Subgroups — Q₈×#16861332 #45656286

Order 32576.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#16861332	C₂	1	3	yes	maximal
2	C₄×#16861332	C₂	1	3	yes	maximal
3	Q₈×Dic₅₀₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#16861332	C₂²	1	3	yes	—
5	C₂×#16861332	C₂²	1	1	yes	characteristic
6	C₄⋊Dic₅₀₉	C₄	1	3	yes	—
7	C₄×Dic₅₀₉	C₂²	1	3	yes	—
8	C₂×Q₈×C₅₀₉	C₄	1	1	yes	characteristic
9	#16861332	—	2	3	no	—
10	#16861332	Q₈	1	2	yes	direct product factor
11	C₂×Dic₅₀₉	C₂×C₄	1	3	yes	—
12	C₂×Dic₅₀₉	C₂³	1	1	yes	characteristic
13	Q₈×C₅₀₉	C₈	1	4	yes	complemented
14	C₂×C₄×C₅₀₉	C₂×C₄	1	3	yes	—
15	C₂²×C₅₀₉	C₂²×C₄	1	1	yes	characteristic
16	C₄×C₅₀₉	C₂×C₈	1	6	yes	—
17	Dic₅₀₉	—	2	3	no	—
18	Dic₅₀₉	C₂×Q₈	1	1	yes	characteristic
19	Dic₅₀₉	#41	1	1	yes	characteristic
20	C₂×C₅₀₉	C₄×Q₈	1	1	yes	characteristic
21	C₂×C₅₀₉	C₂²×C₈	1	1	yes	characteristic
22	C₂×C₅₀₉	#131	1	1	yes	characteristic
23	C₅₀₉	C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
24	C₈×Q₈	—	509	1	no	maximal, Sylow
25	C₄×Q₈	—	509	1	no	—
26	C₄⋊C₈	—	509	3	no	—
27	C₄×C₈	—	509	3	no	—
28	C₂×Q₈	#14523266	1	1	yes	characteristic
29	C₂×C₈	—	509	3	no	—
30	C₂×C₈	—	509	1	no	—
31	C₄⋊C₄	—	509	3	no	—
32	C₄²	—	509	3	no	—
33	Q₈	#16861332	1	4	yes	direct product factor
34	C₂×C₄	—	509	3	no	—
35	C₂×C₄	—	509	1	no	—
36	C₂×C₄	C₂×#14523266	1	3	yes	—
37	C₈	—	1018	3	no	—
38	C₈	—	509	2	no	—
39	C₂²	C₂²×#14523266	1	1	yes	characteristic
40	C₄	—	1018	3	no	—
41	C₄	—	509	1	no	—
42	C₄	—	509	1	no	—
43	C₄	C₂×#16861332	1	6	yes	—
44	C₂	Q₈×#14523266	1	1	yes	characteristic
45	C₂	C₂²×#16861332	1	1	yes	characteristic
46	C₂	#30654303	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#14523266
commutator	C₂×C₅₀₉	C₂²×C₈
Fitting	C₂×Q₈×C₅₀₉	C₄
socle, prime radical	C₂²×C₅₀₉	C₂²×C₄
cosocle	C₂×Dic₅₀₉	C₂³
hypercenter	C₂×Q₈	#14523266
hypocenter	C₅₀₉	C₈×Q₈
LCS	G ▷ C₂×C₅₀₉ ▷ C₅₀₉
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
10. #16861332	Q₈	direct (N × H)
13. Q₈×C₅₀₉	C₈	semidirect
23. C₅₀₉	C₈×Q₈	semidirect
33. Q₈	#16861332	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⋊#16861332	2
C₄×#16861332	2
Q₈×Dic₅₀₉	2
C₈×Q₈	509

Subgroups — Q8×#16861332 #45656286

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#16861332 #45656286