Subgroups — Q₈×#13095525 #27159026

Order 12416.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#13095525	C₂	1	3	yes	maximal
2	C₄×#13095525	C₂	1	3	yes	maximal
3	Q₈×#11715906	C₂	1	1	yes	maximal, characteristic
4	C₂×#13095525	C₂²	1	3	yes	—
5	C₂×#13095525	C₂²	1	1	yes	characteristic
6	C₄⋊#11715906	C₄	1	3	yes	—
7	C₄×#11715906	C₂²	1	3	yes	—
8	Q₈×#87604	C₄	1	1	yes	characteristic
9	#13095525	—	2	3	no	—
10	#13095525	Q₈	1	2	yes	direct product factor
11	C₂×#11715906	C₂×C₄	1	3	yes	—
12	C₂×#11715906	C₂³	1	1	yes	characteristic
13	C₄⋊#87604	C₈	1	3	yes	—
14	C₄×#87604	C₂×C₄	1	3	yes	—
15	Q₈×D₉₇	C₈	1	1	yes	characteristic
16	C₂×#87604	C₂×C₈	1	3	yes	—
17	C₂×#87604	C₂²×C₄	1	1	yes	characteristic
18	#11715906	—	2	3	no	—
19	#11715906	C₂×Q₈	1	1	yes	characteristic
20	#11715906	#41	1	1	yes	characteristic
21	Q₈×C₉₇	C₁₆	1	1	yes	characteristic, complemented
22	C₄×D₉₇	C₂×C₈	1	3	yes	—
23	Dic₁₉₄	C₁₆	1	3	yes	complemented
24	C₂×D₉₇	C₂²×C₈	1	1	yes	characteristic
25	#87604	—	2	3	no	—
26	#87604	C₄×Q₈	1	1	yes	characteristic
27	#87604	#131	1	1	yes	characteristic
28	C₄×C₉₇	C₂×C₁₆	1	3	yes	—
29	Dic₉₇	C₂×C₁₆	1	3	yes	—
30	C₂×C₉₇	C₂²×C₁₆	1	1	yes	characteristic
31	D₉₇	C₈×Q₈	1	1	yes	characteristic
32	D₉₇	#504	1	1	yes	characteristic
33	Q₈×C₁₆	—	97	1	no	maximal, Sylow
34	C₉₇	Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
35	C₈×Q₈	—	97	1	no	—
36	C₄⋊C₁₆	—	97	3	no	—
37	C₄×C₁₆	—	97	3	no	—
38	C₄×Q₈	—	97	1	no	—
39	C₂×C₁₆	—	97	1	no	—
40	C₂×C₁₆	—	97	3	no	—
41	C₄⋊C₈	—	97	3	no	—
42	C₄×C₈	—	97	3	no	—
43	C₂×Q₈	—	97	1	no	—
44	C₂×C₈	—	97	1	no	—
45	C₂×C₈	—	97	3	no	—
46	C₄⋊C₄	—	97	3	no	—
47	C₄²	—	97	3	no	—
48	C₁₆	—	97	2	no	—
49	C₁₆	—	194	3	no	—
50	Q₈	—	97	3	no	—
51	Q₈	#13095525	1	1	yes	characteristic, direct product factor
52	C₂×C₄	—	97	1	no	—
53	C₂×C₄	—	97	3	no	—
54	C₂×C₄	—	97	3	no	—
55	C₈	—	97	1	no	—
56	C₈	—	97	1	no	—
57	C₈	—	194	3	no	—
58	C₂²	—	97	1	no	—
59	C₄	—	97	1	no	—
60	C₄	—	97	1	no	—
61	C₄	—	97	3	no	—
62	C₄	—	194	3	no	—
63	C₄	C₂×#13095525	1	3	yes	—
64	C₂	—	97	1	no	—
65	C₂	—	97	1	no	—
66	C₂	C₂²×#13095525	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×#13095525
commutator, socle	C₂×C₉₇	C₂²×C₁₆
Fitting	Q₈×C₉₇	C₁₆
prime radical	C₂×D₉₇	C₂²×C₈
cosocle	C₂×#11715906	C₂³
hypercenter	Q₈	#13095525
hypocenter	C₉₇	Q₈×C₁₆
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂ ◁ Q₈

Complements

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

N	H ≅ G/N	type
10. #13095525	Q₈	direct (N × H)
21. Q₈×C₉₇	C₁₆	semidirect
23. Dic₁₉₄	C₁₆	semidirect
34. C₉₇	Q₈×C₁₆	semidirect
51. Q₈	#13095525	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⋊#13095525	2
C₄×#13095525	2
Q₈×#11715906	2
Q₈×C₁₆	97

Subgroups — Q8×#13095525 #27159026

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#13095525 #27159026