Subgroups — #560

Order 64.

126 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#127	C₂	1	1	yes	maximal, characteristic, complemented
2	#126	C₂	1	2	yes	maximal, complemented
3	#123	C₂	1	3	yes	maximal, complemented
4	#121	C₂	1	6	yes	maximal, complemented
5	#120	C₂	1	3	yes	maximal, complemented
6	C₂⁴	C₂²	1	1	yes	characteristic
7	C₂×D₄	C₂²	1	6	yes	—
8	C₂×D₄	C₂²	1	3	yes	—
9	C₂×D₄	C₂²	1	3	yes	complemented
10	C₂²×C₄	C₂²	1	3	yes	complemented
11	C₄⋊C₄	C₂²	1	6	yes	complemented
12	C₂²⋊C₄	C₂²	1	6	yes	—
13	C₂²⋊C₄	C₂²	1	3	yes	—
14	C₂²⋊C₄	C₂²	1	3	yes	complemented
15	C₄²	C₂²	1	1	yes	characteristic, complemented
16	C₂³	—	4	3	no	—
17	C₂³	C₂³	1	2	yes	—
18	C₂³	C₂³	1	1	yes	characteristic
19	C₂³	C₂³	1	3	yes	—
20	D₄	—	4	6	no	—
21	D₄	—	4	3	no	—
22	D₄	—	4	3	no	—
23	C₂×C₄	—	4	3	no	—
24	C₂×C₄	C₂³	1	3	yes	—
25	C₂×C₄	C₂³	1	6	yes	—
26	C₂²	—	2	3	no	—
27	C₂²	—	2	6	no	—
28	C₂²	—	2	6	no	—
29	C₂²	—	2	3	no	—
30	C₂²	—	4	4	no	—
31	C₂²	C₂⁴	1	1	yes	characteristic
32	C₄	—	2	6	no	—
33	C₄	—	2	3	no	—
34	C₂	—	4	3	no	—
35	C₂	—	4	2	no	—
36	C₂	—	4	1	no	—
37	C₂	H(5,𝔽₂)	1	3	yes	—

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁴
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.