Subgroups — Q₈×#14519827 #30608584

Order 15616.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#14519827	C₂	1	3	yes	maximal
2	C₄×#14519827	C₂	1	3	yes	maximal
3	Q₈×#11758185	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519827	C₂²	1	3	yes	—
5	C₂×#14519827	C₂²	1	1	yes	characteristic
6	C₄⋊#11758185	C₄	1	3	yes	—
7	C₄×#11758185	C₂²	1	3	yes	—
8	C₈×Q₈×C₆₁	C₄	1	1	yes	characteristic
9	#14519827	—	2	3	no	—
10	#14519827	Q₈	1	2	yes	direct product factor
11	C₂×#11758185	C₂×C₄	1	3	yes	—
12	C₂×#11758185	C₂³	1	1	yes	characteristic
13	C₄×Q₈×C₆₁	C₈	1	1	yes	characteristic
14	C₄⋊C₈×C₆₁	C₈	1	3	yes	—
15	C₄×C₈×C₆₁	C₂×C₄	1	3	yes	—
16	C₂×Q₈×C₆₁	C₁₆	1	1	yes	characteristic
17	C₂×C₈×C₆₁	C₂×C₈	1	3	yes	—
18	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
19	C₄⋊C₄×C₆₁	C₁₆	1	3	yes	—
20	C₄²×C₆₁	C₂×C₈	1	3	yes	—
21	#11758185	—	2	3	no	—
22	#11758185	C₂×Q₈	1	1	yes	characteristic
23	#11758185	#41	1	1	yes	characteristic
24	Q₈×C₆₁	C₃₂	1	4	yes	complemented
25	C₂×C₄×C₆₁	C₂×C₁₆	1	3	yes	—
26	C₂×C₄×C₆₁	C₂×C₁₆	1	3	yes	—
27	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
28	C₈×C₆₁	—	2	3	no	—
29	C₈×C₆₁	C₄×Q₈	1	1	yes	characteristic
30	C₈×C₆₁	#131	1	1	yes	characteristic
31	Q₈×C₃₂	—	61	1	no	maximal, Sylow
32	C₂²×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
33	C₄×C₆₁	—	2	3	no	—
34	C₄×C₆₁	C₂×C₃₂	1	6	yes	—
35	C₄×C₆₁	C₈×Q₈	1	1	yes	characteristic
36	C₄×C₆₁	#504	1	1	yes	characteristic
37	Q₈×C₁₆	—	61	1	no	—
38	C₄⋊C₃₂	—	61	3	no	—
39	C₄×C₃₂	—	61	3	no	—
40	C₂×C₆₁	Q₈×C₁₆	1	1	yes	characteristic
41	C₂×C₆₁	C₂²×C₃₂	1	1	yes	characteristic
42	C₂×C₆₁	#2258	1	1	yes	characteristic
43	C₈×Q₈	#6923	1	1	yes	characteristic
44	C₂×C₃₂	—	61	1	no	—
45	C₂×C₃₂	—	61	3	no	—
46	C₄⋊C₁₆	—	61	3	no	—
47	C₄×C₁₆	—	61	3	no	—
48	C₆₁	Q₈×C₃₂	1	1	yes	characteristic, complemented, Sylow
49	C₄×Q₈	#92420	1	1	yes	characteristic
50	C₂×C₁₆	—	61	1	no	—
51	C₂×C₁₆	—	61	3	no	—
52	C₄⋊C₈	#92420	1	3	yes	—
53	C₄×C₈	C₂×#6923	1	3	yes	—
54	C₃₂	—	61	2	no	—
55	C₃₂	—	122	3	no	—
56	C₂×Q₈	#11758187	1	1	yes	characteristic
57	C₂×C₈	C₂×#92420	1	3	yes	—
58	C₂×C₈	C₂²×#6923	1	1	yes	characteristic
59	C₄⋊C₄	#11758187	1	3	yes	—
60	C₄²	C₂×#92420	1	3	yes	—
61	C₁₆	—	61	1	no	—
62	C₁₆	—	61	1	no	—
63	C₁₆	—	122	3	no	—
64	Q₈	#14519827	1	4	yes	direct product factor
65	C₂×C₄	C₂×#11758187	1	3	yes	—
66	C₂×C₄	C₂×#11758187	1	3	yes	—
67	C₂×C₄	C₂²×#92420	1	1	yes	characteristic
68	C₈	—	2	3	no	—
69	C₈	Q₈×#6923	1	1	yes	characteristic
70	C₈	#14519865	1	1	yes	characteristic
71	C₂²	C₂²×#11758187	1	1	yes	characteristic
72	C₄	—	2	3	no	—
73	C₄	C₂×#14519827	1	6	yes	—
74	C₄	Q₈×#92420	1	1	yes	characteristic
75	C₄	#16840820	1	1	yes	characteristic
76	C₂	Q₈×#11758187	1	1	yes	characteristic
77	C₂	C₂²×#14519827	1	1	yes	characteristic
78	C₂	#21462242	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#6923
commutator	C₂×C₆₁	C₂²×C₃₂
Fitting	C₈×Q₈×C₆₁	C₄
socle, prime radical	C₂²×C₆₁	C₂²×C₁₆
cosocle	C₂×#11758185	C₂³
hypercenter	C₈×Q₈	#6923
hypocenter	C₆₁	Q₈×C₃₂
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₈×Q₈

Complements

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

N	H ≅ G/N	type
10. #14519827	Q₈	direct (N × H)
24. Q₈×C₆₁	C₃₂	semidirect
48. C₆₁	Q₈×C₃₂	semidirect
64. Q₈	#14519827	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⋊#14519827	2
C₄×#14519827	2
Q₈×#11758185	2
Q₈×C₃₂	61

Subgroups — Q8×#14519827 #30608584

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#14519827 #30608584