Subgroups — #13048175

Order 1296.

318 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
2	#88907	C₃	1	2	yes	maximal, complemented
3	C₃×C₂²⋊(C₂²⋊C₉)	C₃	1	1	yes	maximal, characteristic, complemented
4	#66361	—	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	C₂²⋊(C₂²⋊C₉)	—	3	1	no	—
9	C₂²⋊(C₂²⋊C₉)	—	3	2	no	—
10	C₂²×H(3,𝔽₃)	—	3	10	no	—
11	C₂²×H(3,𝔽₃)	A₄	1	5	yes	—
12	#1094	—	4	10	no	—
13	C₃×C₂²⋊C₉	—	4	5	no	—
14	#731	—	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×C₃²	—	3	30	no	—
22	C₂²×C₃²	—	3	10	no	—
23	C₂²×C₃²	—	3	5	no	—
24	C₂²×C₃²	C₃×A₄	1	5	yes	—
25	C₂²⋊C₉	—	12	5	no	—
26	C₂²⋊C₉	—	12	10	no	—
27	3₋¹⁺²	—	16	2	no	—
28	H(3,𝔽₃)	C₂²⋊A₄	1	1	yes	characteristic
29	C₃×C₉	—	16	1	no	—
30	C₂³×C₃	—	3	5	no	—
31	C₂³×C₃	—	9	5	no	—
32	C₂³×C₃	—	9	15	no	—
33	C₂×C₃²	—	3	15	no	—
34	C₂×C₃²	—	3	5	no	—
35	C₂⁴	#731	1	1	yes	characteristic, complemented, Sylow
36	C₂²×C₃	—	3	10	no	—
37	C₂²×C₃	—	3	5	no	—
38	C₂²×C₃	—	9	10	no	—
39	C₂²×C₃	—	9	30	no	—
40	C₂²×C₃	—	9	5	no	—
41	C₂²×C₃	C₂²⋊H(3,𝔽₃)	1	5	yes	—
42	C₃²	—	3	1	no	—
43	C₃²	C₃×C₂²⋊A₄	1	1	yes	characteristic
44	C₉	—	48	1	no	—
45	C₉	—	48	2	no	—
46	C₂³	—	3	5	no	—
47	C₂×C₃	—	3	5	no	—
48	C₂×C₃	—	9	5	no	—
49	C₂×C₃	—	9	15	no	—
50	C₂²	—	3	10	no	—
51	C₂²	#66361	1	5	yes	complemented
52	C₃	—	3	1	no	—
53	C₃	—	9	1	no	—
54	C₃	C₃²⋊(C₂²⋊A₄)	1	1	yes	characteristic
55	C₂	—	3	5	no	—

	subgroup	G/H
center	C₃	C₃²⋊(C₂²⋊A₄)
commutator, cosocle	C₂⁴×C₃²	C₃²
Frattini	C₃²	C₃×C₂²⋊A₄
Fitting, prime radical	C₂⁴×H(3,𝔽₃)	C₃
socle	C₂⁴×C₃	H(3,𝔽₃)
hypercenter	H(3,𝔽₃)	C₂²⋊A₄
hypocenter	C₂⁴	#731
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
2. #88907	C₃	semidirect
3. C₃×C₂²⋊(C₂²⋊C₉)	C₃	semidirect
35. C₂⁴	#731	semidirect
51. C₂²	#66361	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
#88907	3
C₃×C₂²⋊(C₂²⋊C₉)	3
#66361	4