Subgroups — C₉×#500 #10589054

Order 576.

147 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×C₄⋊Q₈	C₂	1	2	yes	maximal
2	C₉×#107	C₂	1	4	yes	maximal
3	C₄×C₈×C₉	C₂	1	1	yes	maximal, characteristic
4	C₃×#500	C₃	1	1	yes	maximal, characteristic
5	C₂×Q₈×C₉	—	2	2	no	—
6	C₂×C₈×C₉	C₂²	1	2	yes	—
7	C₉×C₄⋊C₄	—	2	2	no	—
8	C₉×C₄⋊C₄	C₂²	1	4	yes	—
9	C₄²×C₉	C₂²	1	1	yes	characteristic
10	C₃×C₄⋊Q₈	C₂×C₃	1	2	yes	—
11	C₃×#107	C₂×C₃	1	4	yes	—
12	C₃×C₄×C₈	C₂×C₃	1	1	yes	characteristic
13	Q₈×C₉	—	4	4	no	—
14	C₂×C₄×C₉	—	2	4	no	—
15	C₂×C₄×C₉	D₄	1	1	yes	characteristic
16	C₂×C₄×C₉	D₄	1	1	yes	characteristic
17	C₂×C₄×C₉	C₂³	1	1	yes	characteristic
18	C₈×C₉	Q₈	1	4	yes	complemented
19	#500	C₉	1	1	yes	characteristic, direct product factor, Sylow
20	C₂×C₃×Q₈	—	2	2	no	—
21	C₂×C₃×C₈	C₂²×C₃	1	2	yes	—
22	C₃×C₄⋊C₄	—	2	2	no	—
23	C₃×C₄⋊C₄	C₂²×C₃	1	4	yes	—
24	C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
25	C₂²×C₉	C₂×D₄	1	1	yes	characteristic
26	C₄×C₉	—	4	4	no	—
27	C₄×C₉	D₈	1	2	yes	—
28	C₄×C₉	Q₁₆	1	2	yes	—
29	C₄×C₉	C₂×Q₈	1	2	yes	—
30	C₄⋊Q₈	C₂×C₉	1	2	yes	—
31	#107	C₂×C₉	1	4	yes	—
32	C₄×C₈	C₂×C₉	1	1	yes	characteristic
33	C₃×Q₈	—	4	4	no	—
34	C₂×C₃×C₄	—	2	4	no	—
35	C₂×C₃×C₄	C₃×D₄	1	1	yes	characteristic
36	C₂×C₃×C₄	C₃×D₄	1	1	yes	characteristic
37	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
38	C₃×C₈	C₃×Q₈	1	4	yes	—
39	C₂×C₉	C₄⋊Q₈	1	1	yes	characteristic
40	C₂×C₉	C₂×D₈	1	1	yes	characteristic
41	C₂×C₉	C₂×Q₁₆	1	1	yes	characteristic
42	C₂×Q₈	—	2	2	no	—
43	C₂×C₈	C₂²×C₉	1	2	yes	—
44	C₄⋊C₄	—	2	2	no	—
45	C₄⋊C₄	C₂²×C₉	1	4	yes	—
46	C₄²	C₂²×C₉	1	1	yes	characteristic
47	C₂²×C₃	C₂×C₃×D₄	1	1	yes	characteristic
48	C₃×C₄	—	4	4	no	—
49	C₃×C₄	C₃×D₈	1	2	yes	—
50	C₃×C₄	C₃×Q₁₆	1	2	yes	—
51	C₃×C₄	C₂×C₃×Q₈	1	2	yes	—
52	C₉	#500	1	1	yes	characteristic, direct product factor, Sylow
53	Q₈	—	4	4	no	—
54	C₂×C₄	—	2	4	no	—
55	C₂×C₄	D₄×C₉	1	1	yes	characteristic
56	C₂×C₄	D₄×C₉	1	1	yes	characteristic
57	C₂×C₄	C₂³×C₉	1	1	yes	characteristic
58	C₈	Q₈×C₉	1	4	yes	complemented
59	C₂×C₃	C₃×C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₃	C₂×C₃×D₈	1	1	yes	characteristic
61	C₂×C₃	C₂×C₃×Q₁₆	1	1	yes	characteristic
62	C₂²	C₂×D₄×C₉	1	1	yes	characteristic
63	C₄	—	4	4	no	—
64	C₄	C₉×D₈	1	2	yes	—
65	C₄	C₉×Q₁₆	1	2	yes	—
66	C₄	C₂×Q₈×C₉	1	2	yes	—
67	C₃	C₃×#500	1	1	yes	characteristic
68	C₂	C₉×C₄⋊Q₈	1	1	yes	characteristic
69	C₂	C₂×C₉×D₈	1	1	yes	characteristic
70	C₂	C₂×C₉×Q₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₉	C₂×D₄
commutator	C₂×C₄	C₂³×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃
socle, prime radical	C₂²×C₃	C₂×C₃×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₉ ◁ C₄²×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
18. C₈×C₉	Q₈	semidirect
19. #500	C₉	direct (N × H)
52. C₉	#500	direct (N × H)
58. C₈	Q₈×C₉	semidirect

Maximal subgroups

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

subgroup	index
C₉×C₄⋊Q₈	2
C₉×#107	2
C₄×C₈×C₉	2
C₃×#500	3

Subgroups — C9×#500 #10589054

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#500 #10589054