Subgroups — #490

Order 64.

55 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#127	C₂	1	1	yes	maximal, characteristic
2	#125	C₂	1	1	yes	maximal, characteristic, complemented
3	#102	C₂	1	4	yes	maximal, complemented
4	#97	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₄	—	2	2	no	—
6	C₂×D₄	C₂²	1	2	yes	—
7	C₂×C₈	C₂²	1	2	yes	—
8	C₄⋊C₄	—	2	2	no	—
9	C₄⋊C₄	C₂²	1	2	yes	—
10	C₄²	C₂²	1	1	yes	characteristic
11	C₂³	—	2	2	no	—
12	D₄	—	2	4	no	—
13	D₄	—	4	4	no	—
14	C₂×C₄	—	2	2	no	—
15	C₂×C₄	D₄	1	2	yes	—
16	C₂×C₄	C₂³	1	1	yes	characteristic
17	C₈	—	2	2	no	—
18	C₂²	—	4	2	no	—
19	C₂²	—	4	4	no	—
20	C₂²	C₂×D₄	1	1	yes	characteristic
21	C₄	—	2	2	no	—
22	C₄	—	4	2	no	—
23	C₄	#41	1	2	yes	—
24	C₂	—	8	2	no	—
25	C₂	#124	1	1	yes	characteristic
26	C₂	AGL(1,ℤ/8)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³
prime radical	#127	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ G
LECS	G ▷ C₂×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.