Subgroups — C₃²⋊(C₂²⋊(C₂²⋊C₉)) #13048181

Order 1296.

721 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₃³	C₃	1	1	yes	maximal, characteristic
2	C₃×C₂²⋊(C₂²⋊C₉)	C₃	1	3	yes	maximal, complemented
3	C₃²⋊(C₂²⋊C₉)	—	4	5	no	maximal
4	C₂³×C₃³	—	3	5	no	—
5	C₂⁴×C₃²	—	3	1	no	—
6	C₂⁴×C₃²	—	3	2	no	—
7	C₂⁴×C₃²	C₉	1	3	yes	complemented
8	C₂⁴×C₃²	C₃²	1	1	yes	characteristic
9	C₂²⋊(C₂²⋊C₉)	—	3	3	no	—
10	C₂²×C₃³	—	3	10	no	—
11	C₂²×C₃³	A₄	1	5	yes	—
12	C₃×C₂²⋊C₉	—	4	15	no	—
13	C₃²⋊C₉	—	16	1	no	Sylow
14	C₂³×C₃²	—	3	30	no	—
15	C₂³×C₃²	—	3	15	no	—
16	C₂³×C₃²	—	3	15	no	—
17	C₂³×C₃²	—	3	5	no	—
18	C₂×C₃³	—	3	5	no	—
19	C₂⁴×C₃	—	3	3	no	—
20	C₂⁴×C₃	C₃×C₉	1	1	yes	characteristic
21	C₂⁴×C₃	H(3,𝔽₃)	1	1	yes	characteristic
22	C₂⁴×C₃	3₋¹⁺²	1	2	yes	—
23	C₂²×C₃²	—	3	60	no	—
24	C₂²×C₃²	—	3	30	no	—
25	C₂²×C₃²	—	3	10	no	—
26	C₂²×C₃²	—	3	30	no	—
27	C₂²×C₃²	—	3	10	no	—
28	C₂²×C₃²	—	3	5	no	—
29	C₂²×C₃²	C₂²⋊C₉	1	15	yes	complemented
30	C₂²×C₃²	C₃×A₄	1	5	yes	—
31	C₂²⋊C₉	—	12	15	no	—
32	C₃³	C₂²⋊A₄	1	1	yes	characteristic
33	C₃×C₉	—	16	3	no	—
34	C₂³×C₃	—	3	45	no	—
35	C₂³×C₃	—	3	5	no	—
36	C₂³×C₃	—	3	10	no	—
37	C₂³×C₃	—	3	5	no	—
38	C₂×C₃²	—	3	30	no	—
39	C₂×C₃²	—	3	15	no	—
40	C₂×C₃²	—	3	5	no	—
41	C₂×C₃²	—	3	15	no	—
42	C₂⁴	C₃²⋊C₉	1	1	yes	characteristic, complemented, Sylow
43	C₂²×C₃	—	3	90	no	—
44	C₂²×C₃	—	3	10	no	—
45	C₂²×C₃	—	3	20	no	—
46	C₂²×C₃	—	3	10	no	—
47	C₂²×C₃	—	3	15	no	—
48	C₂²×C₃	C₃×C₂²⋊C₉	1	5	yes	—
49	C₂²×C₃	#1094	1	10	yes	—
50	C₂²×C₃	C₂²⋊H(3,𝔽₃)	1	5	yes	—
51	C₃²	—	3	2	no	—
52	C₃²	—	3	1	no	—
53	C₃²	C₂²⋊(C₂²⋊C₉)	1	3	yes	complemented
54	C₃²	C₃×C₂²⋊A₄	1	1	yes	characteristic
55	C₉	—	48	3	no	—
56	C₂³	—	3	5	no	—
57	C₂×C₃	—	3	45	no	—
58	C₂×C₃	—	3	5	no	—
59	C₂×C₃	—	3	10	no	—
60	C₂×C₃	—	3	5	no	—
61	C₂²	—	3	10	no	—
62	C₂²	C₃²⋊(C₂²⋊C₉)	1	5	yes	complemented
63	C₃	—	3	3	no	—
64	C₃	C₃×C₂²⋊(C₂²⋊C₉)	1	1	yes	characteristic
65	C₃	#88907	1	2	yes	—
66	C₃	C₃²⋊(C₂²⋊A₄)	1	1	yes	characteristic
67	C₂	—	3	5	no	—

	subgroup	G/H
center, Frattini	C₃²	C₃×C₂²⋊A₄
commutator	C₂⁴×C₃	C₃×C₉
Fitting, prime radical	C₂⁴×C₃³	C₃
socle, cosocle	C₂⁴×C₃²	C₃²
hypercenter	C₃³	C₂²⋊A₄
hypocenter	C₂⁴	C₃²⋊C₉
LCS	G ▷ C₂⁴×C₃ ▷ C₂⁴
UCS	C₁ ◁ C₃² ◁ C₃³

Complements

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

N	H ≅ G/N	type
2. C₃×C₂²⋊(C₂²⋊C₉)	C₃	semidirect
7. C₂⁴×C₃²	C₉	semidirect
29. C₂²×C₃²	C₂²⋊C₉	semidirect
42. C₂⁴	C₃²⋊C₉	semidirect
53. C₃²	C₂²⋊(C₂²⋊C₉)	semidirect
62. C₂²	C₃²⋊(C₂²⋊C₉)	semidirect

Maximal subgroups

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

subgroup	index
C₂⁴×C₃³	3
C₃×C₂²⋊(C₂²⋊C₉)	3
C₃²⋊(C₂²⋊C₉)	4

Subgroups — C32⋊(C22⋊(C22⋊C9)) #13048181

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²⋊(C₂²⋊(C₂²⋊C₉)) #13048181