Subgroups — #469

Order 64.

65 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#121	C₂	1	2	yes	maximal, complemented
4	#106	C₂	1	1	yes	maximal, characteristic, complemented
5	#102	C₂	1	2	yes	maximal, complemented
6	C₂×D₄	—	2	2	no	—
7	C₂×D₄	C₂²	1	2	yes	—
8	C₂²×C₄	C₂²	1	1	yes	characteristic
9	D₈	—	2	2	no	—
10	M₁₆	—	2	2	no	—
11	C₂×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₈	C₂²	1	1	yes	characteristic
13	C₄⋊C₄	C₂²	1	2	yes	—
14	C₂²⋊C₄	—	2	2	no	—
15	C₂³	—	2	2	no	—
16	C₂³	D₄	1	1	yes	characteristic
17	D₄	—	2	2	no	—
18	D₄	—	2	2	no	—
19	D₄	—	4	4	no	—
20	C₂×C₄	—	2	1	no	—
21	C₂×C₄	—	2	1	no	—
22	C₂×C₄	—	2	2	no	—
23	C₂×C₄	D₄	1	1	yes	characteristic
24	C₂×C₄	C₂³	1	1	yes	characteristic
25	C₈	—	2	1	no	—
26	C₈	D₄	1	2	yes	complemented
27	C₂²	—	2	1	no	—
28	C₂²	—	2	2	no	—
29	C₂²	—	4	2	no	—
30	C₂²	—	4	2	no	—
31	C₂²	—	4	2	no	—
32	C₂²	C₂×D₄	1	1	yes	characteristic
33	C₄	—	2	1	no	—
34	C₄	—	4	2	no	—
35	C₄	C₂×D₄	1	1	yes	characteristic
36	C₄	#41	1	1	yes	characteristic
37	C₂	—	4	1	no	—
38	C₂	—	8	2	no	—
39	C₂	#121	1	1	yes	characteristic
40	C₂	AGL(1,ℤ/8)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ 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.