Subgroups — C₉⋊(C₂²⋊(C₄²⋊C₃)) #13152242

Order 1728.

333 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₂²⋊(C₄²⋊C₃)	C₃	1	1	yes	maximal, characteristic
2	C₂²×C₄²×C₉	C₃	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊(C₄²⋊C₉)	C₃	1	2	yes	maximal, complemented
4	C₉⋊(C₂²⋊A₄)	—	4	1	no	maximal
5	C₉⋊(C₄²⋊C₃)	—	4	4	no	maximal
6	C₂³×C₄×C₉	—	3	1	no	—
7	C₂×C₄²×C₉	—	3	4	no	—
8	C₂²×C₃×C₄²	C₃²	1	1	yes	characteristic
9	C₂²⋊(C₄²⋊C₃)	—	3	1	no	—
10	C₃×C₂²⋊A₄	—	4	1	no	—
11	C₂⁴×C₉	A₄	1	1	yes	characteristic
12	C₂²⋊(C₂²⋊C₉)	—	4	2	no	—
13	C₃×C₄²⋊C₃	—	4	4	no	—
14	C₂²×C₄×C₉	—	3	6	no	—
15	C₂²×C₄×C₉	—	3	8	no	—
16	C₄²×C₉	—	3	4	no	—
17	C₄²×C₉	A₄	1	4	yes	complemented
18	C₄²⋊C₉	—	4	8	no	—
19	C₉⋊A₄	—	16	4	no	—
20	C₉⋊A₄	—	16	1	no	—
21	C₂³×C₃×C₄	—	3	1	no	—
22	C₂×C₃×C₄²	—	3	4	no	—
23	C₂³×C₉	—	3	4	no	—
24	C₂³×C₉	—	3	1	no	—
25	C₂×C₄×C₉	—	3	24	no	—
26	C₂×C₄×C₉	—	3	4	no	—
27	C₂²×C₄²	3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
28	C₂⁴×C₃	C₃×A₄	1	1	yes	characteristic
29	C₂²⋊A₄	—	12	1	no	—
30	C₂²×C₃×C₄	—	3	6	no	—
31	C₂²×C₃×C₄	—	3	8	no	—
32	C₃×C₄²	—	3	4	no	—
33	C₃×C₄²	C₃×A₄	1	4	yes	—
34	C₄²⋊C₃	—	12	4	no	—
35	C₃×A₄	—	16	1	no	—
36	C₃×A₄	—	16	4	no	—
37	C₂²×C₉	—	3	6	no	—
38	C₂²×C₉	—	3	4	no	—
39	C₂²×C₉	C₄²⋊C₃	1	4	yes	complemented
40	C₂²×C₉	C₂²⋊A₄	1	1	yes	characteristic
41	C₂²⋊C₉	—	16	2	no	—
42	C₂²⋊C₉	—	16	8	no	—
43	C₄×C₉	—	3	8	no	—
44	C₂³×C₄	—	3	1	no	—
45	C₂×C₄²	—	3	4	no	—
46	3₋¹⁺²	—	64	1	no	Sylow
47	C₂³×C₃	—	3	1	no	—
48	C₂³×C₃	—	3	4	no	—
49	C₂×C₃×C₄	—	3	24	no	—
50	C₂×C₃×C₄	—	3	4	no	—
51	C₂×C₉	—	3	1	no	—
52	C₂×C₉	—	3	4	no	—
53	C₂⁴	C₉⋊A₄	1	1	yes	characteristic
54	C₂²×C₄	—	3	6	no	—
55	C₂²×C₄	—	3	8	no	—
56	C₄²	—	3	4	no	—
57	C₄²	C₉⋊A₄	1	4	yes	complemented
58	C₂²×C₃	—	3	6	no	—
59	C₂²×C₃	—	3	4	no	—
60	C₂²×C₃	C₃×C₄²⋊C₃	1	4	yes	—
61	C₂²×C₃	C₃×C₂²⋊A₄	1	1	yes	characteristic
62	A₄	—	48	1	no	—
63	A₄	—	48	4	no	—
64	C₃×C₄	—	3	8	no	—
65	C₃²	—	64	1	no	—
66	C₉	—	64	2	no	—
67	C₉	C₂²⋊(C₄²⋊C₃)	1	1	yes	characteristic, complemented
68	C₂³	—	3	1	no	—
69	C₂³	—	3	4	no	—
70	C₂×C₄	—	3	24	no	—
71	C₂×C₄	—	3	4	no	—
72	C₂×C₃	—	3	1	no	—
73	C₂×C₃	—	3	4	no	—
74	C₂²	—	3	6	no	—
75	C₂²	—	3	4	no	—
76	C₂²	C₉⋊(C₄²⋊C₃)	1	4	yes	complemented
77	C₂²	C₉⋊(C₂²⋊A₄)	1	1	yes	characteristic
78	C₄	—	3	8	no	—
79	C₃	—	192	1	no	—
80	C₃	C₃×C₂²⋊(C₄²⋊C₃)	1	1	yes	characteristic
81	C₂	—	3	1	no	—
82	C₂	—	3	4	no	—

	subgroup	G/H
center	C₃	C₃×C₂²⋊(C₄²⋊C₃)
commutator, cosocle	C₂²×C₃×C₄²	C₃²
Frattini	C₂²×C₃	C₃×C₂²⋊A₄
Fitting	C₂²×C₄²×C₉	C₃
socle	C₂⁴×C₃	C₃×A₄
prime radical	C₃×C₂²⋊(C₄²⋊C₃)	C₃
hypercenter	C₉	C₂²⋊(C₄²⋊C₃)
hypocenter	C₂²×C₄²	3₋¹⁺²
LCS	G ▷ C₂²×C₃×C₄² ▷ C₂²×C₄²
UCS	C₁ ◁ C₃ ◁ C₉

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄²×C₉	C₃	semidirect
3. C₂²⋊(C₄²⋊C₉)	C₃	semidirect
17. C₄²×C₉	A₄	semidirect
27. C₂²×C₄²	3₋¹⁺²	semidirect
39. C₂²×C₉	C₄²⋊C₃	semidirect
57. C₄²	C₉⋊A₄	semidirect
67. C₉	C₂²⋊(C₄²⋊C₃)	semidirect
76. C₂²	C₉⋊(C₄²⋊C₃)	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₂²⋊(C₄²⋊C₃)	3
C₂²×C₄²×C₉	3
C₂²⋊(C₄²⋊C₉)	3
C₉⋊(C₂²⋊A₄)	4
C₉⋊(C₄²⋊C₃)	4

Subgroups — C9⋊(C22⋊(C42⋊C3)) #13152242

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉⋊(C₂²⋊(C₄²⋊C₃)) #13152242