Subgroups — Q₈×#13095514 #27158149

Order 12416.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#13095514	C₂	1	3	yes	maximal
2	C₄×#13095514	C₂	1	3	yes	maximal
3	Q₈×#11715895	C₂	1	1	yes	maximal, characteristic
4	C₂×#13095514	C₂²	1	3	yes	—
5	C₂×#13095514	C₂²	1	1	yes	characteristic
6	C₄⋊#11715895	C₄	1	3	yes	—
7	C₄×#11715895	C₂²	1	3	yes	—
8	C₄×Q₈×C₉₇	C₄	1	1	yes	characteristic
9	#13095514	—	2	3	no	—
10	#13095514	Q₈	1	2	yes	direct product factor
11	C₂×#11715895	C₂×C₄	1	3	yes	—
12	C₂×#11715895	C₂³	1	1	yes	characteristic
13	C₂×Q₈×C₉₇	C₈	1	1	yes	characteristic
14	C₄⋊C₄×C₉₇	C₈	1	3	yes	—
15	C₄²×C₉₇	C₂×C₄	1	3	yes	—
16	Q₈×C₉₇	C₁₆	1	4	yes	complemented
17	C₂×C₄×C₉₇	C₂×C₈	1	3	yes	—
18	C₂×C₄×C₉₇	C₂×C₈	1	3	yes	—
19	C₂×C₄×C₉₇	C₂²×C₄	1	1	yes	characteristic
20	#11715895	—	2	3	no	—
21	#11715895	C₂×Q₈	1	1	yes	characteristic
22	#11715895	#41	1	1	yes	characteristic
23	C₂²×C₉₇	C₂²×C₈	1	1	yes	characteristic
24	C₄×C₉₇	—	2	3	no	—
25	C₄×C₉₇	C₂×C₁₆	1	6	yes	—
26	C₄×C₉₇	C₄×Q₈	1	1	yes	characteristic
27	C₄×C₉₇	#131	1	1	yes	characteristic
28	C₂×C₉₇	C₈×Q₈	1	1	yes	characteristic
29	C₂×C₉₇	C₂²×C₁₆	1	1	yes	characteristic
30	C₂×C₉₇	#504	1	1	yes	characteristic
31	Q₈×C₁₆	—	97	1	no	maximal, Sylow
32	C₉₇	Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
33	C₈×Q₈	—	97	1	no	—
34	C₄⋊C₁₆	—	97	3	no	—
35	C₄×C₁₆	—	97	3	no	—
36	C₄×Q₈	#87604	1	1	yes	characteristic
37	C₂×C₁₆	—	97	1	no	—
38	C₂×C₁₆	—	97	3	no	—
39	C₄⋊C₈	—	97	3	no	—
40	C₄×C₈	—	97	3	no	—
41	C₂×Q₈	#11715897	1	1	yes	characteristic
42	C₂×C₈	—	97	1	no	—
43	C₂×C₈	—	97	3	no	—
44	C₄⋊C₄	#11715897	1	3	yes	—
45	C₄²	C₂×#87604	1	3	yes	—
46	C₁₆	—	97	2	no	—
47	C₁₆	—	194	3	no	—
48	Q₈	#13095514	1	4	yes	direct product factor
49	C₂×C₄	C₂×#11715897	1	3	yes	—
50	C₂×C₄	C₂×#11715897	1	3	yes	—
51	C₂×C₄	C₂²×#87604	1	1	yes	characteristic
52	C₈	—	97	1	no	—
53	C₈	—	97	1	no	—
54	C₈	—	194	3	no	—
55	C₂²	C₂²×#11715897	1	1	yes	characteristic
56	C₄	—	2	3	no	—
57	C₄	C₂×#13095514	1	6	yes	—
58	C₄	Q₈×#87604	1	1	yes	characteristic
59	C₄	#15690898	1	1	yes	characteristic
60	C₂	Q₈×#11715897	1	1	yes	characteristic
61	C₂	C₂²×#13095514	1	1	yes	characteristic
62	C₂	#20261793	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#87604
commutator	C₂×C₉₇	C₂²×C₁₆
Fitting	C₄×Q₈×C₉₇	C₄
socle, prime radical	C₂²×C₉₇	C₂²×C₈
cosocle	C₂×#11715895	C₂³
hypercenter	C₄×Q₈	#87604
hypocenter	C₉₇	Q₈×C₁₆
LCS	G ▷ C₂×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
10. #13095514	Q₈	direct (N × H)
16. Q₈×C₉₇	C₁₆	semidirect
32. C₉₇	Q₈×C₁₆	semidirect
48. Q₈	#13095514	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⋊#13095514	2
C₄×#13095514	2
Q₈×#11715895	2
Q₈×C₁₆	97

Subgroups — Q8×#13095514 #27158149

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#13095514 #27158149