Subgroups — #563

Order 64.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊Q₈	C₂	1	2	yes	maximal, complemented
2	#126	C₂	1	4	yes	maximal, complemented
3	#125	C₂	1	2	yes	maximal, complemented
4	#124	C₂	1	1	yes	maximal, characteristic
5	#123	C₂	1	2	yes	maximal
6	C₂²⋊Q₈	C₂	1	4	yes	maximal, complemented
7	C₂×Q₈	C₂²	1	2	yes	—
8	C₂×Q₈	C₂²	1	1	yes	characteristic
9	C₂×D₄	C₂²	1	1	yes	characteristic
10	C₂²×C₄	C₂²	1	2	yes	—
11	C₄⋊C₄	C₂²	1	2	yes	—
12	C₄⋊C₄	C₂²	1	4	yes	—
13	C₄⋊C₄	C₂²	1	4	yes	—
14	C₄⋊C₄	C₂²	1	4	yes	—
15	C₄⋊C₄	C₂²	1	2	yes	—
16	C₂²⋊C₄	C₂²	1	4	yes	—
17	C₂²⋊C₄	C₂²	1	4	yes	—
18	C₂²⋊C₄	C₂²	1	2	yes	—
19	C₄²	C₂²	1	1	yes	characteristic
20	C₄²	C₂²	1	2	yes	—
21	C₂³	C₂³	1	2	yes	—
22	Q₈	—	4	2	no	—
23	Q₈	—	4	1	no	—
24	D₄	—	4	1	no	—
25	C₂×C₄	—	4	2	no	—
26	C₂×C₄	C₂³	1	2	yes	—
27	C₂×C₄	C₂³	1	4	yes	—
28	C₂×C₄	C₂³	1	2	yes	—
29	C₂×C₄	C₂³	1	1	yes	characteristic
30	C₂×C₄	C₂³	1	4	yes	—
31	C₂²	—	2	2	no	—
32	C₂²	—	2	4	no	—
33	C₂²	C₂⁴	1	1	yes	characteristic
34	C₄	—	2	4	no	—
35	C₄	—	2	1	no	—
36	C₄	—	2	2	no	—
37	C₄	—	2	2	no	—
38	C₄	—	2	4	no	—
39	C₂	—	4	2	no	—
40	C₂	H(5,𝔽₂)	1	1	yes	characteristic
41	C₂	2₋¹⁺⁴	1	2	yes	—

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