Subgroups — Q₈×Q₃₂ #33949

Order 256.

230 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Q₁₆	C₂	1	2	yes	maximal
2	#2252	C₂	1	3	yes	maximal
3	#2227	C₂	1	6	yes	maximal
4	Q₈×C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₄×Q₃₂	C₂	1	3	yes	maximal
6	Q₈²	—	2	2	no	—
7	C₂×Q₃₂	C₂²	1	1	yes	characteristic
8	#500	C₂²	1	6	yes	—
9	#477	—	2	6	no	—
10	C₈×Q₈	C₂²	1	1	yes	characteristic
11	C₄×Q₁₆	C₂²	1	6	yes	—
12	#366	C₂²	1	3	yes	—
13	#366	C₂²	1	6	yes	—
14	C₄⋊C₁₆	C₂²	1	3	yes	—
15	#358	C₂²	1	6	yes	—
16	C₄×C₁₆	C₂²	1	3	yes	—
17	C₂×Q₁₆	C₂³	1	2	yes	—
18	C₄⋊Q₈	—	2	6	no	—
19	C₄⋊Q₈	—	4	6	no	—
20	C₄×Q₈	—	2	6	no	—
21	C₄×Q₈	—	4	2	no	—
22	C₄×Q₈	D₄	1	1	yes	characteristic
23	Q₃₂	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	#103	—	2	6	no	—
30	C₄×C₈	C₂³	1	3	yes	—
31	C₂×Q₈	—	2	2	no	—
32	C₂×Q₈	—	4	6	no	—
33	C₂×Q₈	D₈	1	1	yes	characteristic
34	Q₁₆	—	2	2	no	—
35	Q₁₆	C₂×Q₈	1	4	yes	—
36	C₂×C₈	C₂×D₄	1	3	yes	—
37	C₂×C₈	C₂⁴	1	1	yes	characteristic
38	C₄⋊C₄	—	2	6	no	—
39	C₄⋊C₄	—	4	6	no	—
40	C₄⋊C₄	—	4	6	no	—
41	C₄⋊C₄	D₈	1	3	yes	—
42	C₄²	—	4	6	no	—
43	C₄²	C₂×D₄	1	3	yes	—
44	C₁₆	—	2	3	no	—
45	C₁₆	C₂×Q₈	1	2	yes	—
46	Q₈	—	2	4	no	—
47	Q₈	—	4	2	no	—
48	Q₈	—	16	6	no	—
49	Q₈	Q₃₂	1	4	yes	direct product factor
50	C₂×C₄	—	4	2	no	—
51	C₂×C₄	—	4	6	no	—
52	C₂×C₄	C₂×D₈	1	3	yes	—
53	C₂×C₄	C₂×D₈	1	3	yes	—
54	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
55	C₈	—	2	3	no	—
56	C₈	C₂²×Q₈	1	1	yes	characteristic
57	C₈	2₋¹⁺⁴	1	1	yes	characteristic
58	C₂²	C₂²×D₈	1	1	yes	characteristic
59	C₄	—	2	3	no	—
60	C₄	—	4	4	no	—
61	C₄	—	8	6	no	—
62	C₄	C₂×Q₃₂	1	6	yes	—
63	C₄	D₄×Q₈	1	1	yes	characteristic
64	C₄	#578	1	1	yes	characteristic
65	C₂	Q₈×D₈	1	1	yes	characteristic
66	C₂	C₂²×Q₃₂	1	1	yes	characteristic
67	C₂	#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴
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
23. Q₃₂	Q₈	direct (N × H)
49. Q₈	Q₃₂	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×Q₁₆	2
#2252	2
#2227	2
Q₈×C₁₆	2
C₄×Q₃₂	2

Subgroups — Q8×Q32 #33949

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×Q₃₂ #33949