Subgroups — Q₈×D₈₄₄ #27213813

Order 13504.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×D₂₁₁	C₂	1	2	yes	maximal, complemented
2	#20269977	C₂	1	3	yes	maximal, complemented
3	#20269957	C₂	1	6	yes	maximal, complemented
4	C₄×D₈₄₄	C₂	1	3	yes	maximal
5	C₄×Q₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₄₂₂	C₂²	1	6	yes	—
7	Q₈×D₂₁₁	—	2	8	no	—
8	C₂×D₈₄₄	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₂₁₁	C₂²	1	6	yes	—
10	C₄⋊Dic₂₁₁	C₂²	1	3	yes	—
11	C₄⋊Dic₂₁₁	C₂²	1	6	yes	—
12	(C₂×D₂₁₁)⋊C₄	C₂²	1	6	yes	—
13	C₂×Q₈×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₂₁₁	C₂²	1	3	yes	—
15	C₄²×C₂₁₁	C₂²	1	3	yes	—
16	C₂²×D₂₁₁	C₂³	1	2	yes	—
17	Dic₄₂₂	—	2	12	no	—
18	D₈₄₄	Q₈	1	4	yes	direct product factor
19	C₂×Dic₂₁₁	C₂³	1	6	yes	—
20	C₄×D₂₁₁	—	2	12	no	—
21	Q₈×C₂₁₁	D₄	1	4	yes	complemented
22	C₂×C₄×C₂₁₁	C₂³	1	3	yes	—
23	C₂×C₄×C₂₁₁	C₂³	1	3	yes	—
24	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
25	C₂²×C₂₁₁	C₂⁴	1	1	yes	characteristic
26	C₂×D₂₁₁	—	2	2	no	—
27	C₂×D₂₁₁	—	2	2	no	—
28	C₂×D₂₁₁	C₂×Q₈	1	4	yes	complemented
29	C₄×C₂₁₁	—	2	3	no	—
30	C₄×C₂₁₁	C₂×D₄	1	6	yes	—
31	C₄×C₂₁₁	C₂×Q₈	1	2	yes	complemented
32	Dic₂₁₁	—	2	6	no	—
33	C₂×C₂₁₁	C₂²×D₄	1	1	yes	characteristic
34	C₂×C₂₁₁	C₂²×Q₈	1	1	yes	characteristic
35	C₂×C₂₁₁	2₋¹⁺⁴	1	1	yes	characteristic
36	D₂₁₁	—	2	4	no	—
37	C₂₁₁	D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
38	D₄×Q₈	—	211	1	no	maximal, Sylow
39	C₂²×Q₈	—	211	2	no	—
40	C₄⋊Q₈	—	211	3	no	—
41	C₂²⋊Q₈	—	211	6	no	—
42	C₄×Q₈	D₂₁₁	1	1	yes	characteristic, complemented
43	C₄×D₄	—	211	3	no	—
44	C₂×Q₈	—	422	8	no	—
45	C₂×Q₈	—	211	6	no	—
46	C₂×Q₈	C₂×D₂₁₁	1	1	yes	characteristic
47	C₂×D₄	—	211	1	no	—
48	C₂²×C₄	—	211	6	no	—
49	C₄⋊C₄	—	211	3	no	—
50	C₄⋊C₄	—	211	6	no	—
51	C₄⋊C₄	C₂×D₂₁₁	1	3	yes	—
52	C₂²⋊C₄	—	211	6	no	—
53	C₄²	C₂×D₂₁₁	1	3	yes	—
54	C₂³	—	211	2	no	—
55	Q₈	—	422	12	no	—
56	Q₈	D₈₄₄	1	4	yes	direct product factor
57	D₄	—	211	4	no	—
58	C₂×C₄	—	422	12	no	—
59	C₂×C₄	—	211	6	no	—
60	C₂×C₄	C₂²×D₂₁₁	1	3	yes	—
61	C₂×C₄	C₂²×D₂₁₁	1	3	yes	—
62	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
63	C₂²	—	422	2	no	—
64	C₂²	—	422	2	no	—
65	C₂²	—	211	4	no	—
66	C₂²	C₂³×D₂₁₁	1	1	yes	characteristic
67	C₄	—	2	3	no	—
68	C₄	—	422	6	no	—
69	C₄	C₂×D₈₄₄	1	6	yes	—
70	C₄	Q₈×D₂₁₁	1	2	yes	complemented
71	C₂	—	422	4	no	—
72	C₂	C₂²×D₈₄₄	1	1	yes	characteristic
73	C₂	C₂×Q₈×D₂₁₁	1	1	yes	characteristic
74	C₂	#20270024	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₁₁
commutator, socle, cosocle	C₂²×C₂₁₁	C₂⁴
Fitting	C₄×Q₈×C₂₁₁	C₂
prime radical	C₂×D₈₄₄	C₂²
hypercenter	C₄×Q₈	D₂₁₁
hypocenter	C₂₁₁	D₄×Q₈
LCS	G ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×D₂₁₁	C₂	semidirect
2. #20269977	C₂	semidirect
3. #20269957	C₂	semidirect
5. C₄×Q₈×C₂₁₁	C₂	semidirect
18. D₈₄₄	Q₈	direct (N × H)
21. Q₈×C₂₁₁	D₄	semidirect
28. C₂×D₂₁₁	C₂×Q₈	semidirect
31. C₄×C₂₁₁	C₂×Q₈	semidirect
37. C₂₁₁	D₄×Q₈	semidirect
42. C₄×Q₈	D₂₁₁	semidirect
56. Q₈	D₈₄₄	direct (N × H)
70. C₄	Q₈×D₂₁₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×D₂₁₁	2
#20269977	2
#20269957	2
C₄×D₈₄₄	2
C₄×Q₈×C₂₁₁	2
D₄×Q₈	211

Subgroups — Q8×D844 #27213813

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×D₈₄₄ #27213813