Subgroups — #453

Order 64.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	AGL(1,ℤ/8)	C₂	1	2	yes	maximal, complemented
3	#127	C₂	1	1	yes	maximal, characteristic, complemented
4	#104	C₂	1	2	yes	maximal, complemented
5	#100	C₂	1	1	yes	maximal, characteristic, complemented
6	#41	—	2	2	no	—
7	#41	C₂²	1	2	yes	—
8	C₂×D₄	—	2	2	no	—
9	C₂×D₄	—	2	2	no	—
10	C₂×D₄	—	2	2	no	—
11	C₂×D₄	C₂²	1	1	yes	characteristic
12	C₂×D₄	C₂²	1	1	yes	characteristic, complemented
13	SD₁₆	—	2	2	no	—
14	D₈	—	2	2	no	—
15	M₁₆	C₂²	1	2	yes	complemented
16	C₄²	C₂²	1	1	yes	characteristic, complemented
17	C₂³	—	2	2	no	—
18	C₂³	—	2	1	no	—
19	C₂³	D₄	1	2	yes	—
20	Q₈	D₄	1	2	yes	complemented
21	D₄	—	2	4	no	—
22	D₄	—	2	2	no	—
23	D₄	—	2	2	no	—
24	D₄	—	2	2	no	—
25	D₄	—	4	4	no	—
26	D₄	D₄	1	2	yes	complemented
27	C₂×C₄	—	2	2	no	—
28	C₂×C₄	—	2	1	no	—
29	C₂×C₄	—	2	2	no	—
30	C₂×C₄	C₂³	1	1	yes	characteristic
31	C₈	—	2	2	no	—
32	C₂²	—	2	2	no	—
33	C₂²	—	2	2	no	—
34	C₂²	—	4	2	no	—
35	C₂²	—	4	2	no	—
36	C₂²	—	4	1	no	—
37	C₂²	—	8	2	no	—
38	C₂²	C₂×D₄	1	1	yes	characteristic
39	C₄	—	2	2	no	—
40	C₄	—	2	2	no	—
41	C₄	C₂×D₄	1	2	yes	—
42	C₂	—	2	1	no	—
43	C₂	—	4	2	no	—
44	C₂	—	4	2	no	—
45	C₂	—	8	1	no	—
46	C₂	#120	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#120
commutator, Frattini, cosocle	C₂×C₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×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.