Subgroups — C₃×C₉⋊(C₂²⋊A₄) #13049335

Order 1296.

466 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×C₂²⋊A₄	C₃	1	1	yes	maximal, characteristic
2	C₂⁴×C₃×C₉	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×C₂²⋊(C₂²⋊C₉)	C₃	1	2	yes	maximal, complemented
4	C₉⋊(C₂²⋊A₄)	C₃	1	9	yes	maximal, direct product factor
5	C₃×C₉⋊A₄	—	4	5	no	maximal
6	C₂³×C₃×C₉	—	3	5	no	—
7	C₂⁴×C₃²	C₃²	1	1	yes	characteristic
8	C₃×C₂²⋊A₄	—	3	3	no	—
9	C₃×C₂²⋊A₄	C₃²	1	3	yes	—
10	C₂⁴×C₉	C₃²	1	3	yes	complemented
11	C₂²⋊(C₂²⋊C₉)	C₃²	1	6	yes	complemented
12	C₃²×A₄	—	4	5	no	—
13	C₂²×C₃×C₉	—	3	10	no	—
14	C₂²×C₃×C₉	A₄	1	5	yes	complemented
15	C₃×C₂²⋊C₉	—	4	10	no	—
16	C₉⋊A₄	—	4	45	no	—
17	C₃×3₋¹⁺²	—	16	1	no	Sylow
18	C₂³×C₃²	—	3	5	no	—
19	C₂³×C₉	—	3	15	no	—
20	C₂×C₃×C₉	—	3	5	no	—
21	C₂⁴×C₃	3₋¹⁺²	1	3	yes	complemented
22	C₂⁴×C₃	C₃³	1	1	yes	characteristic
23	C₂²⋊A₄	—	3	3	no	—
24	C₂²×C₃²	—	3	10	no	—
25	C₂²×C₃²	C₃×A₄	1	5	yes	—
26	C₃×A₄	—	4	15	no	—
27	C₃×A₄	—	12	15	no	—
28	C₂²×C₉	—	3	30	no	—
29	C₂²×C₉	C₃×A₄	1	15	yes	complemented
30	C₂²⋊C₉	—	4	30	no	—
31	C₃³	—	16	1	no	—
32	3₋¹⁺²	—	16	9	no	—
33	C₃×C₉	—	16	2	no	—
34	C₃×C₉	C₂²⋊A₄	1	1	yes	characteristic, complemented
35	C₂³×C₃	—	3	15	no	—
36	C₂³×C₃	—	3	5	no	—
37	C₂×C₃²	—	3	5	no	—
38	C₂×C₉	—	3	15	no	—
39	C₂⁴	C₃×3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
40	C₂²×C₃	—	3	10	no	—
41	C₂²×C₃	—	3	30	no	—
42	C₂²×C₃	C₉⋊A₄	1	15	yes	complemented
43	C₂²×C₃	C₃²×A₄	1	5	yes	—
44	A₄	—	12	15	no	—
45	C₃²	—	16	3	no	—
46	C₃²	—	48	3	no	—
47	C₃²	C₃×C₂²⋊A₄	1	1	yes	characteristic
48	C₉	—	16	6	no	—
49	C₉	C₃×C₂²⋊A₄	1	3	yes	complemented
50	C₂³	—	3	5	no	—
51	C₂×C₃	—	3	5	no	—
52	C₂×C₃	—	3	15	no	—
53	C₂²	—	3	10	no	—
54	C₂²	C₃×C₉⋊A₄	1	5	yes	complemented
55	C₃	—	48	3	no	—
56	C₃	C₉⋊(C₂²⋊A₄)	1	3	yes	direct product factor
57	C₃	C₃²×C₂²⋊A₄	1	1	yes	characteristic
58	C₂	—	3	5	no	—

	subgroup	G/H
center	C₃²	C₃×C₂²⋊A₄
commutator, cosocle	C₂⁴×C₃	C₃³
Frattini	C₃	C₃²×C₂²⋊A₄
Fitting	C₂⁴×C₃×C₉	C₃
socle	C₂⁴×C₃²	C₃²
prime radical	C₃²×C₂²⋊A₄	C₃
hypercenter	C₃×C₉	C₂²⋊A₄
hypocenter	C₂⁴	C₃×3₋¹⁺²
LCS	G ▷ C₂⁴×C₃ ▷ C₂⁴
UCS	C₁ ◁ C₃² ◁ C₃×C₉

Complements

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

N	H ≅ G/N	type
2. C₂⁴×C₃×C₉	C₃	semidirect
3. C₃×C₂²⋊(C₂²⋊C₉)	C₃	semidirect
4. C₉⋊(C₂²⋊A₄)	C₃	direct (N × H)
10. C₂⁴×C₉	C₃²	semidirect
11. C₂²⋊(C₂²⋊C₉)	C₃²	semidirect
14. C₂²×C₃×C₉	A₄	semidirect
21. C₂⁴×C₃	3₋¹⁺²	semidirect
29. C₂²×C₉	C₃×A₄	semidirect
34. C₃×C₉	C₂²⋊A₄	semidirect
39. C₂⁴	C₃×3₋¹⁺²	semidirect
42. C₂²×C₃	C₉⋊A₄	semidirect
49. C₉	C₃×C₂²⋊A₄	semidirect
54. C₂²	C₃×C₉⋊A₄	semidirect
56. C₃	C₉⋊(C₂²⋊A₄)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃²×C₂²⋊A₄	3
C₂⁴×C₃×C₉	3
C₃×C₂²⋊(C₂²⋊C₉)	3
C₉⋊(C₂²⋊A₄)	3
C₃×C₉⋊A₄	4

Subgroups — C3×C9⋊(C22⋊A4) #13049335

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₉⋊(C₂²⋊A₄) #13049335