Subgroups — #499

Order 64.

43 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#125	C₂	1	2	yes	maximal
2	#107	C₂	1	2	yes	maximal
3	#106	C₂	1	2	yes	maximal
4	C₄×C₈	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈	C₂²	1	1	yes	characteristic
6	C₂×C₈	C₂²	1	1	yes	characteristic
7	C₄⋊C₄	—	2	4	no	—
8	C₄⋊C₄	C₂²	1	4	yes	—
9	C₄²	C₂²	1	1	yes	characteristic
10	C₂×C₄	—	2	4	no	—
11	C₂×C₄	D₄	1	2	yes	—
12	C₂×C₄	C₂³	1	1	yes	characteristic
13	C₈	Q₈	1	2	yes	—
14	C₈	Q₈	1	2	yes	—
15	C₂²	C₂×D₄	1	1	yes	characteristic
16	C₄	—	2	2	no	—
17	C₄	—	4	4	no	—
18	C₄	C₂×Q₈	1	1	yes	characteristic
19	C₄	C₂×Q₈	1	1	yes	characteristic
20	C₂	C₄⋊Q₈	1	1	yes	characteristic
21	C₂	#135	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

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