Subgroups — #456

Order 64.

71 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
2	#137	C₂	1	2	yes	maximal, complemented
3	C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#104	C₂	1	2	yes	maximal, complemented
5	#101	C₂	1	1	yes	maximal, characteristic, complemented
6	#41	—	2	2	no	—
7	#41	—	2	2	no	—
8	#41	C₂²	1	2	yes	—
9	C₂×Q₈	—	2	2	no	—
10	C₂×Q₈	C₂²	1	1	yes	characteristic
11	C₂×Q₈	C₂²	1	1	yes	characteristic
12	Q₁₆	—	2	2	no	—
13	SD₁₆	—	2	2	no	—
14	M₁₆	C₂²	1	2	yes	—
15	C₄⋊C₄	—	2	2	no	—
16	C₄²	C₂²	1	1	yes	characteristic
17	Q₈	—	2	2	no	—
18	Q₈	—	2	2	no	—
19	Q₈	—	2	2	no	—
20	Q₈	D₄	1	2	yes	—
21	D₄	—	2	2	no	—
22	D₄	—	2	2	no	—
23	D₄	D₄	1	2	yes	—
24	C₂×C₄	—	2	4	no	—
25	C₂×C₄	—	2	1	no	—
26	C₂×C₄	—	2	2	no	—
27	C₂×C₄	—	2	1	no	—
28	C₂×C₄	D₄	1	2	yes	—
29	C₂×C₄	C₂³	1	1	yes	characteristic
30	C₈	—	2	2	no	—
31	C₂²	—	2	2	no	—
32	C₂²	C₂×D₄	1	1	yes	characteristic
33	C₄	—	2	2	no	—
34	C₄	—	2	2	no	—
35	C₄	—	2	2	no	—
36	C₄	—	4	1	no	—
37	C₄	C₂×D₄	1	2	yes	—
38	C₂	—	2	1	no	—
39	C₂	—	4	2	no	—
40	C₂	#120	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#120
commutator, Frattini, cosocle	C₂×C₄	C₂³
prime radical	2₋¹⁺⁴	C₂
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.