Subgroups — C₈×Q₈⋊C₉ #10588321

Order 576.

69 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Q₈⋊C₉	C₂	1	1	yes	maximal, characteristic
2	C₃×C₈×Q₈	C₃	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊C₉	C₄	1	1	yes	characteristic
4	C₂×C₈×C₉	—	4	1	no	maximal
5	C₃×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
6	C₃×C₄⋊C₈	—	3	1	no	—
7	C₃×C₄×C₈	—	3	1	no	—
8	C₂×C₄×C₉	—	4	1	no	—
9	Q₈⋊C₉	C₈	1	1	yes	characteristic, direct product factor
10	C₈×C₉	—	4	2	no	—
11	C₈×Q₈	C₉	1	1	yes	characteristic, complemented, Sylow
12	C₂×C₃×Q₈	C₃×C₄	1	1	yes	characteristic
13	C₂×C₃×C₈	—	3	1	no	—
14	C₂×C₃×C₈	A₄	1	1	yes	characteristic
15	C₃×C₄⋊C₄	—	3	1	no	—
16	C₃×C₄²	—	3	1	no	—
17	C₂²×C₉	—	4	1	no	—
18	C₄×C₉	—	4	1	no	—
19	C₄×C₉	—	4	1	no	—
20	C₄×Q₈	C₂×C₉	1	1	yes	characteristic
21	C₄⋊C₈	—	3	1	no	—
22	C₄×C₈	—	3	1	no	—
23	C₃×Q₈	—	3	1	no	—
24	C₃×Q₈	C₃×C₈	1	1	yes	characteristic
25	C₂×C₃×C₄	—	3	1	no	—
26	C₂×C₃×C₄	—	3	1	no	—
27	C₂×C₃×C₄	C₂×A₄	1	1	yes	characteristic
28	C₃×C₈	—	6	1	no	—
29	C₃×C₈	SL(2,𝔽₃)	1	2	yes	—
30	C₂×C₉	—	4	1	no	—
31	C₂×C₉	—	4	1	no	—
32	C₂×C₉	—	4	1	no	—
33	C₂×Q₈	C₄×C₉	1	1	yes	characteristic
34	C₂×C₈	—	3	1	no	—
35	C₂×C₈	C₂²⋊C₉	1	1	yes	characteristic
36	C₄⋊C₄	—	3	1	no	—
37	C₄²	—	3	1	no	—
38	C₂²×C₃	C₄×A₄	1	1	yes	characteristic
39	C₃×C₄	—	3	1	no	—
40	C₃×C₄	—	3	1	no	—
41	C₃×C₄	—	6	1	no	—
42	C₃×C₄	C₂×SL(2,𝔽₃)	1	1	yes	characteristic
43	C₃×C₄	#41⋊C₃	1	1	yes	characteristic
44	C₉	—	4	1	no	Sylow
45	Q₈	—	3	1	no	—
46	Q₈	C₈×C₉	1	1	yes	characteristic, complemented
47	C₂×C₄	—	3	1	no	—
48	C₂×C₄	—	3	1	no	—
49	C₂×C₄	C₂×C₂²⋊C₉	1	1	yes	characteristic
50	C₈	—	6	1	no	—
51	C₈	Q₈⋊C₉	1	2	yes	direct product factor
52	C₂×C₃	C₄×SL(2,𝔽₃)	1	1	yes	characteristic
53	C₂×C₃	C₈×A₄	1	1	yes	characteristic
54	C₂×C₃	#131⋊C₃	1	1	yes	characteristic
55	C₂²	C₄×C₂²⋊C₉	1	1	yes	characteristic
56	C₄	—	3	1	no	—
57	C₄	—	3	1	no	—
58	C₄	—	6	1	no	—
59	C₄	C₂×Q₈⋊C₉	1	1	yes	characteristic
60	C₄	#41⋊C₉	1	1	yes	characteristic
61	C₃	C₈×SL(2,𝔽₃)	1	1	yes	characteristic
62	C₂	C₄×Q₈⋊C₉	1	1	yes	characteristic
63	C₂	C₈×C₂²⋊C₉	1	1	yes	characteristic
64	C₂	#131⋊C₉	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₃×C₈	A₄
commutator, hypocenter	Q₈	C₈×C₉
Frattini	C₂×C₃×C₄	C₂×A₄
Fitting	C₃×C₈×Q₈	C₃
socle, prime radical	C₂²×C₃	C₄×A₄
cosocle	C₃×C₄×Q₈	C₂×C₃
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂×C₃×C₈

Complements

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

N	H ≅ G/N	type
9. Q₈⋊C₉	C₈	direct (N × H)
11. C₈×Q₈	C₉	semidirect
46. Q₈	C₈×C₉	semidirect
51. C₈	Q₈⋊C₉	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×Q₈⋊C₉	2
C₃×C₈×Q₈	3
C₂×C₈×C₉	4

Subgroups — C8×Q8⋊C9 #10588321

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×Q₈⋊C₉ #10588321