Subgroups — C₉×#504 #10589058

Order 576.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₉×#131	C₂	1	1	yes	maximal, characteristic
2	C₉×M₃₂	C₂	1	3	yes	maximal, complemented
3	C₂×C₉×C₁₆	C₂	1	3	yes	maximal, complemented
4	C₃×#504	C₃	1	1	yes	maximal, characteristic
5	C₉×#41	C₄	1	1	yes	characteristic
6	C₉×M₁₆	C₄	1	3	yes	—
7	C₂×C₈×C₉	C₂²	1	3	yes	—
8	C₉×C₁₆	C₂²	1	3	yes	—
9	C₉×C₁₆	C₂²	1	1	yes	characteristic
10	C₃×#131	C₂×C₃	1	1	yes	characteristic
11	C₃×M₃₂	C₂×C₃	1	3	yes	—
12	C₂×C₃×C₁₆	C₂×C₃	1	3	yes	—
13	Q₈×C₉	C₈	1	1	yes	characteristic
14	D₄×C₉	C₈	1	3	yes	—
15	C₂×C₄×C₉	C₂×C₄	1	3	yes	—
16	C₈×C₉	C₂×C₄	1	3	yes	—
17	C₈×C₉	C₂³	1	1	yes	characteristic
18	#504	C₉	1	1	yes	characteristic, direct product factor, Sylow
19	C₃×#41	C₃×C₄	1	1	yes	characteristic
20	C₃×M₁₆	C₃×C₄	1	3	yes	—
21	C₂×C₃×C₈	C₂²×C₃	1	3	yes	—
22	C₃×C₁₆	C₂²×C₃	1	3	yes	—
23	C₃×C₁₆	C₂²×C₃	1	1	yes	characteristic
24	C₂²×C₉	C₂×C₈	1	3	yes	—
25	C₄×C₉	C₂×C₈	1	3	yes	—
26	C₄×C₉	C₂²×C₄	1	1	yes	characteristic
27	#131	C₂×C₉	1	1	yes	characteristic
28	M₃₂	C₂×C₉	1	3	yes	complemented
29	C₂×C₁₆	C₂×C₉	1	3	yes	complemented
30	C₃×Q₈	C₃×C₈	1	1	yes	characteristic
31	C₃×D₄	C₃×C₈	1	3	yes	—
32	C₂×C₃×C₄	C₂×C₃×C₄	1	3	yes	—
33	C₃×C₈	C₂×C₃×C₄	1	3	yes	—
34	C₃×C₈	C₂³×C₃	1	1	yes	characteristic
35	C₂×C₉	—	2	3	no	—
36	C₂×C₉	C₂²×C₈	1	1	yes	characteristic
37	#41	C₄×C₉	1	1	yes	characteristic
38	M₁₆	C₄×C₉	1	3	yes	—
39	C₂×C₈	C₂²×C₉	1	3	yes	—
40	C₁₆	C₂²×C₉	1	1	yes	characteristic
41	C₁₆	C₂²×C₉	1	3	yes	—
42	C₂²×C₃	C₂×C₃×C₈	1	3	yes	—
43	C₃×C₄	C₂×C₃×C₈	1	3	yes	—
44	C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
45	C₉	#504	1	1	yes	characteristic, direct product factor, Sylow
46	Q₈	C₈×C₉	1	1	yes	characteristic
47	D₄	C₈×C₉	1	3	yes	—
48	C₂×C₄	C₂×C₄×C₉	1	3	yes	—
49	C₈	C₂×C₄×C₉	1	3	yes	—
50	C₈	C₂³×C₉	1	1	yes	characteristic
51	C₂×C₃	—	2	3	no	—
52	C₂×C₃	C₂²×C₃×C₈	1	1	yes	characteristic
53	C₂²	C₂×C₈×C₉	1	3	yes	—
54	C₄	C₂×C₈×C₉	1	3	yes	—
55	C₄	C₂²×C₄×C₉	1	1	yes	characteristic
56	C₃	C₃×#504	1	1	yes	characteristic
57	C₂	—	2	3	no	—
58	C₂	C₂²×C₈×C₉	1	1	yes	characteristic

	subgroup	G/H
center	C₉×C₁₆	C₂²
commutator	C₂	C₂²×C₈×C₉
Frattini, cosocle	C₃×C₈	C₂³×C₃
socle	C₂×C₃	C₂²×C₃×C₈
prime radical	C₃×#41	C₃×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₉×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₉×M₃₂	C₂	semidirect
3. C₂×C₉×C₁₆	C₂	semidirect
18. #504	C₉	direct (N × H)
28. M₃₂	C₂×C₉	semidirect
29. C₂×C₁₆	C₂×C₉	semidirect
45. C₉	#504	direct (N × H)

Maximal subgroups

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

subgroup	index
C₉×#131	2
C₉×M₃₂	2
C₂×C₉×C₁₆	2
C₃×#504	3

Subgroups — C9×#504 #10589058

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#504 #10589058