Subgroups — Q₈×#564 #8092022

Order 512.

4089 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#46153	C₂	1	45	yes	maximal
2	Q₈×#125	C₂	1	15	yes	maximal
3	C₄×#564	C₂	1	3	yes	maximal
4	C₂×#564	C₂²	1	1	yes	characteristic
5	#2859	C₂²	1	45	yes	—
6	#2845	C₂²	1	30	yes	—
7	#2726	C₂²	1	180	yes	—
8	#2552	C₂²	1	90	yes	—
9	#2507	C₂²	1	180	yes	—
10	#2382	C₂²	1	45	yes	—
11	Q₈×C₄⋊C₄	C₂²	1	30	yes	—
12	C₄×#125	C₂²	1	45	yes	—
13	C₄²×Q₈	C₂²	1	5	yes	—
14	#564	Q₈	1	16	yes	direct product factor
15	C₂×#125	C₂³	1	90	yes	—
16	C₂×#125	C₂³	1	15	yes	—
17	C₂×C₄×Q₈	C₂³	1	15	yes	—
18	#398	C₂³	1	360	yes	—
19	#389	C₂³	1	90	yes	—
20	#389	C₂³	1	180	yes	—
21	#389	C₂³	1	180	yes	—
22	#387	C₂³	1	180	yes	—
23	#384	C₂³	1	90	yes	—
24	#382	C₂³	1	45	yes	—
25	C₄×C₄⋊C₄	C₂³	1	90	yes	—
26	C₄×C₄⋊C₄	C₂³	1	45	yes	—
27	C₄³	C₂³	1	15	yes	—
28	C₂²×Q₈	C₂⁴	1	1	yes	characteristic
29	#125	—	8	90	no	—
30	#125	C₂×Q₈	1	120	yes	—
31	C₄×Q₈	—	2	60	no	—
32	C₂×C₄⋊C₄	C₂⁴	1	45	yes	—
33	C₂×C₄⋊C₄	C₂⁴	1	90	yes	—
34	C₂×C₄⋊C₄	C₂⁴	1	180	yes	—
35	C₂×C₄⋊C₄	C₂⁴	1	30	yes	—
36	C₂×C₄²	C₂⁴	1	45	yes	—
37	C₂×C₄²	C₂⁴	1	45	yes	—
38	C₂×C₄²	C₂⁴	1	5	yes	—
39	#95	C₂⁴	1	30	yes	—
40	#95	C₂⁴	1	180	yes	—
41	C₂×Q₈	2₋¹⁺⁴	1	12	yes	—
42	C₂²×C₄	C₂⁵	1	45	yes	—
43	C₂²×C₄	C₂⁵	1	15	yes	—
44	C₂²×C₄	C₂⁵	1	3	yes	—
45	C₄⋊C₄	—	2	90	no	—
46	C₄⋊C₄	—	4	90	no	—
47	C₄⋊C₄	—	4	180	no	—
48	C₄⋊C₄	C₂²×Q₈	1	120	yes	—
49	C₄²	—	2	90	no	—
50	C₄²	—	4	45	no	—
51	C₄²	C₂²×Q₈	1	20	yes	—
52	C₂³	C₂⁶	1	1	yes	characteristic
53	Q₈	#564	1	16	yes	direct product factor
54	C₂×C₄	—	2	15	no	—
55	C₂×C₄	—	2	15	no	—
56	C₂×C₄	—	2	45	no	—
57	C₂×C₄	—	2	90	no	—
58	C₂×C₄	C₂³×Q₈	1	30	yes	—
59	C₂×C₄	C₂×2₋¹⁺⁴	1	18	yes	—
60	C₂²	C₂⁴×Q₈	1	1	yes	characteristic
61	C₂²	C₂²×2₋¹⁺⁴	1	3	yes	—
62	C₂²	H(7,𝔽₂)	1	3	yes	—
63	C₄	—	2	30	no	—
64	C₄	—	4	45	no	—
65	C₄	C₂×#564	1	12	yes	—
66	C₂	C₂²×#564	1	1	yes	characteristic
67	C₂	Q₈×2₋¹⁺⁴	1	3	yes	—
68	C₂	#63070	1	3	yes	—

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
14. #564	Q₈	direct (N × H)
53. Q₈	#564	direct (N × H)

Maximal subgroups

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

subgroup	index
#46153	2
Q₈×#125	2
C₄×#564	2

Subgroups — Q8×#564 #8092022

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#564 #8092022