Subgroups — C₂³×Q₃₂ #62627

Order 256.

1039 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₁₆	C₂	1	2	yes	maximal
2	C₂²×Q₃₂	C₂	1	28	yes	maximal, direct product factor
3	C₂³×C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂³×Q₈	—	2	2	no	—
5	C₂²×Q₁₆	—	2	14	no	—
6	C₂²×Q₁₆	C₂²	1	28	yes	—
7	C₂³×C₈	C₂²	1	1	yes	characteristic
8	C₂×Q₃₂	C₂²	1	112	yes	direct product factor
9	C₂²×C₁₆	C₂²	1	14	yes	—
10	C₂²×Q₈	—	2	28	no	—
11	C₂²×Q₈	—	4	14	no	—
12	C₂³×C₄	—	4	2	no	—
13	C₂³×C₄	D₄	1	1	yes	characteristic
14	C₂×Q₁₆	—	2	84	no	—
15	C₂×Q₁₆	C₂³	1	56	yes	—
16	C₂²×C₈	D₄	1	7	yes	—
17	C₂²×C₈	C₂³	1	7	yes	—
18	Q₃₂	C₂³	1	64	yes	direct product factor
19	C₂×C₁₆	C₂³	1	28	yes	—
20	C₂⁴	D₈	1	1	yes	characteristic
21	C₂×Q₈	—	2	56	no	—
22	C₂×Q₈	—	4	84	no	—
23	C₂²×C₄	—	4	28	no	—
24	C₂²×C₄	D₈	1	7	yes	—
25	C₂²×C₄	C₂×D₄	1	7	yes	—
26	Q₁₆	—	2	56	no	—
27	Q₁₆	C₂⁴	1	16	yes	—
28	C₂×C₈	C₂×D₄	1	21	yes	—
29	C₂×C₈	C₂⁴	1	7	yes	—
30	C₁₆	C₂⁴	1	8	yes	—
31	C₂³	Q₃₂	1	8	yes	direct product factor
32	C₂³	C₂×D₈	1	7	yes	—
33	Q₈	—	2	16	no	—
34	Q₈	—	4	56	no	—
35	C₂×C₄	—	4	56	no	—
36	C₂×C₄	C₂×D₈	1	21	yes	—
37	C₂×C₄	C₂²×D₄	1	7	yes	—
38	C₈	C₂²×D₄	1	7	yes	—
39	C₈	C₂⁵	1	1	yes	characteristic
40	C₂²	C₂×Q₃₂	1	28	yes	direct product factor
41	C₂²	C₂²×D₈	1	7	yes	—
42	C₄	—	4	16	no	—
43	C₄	C₂²×D₈	1	7	yes	—
44	C₄	C₂³×D₄	1	1	yes	characteristic
45	C₂	C₂²×Q₃₂	1	14	yes	direct product factor
46	C₂	C₂³×D₈	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂⁴	D₈
commutator, Frattini, cosocle	C₈	C₂⁵
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

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