Subgroups — C₉×#239 #88720

Order 432.

132 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	3	yes	maximal, complemented
3	C₄×D₃×C₉	C₂	1	3	yes	maximal, complemented
4	C₃×#239	C₃	1	1	yes	maximal, characteristic
5	C₉×#41	—	3	1	no	maximal
6	C₂×D₃×C₉	C₂²	1	3	yes	—
7	C₃×C₄×C₉	C₂²	1	3	yes	—
8	C₉×Dic₃	C₂²	1	1	yes	characteristic
9	C₃²×Q₈	C₂×C₃	1	1	yes	characteristic
10	C₃×D₁₂	C₂×C₃	1	3	yes	—
11	C₃×C₄×D₃	C₂×C₃	1	3	yes	—
12	Q₈×C₉	—	2	1	no	—
13	Q₈×C₉	D₃	1	1	yes	characteristic, complemented
14	D₄×C₉	—	3	3	no	—
15	C₂×C₄×C₉	—	3	3	no	—
16	C₂×C₃×C₉	C₂³	1	1	yes	characteristic
17	D₃×C₉	—	2	3	no	—
18	C₃×#41	—	3	1	no	—
19	#239	C₉	1	1	yes	characteristic, direct product factor
20	C₂×C₃×D₃	C₂²×C₃	1	3	yes	—
21	C₃²×C₄	C₂²×C₃	1	3	yes	—
22	C₃×Dic₃	C₂²×C₃	1	1	yes	characteristic
23	C₂²×C₉	—	3	3	no	—
24	C₄×C₉	—	2	3	no	—
25	C₄×C₉	—	3	1	no	—
26	C₄×C₉	C₂×D₃	1	3	yes	—
27	C₃×C₉	#41	1	1	yes	characteristic, complemented, Sylow
28	C₃×Q₈	—	2	1	no	—
29	C₃×Q₈	C₂×C₉	1	1	yes	characteristic, complemented
30	C₃×Q₈	C₃×D₃	1	1	yes	characteristic
31	C₃×D₄	—	3	3	no	—
32	C₂×C₃×C₄	—	3	3	no	—
33	D₁₂	C₂×C₉	1	3	yes	complemented
34	C₄×D₃	C₂×C₉	1	3	yes	complemented
35	C₂×C₃²	C₂³×C₃	1	1	yes	characteristic
36	C₃×D₃	—	2	3	no	—
37	C₂×C₉	—	2	1	no	—
38	C₂×C₉	—	6	3	no	—
39	C₂×C₉	C₂²×D₃	1	1	yes	characteristic
40	#41	—	3	1	no	Sylow
41	C₂²×C₃	—	3	3	no	—
42	C₂×D₃	C₂²×C₉	1	3	yes	—
43	C₃×C₄	—	2	3	no	—
44	C₃×C₄	—	3	1	no	—
45	C₃×C₄	C₂²×C₉	1	3	yes	—
46	C₃×C₄	C₂×C₃×D₃	1	3	yes	—
47	Dic₃	C₂²×C₉	1	1	yes	characteristic
48	C₃²	C₃×#41	1	1	yes	characteristic
49	C₉	—	2	1	no	—
50	C₉	#239	1	1	yes	characteristic, direct product factor
51	Q₈	D₃×C₉	1	1	yes	characteristic, complemented
52	D₄	—	3	3	no	—
53	C₂×C₄	—	3	3	no	—
54	C₂×C₃	—	2	1	no	—
55	C₂×C₃	—	6	3	no	—
56	C₂×C₃	C₂³×C₉	1	1	yes	characteristic
57	C₂×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
58	D₃	—	2	3	no	—
59	C₂²	—	3	3	no	—
60	C₄	—	3	1	no	—
61	C₄	C₂×D₃×C₉	1	3	yes	—
62	C₃	—	2	1	no	—
63	C₃	C₉×#41	1	1	yes	characteristic, complemented
64	C₃	C₃×#239	1	1	yes	characteristic
65	C₂	—	6	3	no	—
66	C₂	C₂²×D₃×C₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₉	C₂²×D₃
commutator	C₂×C₃	C₂³×C₉
Frattini	C₂×C₃	C₂²×C₃×D₃
Fitting	C₃×Q₈×C₉	C₂
socle, cosocle	C₂×C₃²	C₂³×C₃
prime radical	C₃×#239	C₃
hypercenter	Q₈×C₉	D₃
hypocenter	C₃	C₉×#41
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×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
2. C₉×D₁₂	C₂	semidirect
3. C₄×D₃×C₉	C₂	semidirect
13. Q₈×C₉	D₃	semidirect
19. #239	C₉	direct (N × H)
27. C₃×C₉	#41	semidirect
29. C₃×Q₈	C₂×C₉	semidirect
33. D₁₂	C₂×C₉	semidirect
34. C₄×D₃	C₂×C₉	semidirect
50. C₉	#239	direct (N × H)
51. Q₈	D₃×C₉	semidirect
63. C₃	C₉×#41	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
C₄×D₃×C₉	2
C₃×#239	3
C₉×#41	3

Subgroups — C9×#239 #88720

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#239 #88720