Subgroups — 3₋¹⁺²×C₂²⋊A₄ #13049339

Order 1296.

426 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	C₂⁴×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
3	#88907	C₃	1	2	yes	maximal, complemented
4	C₉⋊(C₂²⋊A₄)	C₃	1	6	yes	maximal, complemented
5	C₉×C₂²⋊A₄	C₃	1	3	yes	maximal, complemented
6	A₄×3₋¹⁺²	—	4	5	no	maximal
7	C₂³×3₋¹⁺²	—	3	5	no	—
8	C₂⁴×C₃²	C₃²	1	1	yes	characteristic
9	C₃×C₂²⋊A₄	—	3	3	no	—
10	C₃×C₂²⋊A₄	C₃²	1	2	yes	—
11	C₃×C₂²⋊A₄	C₃²	1	1	yes	characteristic
12	C₂⁴×C₉	C₃²	1	3	yes	complemented
13	C₂²⋊(C₂²⋊C₉)	C₃²	1	6	yes	complemented
14	C₃²×A₄	—	4	5	no	—
15	C₂²×3₋¹⁺²	—	3	10	no	—
16	C₂²×3₋¹⁺²	A₄	1	5	yes	complemented
17	#1094	—	4	10	no	—
18	C₉⋊A₄	—	4	30	no	—
19	C₉×A₄	—	4	15	no	—
20	C₃×3₋¹⁺²	—	16	1	no	Sylow
21	C₂³×C₃²	—	3	5	no	—
22	C₂³×C₉	—	3	15	no	—
23	C₂×3₋¹⁺²	—	3	5	no	—
24	C₂⁴×C₃	—	3	1	no	—
25	C₂⁴×C₃	C₃³	1	1	yes	characteristic
26	C₂²⋊A₄	—	3	2	no	—
27	C₂²⋊A₄	3₋¹⁺²	1	3	yes	direct product factor
28	C₂²×C₃²	—	3	10	no	—
29	C₂²×C₃²	C₃×A₄	1	5	yes	—
30	C₃×A₄	—	4	5	no	—
31	C₃×A₄	—	4	10	no	—
32	C₃×A₄	—	12	15	no	—
33	C₂²×C₉	—	3	30	no	—
34	C₂²×C₉	C₃×A₄	1	15	yes	complemented
35	C₂²⋊C₉	—	4	30	no	—
36	C₃³	—	16	1	no	—
37	3₋¹⁺²	—	16	2	no	—
38	3₋¹⁺²	—	16	6	no	—
39	3₋¹⁺²	C₂²⋊A₄	1	1	yes	characteristic, direct product factor
40	C₃×C₉	—	16	3	no	—
41	C₂³×C₃	—	3	5	no	—
42	C₂³×C₃	—	9	5	no	—
43	C₂×C₃²	—	3	5	no	—
44	C₂×C₉	—	3	15	no	—
45	C₂⁴	C₃×3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
46	C₂²×C₃	—	3	10	no	—
47	C₂²×C₃	—	3	5	no	—
48	C₂²×C₃	—	9	10	no	—
49	C₂²×C₃	C₃²×A₄	1	5	yes	—
50	A₄	—	4	15	no	—
51	A₄	—	12	10	no	—
52	C₃²	—	16	1	no	—
53	C₃²	—	16	2	no	—
54	C₃²	—	48	3	no	—
55	C₃²	C₃×C₂²⋊A₄	1	1	yes	characteristic
56	C₉	—	16	6	no	—
57	C₉	C₃×C₂²⋊A₄	1	3	yes	complemented
58	C₂³	—	3	5	no	—
59	C₂×C₃	—	3	5	no	—
60	C₂×C₃	—	9	5	no	—
61	C₂²	—	3	10	no	—
62	C₂²	A₄×3₋¹⁺²	1	5	yes	complemented
63	C₃	—	3	1	no	—
64	C₃	—	16	3	no	—
65	C₃	—	48	2	no	—
66	C₃	C₃²×C₂²⋊A₄	1	1	yes	characteristic
67	C₂	—	3	5	no	—

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

Complements

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

N	H ≅ G/N	type
2. C₂⁴×3₋¹⁺²	C₃	semidirect
3. #88907	C₃	semidirect
4. C₉⋊(C₂²⋊A₄)	C₃	semidirect
5. C₉×C₂²⋊A₄	C₃	semidirect
12. C₂⁴×C₉	C₃²	semidirect
13. C₂²⋊(C₂²⋊C₉)	C₃²	semidirect
16. C₂²×3₋¹⁺²	A₄	semidirect
27. C₂²⋊A₄	3₋¹⁺²	direct (N × H)
34. C₂²×C₉	C₃×A₄	semidirect
39. 3₋¹⁺²	C₂²⋊A₄	direct (N × H)
45. C₂⁴	C₃×3₋¹⁺²	semidirect
57. C₉	C₃×C₂²⋊A₄	semidirect
62. C₂²	A₄×3₋¹⁺²	semidirect

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
C₂⁴×3₋¹⁺²	3
#88907	3
C₉⋊(C₂²⋊A₄)	3
C₉×C₂²⋊A₄	3
A₄×3₋¹⁺²	4

Subgroups — 3−1+2×C22⋊A4 #13049339

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×C₂²⋊A₄ #13049339