Subgroups — Q₈×#16864377 #46812789

Order 34432.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#16864377	C₂	1	3	yes	maximal
2	C₄×#16864377	C₂	1	3	yes	maximal
3	C₈×Q₈×C₂₆₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#16864377	C₂²	1	3	yes	—
5	C₂×#16864377	C₂²	1	1	yes	characteristic
6	C₄×Q₈×C₂₆₉	C₄	1	1	yes	characteristic
7	C₄⋊C₈×C₂₆₉	C₄	1	3	yes	—
8	C₄×C₈×C₂₆₉	C₂²	1	3	yes	—
9	#16864377	—	2	3	no	—
10	#16864377	Q₈	1	2	yes	direct product factor
11	C₂×Q₈×C₂₆₉	C₈	1	1	yes	characteristic
12	C₂×C₈×C₂₆₉	C₂×C₄	1	3	yes	—
13	C₂×C₈×C₂₆₉	C₂³	1	1	yes	characteristic
14	C₄⋊C₄×C₂₆₉	C₈	1	3	yes	—
15	C₄²×C₂₆₉	C₂×C₄	1	3	yes	—
16	Q₈×C₂₆₉	C₁₆	1	4	yes	complemented
17	C₂×C₄×C₂₆₉	C₂×C₈	1	3	yes	—
18	C₂×C₄×C₂₆₉	C₂×C₈	1	3	yes	—
19	C₂×C₄×C₂₆₉	C₂²×C₄	1	1	yes	characteristic
20	C₈×C₂₆₉	—	2	3	no	—
21	C₈×C₂₆₉	C₂×Q₈	1	1	yes	characteristic
22	C₈×C₂₆₉	#41	1	1	yes	characteristic
23	C₂²×C₂₆₉	C₂²×C₈	1	1	yes	characteristic
24	C₄×C₂₆₉	—	2	3	no	—
25	C₄×C₂₆₉	C₂×C₁₆	1	6	yes	—
26	C₄×C₂₆₉	C₄×Q₈	1	1	yes	characteristic
27	C₄×C₂₆₉	#131	1	1	yes	characteristic
28	C₂×C₂₆₉	C₈×Q₈	1	1	yes	characteristic
29	C₂×C₂₆₉	C₂²×C₁₆	1	1	yes	characteristic
30	C₂×C₂₆₉	#504	1	1	yes	characteristic
31	C₂₆₉	Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
32	Q₈×C₁₆	—	269	1	no	maximal, Sylow
33	C₈×Q₈	D₂₆₉	1	1	yes	characteristic
34	C₄⋊C₁₆	—	269	3	no	—
35	C₄×C₁₆	—	269	3	no	—
36	C₄×Q₈	Dic₂₆₉	1	1	yes	characteristic
37	C₂×C₁₆	—	269	1	no	—
38	C₂×C₁₆	—	269	3	no	—
39	C₄⋊C₈	Dic₂₆₉	1	3	yes	—
40	C₄×C₈	C₂×D₂₆₉	1	3	yes	—
41	C₂×Q₈	#14524130	1	1	yes	characteristic
42	C₂×C₈	C₂×Dic₂₆₉	1	3	yes	—
43	C₂×C₈	C₂²×D₂₆₉	1	1	yes	characteristic
44	C₄⋊C₄	#14524130	1	3	yes	—
45	C₄²	C₂×Dic₂₆₉	1	3	yes	—
46	C₁₆	—	269	2	no	—
47	C₁₆	—	538	3	no	—
48	Q₈	#16864377	1	4	yes	direct product factor
49	C₂×C₄	C₂×#14524130	1	3	yes	—
50	C₂×C₄	C₂×#14524130	1	3	yes	—
51	C₂×C₄	C₂²×Dic₂₆₉	1	1	yes	characteristic
52	C₈	—	2	3	no	—
53	C₈	Q₈×D₂₆₉	1	1	yes	characteristic
54	C₈	#16864403	1	1	yes	characteristic
55	C₂²	C₂²×#14524130	1	1	yes	characteristic
56	C₄	—	2	3	no	—
57	C₄	C₂×#16864377	1	6	yes	—
58	C₄	Q₈×Dic₂₆₉	1	1	yes	characteristic
59	C₄	#22575460	1	1	yes	characteristic
60	C₂	Q₈×#14524130	1	1	yes	characteristic
61	C₂	C₂²×#16864377	1	1	yes	characteristic
62	C₂	#31771022	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₂₆₉
commutator	C₂×C₂₆₉	C₂²×C₁₆
Fitting	C₈×Q₈×C₂₆₉	C₂
socle, prime radical	C₂²×C₂₆₉	C₂²×C₈
cosocle	C₂×C₈×C₂₆₉	C₂³
hypercenter	C₈×Q₈	D₂₆₉
hypocenter	C₂₆₉	Q₈×C₁₆
LCS	G ▷ C₂×C₂₆₉ ▷ C₂₆₉
UCS	C₁ ◁ C₂×C₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
10. #16864377	Q₈	direct (N × H)
16. Q₈×C₂₆₉	C₁₆	semidirect
31. C₂₆₉	Q₈×C₁₆	semidirect
48. Q₈	#16864377	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⋊#16864377	2
C₄×#16864377	2
C₈×Q₈×C₂₆₉	2
Q₈×C₁₆	269

Subgroups — Q8×#16864377 #46812789

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#16864377 #46812789