Subgroups — #13048180

Order 1296.

300 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, complemented
2	C₂⁴×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃²⋊(C₂²⋊A₄)	C₃	1	1	yes	maximal, characteristic, complemented
4	#88907	C₃	1	1	yes	maximal, characteristic, complemented
5	#66366	—	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	C₃×C₂²⋊A₄	—	3	3	no	—
10	C₃×C₂²⋊A₄	—	3	1	no	—
11	C₂⁴×C₉	—	3	1	no	—
12	C₂²⋊(C₂²⋊C₉)	—	3	1	no	—
13	C₃²×A₄	—	4	5	no	—
14	C₂²×3₋¹⁺²	—	3	10	no	—
15	C₂²×3₋¹⁺²	A₄	1	5	yes	complemented
16	C₂²⋊H(3,𝔽₃)	—	4	5	no	—
17	#1094	—	4	5	no	—
18	AΣL(1,𝔽₂₇)	—	16	1	no	Sylow
19	C₂³×C₃²	—	3	5	no	—
20	C₂³×C₉	—	3	15	no	—
21	C₂×3₋¹⁺²	—	3	5	no	—
22	C₂⁴×C₃	—	3	1	no	—
23	C₂⁴×C₃	H(3,𝔽₃)	1	1	yes	characteristic
24	C₂²⋊A₄	—	3	3	no	—
25	C₂²⋊A₄	—	9	1	no	—
26	C₂²×C₃²	—	3	10	no	—
27	C₂²×C₃²	C₃×A₄	1	5	yes	—
28	C₃×A₄	—	12	5	no	—
29	C₃×A₄	—	12	15	no	—
30	C₃×A₄	—	12	5	no	—
31	C₂²×C₉	—	3	30	no	—
32	C₂²×C₉	—	3	5	no	—
33	C₂²⋊C₉	—	12	5	no	—
34	C₃³	—	16	1	no	—
35	3₋¹⁺²	—	16	1	no	—
36	3₋¹⁺²	C₂²⋊A₄	1	1	yes	characteristic, complemented
37	H(3,𝔽₃)	—	16	1	no	—
38	C₂³×C₃	—	3	5	no	—
39	C₂³×C₃	—	9	5	no	—
40	C₂×C₃²	—	3	5	no	—
41	C₂×C₉	—	3	15	no	—
42	C₂⁴	AΣL(1,𝔽₂₇)	1	1	yes	characteristic, complemented, Sylow
43	C₂²×C₃	—	3	10	no	—
44	C₂²×C₃	—	3	5	no	—
45	C₂²×C₃	—	9	10	no	—
46	C₂²×C₃	C₂²⋊H(3,𝔽₃)	1	5	yes	—
47	A₄	—	12	15	no	—
48	A₄	—	36	5	no	—
49	C₃²	—	48	3	no	—
50	C₃²	—	48	1	no	—
51	C₃²	—	48	1	no	—
52	C₃²	C₃×C₂²⋊A₄	1	1	yes	characteristic
53	C₉	—	3	1	no	—
54	C₉	—	48	1	no	—
55	C₂³	—	3	5	no	—
56	C₂×C₃	—	3	5	no	—
57	C₂×C₃	—	9	5	no	—
58	C₂²	—	3	10	no	—
59	C₂²	#66366	1	5	yes	complemented
60	C₃	—	3	1	no	—
61	C₃	—	48	3	no	—
62	C₃	—	144	1	no	—
63	C₃	C₃²⋊(C₂²⋊A₄)	1	1	yes	characteristic
64	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)
hypercenter	3₋¹⁺²	C₂²⋊A₄
hypocenter	C₂⁴	AΣL(1,𝔽₂₇)
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. C₃²×C₂²⋊A₄	C₃	semidirect
2. C₂⁴×3₋¹⁺²	C₃	semidirect
3. C₃²⋊(C₂²⋊A₄)	C₃	semidirect
4. #88907	C₃	semidirect
15. C₂²×3₋¹⁺²	A₄	semidirect
36. 3₋¹⁺²	C₂²⋊A₄	semidirect
42. C₂⁴	AΣL(1,𝔽₂₇)	semidirect
59. C₂²	#66366	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
C₃²⋊(C₂²⋊A₄)	3
#88907	3
#66366	4