Subgroups — #496

Order 64.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×SD₁₆	C₂	1	2	yes	maximal, complemented
2	C₂×D₈	C₂	1	2	yes	maximal, complemented
3	#127	C₂	1	1	yes	maximal, characteristic, complemented
4	#124	C₂	1	1	yes	maximal, characteristic, complemented
5	#97	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈	C₂²	1	1	yes	characteristic
7	C₂×D₄	—	2	2	no	—
8	C₂×D₄	C₂²	1	1	yes	characteristic
9	C₂×D₄	C₂²	1	2	yes	—
10	SD₁₆	—	2	4	no	—
11	D₈	—	2	4	no	—
12	C₂×C₈	C₂²	1	2	yes	—
13	C₂²⋊C₄	—	2	2	no	—
14	C₄²	C₂²	1	1	yes	characteristic
15	C₂³	—	2	1	no	—
16	C₂³	—	2	2	no	—
17	Q₈	—	2	2	no	—
18	D₄	—	2	2	no	—
19	D₄	—	2	2	no	—
20	D₄	—	2	2	no	—
21	D₄	—	4	4	no	—
22	C₂×C₄	—	2	1	no	—
23	C₂×C₄	D₄	1	2	yes	—
24	C₂×C₄	C₂³	1	1	yes	characteristic
25	C₈	D₄	1	4	yes	complemented
26	C₂²	—	4	1	no	—
27	C₂²	—	4	2	no	—
28	C₂²	—	4	4	no	—
29	C₂²	—	4	2	no	—
30	C₂²	C₂×D₄	1	1	yes	characteristic
31	C₄	—	2	2	no	—
32	C₄	—	4	1	no	—
33	C₄	C₂×D₄	1	2	yes	—
34	C₂	—	8	2	no	—
35	C₂	—	8	1	no	—
36	C₂	#127	1	1	yes	characteristic
37	C₂	AGL(1,ℤ/8)	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ 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.