Subgroups — C₂³×Q₈² #10585935

Order 512.

37970 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈²	C₂	1	112	yes	maximal, direct product factor
2	C₂³×C₄⋊Q₈	C₂	1	9	yes	maximal
3	C₂³×C₄×Q₈	C₂	1	6	yes	maximal
4	C₂⁴×Q₈	C₂²	1	6	yes	—
5	C₂⁴×Q₈	C₂²	1	2	yes	—
6	C₂×Q₈²	C₂²	1	1792	yes	direct product factor
7	C₂²×C₄⋊Q₈	C₂²	1	504	yes	—
8	C₂²×C₄×Q₈	C₂²	1	336	yes	—
9	C₂³×C₄⋊C₄	C₂²	1	18	yes	—
10	C₂³×C₄²	C₂²	1	9	yes	—
11	C₂³×Q₈	—	4	48	no	—
12	C₂³×Q₈	Q₈	1	64	yes	direct product factor
13	C₂³×Q₈	C₂³	1	168	yes	—
14	C₂³×Q₈	C₂³	1	56	yes	—
15	C₂⁴×C₄	C₂³	1	9	yes	—
16	C₂⁴×C₄	C₂³	1	6	yes	—
17	Q₈²	C₂³	1	4096	yes	direct product factor
18	C₂×C₄⋊Q₈	C₂³	1	4032	yes	—
19	C₂×C₄×Q₈	C₂³	1	2688	yes	—
20	C₂²×C₄⋊C₄	C₂³	1	504	yes	—
21	C₂²×C₄²	C₂³	1	252	yes	—
22	C₂⁵	C₂⁴	1	1	yes	characteristic
23	C₂²×Q₈	—	4	672	no	—
24	C₂²×Q₈	C₂×Q₈	1	896	yes	direct product factor
25	C₂²×Q₈	C₂⁴	1	672	yes	—
26	C₂²×Q₈	C₂⁴	1	224	yes	—
27	C₂³×C₄	—	2	72	no	—
28	C₂³×C₄	C₂×Q₈	1	96	yes	—
29	C₂³×C₄	C₂⁴	1	126	yes	—
30	C₂³×C₄	C₂⁴	1	84	yes	—
31	C₄⋊Q₈	C₂⁴	1	4608	yes	—
32	C₄×Q₈	C₂⁴	1	3072	yes	—
33	C₂×C₄⋊C₄	C₂⁴	1	2016	yes	—
34	C₂×C₄²	C₂⁴	1	1008	yes	—
35	C₂⁴	C₂²×Q₈	1	16	yes	—
36	C₂⁴	H(5,𝔽₂)	1	8	yes	—
37	C₂⁴	C₂⁵	1	7	yes	—
38	C₂×Q₈	—	4	1344	no	—
39	C₂×Q₈	C₂²×Q₈	1	1792	yes	direct product factor
40	C₂×Q₈	C₂⁵	1	384	yes	—
41	C₂×Q₈	C₂⁵	1	128	yes	—
42	C₂²×C₄	—	2	504	no	—
43	C₂²×C₄	C₂²×Q₈	1	672	yes	—
44	C₂²×C₄	C₂⁵	1	252	yes	—
45	C₂²×C₄	C₂⁵	1	168	yes	—
46	C₄⋊C₄	C₂⁵	1	1152	yes	—
47	C₄²	C₂⁵	1	576	yes	—
48	C₂³	Q₈²	1	64	yes	direct product factor
49	C₂³	C₂³×Q₈	1	56	yes	—
50	C₂³	C₂×H(5,𝔽₂)	1	28	yes	—
51	C₂³	C₂⁶	1	7	yes	—
52	Q₈	—	4	384	no	—
53	Q₈	C₂³×Q₈	1	512	yes	direct product factor
54	C₂×C₄	—	2	504	no	—
55	C₂×C₄	C₂³×Q₈	1	672	yes	—
56	C₂×C₄	C₂⁶	1	72	yes	—
57	C₂×C₄	C₂⁶	1	48	yes	—
58	C₂²	C₂×Q₈²	1	112	yes	direct product factor
59	C₂²	C₂⁴×Q₈	1	28	yes	—
60	C₂²	C₂²×H(5,𝔽₂)	1	14	yes	—
61	C₂²	C₂⁷	1	1	yes	characteristic
62	C₄	—	2	72	no	—
63	C₄	C₂⁴×Q₈	1	96	yes	—
64	C₂	C₂²×Q₈²	1	28	yes	direct product factor
65	C₂	C₂⁵×Q₈	1	2	yes	—
66	C₂	C₂³×H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁵	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁷
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂²×Q₈²	C₂	direct (N × H)
6. C₂×Q₈²	C₂²	direct (N × H)
12. C₂³×Q₈	Q₈	direct (N × H)
17. Q₈²	C₂³	direct (N × H)
24. C₂²×Q₈	C₂×Q₈	direct (N × H)
39. C₂×Q₈	C₂²×Q₈	direct (N × H)
48. C₂³	Q₈²	direct (N × H)
53. Q₈	C₂³×Q₈	direct (N × H)
58. C₂²	C₂×Q₈²	direct (N × H)
64. C₂	C₂²×Q₈²	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×Q₈²	2
C₂³×C₄⋊Q₈	2
C₂³×C₄×Q₈	2

Subgroups — C23×Q82 #10585935

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×Q₈² #10585935