Subgroups — C₂×#120 #521

Order 64.

331 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×D₄	C₂	1	3	yes	maximal, complemented
3	#120	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₂²⋊C₄	C₂	1	3	yes	maximal, complemented
5	C₂⁴	—	2	12	no	—
6	C₂⁴	C₂²	1	3	yes	—
7	C₂⁴	C₂²	1	4	yes	complemented
8	C₂⁴	C₂²	1	1	yes	characteristic, complemented
9	C₂×D₄	—	2	12	no	—
10	C₂×D₄	C₂²	1	12	yes	complemented
11	C₂²×C₄	C₂²	1	3	yes	complemented
12	C₂²⋊C₄	C₂²	1	12	yes	complemented
13	C₂³	—	2	32	no	—
14	C₂³	—	2	12	no	—
15	C₂³	—	2	24	no	—
16	C₂³	—	2	6	no	—
17	C₂³	D₄	1	12	yes	complemented
18	C₂³	C₂³	1	6	yes	—
19	C₂³	C₂³	1	1	yes	characteristic
20	C₂³	C₂³	1	2	yes	complemented
21	D₄	—	2	24	no	—
22	C₂×C₄	—	2	6	no	—
23	C₂×C₄	C₂³	1	6	yes	complemented
24	C₂²	—	2	32	no	—
25	C₂²	—	2	24	no	—
26	C₂²	—	2	12	no	—
27	C₂²	—	2	6	no	—
28	C₂²	—	4	4	no	—
29	C₂²	C₂×D₄	1	6	yes	—
30	C₂²	C₂×D₄	1	12	yes	complemented
31	C₂²	C₂⁴	1	1	yes	characteristic
32	C₄	—	2	6	no	—
33	C₂	—	2	12	no	—
34	C₂	—	4	2	no	—
35	C₂	#120	1	4	yes	direct product factor
36	C₂	C₂²×D₄	1	3	yes	—

	subgroup	G/H
center, socle	C₂³	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ 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.