Subgroups — Q₈⋊#13095525 #27158999

Order 12416.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#13095525	C₂	1	1	yes	maximal, characteristic
2	C₄×#13095525	C₂	1	1	yes	maximal, characteristic
3	Q₈×#11715906	C₂	1	1	yes	maximal, characteristic
4	C₂×#13095525	—	2	1	no	—
5	C₂×#13095525	—	2	1	no	—
6	C₄⋊#11715906	—	2	1	no	—
7	C₄⋊#11715906	C₄	1	1	yes	characteristic
8	C₄×#11715906	—	2	1	no	—
9	C₄×#11715906	C₂²	1	1	yes	characteristic
10	Q₈×#87604	C₄	1	1	yes	characteristic
11	#13095525	—	2	2	no	—
12	#13095525	—	4	1	no	—
13	C₂×#11715906	—	2	1	no	—
14	C₂×#11715906	D₄	1	1	yes	characteristic
15	C₂×#11715906	D₄	1	1	yes	characteristic
16	C₄⋊#87604	—	2	1	no	—
17	C₄⋊#87604	C₈	1	1	yes	characteristic
18	C₄×#87604	—	2	1	no	—
19	C₄×#87604	C₂×C₄	1	1	yes	characteristic
20	Q₈×D₉₇	C₈	1	1	yes	characteristic
21	C₂×#87604	—	2	1	no	—
22	C₂×#87604	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×#87604	M₁₆	1	1	yes	characteristic
24	#11715906	—	2	1	no	—
25	#11715906	—	4	1	no	—
26	#11715906	SD₁₆	1	1	yes	characteristic
27	#11715906	Q₁₆	1	1	yes	characteristic
28	Q₈×C₉₇	C₁₆	1	1	yes	characteristic, complemented
29	C₄×D₉₇	—	2	1	no	—
30	C₄×D₉₇	C₂×C₈	1	1	yes	characteristic
31	Dic₁₉₄	—	2	1	no	—
32	Dic₁₉₄	C₁₆	1	1	yes	characteristic, complemented
33	C₂×D₉₇	C₂²⋊C₈	1	1	yes	characteristic
34	#87604	—	2	1	no	—
35	#87604	—	4	1	no	—
36	#87604	#103	1	1	yes	characteristic
37	#87604	#104	1	1	yes	characteristic
38	C₄×C₉₇	—	2	1	no	—
39	C₄×C₉₇	C₂×C₁₆	1	1	yes	characteristic
40	Dic₉₇	—	2	1	no	—
41	Dic₉₇	M₃₂	1	1	yes	characteristic
42	C₂×C₉₇	C₂²⋊C₁₆	1	1	yes	characteristic
43	D₉₇	Q₈⋊C₈	1	1	yes	characteristic
44	D₉₇	#350	1	1	yes	characteristic
45	Q₈⋊C₁₆	—	97	1	no	maximal, Sylow
46	C₉₇	Q₈⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
47	C₈×Q₈	—	97	1	no	—
48	C₄⋊C₁₆	—	97	1	no	—
49	C₄×C₁₆	—	97	1	no	—
50	C₄×Q₈	—	97	1	no	—
51	C₂×C₁₆	—	194	1	no	—
52	C₂×C₁₆	—	194	1	no	—
53	C₄⋊C₈	—	97	1	no	—
54	C₄⋊C₈	—	194	1	no	—
55	C₄×C₈	—	97	1	no	—
56	C₄×C₈	—	194	1	no	—
57	C₂×Q₈	—	97	1	no	—
58	C₂×C₈	—	97	1	no	—
59	C₂×C₈	—	97	1	no	—
60	C₂×C₈	—	194	1	no	—
61	C₄⋊C₄	—	97	1	no	—
62	C₄⋊C₄	—	194	1	no	—
63	C₄²	—	97	1	no	—
64	C₄²	—	194	1	no	—
65	C₁₆	—	388	1	no	—
66	C₁₆	—	194	2	no	—
67	Q₈	—	97	1	no	—
68	Q₈	—	194	1	no	—
69	Q₈	#13095525	1	1	yes	characteristic, complemented
70	C₂×C₄	—	97	1	no	—
71	C₂×C₄	—	97	1	no	—
72	C₂×C₄	—	97	1	no	—
73	C₂×C₄	—	194	1	no	—
74	C₂×C₄	—	194	1	no	—
75	C₈	—	97	1	no	—
76	C₈	—	97	1	no	—
77	C₈	—	388	1	no	—
78	C₈	—	194	1	no	—
79	C₂²	—	97	1	no	—
80	C₄	—	2	1	no	—
81	C₄	—	97	1	no	—
82	C₄	—	97	1	no	—
83	C₄	—	97	1	no	—
84	C₄	—	388	1	no	—
85	C₄	—	194	1	no	—
86	C₄	—	194	1	no	—
87	C₄	C₂×#13095525	1	1	yes	characteristic
88	C₂	—	97	1	no	—
89	C₂	—	97	1	no	—
90	C₂	C₂²⋊#13095525	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²⋊#13095525
commutator	C₄×C₉₇	C₂×C₁₆
Frattini	C₄	C₂×#13095525
Fitting	Q₈×C₉₇	C₁₆
socle	C₂×C₉₇	C₂²⋊C₁₆
prime radical	C₂×D₉₇	C₂²⋊C₈
cosocle	C₄×#11715906	C₂²
hypercenter	Q₈	#13095525
hypocenter	C₉₇	Q₈⋊C₁₆
LCS	G ▷ C₄×C₉₇ ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₈

Complements

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

N	H ≅ G/N	type
28. Q₈×C₉₇	C₁₆	semidirect
32. Dic₁₉₄	C₁₆	semidirect
46. C₉₇	Q₈⋊C₁₆	semidirect
69. Q₈	#13095525	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#13095525	2
C₄×#13095525	2
Q₈×#11715906	2
Q₈⋊C₁₆	97

Subgroups — Q8⋊#13095525 #27158999

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈⋊#13095525 #27158999