Subgroups — C₉×#215 #88511

Order 432.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×Q₈×C₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₉×D₁₂	C₂	1	1	yes	maximal, characteristic
3	C₉×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#215	C₃	1	1	yes	maximal, characteristic
5	C₉×SD₁₆	—	3	1	no	maximal
6	C₂×D₃×C₉	—	2	1	no	—
7	C₃×C₄×C₉	—	2	1	no	—
8	C₃×C₄×C₉	C₂²	1	1	yes	characteristic
9	C₃²×Q₈	C₂×C₃	1	1	yes	characteristic
10	C₃×D₁₂	C₂×C₃	1	1	yes	characteristic
11	C₃×C₃⋊C₈	C₂×C₃	1	1	yes	characteristic
12	Q₈×C₉	—	2	1	no	—
13	Q₈×C₉	D₃	1	1	yes	characteristic, complemented
14	D₄×C₉	—	3	1	no	—
15	C₈×C₉	—	3	1	no	—
16	C₂×C₃×C₉	D₄	1	1	yes	characteristic
17	D₃×C₉	—	4	1	no	—
18	C₃×SD₁₆	—	3	1	no	—
19	#215	C₉	1	1	yes	characteristic, direct product factor
20	C₂×C₃×D₃	—	2	1	no	—
21	C₃²×C₄	—	2	1	no	—
22	C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
23	C₂²×C₉	—	6	1	no	—
24	C₄×C₉	—	2	2	no	—
25	C₄×C₉	—	2	1	no	—
26	C₄×C₉	—	2	1	no	—
27	C₄×C₉	C₂×D₃	1	1	yes	characteristic
28	C₃×C₉	SD₁₆	1	1	yes	characteristic, complemented, Sylow
29	C₃×Q₈	—	2	1	no	—
30	C₃×Q₈	C₂×C₉	1	1	yes	characteristic, complemented
31	C₃×Q₈	C₃×D₃	1	1	yes	characteristic
32	C₃×D₄	—	3	1	no	—
33	D₁₂	C₂×C₉	1	1	yes	characteristic
34	C₃×C₈	—	3	1	no	—
35	C₃⋊C₈	C₂×C₉	1	1	yes	characteristic, complemented
36	C₂×C₃²	C₃×D₄	1	1	yes	characteristic
37	C₃×D₃	—	4	1	no	—
38	C₂×C₉	—	2	1	no	—
39	C₂×C₉	—	12	1	no	—
40	C₂×C₉	#67	1	1	yes	characteristic
41	SD₁₆	—	3	1	no	Sylow
42	C₂²×C₃	—	6	1	no	—
43	C₂×D₃	—	2	1	no	—
44	C₃×C₄	—	2	2	no	—
45	C₃×C₄	—	2	1	no	—
46	C₃×C₄	—	2	1	no	—
47	C₃×C₄	—	2	1	no	—
48	C₃×C₄	C₂²×C₉	1	1	yes	characteristic
49	C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
50	C₃²	C₃×SD₁₆	1	1	yes	characteristic
51	C₉	—	2	1	no	—
52	C₉	#215	1	1	yes	characteristic, direct product factor
53	Q₈	D₃×C₉	1	1	yes	characteristic, complemented
54	D₄	—	3	1	no	—
55	C₈	—	3	1	no	—
56	C₂×C₃	—	2	1	no	—
57	C₂×C₃	—	12	1	no	—
58	C₂×C₃	D₄×C₉	1	1	yes	characteristic
59	C₂×C₃	C₃×#67	1	1	yes	characteristic
60	D₃	—	4	1	no	—
61	C₂²	—	6	1	no	—
62	C₄	—	2	1	no	—
63	C₄	C₂×D₃×C₉	1	1	yes	characteristic
64	C₃	—	2	1	no	—
65	C₃	C₉×SD₁₆	1	1	yes	characteristic, complemented
66	C₃	C₃×#215	1	1	yes	characteristic
67	C₂	—	12	1	no	—
68	C₂	C₉×#67	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₉	#67
commutator	C₃×C₄	C₂²×C₉
Frattini	C₃×C₄	C₂×C₃×D₃
Fitting	C₃×Q₈×C₉	C₂
socle	C₂×C₃²	C₃×D₄
prime radical	C₃×D₁₂	C₂×C₃
cosocle	C₃²×C₄	C₂²×C₃
hypercenter	Q₈×C₉	D₃
hypocenter	C₃	C₉×SD₁₆
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₉ ◁ C₄×C₉ ◁ Q₈×C₉

Complements

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

N	H ≅ G/N	type
1. C₃×Q₈×C₉	C₂	semidirect
3. C₉×C₃⋊C₈	C₂	semidirect
13. Q₈×C₉	D₃	semidirect
19. #215	C₉	direct (N × H)
28. C₃×C₉	SD₁₆	semidirect
30. C₃×Q₈	C₂×C₉	semidirect
35. C₃⋊C₈	C₂×C₉	semidirect
52. C₉	#215	direct (N × H)
53. Q₈	D₃×C₉	semidirect
65. C₃	C₉×SD₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₃×Q₈×C₉	2
C₉×D₁₂	2
C₉×C₃⋊C₈	2
C₃×#215	3
C₉×SD₁₆	3

Subgroups — C9×#215 #88511

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#215 #88511