Subgroups — #12511

Order 256.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2241	C₂	1	2	yes	maximal, complemented
2	#2240	C₂	1	2	yes	maximal
3	#1710	C₂	1	1	yes	maximal, characteristic, complemented
4	#1709	C₂	1	1	yes	maximal, characteristic
5	C₈×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#500	C₂²	1	2	yes	—
7	#494	C₂²	1	1	yes	characteristic
8	#493	C₂²	1	1	yes	characteristic
9	#487	—	2	2	no	—
10	#486	—	2	2	no	—
11	#358	—	2	4	no	—
12	#357	—	2	4	no	—
13	C₄×C₁₆	C₂²	1	2	yes	—
14	C₈²	C₂²	1	1	yes	characteristic
15	C₂×Q₁₆	—	2	2	no	—
16	C₂×D₈	—	2	2	no	—
17	C₄⋊Q₈	—	2	1	no	—
18	C₄⋊Q₈	—	2	2	no	—
19	#127	—	2	1	no	—
20	C₂×C₁₆	D₄	1	4	yes	—
21	#107	—	2	2	no	—
22	#107	—	2	2	no	—
23	#103	—	4	4	no	—
24	#102	—	4	4	no	—
25	C₄×C₈	D₄	1	2	yes	—
26	C₄×C₈	C₂³	1	1	yes	characteristic
27	C₂×Q₈	—	4	1	no	—
28	C₂×Q₈	—	4	2	no	—
29	C₂×D₄	—	4	1	no	—
30	C₂×D₄	—	4	2	no	—
31	Q₁₆	—	4	4	no	—
32	D₈	—	4	4	no	—
33	C₂×C₈	D₈	1	4	yes	—
34	C₂×C₈	C₂×D₄	1	2	yes	—
35	C₄⋊C₄	—	4	2	no	—
36	C₄⋊C₄	—	4	2	no	—
37	C₄⋊C₄	—	4	2	no	—
38	C₄²	C₂×D₄	1	1	yes	characteristic
39	C₁₆	—	2	4	no	—
40	C₂³	—	8	1	no	—
41	Q₈	—	8	2	no	—
42	Q₈	—	8	4	no	—
43	D₄	—	8	2	no	—
44	D₄	—	8	4	no	—
45	C₂×C₄	—	8	1	no	—
46	C₂×C₄	—	8	2	no	—
47	C₂×C₄	#127	1	1	yes	characteristic
48	C₂×C₄	C₂×D₈	1	2	yes	—
49	C₈	—	2	4	no	—
50	C₈	#135	1	4	yes	—
51	C₂²	—	16	1	no	—
52	C₂²	—	16	2	no	—
53	C₂²	#493	1	1	yes	characteristic
54	C₄	—	16	1	no	—
55	C₄	—	16	2	no	—
56	C₄	#495	1	2	yes	—
57	C₄	#508	1	4	yes	—
58	C₂	—	32	1	no	—
59	C₂	#1709	1	1	yes	characteristic
60	C₂	#2249	1	2	yes	—

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