complemented
44	C₃³	—	16	1	no	—
45	C₃×C₉	—	4	2	no	—
46	C₃×C₉	—	16	4	no	—
47	C₃×C₉	—	16	4	no	—
48	C₃×C₉	—	16	2	no	—
49	C₂³×C₃	—	3	2	no	—
50	C₂³×C₃	—	9	1	no	—
51	C₂×A₄	—	3	6	no	—
52	C₂×C₃²	—	12	2	no	—
53	C₂×C₉	—	3	2	no	—
54	C₂×C₉	—	9	1	no	—
55	C₂×C₉	—	12	4	no	—
56	C₂⁴	C₃²×C₉	1	1	yes	characteristic, complemented, Sylow
57	C₂²×C₃	—	3	2	no	—
58	C₂²×C₃	—	4	6	no	—
59	C₂²×C₃	—	9	1	no	—
60	C₂²×C₃	—	9	2	no	—
61	C₂²×C₃	C₃²×A₄	1	2	yes	—
62	A₄	—	4	12	no	—
63	A₄	—	12	6	no	—
64	A₄	C₉×A₄	1	6	yes	direct product factor
65	C₃²	—	4	2	no	—
66	C₃²	—	16	9	no	—
67	C₃²	—	16	2	no	—
68	C₉	—	4	4	no	—
69	C₉	—	16	4	no	—
70	C₉	A₄²	1	1	yes	characteristic, direct product factor
71	C₂³	—	3	2	no	—
72	C₂³	—	9	1	no	—
73	C₂×C₃	—	3	2	no	—
74	C₂×C₃	—	9	1	no	—
75	C₂×C₃	—	12	6	no	—
76	C₂²	—	3	2	no	—
77	C₂²	—	9	1	no	—
78	C₂²	—	9	2	no	—
79	C₂²	C₃×C₉×A₄	1	2	yes	complemented
80	C₃	—	4	6	no	—
81	C₃	—	16	6	no	—
82	C₃	C₃×A₄²	1	1	yes	characteristic
83	C₂	—	3	2	no	—
84	C₂	—	9	1	no	—

	subgroup	G/H
center, hypercenter	C₉	A₄²
commutator, hypocenter	C₂⁴	C₃²×C₉
Frattini	C₃	C₃×A₄²
Fitting	C₂⁴×C₉	C₃²
socle, cosocle	C₂⁴×C₃	C₃³
prime radical	C₃×A₄²	C₃
LCS	G ▷ C₂⁴
UCS	C₁ ◁ C₉

Complements

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

N	H ≅ G/N	type
2. C₉×C₂²⋊A₄	C₃	semidirect
3. C₂²×C₉×A₄	C₃	semidirect
4. (C₂²⋊A₄)⋊C₉	C₃	semidirect
5. A₄×C₂²⋊C₉	C₃	semidirect
10. A₄²	C₉	direct (N × H)
11. C₂⁴×C₉	C₃²	semidirect
12. C₂²⋊(C₂²⋊C₉)	C₃²	semidirect
13. C₂²×C₂²⋊C₉	C₃²	semidirect
21. C₉×A₄	A₄	direct (N × H)
29. C₂²⋊A₄	C₃×C₉	semidirect
30. C₂²×A₄	C₃×C₉	semidirect
40. C₂²×C₉	C₃×A₄	semidirect
43. C₂²⋊C₉	C₃×A₄	semidirect
56. C₂⁴	C₃²×C₉	semidirect
64. A₄	C₉×A₄	direct (N × H)
70. C₉	A₄²	direct (N × H)
79. C₂²	C₃×C₉×A₄	semidirect

Maximal subgroups

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

subgroup	index
C₃×A₄²	3
C₉×C₂²⋊A₄	3
C₂²×C₉×A₄	3
(C₂²⋊A₄)⋊C₉	3
A₄×C₂²⋊C₉	3
C₃×C₉×A₄	4

Subgroups — C9×A42 #13049231

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×A₄² #13049231