Subgroups — #13049336

Order 1296.

431 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	C₉⋊(C₂²⋊A₄)	C₃	1	6	yes	maximal, complemented
5	C₉×C₂²⋊A₄	C₃	1	3	yes	maximal, complemented
6	#66436	—	4	5	no	maximal
7	C₂³×C₃×C₉	—	3	5	no	—
8	C₂⁴×C₃²	C₃²	1	1	yes	characteristic
9	C₃×C₂²⋊A₄	C₃²	1	3	yes	—
10	C₂⁴×C₉	C₃²	1	1	yes	characteristic
11	C₂⁴×C₉	C₃²	1	2	yes	—
12	C₂²⋊(C₂²⋊C₉)	C₃²	1	6	yes	—
13	C₂²×C₃×C₉	—	3	10	no	—
14	C₂²×C₃×C₉	A₄	1	5	yes	complemented
15	C₂²⋊H(3,𝔽₃)	—	4	5	no	—
16	#1094	—	4	10	no	—
17	C₉⋊A₄	—	4	30	no	—
18	C₉×A₄	—	4	15	no	—
19	#737	—	16	1	no	Sylow
20	C₂³×C₃²	—	3	5	no	—
21	C₂³×C₉	—	3	5	no	—
22	C₂³×C₉	—	3	10	no	—
23	C₂×C₃×C₉	—	3	5	no	—
24	C₂⁴×C₃	—	3	1	no	—
25	C₂⁴×C₃	C₃³	1	1	yes	characteristic
26	C₂²⋊A₄	—	3	3	no	—
27	C₂²×C₃²	—	3	10	no	—
28	C₂²×C₃²	C₃×A₄	1	5	yes	—
29	C₃×A₄	—	4	15	no	—
30	C₂²×C₉	—	3	20	no	—
31	C₂²×C₉	—	3	10	no	—
32	C₂²×C₉	C₃×A₄	1	5	yes	—
33	C₂²×C₉	C₃×A₄	1	10	yes	—
34	C₂²⋊C₉	—	4	30	no	—
35	3₋¹⁺²	—	16	2	no	—
36	3₋¹⁺²	—	16	6	no	—
37	H(3,𝔽₃)	—	16	1	no	—
38	C₃×C₉	—	16	3	no	—
39	C₃×C₉	C₂²⋊A₄	1	1	yes	characteristic, complemented
40	C₂³×C₃	—	3	15	no	—
41	C₂³×C₃	—	3	5	no	—
42	C₂×C₃²	—	3	5	no	—
43	C₂×C₉	—	3	10	no	—
44	C₂×C₉	—	3	5	no	—
45	C₂⁴	#737	1	1	yes	characteristic, complemented, Sylow
46	C₂²×C₃	—	3	10	no	—
47	C₂²×C₃	—	3	30	no	—
48	C₂²×C₃	—	3	5	no	—
49	C₂²×C₃	C₃²×A₄	1	5	yes	—
50	A₄	—	12	15	no	—
51	C₃²	—	16	3	no	—
52	C₃²	C₃×C₂²⋊A₄	1	1	yes	characteristic
53	C₉	—	16	6	no	—
54	C₉	C₃×C₂²⋊A₄	1	2	yes	—
55	C₉	C₃×C₂²⋊A₄	1	1	yes	characteristic
56	C₂³	—	3	5	no	—
57	C₂×C₃	—	3	15	no	—
58	C₂×C₃	—	3	5	no	—
59	C₂²	—	3	10	no	—
60	C₂²	#66436	1	5	yes	complemented
61	C₃	—	3	1	no	—
62	C₃	—	48	3	no	—
63	C₃	C₃²×C₂²⋊A₄	1	1	yes	characteristic
64	C₂	—	3	5	no	—

	subgroup	G/H
center	C₉	C₃×C₂²⋊A₄
commutator, socle, cosocle	C₂⁴×C₃	C₃³
Frattini	C₃	C₃²×C₂²⋊A₄
Fitting	C₂⁴×C₃×C₉	C₃
prime radical	C₃²⋊(C₂²⋊A₄)	C₃
hypercenter	C₃×C₉	C₂²⋊A₄
hypocenter	C₂⁴	#737
LCS	G ▷ C₂⁴×C₃ ▷ C₂⁴
UCS	C₁ ◁ C₉ ◁ 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
4. C₉⋊(C₂²⋊A₄)	C₃	semidirect
5. C₉×C₂²⋊A₄	C₃	semidirect
14. C₂²×C₃×C₉	A₄	semidirect
39. C₃×C₉	C₂²⋊A₄	semidirect
45. C₂⁴	#737	semidirect
60. C₂²	#66436	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
C₉⋊(C₂²⋊A₄)	3
C₉×C₂²⋊A₄	3
#66436	4