Subgroups — #518

Order 64.

169 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₄	C₂	1	1	yes	maximal, characteristic
2	C₄×D₄	C₂	1	8	yes	maximal, complemented
3	#117	C₂	1	2	yes	maximal, complemented
4	C₂×C₂²⋊C₄	C₂	1	4	yes	maximal, complemented
5	C₂⁴	C₂²	1	2	yes	—
6	C₂×D₄	C₄	1	8	yes	complemented
7	C₂×D₄	C₂²	1	4	yes	—
8	C₂²×C₄	C₂²	1	1	yes	characteristic
9	C₂²×C₄	C₂²	1	8	yes	complemented
10	C₄⋊C₄	C₂²	1	4	yes	complemented
11	C₂²⋊C₄	C₂²	1	4	yes	—
12	C₂²⋊C₄	C₂²	1	8	yes	complemented
13	C₄²	C₂²	1	4	yes	complemented
14	C₂³	—	4	4	no	—
15	C₂³	C₂×C₄	1	8	yes	complemented
16	C₂³	C₂³	1	1	yes	characteristic
17	C₂³	C₂³	1	4	yes	—
18	D₄	C₂×C₄	1	16	yes	complemented
19	C₂×C₄	—	4	8	no	—
20	C₂×C₄	C₂×C₄	1	4	yes	complemented
21	C₂×C₄	C₂³	1	8	yes	—
22	C₂×C₄	C₂³	1	2	yes	—
23	C₂²	—	2	8	no	—
24	C₂²	—	2	2	no	—
25	C₂²	—	4	8	no	—
26	C₂²	C₂²×C₄	1	2	yes	—
27	C₂²	C₂²×C₄	1	8	yes	—
28	C₂²	C₂⁴	1	1	yes	characteristic
29	C₄	—	2	8	no	—
30	C₄	C₂²×C₄	1	4	yes	—
31	C₂	—	2	2	no	—
32	C₂	—	2	8	no	—
33	C₂	C₂³×C₄	1	1	yes	characteristic
34	C₂	H(5,𝔽₂)	1	2	yes	—

	subgroup	G/H
center, Frattini, socle, cosocle	C₂²	C₂⁴
commutator	C₂	C₂³×C₄
prime radical	C₂²×D₄	C₂
LCS	G ▷ C₂ ▷ 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.