Subgroups — #13048179

Order 1296.

258 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	#66365	—	4	5	no	maximal
6	C₂³×3₋¹⁺²	—	3	5	no	—
7	C₂⁴×C₃²	C₃²	1	1	yes	characteristic
8	C₃×C₂²⋊A₄	—	3	1	no	—
9	C₂⁴×C₉	—	3	1	no	—
10	C₂²⋊(C₂²⋊C₉)	—	3	1	no	—
11	C₂²⋊(C₂²⋊C₉)	—	3	1	no	—
12	C₂²×3₋¹⁺²	—	3	10	no	—
13	C₂²×3₋¹⁺²	A₄	1	5	yes	complemented
14	C₂²⋊H(3,𝔽₃)	—	4	5	no	—
15	#1094	—	4	5	no	—
16	C₃×C₂²⋊C₉	—	4	5	no	—
17	#731	—	16	1	no	Sylow
18	C₂³×C₃²	—	3	5	no	—
19	C₂³×C₉	—	3	15	no	—
20	C₂×3₋¹⁺²	—	3	5	no	—
21	C₂⁴×C₃	—	3	1	no	—
22	C₂⁴×C₃	H(3,𝔽₃)	1	1	yes	characteristic
23	C₂²⋊A₄	—	9	1	no	—
24	C₂²×C₃²	—	3	10	no	—
25	C₂²×C₃²	C₃×A₄	1	5	yes	—
26	C₃×A₄	—	12	5	no	—
27	C₂²×C₉	—	3	30	no	—
28	C₂²×C₉	—	3	5	no	—
29	C₂²⋊C₉	—	12	5	no	—
30	C₂²⋊C₉	—	12	5	no	—
31	3₋¹⁺²	—	16	1	no	—
32	3₋¹⁺²	C₂²⋊A₄	1	1	yes	characteristic, complemented
33	H(3,𝔽₃)	—	16	1	no	—
34	C₃×C₉	—	16	1	no	—
35	C₂³×C₃	—	3	5	no	—
36	C₂³×C₃	—	9	5	no	—
37	C₂×C₃²	—	3	5	no	—
38	C₂×C₉	—	3	15	no	—
39	C₂⁴	#731	1	1	yes	characteristic, complemented, Sylow
40	C₂²×C₃	—	3	10	no	—
41	C₂²×C₃	—	3	5	no	—
42	C₂²×C₃	—	9	10	no	—
43	C₂²×C₃	C₂²⋊H(3,𝔽₃)	1	5	yes	—
44	A₄	—	36	5	no	—
45	C₃²	—	48	1	no	—
46	C₃²	C₃×C₂²⋊A₄	1	1	yes	characteristic
47	C₉	—	3	1	no	—
48	C₉	—	48	1	no	—
49	C₉	—	48	1	no	—
50	C₂³	—	3	5	no	—
51	C₂×C₃	—	3	5	no	—
52	C₂×C₃	—	9	5	no	—
53	C₂²	—	3	10	no	—
54	C₂²	#66365	1	5	yes	complemented
55	C₃	—	3	1	no	—
56	C₃	—	144	1	no	—
57	C₃	C₃²⋊(C₂²⋊A₄)	1	1	yes	characteristic
58	C₂	—	3	5	no	—

	subgroup	G/H
center	C₃	C₃²⋊(C₂²⋊A₄)
commutator, cosocle	C₂⁴×C₃²	C₃²
Frattini	C₃²	C₃×C₂²⋊A₄
Fitting	C₂⁴×3₋¹⁺²	C₃
socle	C₂⁴×C₃	H(3,𝔽₃)
prime radical	C₃²⋊(C₂²⋊A₄)	C₃
hypercenter	3₋¹⁺²	C₂²⋊A₄
hypocenter	C₂⁴	#731
LCS	G ▷ C₂⁴×C₃² ▷ C₂⁴×C₃ ▷ C₂⁴
UCS	C₁ ◁ C₃ ◁ C₃² ◁ 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
13. C₂²×3₋¹⁺²	A₄	semidirect
32. 3₋¹⁺²	C₂²⋊A₄	semidirect
39. C₂⁴	#731	semidirect
54. C₂²	#66365	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
#66365	4