Subgroups — Q₈×D₁₆ #33945

Order 256.

264 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×D₈	C₂	1	2	yes	maximal, complemented
2	#2252	C₂	1	3	yes	maximal, complemented
3	#2224	C₂	1	6	yes	maximal, complemented
4	Q₈×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×D₁₆	C₂	1	3	yes	maximal
6	D₄×Q₈	—	2	2	no	—
7	C₂×D₁₆	C₂²	1	1	yes	characteristic
8	#500	C₂²	1	6	yes	—
9	#474	—	2	6	no	—
10	C₈×Q₈	C₂²	1	1	yes	characteristic
11	C₄×D₈	C₂²	1	6	yes	—
12	#366	C₂²	1	3	yes	—
13	#366	C₂²	1	6	yes	—
14	C₄⋊C₁₆	C₂²	1	3	yes	—
15	#357	C₂²	1	6	yes	—
16	C₄×C₁₆	C₂²	1	3	yes	—
17	C₂²×Q₈	—	4	2	no	—
18	C₂×D₈	C₂³	1	2	yes	—
19	C₄⋊Q₈	—	2	6	no	—
20	C₂²⋊Q₈	—	4	6	no	—
21	C₄×Q₈	D₄	1	1	yes	characteristic
22	C₄×D₄	—	2	6	no	—
23	D₁₆	Q₈	1	4	yes	direct product factor
24	C₂×C₁₆	C₂³	1	3	yes	—
25	C₂×C₁₆	C₂³	1	1	yes	characteristic
26	#107	—	2	6	no	—
27	#107	C₂³	1	6	yes	—
28	C₄⋊C₈	D₄	1	3	yes	—
29	#102	—	2	6	no	—
30	C₄×C₈	C₂³	1	3	yes	—
31	C₂×Q₈	—	4	6	no	—
32	C₂×Q₈	—	8	8	no	—
33	C₂×Q₈	D₈	1	1	yes	characteristic
34	C₂×D₄	—	2	2	no	—
35	C₂²×C₄	—	4	6	no	—
36	D₈	—	2	2	no	—
37	D₈	C₂×Q₈	1	4	yes	complemented
38	C₂×C₈	C₂×D₄	1	3	yes	—
39	C₂×C₈	C₂⁴	1	1	yes	characteristic
40	C₄⋊C₄	—	2	6	no	—
41	C₄⋊C₄	—	4	6	no	—
42	C₄⋊C₄	D₈	1	3	yes	—
43	C₂²⋊C₄	—	4	6	no	—
44	C₄²	C₂×D₄	1	3	yes	—
45	C₁₆	—	2	3	no	—
46	C₁₆	C₂×Q₈	1	2	yes	complemented
47	C₂³	—	4	2	no	—
48	Q₈	—	8	12	no	—
49	Q₈	D₁₆	1	4	yes	direct product factor
50	D₄	—	2	4	no	—
51	D₄	—	4	2	no	—
52	C₂×C₄	—	4	6	no	—
53	C₂×C₄	—	8	12	no	—
54	C₂×C₄	C₂×D₈	1	3	yes	—
55	C₂×C₄	C₂×D₈	1	3	yes	—
56	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
57	C₈	—	2	3	no	—
58	C₈	C₂²×Q₈	1	1	yes	characteristic
59	C₈	2₋¹⁺⁴	1	1	yes	characteristic
60	C₂²	—	4	4	no	—
61	C₂²	—	8	2	no	—
62	C₂²	—	8	2	no	—
63	C₂²	C₂²×D₈	1	1	yes	characteristic
64	C₄	—	2	3	no	—
65	C₄	—	8	6	no	—
66	C₄	C₂×D₁₆	1	6	yes	—
67	C₄	D₄×Q₈	1	1	yes	characteristic
68	C₄	#578	1	1	yes	characteristic
69	C₂	—	8	4	no	—
70	C₂	Q₈×D₈	1	1	yes	characteristic
71	C₂	C₂²×D₁₆	1	1	yes	characteristic
72	C₂	#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂×D₁₆	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. Q₈×D₈	C₂	semidirect
2. #2252	C₂	semidirect
3. #2224	C₂	semidirect
4. Q₈×C₁₆	C₂	semidirect
23. D₁₆	Q₈	direct (N × H)
37. D₈	C₂×Q₈	semidirect
46. C₁₆	C₂×Q₈	semidirect
49. Q₈	D₁₆	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×D₈	2
#2252	2
#2224	2
Q₈×C₁₆	2
C₄×D₁₆	2

Subgroups — Q8×D16 #33945

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×D₁₆ #33945