Subgroups — #577

Order 64.

129 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	2₋¹⁺⁴	C₂	1	1	yes	maximal, characteristic, complemented
2	H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
3	#137	C₂	1	3	yes	maximal, complemented
4	AGL(1,ℤ/8)	C₂	1	3	yes	maximal, complemented
5	#135	C₂	1	3	yes	maximal, complemented
6	C₂×SD₁₆	C₂	1	3	yes	maximal, complemented
7	#131	C₂	1	1	yes	maximal, characteristic, complemented
8	#41	—	2	1	no	—
9	#41	—	2	3	no	—
10	#41	C₂²	1	3	yes	—
11	#41	C₂²	1	1	yes	characteristic
12	#41	C₂²	1	3	yes	complemented
13	C₂×Q₈	—	2	1	no	—
14	C₂×Q₈	C₂²	1	3	yes	complemented
15	C₂×D₄	—	2	3	no	—
16	C₂×D₄	C₂²	1	3	yes	—
17	Q₁₆	C₂²	1	3	yes	complemented
18	SD₁₆	C₂²	1	3	yes	complemented
19	SD₁₆	C₂²	1	6	yes	complemented
20	SD₁₆	C₂²	1	1	yes	characteristic, complemented
21	D₈	C₂²	1	3	yes	complemented
22	M₁₆	C₂²	1	3	yes	complemented
23	C₂×C₈	C₂²	1	3	yes	complemented
24	C₂³	—	2	3	no	—
25	Q₈	—	2	3	no	—
26	Q₈	D₄	1	1	yes	characteristic
27	Q₈	C₂³	1	3	yes	—
28	Q₈	C₂³	1	1	yes	characteristic
29	D₄	—	2	3	no	—
30	D₄	—	2	3	no	—
31	D₄	—	2	3	no	—
32	D₄	D₄	1	3	yes	—
33	D₄	C₂³	1	1	yes	characteristic
34	D₄	C₂³	1	3	yes	—
35	C₂×C₄	—	2	3	no	—
36	C₂×C₄	—	2	3	no	—
37	C₂×C₄	—	2	3	no	—
38	C₂×C₄	C₂³	1	3	yes	—
39	C₈	C₂³	1	1	yes	characteristic
40	C₈	C₂³	1	3	yes	—
41	C₂²	—	2	1	no	—
42	C₂²	—	2	3	no	—
43	C₂²	—	8	3	no	—
44	C₂²	C₂×D₄	1	3	yes	—
45	C₄	—	2	1	no	—
46	C₄	—	2	3	no	—
47	C₄	C₂×D₄	1	3	yes	—
48	C₄	C₂⁴	1	1	yes	characteristic
49	C₂	—	2	3	no	—
50	C₂	—	4	1	no	—
51	C₂	—	4	3	no	—
52	C₂	C₂²×D₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	C₂²×D₄
commutator, Frattini, cosocle	C₄	C₂⁴
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ #41 ◁ 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.