Subgroups — C₉×#216 #88512

Order 432.

66 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
2	C₉×Dic₆	C₂	1	1	yes	maximal, characteristic
3	C₉×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₃×#216	C₃	1	1	yes	maximal, characteristic
5	C₉×Q₁₆	—	3	1	no	maximal
6	C₃×C₄×C₉	—	2	1	no	—
7	C₃×C₄×C₉	C₂²	1	1	yes	characteristic
8	C₉×Dic₃	—	2	1	no	—
9	C₃²×Q₈	C₂×C₃	1	1	yes	characteristic
10	C₃×Dic₆	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	—	3	1	no	—
14	Q₈×C₉	D₃	1	1	yes	characteristic
15	C₈×C₉	—	3	1	no	—
16	C₂×C₃×C₉	D₄	1	1	yes	characteristic
17	C₃×Q₁₆	—	3	1	no	—
18	#216	C₉	1	1	yes	characteristic, direct product factor
19	C₃²×C₄	—	2	1	no	—
20	C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
21	C₃×Dic₃	—	2	1	no	—
22	C₄×C₉	—	2	2	no	—
23	C₄×C₉	—	2	1	no	—
24	C₄×C₉	—	2	1	no	—
25	C₄×C₉	—	6	1	no	—
26	C₄×C₉	C₂×D₃	1	1	yes	characteristic
27	C₃×C₉	Q₁₆	1	1	yes	characteristic, complemented, Sylow
28	C₃×Q₈	—	2	1	no	—
29	C₃×Q₈	—	3	1	no	—
30	C₃×Q₈	C₂×C₉	1	1	yes	characteristic
31	C₃×Q₈	C₃×D₃	1	1	yes	characteristic
32	Dic₆	C₂×C₉	1	1	yes	characteristic
33	C₃×C₈	—	3	1	no	—
34	C₃⋊C₈	C₂×C₉	1	1	yes	characteristic
35	C₂×C₃²	C₃×D₄	1	1	yes	characteristic
36	C₂×C₉	—	2	1	no	—
37	C₂×C₉	#67	1	1	yes	characteristic
38	Q₁₆	—	3	1	no	Sylow
39	C₃×C₄	—	2	2	no	—
40	C₃×C₄	—	2	1	no	—
41	C₃×C₄	—	2	1	no	—
42	C₃×C₄	—	2	1	no	—
43	C₃×C₄	—	6	1	no	—
44	C₃×C₄	C₂²×C₉	1	1	yes	characteristic
45	C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
46	Dic₃	—	2	1	no	—
47	C₃²	C₃×Q₁₆	1	1	yes	characteristic
48	C₉	—	2	1	no	—
49	C₉	#216	1	1	yes	characteristic, direct product factor
50	Q₈	—	3	1	no	—
51	Q₈	D₃×C₉	1	1	yes	characteristic
52	C₈	—	3	1	no	—
53	C₂×C₃	—	2	1	no	—
54	C₂×C₃	D₄×C₉	1	1	yes	characteristic
55	C₂×C₃	C₃×#67	1	1	yes	characteristic
56	C₄	—	2	1	no	—
57	C₄	—	6	1	no	—
58	C₄	C₂×D₃×C₉	1	1	yes	characteristic
59	C₃	—	2	1	no	—
60	C₃	C₉×Q₁₆	1	1	yes	characteristic, complemented
61	C₃	C₃×#216	1	1	yes	characteristic
62	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, prime radical	C₂×C₃²	C₃×D₄
cosocle	C₃²×C₄	C₂²×C₃
hypercenter	Q₈×C₉	D₃
hypocenter	C₃	C₉×Q₁₆
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
18. #216	C₉	direct (N × H)
27. C₃×C₉	Q₁₆	semidirect
49. C₉	#216	direct (N × H)
60. C₃	C₉×Q₁₆	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×Dic₆	2
C₉×C₃⋊C₈	2
C₃×#216	3
C₉×Q₁₆	3

Subgroups — C9×#216 #88512

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#216 #88512