Subgroups — #13049340

Order 1296.

391 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃²⋊(C₂²⋊A₄)	C₃	1	1	yes	maximal, characteristic
3	#88907	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₂²⋊(C₂²⋊C₉)	C₃	1	1	yes	maximal, characteristic, complemented
5	C₉⋊(C₂²⋊A₄)	C₃	1	6	yes	maximal, complemented
6	C₉×C₂²⋊A₄	C₃	1	3	yes	maximal, complemented
7	#66440	—	4	5	no	maximal
8	C₂³×3₋¹⁺²	—	3	5	no	—
9	C₂⁴×C₃²	C₃²	1	1	yes	characteristic
10	C₃×C₂²⋊A₄	C₃²	1	3	yes	—
11	C₂⁴×C₉	C₃²	1	3	yes	—
12	C₂²⋊(C₂²⋊C₉)	C₃²	1	1	yes	characteristic
13	C₂²⋊(C₂²⋊C₉)	C₃²	1	3	yes	—
14	C₂²⋊(C₂²⋊C₉)	C₃²	1	2	yes	—
15	C₂²×3₋¹⁺²	—	3	10	no	—
16	C₂²×3₋¹⁺²	A₄	1	5	yes	complemented
17	C₂²⋊H(3,𝔽₃)	—	4	5	no	—
18	#1094	—	4	5	no	—
19	C₃×C₂²⋊C₉	—	4	5	no	—
20	C₉⋊A₄	—	4	30	no	—
21	C₉×A₄	—	4	15	no	—
22	#737	—	16	1	no	Sylow
23	C₂³×C₃²	—	3	5	no	—
24	C₂³×C₉	—	3	15	no	—
25	C₂×3₋¹⁺²	—	3	5	no	—
26	C₂⁴×C₃	—	3	1	no	—
27	C₂⁴×C₃	C₃³	1	1	yes	characteristic
28	C₂²⋊A₄	—	3	3	no	—
29	C₂²×C₃²	—	3	10	no	—
30	C₂²×C₃²	C₃×A₄	1	5	yes	—
31	C₃×A₄	—	4	15	no	—
32	C₂²×C₉	—	3	30	no	—
33	C₂²×C₉	C₃×A₄	1	15	yes	—
34	C₂²⋊C₉	—	4	5	no	—
35	C₂²⋊C₉	—	4	15	no	—
36	C₂²⋊C₉	—	4	10	no	—
37	3₋¹⁺²	—	16	1	no	—
38	3₋¹⁺²	—	16	6	no	—
39	3₋¹⁺²	C₂²⋊A₄	1	1	yes	characteristic, complemented
40	H(3,𝔽₃)	—	16	1	no	—
41	C₃×C₉	—	16	1	no	—
42	C₃×C₉	—	16	3	no	—
43	C₂³×C₃	—	3	5	no	—
44	C₂³×C₃	—	9	5	no	—
45	C₂×C₃²	—	3	5	no	—
46	C₂×C₉	—	3	15	no	—
47	C₂⁴	#737	1	1	yes	characteristic, complemented, Sylow
48	C₂²×C₃	—	3	10	no	—
49	C₂²×C₃	—	3	5	no	—
50	C₂²×C₃	—	9	10	no	—
51	C₂²×C₃	C₃²×A₄	1	5	yes	—
52	A₄	—	12	15	no	—
53	C₃²	—	16	3	no	—
54	C₃²	C₃×C₂²⋊A₄	1	1	yes	characteristic
55	C₉	—	16	1	no	—
56	C₉	—	16	2	no	—
57	C₉	—	16	3	no	—
58	C₉	C₃×C₂²⋊A₄	1	3	yes	—
59	C₂³	—	3	5	no	—
60	C₂×C₃	—	3	5	no	—
61	C₂×C₃	—	9	5	no	—
62	C₂²	—	3	10	no	—
63	C₂²	#66440	1	5	yes	complemented
64	C₃	—	3	1	no	—
65	C₃	—	48	3	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	#737
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
1. C₂⁴×3₋¹⁺²	C₃	semidirect
3. #88907	C₃	semidirect
4. C₃×C₂²⋊(C₂²⋊C₉)	C₃	semidirect
5. C₉⋊(C₂²⋊A₄)	C₃	semidirect
6. C₉×C₂²⋊A₄	C₃	semidirect
16. C₂²×3₋¹⁺²	A₄	semidirect
39. 3₋¹⁺²	C₂²⋊A₄	semidirect
47. C₂⁴	#737	semidirect
63. C₂²	#66440	semidirect

Maximal subgroups

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

subgroup	index
C₂⁴×3₋¹⁺²	3
C₃²⋊(C₂²⋊A₄)	3
#88907	3
C₃×C₂²⋊(C₂²⋊C₉)	3
C₉⋊(C₂²⋊A₄)	3
C₉×C₂²⋊A₄	3
#66440	4