Subgroups — C₂⁴×D₄ #3588

Order 128.

5276 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁶	C₂	1	2	yes	maximal, complemented
2	C₂³×D₄	C₂	1	60	yes	maximal, direct product factor
3	C₂⁴×C₄	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁵	—	2	32	no	—
5	C₂⁵	C₂²	1	1	yes	characteristic
6	C₂⁵	C₂²	1	60	yes	complemented
7	C₂²×D₄	C₂²	1	560	yes	direct product factor
8	C₂³×C₄	C₂²	1	30	yes	complemented
9	C₂⁴	—	2	480	no	—
10	C₂⁴	D₄	1	16	yes	direct product factor
11	C₂⁴	C₂³	1	15	yes	—
12	C₂⁴	C₂³	1	280	yes	complemented
13	C₂×D₄	C₂³	1	960	yes	direct product factor
14	C₂²×C₄	C₂³	1	140	yes	complemented
15	C₂³	—	2	1120	no	—
16	C₂³	C₂×D₄	1	120	yes	direct product factor
17	C₂³	C₂⁴	1	35	yes	—
18	C₂³	C₂⁴	1	240	yes	complemented
19	D₄	C₂⁴	1	256	yes	direct product factor
20	C₂×C₄	C₂⁴	1	120	yes	complemented
21	C₂²	—	2	480	no	—
22	C₂²	C₂²×D₄	1	140	yes	direct product factor
23	C₂²	C₂⁵	1	15	yes	—
24	C₂²	C₂⁵	1	32	yes	complemented
25	C₄	C₂⁵	1	16	yes	complemented
26	C₂	—	2	32	no	—
27	C₂	C₂³×D₄	1	30	yes	direct product factor
28	C₂	C₂⁶	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁵	C₂²
commutator, Frattini, cosocle	C₂	C₂⁶
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ 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.