Subgroups — Q₈×Dic₈₄₂ #41003571

Order 26944.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27212366	C₂	1	3	yes	maximal
2	#27212365	C₂	1	6	yes	maximal
3	Q₈×Dic₄₂₁	C₂	1	2	yes	maximal
4	C₄×Dic₈₄₂	C₂	1	3	yes	maximal
5	C₄×Q₈×C₄₂₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₈₄₂	C₂²	1	6	yes	—
7	C₂×Dic₈₄₂	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₄₂₁	C₂²	1	3	yes	—
9	C₄⋊Dic₄₂₁	C₂²	1	6	yes	—
10	#20269750	C₂²	1	6	yes	—
11	C₄×Dic₄₂₁	C₂²	1	6	yes	—
12	C₂×Q₈×C₄₂₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₄₂₁	C₂²	1	3	yes	—
14	C₄²×C₄₂₁	C₂²	1	3	yes	—
15	Dic₈₄₂	—	4	6	no	—
16	Dic₈₄₂	Q₈	1	4	yes	direct product factor
17	C₂×Dic₄₂₁	C₂³	1	6	yes	—
18	C₂×Dic₄₂₁	C₂³	1	2	yes	—
19	Q₈×C₄₂₁	Q₈	1	4	yes	complemented
20	C₂×C₄×C₄₂₁	C₂³	1	3	yes	—
21	C₂×C₄×C₄₂₁	C₂³	1	1	yes	characteristic
22	C₂×C₄×C₄₂₁	C₂³	1	3	yes	—
23	C₂²×C₄₂₁	C₂⁴	1	1	yes	characteristic
24	C₄×C₄₂₁	—	2	3	no	—
25	C₄×C₄₂₁	C₂×Q₈	1	2	yes	—
26	C₄×C₄₂₁	C₂×Q₈	1	6	yes	—
27	Dic₄₂₁	—	2	6	no	—
28	Dic₄₂₁	C₂×Q₈	1	4	yes	—
29	C₂×C₄₂₁	C₂²×Q₈	1	1	yes	characteristic
30	C₂×C₄₂₁	C₂²×Q₈	1	1	yes	characteristic
31	C₂×C₄₂₁	H(5,𝔽₂)	1	1	yes	characteristic
32	C₄₂₁	Q₈²	1	1	yes	characteristic, complemented, Sylow
33	Q₈²	—	421	1	no	maximal, Sylow
34	C₄⋊Q₈	—	421	6	no	—
35	C₄⋊Q₈	—	421	3	no	—
36	C₄×Q₈	—	421	2	no	—
37	C₄×Q₈	—	421	3	no	—
38	C₄×Q₈	D₄₂₁	1	1	yes	characteristic
39	C₂×Q₈	—	421	6	no	—
40	C₂×Q₈	—	421	1	no	—
41	C₂×Q₈	C₂×D₄₂₁	1	1	yes	characteristic
42	C₄⋊C₄	—	421	6	no	—
43	C₄⋊C₄	—	421	3	no	—
44	C₄⋊C₄	—	421	6	no	—
45	C₄⋊C₄	C₂×D₄₂₁	1	3	yes	—
46	C₄²	—	421	6	no	—
47	C₄²	C₂×D₄₂₁	1	3	yes	—
48	Q₈	—	1684	6	no	—
49	Q₈	—	421	4	no	—
50	Q₈	Dic₈₄₂	1	4	yes	direct product factor
51	C₂×C₄	—	421	6	no	—
52	C₂×C₄	—	421	2	no	—
53	C₂×C₄	C₂²×D₄₂₁	1	3	yes	—
54	C₂×C₄	C₂²×D₄₂₁	1	1	yes	characteristic
55	C₂×C₄	C₂²×D₄₂₁	1	3	yes	—
56	C₂²	C₂³×D₄₂₁	1	1	yes	characteristic
57	C₄	—	2	3	no	—
58	C₄	—	842	6	no	—
59	C₄	—	421	4	no	—
60	C₄	Q₈×D₄₂₁	1	2	yes	—
61	C₄	C₂×Dic₈₄₂	1	6	yes	—
62	C₂	C₂×Q₈×D₄₂₁	1	1	yes	characteristic
63	C₂	C₂²×Dic₈₄₂	1	1	yes	characteristic
64	C₂	#27212412	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₂₁
commutator, socle, prime radical, cosocle	C₂²×C₄₂₁	C₂⁴
Fitting	C₄×Q₈×C₄₂₁	C₂
hypercenter	C₄×Q₈	D₄₂₁
hypocenter	C₄₂₁	Q₈²
LCS	G ▷ C₂²×C₄₂₁ ▷ C₄₂₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
16. Dic₈₄₂	Q₈	direct (N × H)
19. Q₈×C₄₂₁	Q₈	semidirect
32. C₄₂₁	Q₈²	semidirect
50. Q₈	Dic₈₄₂	direct (N × H)

Maximal subgroups

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

subgroup	index
#27212366	2
#27212365	2
Q₈×Dic₄₂₁	2
C₄×Dic₈₄₂	2
C₄×Q₈×C₄₂₁	2
Q₈²	421

Subgroups — Q8×Dic842 #41003571

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×Dic₈₄₂ #41003571