Subgroups — #462

Order 64.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₁₆	C₂	1	2	yes	maximal
2	C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₄×Q₈	C₂	1	1	yes	maximal, characteristic
4	C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#103	C₂	1	2	yes	maximal
6	C₂×Q₈	C₂²	1	2	yes	—
7	C₂×Q₈	C₂²	1	1	yes	characteristic
8	Q₁₆	—	2	4	no	—
9	C₂×C₈	C₂²	1	2	yes	—
10	C₄⋊C₄	—	2	1	no	—
11	C₄⋊C₄	—	2	1	no	—
12	C₄⋊C₄	—	2	1	no	—
13	C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₄²	—	2	1	no	—
15	C₄²	C₂²	1	1	yes	characteristic
16	Q₈	—	2	1	no	—
17	Q₈	—	2	2	no	—
18	Q₈	—	2	2	no	—
19	Q₈	D₄	1	2	yes	—
20	C₂×C₄	—	2	1	no	—
21	C₂×C₄	—	2	1	no	—
22	C₂×C₄	—	2	2	no	—
23	C₂×C₄	D₄	1	1	yes	characteristic
24	C₂×C₄	D₄	1	1	yes	characteristic
25	C₂×C₄	C₂³	1	1	yes	characteristic
26	C₈	—	2	2	no	—
27	C₂²	C₂×D₄	1	1	yes	characteristic
28	C₄	—	2	2	no	—
29	C₄	—	2	1	no	—
30	C₄	—	4	2	no	—
31	C₄	—	4	1	no	—
32	C₄	Q₁₆	1	2	yes	complemented
33	C₄	C₂×D₄	1	1	yes	characteristic
34	C₄	#41	1	1	yes	characteristic
35	C₂	#121	1	1	yes	characteristic
36	C₂	C₂×Q₁₆	1	1	yes	characteristic
37	C₂	#137	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	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.

N	H ≅ G/N	type
32. C₄	Q₁₆	semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.