Subgroups — C₂³×D₈ #3574

Order 128.

988 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, socle	C₂⁴	D₄
commutator, Frattini, cosocle	C₄	C₂⁵
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ G
LECS	G ▷ 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.