Subgroups — #13732

Order 256.

73 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2253	C₂	1	2	yes	maximal
2	#1424	C₂	1	2	yes	maximal
3	#1423	C₂	1	2	yes	maximal
4	C₄×C₃₂	C₂	1	1	yes	maximal, characteristic
5	#499	—	2	2	no	—
6	C₂×C₃₂	C₂²	1	1	yes	characteristic
7	C₂×C₃₂	C₂²	1	1	yes	characteristic
8	#367	—	2	2	no	—
9	#366	C₂²	1	4	yes	—
10	C₄×C₁₆	C₂²	1	1	yes	characteristic
11	#125	—	4	2	no	—
12	C₂×C₁₆	D₄	1	1	yes	characteristic
13	C₂×C₁₆	C₂³	1	1	yes	characteristic
14	#107	—	2	4	no	—
15	#106	—	4	2	no	—
16	C₄×C₈	D₄	1	1	yes	characteristic
17	C₃₂	Q₈	1	2	yes	—
18	C₃₂	Q₈	1	2	yes	—
19	C₂×C₈	D₈	1	1	yes	characteristic
20	C₂×C₈	C₂×D₄	1	1	yes	characteristic
21	C₄⋊C₄	—	4	4	no	—
22	C₄⋊C₄	—	8	4	no	—
23	C₄²	D₈	1	1	yes	characteristic
24	C₁₆	Q₁₆	1	2	yes	—
25	C₁₆	C₂×Q₈	1	1	yes	characteristic
26	C₁₆	C₂×Q₈	1	1	yes	characteristic
27	C₂×C₄	—	8	4	no	—
28	C₂×C₄	D₁₆	1	2	yes	—
29	C₂×C₄	C₂×D₈	1	1	yes	characteristic
30	C₈	Q₃₂	1	2	yes	—
31	C₈	C₄⋊Q₈	1	1	yes	characteristic
32	C₈	C₂×Q₁₆	1	1	yes	characteristic
33	C₂²	C₂×D₁₆	1	1	yes	characteristic
34	C₄	—	2	2	no	—
35	C₄	—	16	4	no	—
36	C₄	#500	1	1	yes	characteristic
37	C₄	C₂×Q₃₂	1	1	yes	characteristic
38	C₂	#2252	1	1	yes	characteristic
39	C₂	#2262	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	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

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