Subgroups — #451

Order 64.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₁₆	C₂	1	2	yes	maximal, complemented
3	C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
4	#103	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₈	—	2	4	no	—
7	C₂×Q₈	—	2	1	no	—
8	C₂×Q₈	C₂²	1	1	yes	characteristic
9	C₂×Q₈	C₂²	1	2	yes	—
10	C₂²×C₄	—	2	1	no	—
11	C₂²×C₄	C₂²	1	1	yes	characteristic
12	Q₁₆	—	2	4	no	—
13	C₂×C₈	C₂²	1	2	yes	—
14	C₄⋊C₄	—	2	1	no	—
15	C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂²⋊C₄	—	2	1	no	—
17	C₂³	D₄	1	1	yes	characteristic
18	Q₈	—	2	4	no	—
19	Q₈	—	2	2	no	—
20	Q₈	—	2	2	no	—
21	Q₈	D₄	1	4	yes	—
22	C₂×C₄	—	2	4	no	—
23	C₂×C₄	—	2	2	no	—
24	C₂×C₄	—	2	2	no	—
25	C₂×C₄	—	2	1	no	—
26	C₂×C₄	—	2	1	no	—
27	C₂×C₄	D₄	1	1	yes	characteristic
28	C₂×C₄	C₂³	1	1	yes	characteristic
29	C₈	—	2	2	no	—
30	C₂²	—	2	1	no	—
31	C₂²	—	2	1	no	—
32	C₂²	Q₁₆	1	2	yes	complemented
33	C₂²	C₂×D₄	1	1	yes	characteristic
34	C₄	—	2	4	no	—
35	C₄	—	2	1	no	—
36	C₄	—	4	1	no	—
37	C₄	—	4	1	no	—
38	C₄	C₂×D₄	1	2	yes	—
39	C₂	—	2	2	no	—
40	C₂	#120	1	1	yes	characteristic
41	C₂	C₂×Q₁₆	1	1	yes	characteristic
42	C₂	#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³
prime radical	C₂³	D₄
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.