Subgroups — #575

Order 64.

131 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41	C₂	1	2	yes	maximal, complemented
2	#137	C₂	1	4	yes	maximal, complemented
3	AGL(1,ℤ/8)	C₂	1	4	yes	maximal, complemented
4	#135	C₂	1	4	yes	maximal, complemented
5	C₂×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#41	—	2	2	no	—
7	#41	—	2	2	no	—
8	#41	C₂²	1	4	yes	—
9	#41	C₂²	1	4	yes	complemented
10	C₂×Q₈	C₂²	1	2	yes	complemented
11	C₂×D₄	—	2	2	no	—
12	C₂×D₄	C₂²	1	2	yes	complemented
13	C₂²×C₄	—	2	2	no	—
14	C₂²×C₄	C₂²	1	1	yes	characteristic
15	Q₁₆	C₂²	1	4	yes	complemented
16	SD₁₆	C₂²	1	8	yes	complemented
17	D₈	C₂²	1	4	yes	complemented
18	M₁₆	C₂²	1	4	yes	complemented
19	C₂×C₈	C₂²	1	2	yes	complemented
20	C₂³	—	2	2	no	—
21	C₂³	D₄	1	1	yes	characteristic
22	Q₈	—	2	2	no	—
23	Q₈	C₂³	1	4	yes	—
24	D₄	—	2	4	no	—
25	D₄	—	2	4	no	—
26	D₄	—	2	2	no	—
27	D₄	C₂³	1	4	yes	—
28	C₂×C₄	—	2	2	no	—
29	C₂×C₄	—	2	4	no	—
30	C₂×C₄	—	2	4	no	—
31	C₂×C₄	D₄	1	1	yes	characteristic
32	C₂×C₄	D₄	1	2	yes	—
33	C₂×C₄	C₂³	1	2	yes	—
34	C₂×C₄	C₂³	1	1	yes	characteristic
35	C₈	C₂³	1	4	yes	—
36	C₂²	—	2	4	no	—
37	C₂²	—	4	1	no	—
38	C₂²	—	4	4	no	—
39	C₂²	C₂×D₄	1	2	yes	—
40	C₂²	C₂×D₄	1	1	yes	characteristic
41	C₄	—	2	4	no	—
42	C₄	C₂×D₄	1	2	yes	—
43	C₄	C₂×D₄	1	1	yes	characteristic
44	C₄	C₂⁴	1	1	yes	characteristic
45	C₂	—	2	2	no	—
46	C₂	—	2	1	no	—
47	C₂	—	4	4	no	—
48	C₂	C₂²×D₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂⁴
socle	C₂	C₂²×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ G
LECS	G ▷ 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.