Subgroups — #559

Order 64.

118 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	#124	C₂	1	4	yes	maximal, complemented
3	#123	C₂	1	4	yes	maximal, complemented
4	C₄×Q₈	C₂	1	2	yes	maximal, complemented
5	C₄×D₄	C₂	1	4	yes	maximal, complemented
6	C₂×Q₈	C₂²	1	2	yes	—
7	C₂×D₄	C₂²	1	4	yes	—
8	C₂×D₄	C₂²	1	2	yes	—
9	C₂²×C₄	C₂²	1	4	yes	—
10	C₄⋊C₄	C₂²	1	2	yes	—
11	C₄⋊C₄	C₂²	1	4	yes	—
12	C₂²⋊C₄	C₂²	1	8	yes	—
13	C₂²⋊C₄	C₂²	1	4	yes	—
14	C₄²	C₂²	1	4	yes	—
15	C₄²	C₂²	1	1	yes	characteristic
16	C₂³	C₂³	1	4	yes	—
17	Q₈	—	2	4	no	—
18	D₄	—	2	8	no	—
19	D₄	—	4	2	no	—
20	C₂×C₄	—	4	4	no	—
21	C₂×C₄	C₂³	1	4	yes	—
22	C₂×C₄	C₂³	1	4	yes	—
23	C₂×C₄	C₂³	1	1	yes	characteristic
24	C₂×C₄	C₂³	1	2	yes	—
25	C₂²	—	2	8	no	—
26	C₂²	—	2	4	no	—
27	C₂²	C₂⁴	1	1	yes	characteristic
28	C₄	—	2	4	no	—
29	C₄	—	2	4	no	—
30	C₄	—	2	1	no	—
31	C₄	#41	1	4	yes	—
32	C₂	—	4	4	no	—
33	C₂	C₂×#41	1	2	yes	—
34	C₂	H(5,𝔽₂)	1	1	yes	characteristic

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁴
prime radical	#127	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.