Subgroups — #13048171

Order 1296.

298 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃×C₉	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃²⋊(C₂²⋊A₄)	C₃	1	1	yes	maximal, characteristic
3	#88907	C₃	1	2	yes	maximal, complemented
4	#66357	—	4	5	no	maximal
5	C₂³×C₃×C₉	—	3	5	no	—
6	C₂⁴×C₃²	C₃²	1	1	yes	characteristic
7	C₃×C₂²⋊A₄	—	3	1	no	—
8	C₂⁴×C₉	—	3	1	no	—
9	C₂²⋊(C₂²⋊C₉)	—	3	2	no	—
10	C₂²×C₃×C₉	—	3	10	no	—
11	C₂²×C₃×C₉	A₄	1	5	yes	complemented
12	C₂²⋊H(3,𝔽₃)	—	4	5	no	—
13	#1094	—	4	10	no	—
14	#731	—	16	1	no	Sylow
15	C₂³×C₃²	—	3	5	no	—
16	C₂³×C₉	—	3	15	no	—
17	C₂×C₃×C₉	—	3	5	no	—
18	C₂⁴×C₃	—	3	1	no	—
19	C₂⁴×C₃	H(3,𝔽₃)	1	1	yes	characteristic
20	C₂²⋊A₄	—	9	1	no	—
21	C₂²×C₃²	—	3	10	no	—
22	C₂²×C₃²	C₃×A₄	1	5	yes	—
23	C₃×A₄	—	12	5	no	—
24	C₂²×C₉	—	3	30	no	—
25	C₂²×C₉	—	3	5	no	—
26	C₂²⋊C₉	—	12	10	no	—
27	3₋¹⁺²	—	16	2	no	—
28	H(3,𝔽₃)	—	16	1	no	—
29	C₃×C₉	C₂²⋊A₄	1	1	yes	characteristic, complemented
30	C₂³×C₃	—	3	15	no	—
31	C₂³×C₃	—	3	5	no	—
32	C₂×C₃²	—	3	5	no	—
33	C₂×C₉	—	3	15	no	—
34	C₂⁴	#731	1	1	yes	characteristic, complemented, Sylow
35	C₂²×C₃	—	3	10	no	—
36	C₂²×C₃	—	3	30	no	—
37	C₂²×C₃	—	3	5	no	—
38	C₂²×C₃	C₂²⋊H(3,𝔽₃)	1	5	yes	—
39	A₄	—	36	5	no	—
40	C₃²	—	48	1	no	—
41	C₃²	C₃×C₂²⋊A₄	1	1	yes	characteristic
42	C₉	—	3	1	no	—
43	C₉	—	48	2	no	—
44	C₂³	—	3	5	no	—
45	C₂×C₃	—	3	15	no	—
46	C₂×C₃	—	3	5	no	—
47	C₂²	—	3	10	no	—
48	C₂²	#66357	1	5	yes	complemented
49	C₃	—	3	1	no	—
50	C₃	—	144	1	no	—
51	C₃	C₃²⋊(C₂²⋊A₄)	1	1	yes	characteristic
52	C₂	—	3	5	no	—

	subgroup	G/H
center	C₃	C₃²⋊(C₂²⋊A₄)
commutator, cosocle	C₂⁴×C₃²	C₃²
Frattini	C₃²	C₃×C₂²⋊A₄
Fitting	C₂⁴×C₃×C₉	C₃
socle	C₂⁴×C₃	H(3,𝔽₃)
prime radical	C₃²⋊(C₂²⋊A₄)	C₃
hypercenter	C₃×C₉	C₂²⋊A₄
hypocenter	C₂⁴	#731
LCS	G ▷ C₂⁴×C₃² ▷ C₂⁴×C₃ ▷ C₂⁴
UCS	C₁ ◁ C₃ ◁ C₃² ◁ C₃×C₉

Complements

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

N	H ≅ G/N	type
1. C₂⁴×C₃×C₉	C₃	semidirect
3. #88907	C₃	semidirect
11. C₂²×C₃×C₉	A₄	semidirect
29. C₃×C₉	C₂²⋊A₄	semidirect
34. C₂⁴	#731	semidirect
48. C₂²	#66357	semidirect

Maximal subgroups

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

subgroup	index
C₂⁴×C₃×C₉	3
C₃²⋊(C₂²⋊A₄)	3
#88907	3
#66357	4