Subgroups — #11721332

Order 864.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉⋊(C₄²⋊C₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#126⋊C₃	C₃	1	1	yes	maximal, characteristic
3	C₉×#126	C₃	1	1	yes	maximal, characteristic, complemented
4	#126⋊C₉	C₃	1	2	yes	maximal, complemented
5	C₂×C₉⋊A₄	—	4	1	no	maximal
6	C₃×C₄²⋊C₃	C₂×C₃	1	1	yes	characteristic
7	C₉×C₄⋊C₄	—	3	1	no	—
8	C₉×C₂²⋊C₄	—	3	1	no	—
9	C₄²×C₉	C₂×C₃	1	1	yes	characteristic, complemented
10	C₄²⋊C₉	C₂×C₃	1	2	yes	complemented
11	C₉⋊A₄	—	4	1	no	—
12	C₃×#126	C₃²	1	1	yes	characteristic
13	#126⋊C₃	—	3	1	no	—
14	C₂×C₃×A₄	—	4	1	no	—
15	C₂³×C₉	A₄	1	1	yes	characteristic
16	C₂×C₂²⋊C₉	—	4	2	no	—
17	C₂×C₄×C₉	—	3	1	no	—
18	C₂×C₄×C₉	—	3	1	no	—
19	C₂×3₋¹⁺²	—	16	1	no	—
20	C₃×C₄⋊C₄	—	3	1	no	—
21	C₃×C₂²⋊C₄	—	3	1	no	—
22	C₃×C₄²	C₂×C₃²	1	1	yes	characteristic
23	C₄²⋊C₃	—	3	1	no	—
24	C₃×A₄	—	4	1	no	—
25	C₂²×C₉	—	6	1	no	—
26	C₂²×C₉	C₂×A₄	1	1	yes	characteristic
27	C₂²⋊C₉	—	4	2	no	—
28	C₄×C₉	—	6	1	no	—
29	C₄×C₉	—	6	1	no	—
30	#126	3₋¹⁺²	1	1	yes	characteristic, complemented, Sylow
31	3₋¹⁺²	—	16	1	no	Sylow
32	C₂³×C₃	C₃×A₄	1	1	yes	characteristic
33	C₂×A₄	—	12	1	no	—
34	C₂×C₃×C₄	—	3	1	no	—
35	C₂×C₃×C₄	—	3	1	no	—
36	C₂×C₃²	—	16	1	no	—
37	C₂×C₉	—	3	1	no	—
38	C₂×C₉	—	4	1	no	—
39	C₂×C₉	—	16	2	no	—
40	C₄⋊C₄	—	3	1	no	—
41	C₂²⋊C₄	—	3	1	no	—
42	C₄²	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
43	C₂²×C₃	—	6	1	no	—
44	C₂²×C₃	C₂×C₃×A₄	1	1	yes	characteristic
45	A₄	—	12	1	no	—
46	C₃×C₄	—	6	1	no	—
47	C₃×C₄	—	6	1	no	—
48	C₃²	—	16	1	no	—
49	C₉	—	16	2	no	—
50	C₉	#126⋊C₃	1	1	yes	characteristic, complemented
51	C₂³	C₉⋊A₄	1	1	yes	characteristic
52	C₂×C₄	—	3	1	no	—
53	C₂×C₄	—	3	1	no	—
54	C₂×C₃	—	3	1	no	—
55	C₂×C₃	—	4	1	no	—
56	C₂×C₃	—	48	1	no	—
57	C₂²	—	6	1	no	—
58	C₂²	C₂×C₉⋊A₄	1	1	yes	characteristic
59	C₄	—	6	1	no	—
60	C₄	—	6	1	no	—
61	C₃	—	48	1	no	—
62	C₃	C₃×#126⋊C₃	1	1	yes	characteristic
63	C₂	—	3	1	no	—
64	C₂	—	4	1	no	—

	subgroup	G/H
center	C₃	C₃×#126⋊C₃
commutator, cosocle	C₃×C₄²	C₂×C₃²
Frattini, socle	C₂²×C₃	C₂×C₃×A₄
Fitting	C₉×#126	C₃
prime radical	C₃×#126⋊C₃	C₃
hypercenter	C₉	#126⋊C₃
hypocenter	C₄²	C₂×3₋¹⁺²
LCS	G ▷ C₃×C₄² ▷ C₄²
UCS	C₁ ◁ C₃ ◁ C₉

Complements

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

N	H ≅ G/N	type
1. C₉⋊(C₄²⋊C₃)	C₂	semidirect
3. C₉×#126	C₃	semidirect
4. #126⋊C₉	C₃	semidirect
9. C₄²×C₉	C₂×C₃	semidirect
10. C₄²⋊C₉	C₂×C₃	semidirect
30. #126	3₋¹⁺²	semidirect
42. C₄²	C₂×3₋¹⁺²	semidirect
50. C₉	#126⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₉⋊(C₄²⋊C₃)	2
C₃×#126⋊C₃	3
C₉×#126	3
#126⋊C₉	3
C₂×C₉⋊A₄	4