Subgroups — #540

Order 64.

120 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	#125	C₂	1	1	yes	maximal, characteristic, complemented
3	#123	C₂	1	4	yes	maximal, complemented
4	#121	C₂	1	4	yes	maximal, complemented
5	#121	C₂	1	2	yes	maximal, complemented
6	C₄×D₄	C₂	1	2	yes	maximal, complemented
7	#117	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₄	C₂²	1	2	yes	—
9	C₂×D₄	C₂²	1	4	yes	—
10	C₂×D₄	C₂²	1	4	yes	—
11	C₂²×C₄	C₂²	1	1	yes	characteristic
12	C₂²×C₄	C₂²	1	4	yes	—
13	C₄⋊C₄	C₂²	1	2	yes	—
14	C₄⋊C₄	C₂²	1	2	yes	—
15	C₄⋊C₄	C₂²	1	4	yes	—
16	C₂²⋊C₄	C₂²	1	2	yes	—
17	C₂²⋊C₄	C₂²	1	4	yes	—
18	C₂²⋊C₄	C₂²	1	4	yes	—
19	C₄²	C₂²	1	1	yes	characteristic
20	C₄²	C₂²	1	1	yes	characteristic
21	C₂³	C₂³	1	1	yes	characteristic
22	C₂³	C₂³	1	4	yes	—
23	D₄	—	2	4	no	—
24	D₄	—	4	4	no	—
25	D₄	—	4	4	no	—
26	C₂×C₄	—	2	2	no	—
27	C₂×C₄	—	4	4	no	—
28	C₂×C₄	C₂³	1	2	yes	—
29	C₂×C₄	C₂³	1	2	yes	—
30	C₂×C₄	C₂³	1	1	yes	characteristic
31	C₂×C₄	C₂³	1	4	yes	—
32	C₂×C₄	C₂³	1	1	yes	characteristic
33	C₂²	—	2	1	no	—
34	C₂²	—	2	4	no	—
35	C₂²	—	2	4	no	—
36	C₂²	—	2	2	no	—
37	C₂²	—	2	4	no	—
38	C₂²	C₂⁴	1	1	yes	characteristic
39	C₄	—	2	1	no	—
40	C₄	—	2	2	no	—
41	C₄	—	2	4	no	—
42	C₄	—	2	2	no	—
43	C₄	#41	1	2	yes	—
44	C₂	—	4	1	no	—
45	C₂	—	4	4	no	—
46	C₂	C₂×#41	1	1	yes	characteristic
47	C₂	H(5,𝔽₂)	1	2	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.