Subgroups — #13725

Order 256.

135 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₃₂	C₂	1	4	yes	maximal, complemented
2	#2246	C₂	1	2	yes	maximal, complemented
3	C₄×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₁₆	—	2	2	no	—
5	C₂×D₁₆	C₂²	1	4	yes	—
6	#493	—	2	2	no	—
7	D₃₂	—	2	8	no	—
8	C₂×C₃₂	C₂²	1	2	yes	—
9	C₄×C₁₆	C₂²	1	1	yes	characteristic
10	C₂×D₈	—	2	4	no	—
11	C₂×D₈	—	4	2	no	—
12	#127	—	4	2	no	—
13	D₁₆	—	2	8	no	—
14	D₁₆	—	4	4	no	—
15	C₂×C₁₆	D₄	1	1	yes	characteristic
16	C₂×C₁₆	C₂³	1	1	yes	characteristic
17	C₄×C₈	D₄	1	1	yes	characteristic
18	C₃₂	D₄	1	4	yes	complemented
19	C₂×D₄	—	4	4	no	—
20	C₂×D₄	—	8	4	no	—
21	D₈	—	4	4	no	—
22	D₈	—	4	4	no	—
23	D₈	—	8	4	no	—
24	C₂×C₈	D₈	1	1	yes	characteristic
25	C₂×C₈	C₂×D₄	1	1	yes	characteristic
26	C₄²	D₈	1	1	yes	characteristic
27	C₁₆	D₈	1	2	yes	—
28	C₁₆	C₂×D₄	1	2	yes	—
29	C₂³	—	8	4	no	—
30	D₄	—	8	4	no	—
31	D₄	—	8	4	no	—
32	D₄	—	16	8	no	—
33	C₂×C₄	D₁₆	1	2	yes	—
34	C₂×C₄	C₂×D₈	1	1	yes	characteristic
35	C₈	D₁₆	1	2	yes	—
36	C₈	#127	1	1	yes	characteristic
37	C₈	C₂×D₈	1	1	yes	characteristic
38	C₂²	—	16	4	no	—
39	C₂²	—	16	8	no	—
40	C₂²	C₂×D₁₆	1	1	yes	characteristic
41	C₄	D₃₂	1	4	yes	complemented
42	C₄	#493	1	1	yes	characteristic
43	C₄	C₂×D₁₆	1	1	yes	characteristic
44	C₂	—	32	4	no	—
45	C₂	#2246	1	1	yes	characteristic
46	C₂	C₂×D₃₂	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ 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.