Subgroups — C₂³×C₄×Q₈ #62643

Order 256.

6223 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Q₈	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₄⋊C₄	C₂	1	3	yes	maximal
4	C₂³×C₄²	C₂	1	3	yes	maximal
5	C₂³×Q₈	C₄	1	32	yes	direct product factor
6	C₂³×Q₈	C₂²	1	28	yes	—
7	C₂⁴×C₄	C₂²	1	3	yes	—
8	C₂⁴×C₄	C₂²	1	3	yes	—
9	C₂⁴×C₄	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈	C₂²	1	448	yes	direct product factor
11	C₂²×C₄⋊C₄	C₂²	1	84	yes	—
12	C₂²×C₄²	C₂²	1	84	yes	—
13	C₂⁵	C₂³	1	1	yes	characteristic
14	C₂²×Q₈	C₂×C₄	1	448	yes	direct product factor
15	C₂²×Q₈	C₂³	1	112	yes	—
16	C₂³×C₄	—	2	24	no	—
17	C₂³×C₄	C₂×C₄	1	48	yes	—
18	C₂³×C₄	Q₈	1	16	yes	direct product factor
19	C₂³×C₄	C₂³	1	42	yes	—
20	C₂³×C₄	C₂³	1	42	yes	—
21	C₂³×C₄	C₂³	1	14	yes	—
22	C₄×Q₈	C₂³	1	512	yes	direct product factor
23	C₂×C₄⋊C₄	C₂³	1	336	yes	—
24	C₂×C₄²	C₂³	1	336	yes	—
25	C₂⁴	C₂²×C₄	1	8	yes	—
26	C₂⁴	C₂×Q₈	1	8	yes	—
27	C₂⁴	#41	1	8	yes	—
28	C₂⁴	C₂⁴	1	7	yes	—
29	C₂×Q₈	C₂²×C₄	1	896	yes	direct product factor
30	C₂×Q₈	C₂⁴	1	64	yes	—
31	C₂²×C₄	—	2	168	no	—
32	C₂²×C₄	C₂²×C₄	1	336	yes	—
33	C₂²×C₄	C₂×Q₈	1	112	yes	direct product factor
34	C₂²×C₄	C₂⁴	1	84	yes	—
35	C₂²×C₄	C₂⁴	1	84	yes	—
36	C₂²×C₄	C₂⁴	1	28	yes	—
37	C₄⋊C₄	C₂⁴	1	192	yes	—
38	C₄²	C₂⁴	1	192	yes	—
39	C₂³	C₄×Q₈	1	64	yes	direct product factor
40	C₂³	C₂³×C₄	1	28	yes	—
41	C₂³	C₂²×Q₈	1	28	yes	—
42	C₂³	C₂×#41	1	28	yes	—
43	C₂³	C₂⁵	1	7	yes	—
44	Q₈	C₂³×C₄	1	256	yes	direct product factor
45	C₂×C₄	—	2	168	no	—
46	C₂×C₄	C₂³×C₄	1	336	yes	—
47	C₂×C₄	C₂²×Q₈	1	112	yes	direct product factor
48	C₂×C₄	C₂⁵	1	24	yes	—
49	C₂×C₄	C₂⁵	1	24	yes	—
50	C₂×C₄	C₂⁵	1	8	yes	—
51	C₂²	C₂×C₄×Q₈	1	112	yes	direct product factor
52	C₂²	C₂⁴×C₄	1	14	yes	—
53	C₂²	C₂³×Q₈	1	14	yes	—
54	C₂²	C₂²×#41	1	14	yes	—
55	C₂²	C₂⁶	1	1	yes	characteristic
56	C₄	—	2	24	no	—
57	C₄	C₂⁴×C₄	1	48	yes	—
58	C₄	C₂³×Q₈	1	16	yes	direct product factor
59	C₂	C₂²×C₄×Q₈	1	28	yes	direct product factor
60	C₂	C₂⁵×C₄	1	1	yes	characteristic
61	C₂	C₂⁴×Q₈	1	1	yes	characteristic
62	C₂	C₂³×#41	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴×C₄	C₂²
commutator	C₂	C₂⁵×C₄
Frattini, cosocle	C₂²	C₂⁶
socle, prime radical	C₂⁵	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂²×C₄×Q₈	C₂	direct (N × H)
5. C₂³×Q₈	C₄	direct (N × H)
10. C₂×C₄×Q₈	C₂²	direct (N × H)
14. C₂²×Q₈	C₂×C₄	direct (N × H)
18. C₂³×C₄	Q₈	direct (N × H)
22. C₄×Q₈	C₂³	direct (N × H)
29. C₂×Q₈	C₂²×C₄	direct (N × H)
33. C₂²×C₄	C₂×Q₈	direct (N × H)
39. C₂³	C₄×Q₈	direct (N × H)
44. Q₈	C₂³×C₄	direct (N × H)
47. C₂×C₄	C₂²×Q₈	direct (N × H)
51. C₂²	C₂×C₄×Q₈	direct (N × H)
58. C₄	C₂³×Q₈	direct (N × H)
59. C₂	C₂²×C₄×Q₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Q₈	2
C₂²×C₄×Q₈	2
C₂³×C₄⋊C₄	2
C₂³×C₄²	2

Subgroups — C23×C4×Q8 #62643

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₄×Q₈ #62643