Subgroups — #13048176

Order 1296.

332 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃²⋊(C₂²⋊A₄)	C₃	1	2	yes	maximal, complemented
3	C₃×C₂²⋊(C₂²⋊C₉)	C₃	1	1	yes	maximal, characteristic, complemented
4	#66362	—	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	2	no	—
9	C₂²⋊(C₂²⋊C₉)	—	3	1	no	—
10	C₂²×H(3,𝔽₃)	—	3	10	no	—
11	C₂²×H(3,𝔽₃)	A₄	1	5	yes	complemented
12	C₂²⋊H(3,𝔽₃)	—	4	10	no	—
13	C₃×C₂²⋊C₉	—	4	5	no	—
14	#732	—	16	1	no	Sylow
15	C₂³×C₃²	—	3	15	no	—
16	C₂³×C₃²	—	3	5	no	—
17	C₂×H(3,𝔽₃)	—	3	5	no	—
18	C₂⁴×C₃	—	3	1	no	—
19	C₂⁴×C₃	—	9	1	no	—
20	C₂⁴×C₃	H(3,𝔽₃)	1	1	yes	characteristic
21	C₂²⋊A₄	—	9	2	no	—
22	C₂²×C₃²	—	3	30	no	—
23	C₂²×C₃²	—	3	10	no	—
24	C₂²×C₃²	—	3	5	no	—
25	C₂²×C₃²	C₃×A₄	1	5	yes	—
26	C₃×A₄	—	12	10	no	—
27	C₂²⋊C₉	—	12	5	no	—
28	H(3,𝔽₃)	—	16	2	no	—
29	H(3,𝔽₃)	C₂²⋊A₄	1	1	yes	characteristic, complemented
30	C₃×C₉	—	16	1	no	—
31	C₂³×C₃	—	3	5	no	—
32	C₂³×C₃	—	9	5	no	—
33	C₂³×C₃	—	9	15	no	—
34	C₂×C₃²	—	3	15	no	—
35	C₂×C₃²	—	3	5	no	—
36	C₂⁴	#732	1	1	yes	characteristic, complemented, Sylow
37	C₂²×C₃	—	3	10	no	—
38	C₂²×C₃	—	3	5	no	—
39	C₂²×C₃	—	9	10	no	—
40	C₂²×C₃	—	9	30	no	—
41	C₂²×C₃	—	9	5	no	—
42	C₂²×C₃	C₂²⋊H(3,𝔽₃)	1	5	yes	—
43	A₄	—	36	10	no	—
44	C₃²	—	3	1	no	—
45	C₃²	—	48	2	no	—
46	C₃²	C₃×C₂²⋊A₄	1	1	yes	characteristic
47	C₉	—	48	1	no	—
48	C₂³	—	3	5	no	—
49	C₂×C₃	—	3	5	no	—
50	C₂×C₃	—	9	5	no	—
51	C₂×C₃	—	9	15	no	—
52	C₂²	—	3	10	no	—
53	C₂²	#66362	1	5	yes	complemented
54	C₃	—	3	1	no	—
55	C₃	—	9	1	no	—
56	C₃	—	144	2	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×H(3,𝔽₃)	C₃
socle	C₂⁴×C₃	H(3,𝔽₃)
hypercenter	H(3,𝔽₃)	C₂²⋊A₄
hypocenter	C₂⁴	#732
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. C₂⁴×H(3,𝔽₃)	C₃	semidirect
2. C₃²⋊(C₂²⋊A₄)	C₃	semidirect
3. C₃×C₂²⋊(C₂²⋊C₉)	C₃	semidirect
11. C₂²×H(3,𝔽₃)	A₄	semidirect
29. H(3,𝔽₃)	C₂²⋊A₄	semidirect
36. C₂⁴	#732	semidirect
53. C₂²	#66362	semidirect

Maximal subgroups

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

subgroup	index
C₂⁴×H(3,𝔽₃)	3
C₃²⋊(C₂²⋊A₄)	3
C₃×C₂²⋊(C₂²⋊C₉)	3
#66362	4