Subgroups — #492

Order 64.

71 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×SD₁₆	C₂	1	4	yes	maximal, complemented
2	C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#127	C₂	1	1	yes	maximal, characteristic
4	C₄×C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Q₈	C₂²	1	2	yes	—
6	C₂×D₄	—	2	2	no	—
7	C₂×D₄	C₂²	1	2	yes	—
8	SD₁₆	—	2	8	no	—
9	C₂×C₈	C₂²	1	2	yes	—
10	C₄⋊C₄	—	2	2	no	—
11	C₄²	C₂²	1	1	yes	characteristic
12	C₂³	—	2	2	no	—
13	Q₈	—	2	4	no	—
14	D₄	—	2	4	no	—
15	D₄	—	4	4	no	—
16	C₂×C₄	—	2	2	no	—
17	C₂×C₄	D₄	1	2	yes	—
18	C₂×C₄	C₂³	1	1	yes	characteristic
19	C₈	D₄	1	4	yes	complemented
20	C₂²	—	4	2	no	—
21	C₂²	—	4	4	no	—
22	C₂²	C₂×D₄	1	1	yes	characteristic
23	C₄	—	4	2	no	—
24	C₄	SD₁₆	1	4	yes	complemented
25	C₄	C₂×D₄	1	2	yes	—
26	C₂	—	8	2	no	—
27	C₂	#127	1	1	yes	characteristic
28	C₂	C₂×SD₁₆	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³
prime radical	#127	C₂
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.