Subgroups — #123

Order 32.

39 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊C₄	C₂	1	2	yes	maximal, complemented
4	C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₂³	C₂²	1	1	yes	characteristic
7	C₂³	C₂²	1	1	yes	characteristic
8	D₄	—	2	2	no	—
9	C₂×C₄	—	2	2	no	—
10	C₂×C₄	C₂²	1	2	yes	—
11	C₂×C₄	C₂²	1	2	yes	—
12	C₂×C₄	C₂²	1	1	yes	characteristic
13	C₂²	—	2	2	no	—
14	C₂²	—	2	1	no	—
15	C₂²	—	2	2	no	—
16	C₂²	D₄	1	2	yes	—
17	C₂²	C₂³	1	1	yes	characteristic
18	C₄	—	2	2	no	—
19	C₄	—	2	1	no	—
20	C₄	—	2	2	no	—
21	C₂	—	2	2	no	—
22	C₂	—	4	1	no	—
23	C₂	C₂×D₄	1	1	yes	characteristic
24	C₂	#41	1	2	yes	—

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂³
prime radical	C₂×D₄	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ 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.