Subgroups — #12522

Order 256.

119 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2254	C₂	1	2	yes	maximal
2	#2252	C₂	1	2	yes	maximal
3	#1710	C₂	1	2	yes	maximal
4	C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
5	#500	C₂²	1	4	yes	—
6	#487	—	2	4	no	—
7	#367	—	2	4	no	—
8	#366	—	2	4	no	—
9	C₄×C₁₆	C₂²	1	1	yes	characteristic
10	C₄×C₁₆	C₂²	1	1	yes	characteristic
11	C₈²	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈	—	2	4	no	—
13	C₂×C₁₆	D₄	1	2	yes	—
14	C₂×C₁₆	D₄	1	2	yes	—
15	#107	—	2	4	no	—
16	#107	—	2	4	no	—
17	#103	—	4	8	no	—
18	C₄×C₈	D₄	1	2	yes	—
19	C₄×C₈	C₂³	1	1	yes	characteristic
20	C₂×Q₈	—	4	4	no	—
21	C₂×C₈	D₈	1	4	yes	—
22	C₂×C₈	C₂×D₄	1	1	yes	characteristic
23	C₂×C₈	C₂×D₄	1	1	yes	characteristic
24	C₄⋊C₄	—	4	4	no	—
25	C₄⋊C₄	—	4	4	no	—
26	C₄²	C₂×D₄	1	1	yes	characteristic
27	C₁₆	Q₁₆	1	4	yes	—
28	C₁₆	Q₁₆	1	4	yes	—
29	Q₈	—	8	8	no	—
30	C₂×C₄	—	8	4	no	—
31	C₂×C₄	#127	1	1	yes	characteristic
32	C₂×C₄	C₂×D₈	1	2	yes	—
33	C₈	—	2	4	no	—
34	C₈	C₂×Q₁₆	1	2	yes	—
35	C₈	C₂×Q₁₆	1	2	yes	—
36	C₂²	#493	1	1	yes	characteristic
37	C₄	—	16	4	no	—
38	C₄	#494	1	1	yes	characteristic
39	C₄	#494	1	1	yes	characteristic
40	C₄	#508	1	4	yes	—
41	C₂	#1713	1	1	yes	characteristic
42	C₂	#2249	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂²	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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