Subgroups — C₂×Q₂₅₆ #153704

Order 512.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₁₂₈	C₂	1	2	yes	maximal
2	Q₂₅₆	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₁₂₈	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₆₄	—	2	2	no	—
5	Q₁₂₈	—	2	2	no	—
6	Q₁₂₈	C₂²	1	4	yes	—
7	C₂×C₆₄	C₂²	1	1	yes	characteristic
8	C₁₂₈	C₂²	1	2	yes	—
9	C₂×Q₃₂	—	4	2	no	—
10	Q₆₄	—	2	4	no	—
11	Q₆₄	—	4	2	no	—
12	C₂×C₃₂	D₄	1	1	yes	characteristic
13	C₆₄	D₄	1	1	yes	characteristic
14	C₆₄	C₂³	1	1	yes	characteristic
15	C₂×Q₁₆	—	8	2	no	—
16	Q₃₂	—	4	4	no	—
17	Q₃₂	—	8	2	no	—
18	C₂×C₁₆	D₈	1	1	yes	characteristic
19	C₃₂	D₈	1	1	yes	characteristic
20	C₃₂	C₂×D₄	1	1	yes	characteristic
21	C₂×Q₈	—	16	2	no	—
22	Q₁₆	—	8	4	no	—
23	Q₁₆	—	16	2	no	—
24	C₂×C₈	D₁₆	1	1	yes	characteristic
25	C₁₆	D₁₆	1	1	yes	characteristic
26	C₁₆	C₂×D₈	1	1	yes	characteristic
27	Q₈	—	16	4	no	—
28	Q₈	—	32	2	no	—
29	C₂×C₄	—	32	2	no	—
30	C₂×C₄	D₃₂	1	1	yes	characteristic
31	C₈	D₃₂	1	1	yes	characteristic
32	C₈	C₂×D₁₆	1	1	yes	characteristic
33	C₂²	D₆₄	1	1	yes	characteristic
34	C₄	—	32	4	no	—
35	C₄	D₆₄	1	1	yes	characteristic
36	C₄	C₂×D₃₂	1	1	yes	characteristic
37	C₂	Q₂₅₆	1	2	yes	direct product factor
38	C₂	C₂×D₆₄	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	D₆₄
commutator, Frattini, cosocle	C₆₄	C₂³
LCS	G ▷ C₆₄ ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ C₂×C₆₄ ◁ G
LECS	G ▷ C₆₄ ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. Q₂₅₆	C₂	direct (N × H)
37. C₂	Q₂₅₆	direct (N × H)

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