Subgroups — Q₈×#15667918 #36475059

Order 22784.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#15667918	C₂	1	3	yes	maximal
2	C₄×#15667918	C₂	1	3	yes	maximal
3	Q₈×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂×#15667918	C₂²	1	3	yes	—
5	C₂×#15667918	C₂²	1	1	yes	characteristic
6	C₈×Q₈×C₈₉	C₄	1	1	yes	characteristic
7	C₄⋊C₁₆×C₈₉	C₄	1	3	yes	—
8	C₄×C₁₆×C₈₉	C₂²	1	3	yes	—
9	#15667918	—	2	3	no	—
10	#15667918	Q₈	1	2	yes	direct product factor
11	C₄×Q₈×C₈₉	C₈	1	1	yes	characteristic
12	C₂×C₁₆×C₈₉	C₂×C₄	1	3	yes	—
13	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
14	C₄⋊C₈×C₈₉	C₈	1	3	yes	—
15	C₄×C₈×C₈₉	C₂×C₄	1	3	yes	—
16	C₂×Q₈×C₈₉	C₁₆	1	1	yes	characteristic
17	C₂×C₈×C₈₉	C₂×C₈	1	3	yes	—
18	C₂×C₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
19	C₄⋊C₄×C₈₉	C₁₆	1	3	yes	—
20	C₄²×C₈₉	C₂×C₈	1	3	yes	—
21	C₁₆×C₈₉	—	2	3	no	—
22	C₁₆×C₈₉	C₂×Q₈	1	1	yes	characteristic
23	C₁₆×C₈₉	#41	1	1	yes	characteristic
24	Q₈×C₈₉	C₃₂	1	4	yes	complemented
25	C₂×C₄×C₈₉	C₂×C₁₆	1	3	yes	—
26	C₂×C₄×C₈₉	C₂×C₁₆	1	3	yes	—
27	C₂×C₄×C₈₉	C₂²×C₈	1	1	yes	characteristic
28	C₈×C₈₉	—	2	3	no	—
29	C₈×C₈₉	C₄×Q₈	1	1	yes	characteristic
30	C₈×C₈₉	#131	1	1	yes	characteristic
31	C₂²×C₈₉	C₂²×C₁₆	1	1	yes	characteristic
32	C₄×C₈₉	—	2	3	no	—
33	C₄×C₈₉	C₂×C₃₂	1	6	yes	—
34	C₄×C₈₉	C₈×Q₈	1	1	yes	characteristic
35	C₄×C₈₉	#504	1	1	yes	characteristic
36	Q₈×C₃₂	—	89	1	no	maximal, Sylow
37	C₂×C₈₉	Q₈×C₁₆	1	1	yes	characteristic
38	C₂×C₈₉	C₂²×C₃₂	1	1	yes	characteristic
39	C₂×C₈₉	#2258	1	1	yes	characteristic
40	Q₈×C₁₆	D₈₉	1	1	yes	characteristic
41	C₄⋊C₃₂	—	89	3	no	—
42	C₄×C₃₂	—	89	3	no	—
43	C₈₉	Q₈×C₃₂	1	1	yes	characteristic, complemented, Sylow
44	C₈×Q₈	Dic₈₉	1	1	yes	characteristic
45	C₂×C₃₂	—	89	1	no	—
46	C₂×C₃₂	—	89	3	no	—
47	C₄⋊C₁₆	Dic₈₉	1	3	yes	—
48	C₄×C₁₆	C₂×D₈₉	1	3	yes	—
49	C₄×Q₈	#10623494	1	1	yes	characteristic
50	C₂×C₁₆	C₂×Dic₈₉	1	3	yes	—
51	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
52	C₄⋊C₈	#10623494	1	3	yes	—
53	C₄×C₈	C₂×Dic₈₉	1	3	yes	—
54	C₃₂	—	89	2	no	—
55	C₃₂	—	178	3	no	—
56	C₂×Q₈	#13083549	1	1	yes	characteristic
57	C₂×C₈	C₂×#10623494	1	3	yes	—
58	C₂×C₈	C₂²×Dic₈₉	1	1	yes	characteristic
59	C₄⋊C₄	#13083549	1	3	yes	—
60	C₄²	C₂×#10623494	1	3	yes	—
61	C₁₆	—	2	3	no	—
62	C₁₆	Q₈×D₈₉	1	1	yes	characteristic
63	C₁₆	#13083575	1	1	yes	characteristic
64	Q₈	#15667918	1	4	yes	direct product factor
65	C₂×C₄	C₂×#13083549	1	3	yes	—
66	C₂×C₄	C₂×#13083549	1	3	yes	—
67	C₂×C₄	C₂²×#10623494	1	1	yes	characteristic
68	C₈	—	2	3	no	—
69	C₈	Q₈×Dic₈₉	1	1	yes	characteristic
70	C₈	#15668019	1	1	yes	characteristic
71	C₂²	C₂²×#13083549	1	1	yes	characteristic
72	C₄	—	2	3	no	—
73	C₄	C₂×#15667918	1	6	yes	—
74	C₄	Q₈×#10623494	1	1	yes	characteristic
75	C₄	#19152015	1	1	yes	characteristic
76	C₂	Q₈×#13083549	1	1	yes	characteristic
77	C₂	C₂²×#15667918	1	1	yes	characteristic
78	C₂	#26022546	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₁₆	C₂²×D₈₉
commutator	C₂×C₈₉	C₂²×C₃₂
Fitting	Q₈×C₁₆×C₈₉	C₂
socle, prime radical	C₂²×C₈₉	C₂²×C₁₆
cosocle	C₂×C₁₆×C₈₉	C₂³
hypercenter	Q₈×C₁₆	D₈₉
hypocenter	C₈₉	Q₈×C₃₂
LCS	G ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₁₆ ◁ Q₈×C₁₆

Complements

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

N	H ≅ G/N	type
10. #15667918	Q₈	direct (N × H)
24. Q₈×C₈₉	C₃₂	semidirect
43. C₈₉	Q₈×C₃₂	semidirect
64. Q₈	#15667918	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⋊#15667918	2
C₄×#15667918	2
Q₈×C₁₆×C₈₉	2
Q₈×C₃₂	89

Subgroups — Q8×#15667918 #36475059

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#15667918 #36475059