Subgroups — Q₈×M₃₂ #33477

Order 256.

265 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈	C₂	1	1	yes	maximal, characteristic
2	#2183	C₂	1	6	yes	maximal, complemented
3	Q₈×C₁₆	C₂	1	2	yes	maximal, complemented
4	#2150	C₂	1	3	yes	maximal
5	C₄×M₃₂	C₂	1	3	yes	maximal
6	C₂×C₄×Q₈	C₄	1	1	yes	characteristic
7	C₂×M₃₂	C₂²	1	3	yes	—
8	C₂×M₃₂	C₂²	1	1	yes	characteristic
9	C₈×Q₈	C₄	1	3	yes	—
10	C₈×Q₈	C₄	1	1	yes	characteristic
11	C₈×Q₈	C₂²	1	1	yes	characteristic
12	C₈×Q₈	C₂²	1	3	yes	—
13	C₂×C₄⋊C₈	C₄	1	3	yes	—
14	C₂×C₄×C₈	C₂²	1	3	yes	—
15	C₄⋊C₁₆	C₂²	1	6	yes	—
16	C₄⋊C₁₆	C₂²	1	6	yes	—
17	#346	C₂²	1	6	yes	—
18	C₄×C₁₆	C₂²	1	6	yes	—
19	C₂²×Q₈	C₈	1	1	yes	characteristic
20	C₂²×C₈	C₂×C₄	1	3	yes	—
21	C₂²×C₈	C₂³	1	1	yes	characteristic
22	C₄×Q₈	C₈	1	3	yes	—
23	C₄×Q₈	C₈	1	1	yes	characteristic
24	C₄×Q₈	C₂×C₄	1	3	yes	—
25	C₄×Q₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄⋊C₄	C₈	1	3	yes	—
27	C₂×C₄²	C₂×C₄	1	3	yes	—
28	M₃₂	—	2	6	no	—
29	M₃₂	Q₈	1	4	yes	direct product factor
30	C₂×C₁₆	C₂³	1	6	yes	—
31	C₂×C₁₆	C₂³	1	2	yes	—
32	C₄⋊C₈	C₂×C₄	1	3	yes	—
33	C₄⋊C₈	C₂×C₄	1	6	yes	—
34	C₄⋊C₈	C₂×C₄	1	3	yes	—
35	C₄×C₈	C₂×C₄	1	3	yes	—
36	C₄×C₈	C₂×C₄	1	3	yes	—
37	C₄×C₈	C₂³	1	3	yes	—
38	C₄×C₈	C₂³	1	3	yes	—
39	C₂×Q₈	—	2	4	no	—
40	C₂×Q₈	C₂×C₈	1	3	yes	—
41	C₂×Q₈	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄	C₂×C₈	1	3	yes	—
43	C₂²×C₄	C₂×C₈	1	3	yes	—
44	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈	—	2	6	no	—
46	C₂×C₈	C₂²×C₄	1	3	yes	—
47	C₂×C₈	C₂²×C₄	1	3	yes	—
48	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₈	C₂×Q₈	1	2	yes	—
50	C₂×C₈	#41	1	2	yes	—
51	C₂×C₈	C₂⁴	1	1	yes	characteristic
52	C₄⋊C₄	C₂×C₈	1	6	yes	—
53	C₄⋊C₄	C₂×C₈	1	3	yes	—
54	C₄⋊C₄	C₂×C₈	1	3	yes	—
55	C₄²	C₂×C₈	1	3	yes	—
56	C₄²	C₂×C₈	1	3	yes	—
57	C₄²	C₂²×C₄	1	3	yes	—
58	C₄²	C₂²×C₄	1	3	yes	—
59	C₁₆	—	2	6	no	—
60	C₁₆	C₂×Q₈	1	4	yes	complemented
61	C₂³	C₂²×C₈	1	1	yes	characteristic
62	Q₈	—	2	6	no	—
63	Q₈	M₃₂	1	4	yes	direct product factor
64	C₂×C₄	—	2	6	no	—
65	C₂×C₄	—	4	3	no	—
66	C₂×C₄	C₄×Q₈	1	2	yes	—
67	C₂×C₄	C₂²×C₈	1	3	yes	—
68	C₂×C₄	C₂²×C₈	1	3	yes	—
69	C₂×C₄	C₂²×C₈	1	3	yes	—
70	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
71	C₂×C₄	C₂²×C₈	1	3	yes	—
72	C₂×C₄	#131	1	2	yes	—
73	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
74	C₈	—	2	3	no	—
75	C₈	—	2	3	no	—
76	C₈	C₄×Q₈	1	2	yes	—
77	C₈	C₂²×Q₈	1	1	yes	characteristic
78	C₈	C₂×#41	1	1	yes	characteristic
79	C₂²	—	2	1	no	—
80	C₂²	—	2	1	no	—
81	C₂²	C₈×Q₈	1	2	yes	—
82	C₂²	C₂³×C₈	1	1	yes	characteristic
83	C₄	—	2	3	no	—
84	C₄	—	2	3	no	—
85	C₄	—	2	3	no	—
86	C₄	C₈×Q₈	1	2	yes	—
87	C₄	C₂×M₃₂	1	6	yes	—
88	C₄	C₂×C₄×Q₈	1	1	yes	characteristic
89	C₄	C₂×#131	1	1	yes	characteristic
90	C₂	—	2	2	no	—
91	C₂	C₂×C₈×Q₈	1	1	yes	characteristic
92	C₂	C₂²×M₃₂	1	1	yes	characteristic
93	C₂	#3407	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂×C₈	C₂⁴
commutator, socle	C₂²	C₂³×C₈
prime radical	C₂³	C₂²×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #2183	C₂	semidirect
3. Q₈×C₁₆	C₂	semidirect
29. M₃₂	Q₈	direct (N × H)
60. C₁₆	C₂×Q₈	semidirect
63. Q₈	M₃₂	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₈×Q₈	2
#2183	2
Q₈×C₁₆	2
#2150	2
C₄×M₃₂	2

Subgroups — Q8×M32 #33477

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×M₃₂ #33477