Subgroups — #13048177

Order 1296.

360 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×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
3	#88907	C₃	1	2	yes	maximal, complemented
4	#66363	—	4	5	no	maximal
5	C₂³×H(3,𝔽₃)	—	3	5	no	—
6	C₂⁴×C₃²	—	3	1	no	—
7	C₂⁴×C₃²	C₃²	1	1	yes	characteristic
8	C₃×C₂²⋊A₄	—	3	1	no	—
9	C₃×C₂²⋊A₄	—	3	3	no	—
10	C₂²⋊(C₂²⋊C₉)	—	3	2	no	—
11	C₃²×A₄	—	4	5	no	—
12	C₂²×H(3,𝔽₃)	—	3	10	no	—
13	C₂²×H(3,𝔽₃)	A₄	1	5	yes	complemented
14	#1094	—	4	10	no	—
15	AΣL(1,𝔽₂₇)	—	16	1	no	Sylow
16	C₂³×C₃²	—	3	15	no	—
17	C₂³×C₃²	—	3	5	no	—
18	C₂×H(3,𝔽₃)	—	3	5	no	—
19	C₂⁴×C₃	—	3	1	no	—
20	C₂⁴×C₃	—	9	1	no	—
21	C₂⁴×C₃	H(3,𝔽₃)	1	1	yes	characteristic
22	C₂²⋊A₄	—	3	3	no	—
23	C₂²×C₃²	—	3	30	no	—
24	C₂²×C₃²	—	3	10	no	—
25	C₂²×C₃²	—	3	5	no	—
26	C₂²×C₃²	C₃×A₄	1	5	yes	—
27	C₃×A₄	—	12	5	no	—
28	C₃×A₄	—	12	15	no	—
29	C₂²⋊C₉	—	12	10	no	—
30	C₃³	—	16	1	no	—
31	3₋¹⁺²	—	16	2	no	—
32	H(3,𝔽₃)	C₂²⋊A₄	1	1	yes	characteristic, complemented
33	C₂³×C₃	—	3	5	no	—
34	C₂³×C₃	—	9	5	no	—
35	C₂³×C₃	—	9	15	no	—
36	C₂×C₃²	—	3	15	no	—
37	C₂×C₃²	—	3	5	no	—
38	C₂⁴	AΣL(1,𝔽₂₇)	1	1	yes	characteristic, complemented, Sylow
39	C₂²×C₃	—	3	10	no	—
40	C₂²×C₃	—	3	5	no	—
41	C₂²×C₃	—	9	10	no	—
42	C₂²×C₃	—	9	30	no	—
43	C₂²×C₃	—	9	5	no	—
44	C₂²×C₃	C₂²⋊H(3,𝔽₃)	1	5	yes	—
45	A₄	—	12	15	no	—
46	C₃²	—	3	1	no	—
47	C₃²	—	48	3	no	—
48	C₃²	—	48	1	no	—
49	C₃²	C₃×C₂²⋊A₄	1	1	yes	characteristic
50	C₉	—	48	2	no	—
51	C₂³	—	3	5	no	—
52	C₂×C₃	—	3	5	no	—
53	C₂×C₃	—	9	5	no	—
54	C₂×C₃	—	9	15	no	—
55	C₂²	—	3	10	no	—
56	C₂²	#66363	1	5	yes	complemented
57	C₃	—	3	1	no	—
58	C₃	—	9	1	no	—
59	C₃	—	48	3	no	—
60	C₃	C₃²⋊(C₂²⋊A₄)	1	1	yes	characteristic
61	C₂	—	3	5	no	—

	subgroup	G/H
center	C₃	C₃²⋊(C₂²⋊A₄)
commutator, cosocle	C₂⁴×C₃²	C₃²
Frattini	C₃²	C₃×C₂²⋊A₄
Fitting	C₂⁴×H(3,𝔽₃)	C₃
socle	C₂⁴×C₃	H(3,𝔽₃)
hypercenter	H(3,𝔽₃)	C₂²⋊A₄
hypocenter	C₂⁴	AΣL(1,𝔽₂₇)
LCS	G ▷ C₂⁴×C₃² ▷ C₂⁴×C₃ ▷ C₂⁴
UCS	C₁ ◁ C₃ ◁ C₃² ◁ H(3,𝔽₃)

Complements

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

N	H ≅ G/N	type
1. C₃²×C₂²⋊A₄	C₃	semidirect
2. C₂⁴×H(3,𝔽₃)	C₃	semidirect
3. #88907	C₃	semidirect
13. C₂²×H(3,𝔽₃)	A₄	semidirect
32. H(3,𝔽₃)	C₂²⋊A₄	semidirect
38. C₂⁴	AΣL(1,𝔽₂₇)	semidirect
56. C₂²	#66363	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×H(3,𝔽₃)	3
#88907	3
#66363	4