Subgroups — Q₈×D₂₄₄ #16840429

Order 3904.

280 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×D₆₁	C₂	1	2	yes	maximal, complemented
2	#14519777	C₂	1	3	yes	maximal, complemented
3	#14519757	C₂	1	6	yes	maximal, complemented
4	C₄×D₂₄₄	C₂	1	3	yes	maximal
5	C₄×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×C₆₁	C₂²	1	1	yes	characteristic
7	Q₈×D₆₁	—	2	8	no	—
8	C₂×D₂₄₄	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₆₁	C₂²	1	6	yes	—
10	C₂×Dic₁₂₂	C₂²	1	6	yes	—
11	C₄⋊C₄×C₆₁	C₂²	1	3	yes	—
12	C₄²×C₆₁	C₂²	1	3	yes	—
13	#11758198	C₂²	1	6	yes	—
14	C₄⋊Dic₆₁	C₂²	1	3	yes	—
15	C₄⋊Dic₆₁	C₂²	1	6	yes	—
16	C₂²×D₆₁	C₂³	1	2	yes	—
17	Q₈×C₆₁	D₄	1	4	yes	complemented
18	C₂×C₄×C₆₁	C₂³	1	3	yes	—
19	C₂×C₄×C₆₁	C₂³	1	3	yes	—
20	C₂×C₄×C₆₁	C₂³	1	1	yes	characteristic
21	C₂×Dic₆₁	C₂³	1	6	yes	—
22	D₂₄₄	Q₈	1	4	yes	direct product factor
23	C₄×D₆₁	—	2	12	no	—
24	Dic₁₂₂	—	2	12	no	—
25	C₂²×C₆₁	C₂⁴	1	1	yes	characteristic
26	C₂×D₆₁	—	2	2	no	—
27	C₂×D₆₁	—	2	2	no	—
28	C₂×D₆₁	C₂×Q₈	1	4	yes	complemented
29	C₄×C₆₁	—	2	3	no	—
30	C₄×C₆₁	C₂×D₄	1	6	yes	—
31	C₄×C₆₁	C₂×Q₈	1	2	yes	complemented
32	Dic₆₁	—	2	6	no	—
33	C₂×C₆₁	C₂²×D₄	1	1	yes	characteristic
34	C₂×C₆₁	C₂²×Q₈	1	1	yes	characteristic
35	C₂×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
36	D₆₁	—	2	4	no	—
37	D₄×Q₈	—	61	1	no	maximal, Sylow
38	C₆₁	D₄×Q₈	1	1	yes	characteristic, complemented, Sylow
39	C₂²×Q₈	—	61	2	no	—
40	C₄⋊Q₈	—	61	3	no	—
41	C₂²⋊Q₈	—	61	6	no	—
42	C₄×Q₈	D₆₁	1	1	yes	characteristic, complemented
43	C₄×D₄	—	61	3	no	—
44	C₂×Q₈	—	61	6	no	—
45	C₂×Q₈	—	122	8	no	—
46	C₂×Q₈	C₂×D₆₁	1	1	yes	characteristic
47	C₂×D₄	—	61	1	no	—
48	C₂²×C₄	—	61	6	no	—
49	C₄⋊C₄	—	61	3	no	—
50	C₄⋊C₄	—	61	6	no	—
51	C₄⋊C₄	C₂×D₆₁	1	3	yes	—
52	C₂²⋊C₄	—	61	6	no	—
53	C₄²	C₂×D₆₁	1	3	yes	—
54	C₂³	—	61	2	no	—
55	Q₈	—	122	12	no	—
56	Q₈	D₂₄₄	1	4	yes	direct product factor
57	D₄	—	61	4	no	—
58	C₂×C₄	—	61	6	no	—
59	C₂×C₄	—	122	12	no	—
60	C₂×C₄	C₂²×D₆₁	1	3	yes	—
61	C₂×C₄	C₂²×D₆₁	1	3	yes	—
62	C₂×C₄	C₂²×D₆₁	1	1	yes	characteristic
63	C₂²	—	61	4	no	—
64	C₂²	—	122	2	no	—
65	C₂²	—	122	2	no	—
66	C₂²	C₂³×D₆₁	1	1	yes	characteristic
67	C₄	—	2	3	no	—
68	C₄	—	122	6	no	—
69	C₄	C₂×D₂₄₄	1	6	yes	—
70	C₄	Q₈×D₆₁	1	2	yes	complemented
71	C₂	—	122	4	no	—
72	C₂	C₂²×D₂₄₄	1	1	yes	characteristic
73	C₂	C₂×Q₈×D₆₁	1	1	yes	characteristic
74	C₂	#14519824	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₆₁
commutator, socle, cosocle	C₂²×C₆₁	C₂⁴
Fitting	C₄×Q₈×C₆₁	C₂
prime radical	C₂×D₂₄₄	C₂²
hypercenter	C₄×Q₈	D₆₁
hypocenter	C₆₁	D₄×Q₈
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×D₆₁	C₂	semidirect
2. #14519777	C₂	semidirect
3. #14519757	C₂	semidirect
5. C₄×Q₈×C₆₁	C₂	semidirect
17. Q₈×C₆₁	D₄	semidirect
22. D₂₄₄	Q₈	direct (N × H)
28. C₂×D₆₁	C₂×Q₈	semidirect
31. C₄×C₆₁	C₂×Q₈	semidirect
38. C₆₁	D₄×Q₈	semidirect
42. C₄×Q₈	D₆₁	semidirect
56. Q₈	D₂₄₄	direct (N × H)
70. C₄	Q₈×D₆₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×D₆₁	2
#14519777	2
#14519757	2
C₄×D₂₄₄	2
C₄×Q₈×C₆₁	2
D₄×Q₈	61

Subgroups — Q8×D244 #16840429

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×D₂₄₄ #16840429