Subgroups — Q₈×D₇₉₆ #27164685

Order 12736.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×D₁₉₉	C₂	1	2	yes	maximal, complemented
2	#20263955	C₂	1	3	yes	maximal, complemented
3	#20263935	C₂	1	6	yes	maximal, complemented
4	C₄×D₇₉₆	C₂	1	3	yes	maximal
5	C₄×Q₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₃₉₈	C₂²	1	6	yes	—
7	Q₈×D₁₉₉	—	2	8	no	—
8	C₂×D₇₉₆	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₁₉₉	C₂²	1	6	yes	—
10	C₄⋊Dic₁₉₉	C₂²	1	3	yes	—
11	C₄⋊Dic₁₉₉	C₂²	1	6	yes	—
12	(C₂×D₁₉₉)⋊C₄	C₂²	1	6	yes	—
13	C₂×Q₈×C₁₉₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₁₉₉	C₂²	1	3	yes	—
15	C₄²×C₁₉₉	C₂²	1	3	yes	—
16	C₂²×D₁₉₉	C₂³	1	2	yes	—
17	Dic₃₉₈	—	2	12	no	—
18	D₇₉₆	Q₈	1	4	yes	direct product factor
19	C₂×Dic₁₉₉	C₂³	1	6	yes	—
20	C₄×D₁₉₉	—	2	12	no	—
21	Q₈×C₁₉₉	D₄	1	4	yes	complemented
22	C₂×C₄×C₁₉₉	C₂³	1	3	yes	—
23	C₂×C₄×C₁₉₉	C₂³	1	3	yes	—
24	C₂×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
25	C₂²×C₁₉₉	C₂⁴	1	1	yes	characteristic
26	C₂×D₁₉₉	—	2	2	no	—
27	C₂×D₁₉₉	—	2	2	no	—
28	C₂×D₁₉₉	C₂×Q₈	1	4	yes	complemented
29	C₄×C₁₉₉	—	2	3	no	—
30	C₄×C₁₉₉	C₂×D₄	1	6	yes	—
31	C₄×C₁₉₉	C₂×Q₈	1	2	yes	complemented
32	Dic₁₉₉	—	2	6	no	—
33	C₂×C₁₉₉	C₂²×D₄	1	1	yes	characteristic
34	C₂×C₁₉₉	C₂²×Q₈	1	1	yes	characteristic
35	C₂×C₁₉₉	2₋¹⁺⁴	1	1	yes	characteristic
36	D₁₉₉	—	2	4	no	—
37	C₁₉₉	D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
38	D₄×Q₈	—	199	1	no	maximal, Sylow
39	C₂²×Q₈	—	199	2	no	—
40	C₄⋊Q₈	—	199	3	no	—
41	C₂²⋊Q₈	—	199	6	no	—
42	C₄×Q₈	D₁₉₉	1	1	yes	characteristic, complemented
43	C₄×D₄	—	199	3	no	—
44	C₂×Q₈	—	398	8	no	—
45	C₂×Q₈	—	199	6	no	—
46	C₂×Q₈	C₂×D₁₉₉	1	1	yes	characteristic
47	C₂×D₄	—	199	1	no	—
48	C₂²×C₄	—	199	6	no	—
49	C₄⋊C₄	—	199	3	no	—
50	C₄⋊C₄	—	199	6	no	—
51	C₄⋊C₄	C₂×D₁₉₉	1	3	yes	—
52	C₂²⋊C₄	—	199	6	no	—
53	C₄²	C₂×D₁₉₉	1	3	yes	—
54	C₂³	—	199	2	no	—
55	Q₈	—	398	12	no	—
56	Q₈	D₇₉₆	1	4	yes	direct product factor
57	D₄	—	199	4	no	—
58	C₂×C₄	—	398	12	no	—
59	C₂×C₄	—	199	6	no	—
60	C₂×C₄	C₂²×D₁₉₉	1	3	yes	—
61	C₂×C₄	C₂²×D₁₉₉	1	3	yes	—
62	C₂×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
63	C₂²	—	398	2	no	—
64	C₂²	—	398	2	no	—
65	C₂²	—	199	4	no	—
66	C₂²	C₂³×D₁₉₉	1	1	yes	characteristic
67	C₄	—	2	3	no	—
68	C₄	—	398	6	no	—
69	C₄	C₂×D₇₉₆	1	6	yes	—
70	C₄	Q₈×D₁₉₉	1	2	yes	complemented
71	C₂	—	398	4	no	—
72	C₂	C₂²×D₇₉₆	1	1	yes	characteristic
73	C₂	C₂×Q₈×D₁₉₉	1	1	yes	characteristic
74	C₂	#20264002	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₉₉
commutator, socle, cosocle	C₂²×C₁₉₉	C₂⁴
Fitting	C₄×Q₈×C₁₉₉	C₂
prime radical	C₂×D₇₉₆	C₂²
hypercenter	C₄×Q₈	D₁₉₉
hypocenter	C₁₉₉	D₄×Q₈
LCS	G ▷ C₂²×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×D₁₉₉	C₂	semidirect
2. #20263955	C₂	semidirect
3. #20263935	C₂	semidirect
5. C₄×Q₈×C₁₉₉	C₂	semidirect
18. D₇₉₆	Q₈	direct (N × H)
21. Q₈×C₁₉₉	D₄	semidirect
28. C₂×D₁₉₉	C₂×Q₈	semidirect
31. C₄×C₁₉₉	C₂×Q₈	semidirect
37. C₁₉₉	D₄×Q₈	semidirect
42. C₄×Q₈	D₁₉₉	semidirect
56. Q₈	D₇₉₆	direct (N × H)
70. C₄	Q₈×D₁₉₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×D₁₉₉	2
#20263955	2
#20263935	2
C₄×D₇₉₆	2
C₄×Q₈×C₁₉₉	2
D₄×Q₈	199

Subgroups — Q8×D796 #27164685

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×D₇₉₆ #27164685