Subgroups — #447

Order 64.

99 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, complemented
2	C₂×D₈	C₂	1	2	yes	maximal, complemented
3	#121	C₂	1	1	yes	maximal, characteristic, complemented
4	#102	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁴	—	2	1	no	—
7	C₂×D₄	—	2	4	no	—
8	C₂×D₄	—	2	1	no	—
9	C₂×D₄	—	2	1	no	—
10	C₂×D₄	C₂²	1	2	yes	—
11	C₂×D₄	C₂²	1	1	yes	characteristic, complemented
12	C₂²×C₄	C₂²	1	1	yes	characteristic
13	D₈	—	2	4	no	—
14	C₂×C₈	C₂²	1	2	yes	complemented
15	C₄⋊C₄	C₂²	1	1	yes	characteristic, complemented
16	C₂²⋊C₄	—	2	1	no	—
17	C₂³	—	2	4	no	—
18	C₂³	—	2	2	no	—
19	C₂³	—	2	1	no	—
20	C₂³	—	4	2	no	—
21	C₂³	—	4	2	no	—
22	C₂³	D₄	1	1	yes	characteristic
23	D₄	—	2	4	no	—
24	D₄	—	2	2	no	—
25	D₄	—	2	2	no	—
26	D₄	—	4	2	no	—
27	D₄	D₄	1	4	yes	complemented
28	C₂×C₄	—	2	2	no	—
29	C₂×C₄	—	2	1	no	—
30	C₂×C₄	D₄	1	1	yes	characteristic
31	C₂×C₄	C₂³	1	1	yes	characteristic
32	C₈	—	2	2	no	—
33	C₂²	—	2	1	no	—
34	C₂²	—	2	1	no	—
35	C₂²	—	2	4	no	—
36	C₂²	—	4	8	no	—
37	C₂²	—	4	2	no	—
38	C₂²	—	4	2	no	—
39	C₂²	—	4	1	no	—
40	C₂²	—	4	1	no	—
41	C₂²	—	4	1	no	—
42	C₂²	D₈	1	2	yes	complemented
43	C₂²	C₂×D₄	1	1	yes	characteristic
44	C₄	—	2	1	no	—
45	C₄	—	4	1	no	—
46	C₄	C₂×D₄	1	2	yes	—
47	C₂	—	2	2	no	—
48	C₂	—	4	4	no	—
49	C₂	—	8	1	no	—
50	C₂	#120	1	1	yes	characteristic
51	C₂	C₂×D₈	1	1	yes	characteristic
52	C₂	AGL(1,ℤ/8)	1	1	yes	characteristic

	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.