Subgroups — Q₈⋊#63216 #14543750

Order 2176.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#63216	C₂	1	1	yes	maximal, characteristic
2	C₄×#63216	C₂	1	1	yes	maximal, characteristic
3	Q₈×#3622	C₂	1	1	yes	maximal, characteristic
4	C₄×Q₈×C₁₇	C₄	1	1	yes	characteristic
5	C₂×#63216	—	2	1	no	—
6	C₂×#63216	—	2	1	no	—
7	C₄⋊#3622	—	2	1	no	—
8	C₄⋊#3622	C₄	1	1	yes	characteristic
9	C₄×#3622	—	2	1	no	—
10	C₄×#3622	C₂²	1	1	yes	characteristic
11	C₂×Q₈×C₁₇	C₈	1	1	yes	characteristic
12	C₄⋊C₄×C₁₇	—	2	1	no	—
13	C₄⋊C₄×C₁₇	C₈	1	1	yes	characteristic
14	C₄²×C₁₇	—	2	1	no	—
15	C₄²×C₁₇	C₂×C₄	1	1	yes	characteristic
16	C₂×#3622	—	2	1	no	—
17	C₂×#3622	D₄	1	1	yes	characteristic
18	C₂×#3622	D₄	1	1	yes	characteristic
19	#63216	—	2	2	no	—
20	#63216	—	4	1	no	—
21	Q₈×C₁₇	—	2	1	no	—
22	Q₈×C₁₇	C₁₆	1	2	yes	complemented
23	C₂×C₄×C₁₇	—	2	1	no	—
24	C₂×C₄×C₁₇	—	2	1	no	—
25	C₂×C₄×C₁₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₁₇	C₂×C₈	1	1	yes	characteristic
27	C₂×C₄×C₁₇	M₁₆	1	1	yes	characteristic
28	#3622	—	2	1	no	—
29	#3622	—	4	1	no	—
30	#3622	SD₁₆	1	1	yes	characteristic
31	#3622	Q₁₆	1	1	yes	characteristic
32	Q₈⋊C₁₆	—	17	1	no	maximal, Sylow
33	C₂²×C₁₇	C₂²⋊C₈	1	1	yes	characteristic
34	C₄×C₁₇	—	2	2	no	—
35	C₄×C₁₇	—	2	1	no	—
36	C₄×C₁₇	—	4	1	no	—
37	C₄×C₁₇	#103	1	1	yes	characteristic
38	C₄×C₁₇	#104	1	1	yes	characteristic
39	C₄×C₁₇	C₂×C₁₆	1	1	yes	characteristic
40	C₄×C₁₇	M₃₂	1	1	yes	characteristic
41	C₈×Q₈	—	17	1	no	—
42	C₄⋊C₁₆	—	17	1	no	—
43	C₄×C₁₆	—	17	1	no	—
44	C₂×C₁₇	Q₈⋊C₈	1	1	yes	characteristic
45	C₂×C₁₇	C₂²⋊C₁₆	1	1	yes	characteristic
46	C₂×C₁₇	#350	1	1	yes	characteristic
47	C₄×Q₈	#595	1	1	yes	characteristic
48	C₂×C₁₆	—	34	1	no	—
49	C₂×C₁₆	—	34	1	no	—
50	C₄⋊C₈	—	17	1	no	—
51	C₄⋊C₈	—	34	1	no	—
52	C₄×C₈	—	17	1	no	—
53	C₄×C₈	—	34	1	no	—
54	C₁₇	Q₈⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
55	C₂×Q₈	#3624	1	1	yes	characteristic
56	C₂×C₈	—	17	1	no	—
57	C₂×C₈	—	17	1	no	—
58	C₂×C₈	—	34	1	no	—
59	C₄⋊C₄	—	2	1	no	—
60	C₄⋊C₄	#3624	1	1	yes	characteristic
61	C₄²	—	2	1	no	—
62	C₄²	C₂×#595	1	1	yes	characteristic
63	C₁₆	—	34	2	no	—
64	C₁₆	—	68	1	no	—
65	Q₈	—	2	1	no	—
66	Q₈	#63216	1	2	yes	complemented
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	C₂×#3624	1	1	yes	characteristic
70	C₂×C₄	#63247	1	1	yes	characteristic
71	C₂×C₄	C₂²⋊#595	1	1	yes	characteristic
72	C₈	—	17	1	no	—
73	C₈	—	17	1	no	—
74	C₈	—	34	1	no	—
75	C₈	—	68	1	no	—
76	C₂²	C₂²⋊#3624	1	1	yes	characteristic
77	C₄	—	2	2	no	—
78	C₄	—	2	1	no	—
79	C₄	—	4	1	no	—
80	C₄	C₂×#63216	1	1	yes	characteristic
81	C₄	#10587538	1	1	yes	characteristic
82	C₄	#10587549	1	1	yes	characteristic
83	C₄	#10587550	1	1	yes	characteristic
84	C₂	Q₈⋊#3624	1	1	yes	characteristic
85	C₂	C₂²⋊#63216	1	1	yes	characteristic
86	C₂	#11763707	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊#595
commutator	C₄×C₁₇	C₂×C₁₆
Frattini	C₄²	C₂×#595
Fitting	C₄×Q₈×C₁₇	C₄
socle, prime radical	C₂²×C₁₇	C₂²⋊C₈
cosocle	C₄×#3622	C₂²
hypercenter	C₄×Q₈	#595
hypocenter	C₁₇	Q₈⋊C₁₆
LCS	G ▷ C₄×C₁₇ ▷ C₂×C₁₇ ▷ C₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
22. Q₈×C₁₇	C₁₆	semidirect
54. C₁₇	Q₈⋊C₁₆	semidirect
66. Q₈	#63216	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#63216	2
C₄×#63216	2
Q₈×#3622	2
Q₈⋊C₁₆	17

Subgroups — Q8⋊#63216 #14543750

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈⋊#63216 #14543750