Subgroups — #12513

Order 256.

145 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2240	C₂	1	4	yes	maximal, complemented
2	#1712	C₂	1	1	yes	maximal, characteristic
3	#1710	C₂	1	1	yes	maximal, characteristic, complemented
4	#1311	C₂	1	1	yes	maximal, characteristic, complemented
5	#500	C₂²	1	2	yes	—
6	#493	—	2	2	no	—
7	#493	C₂²	1	2	yes	—
8	#487	—	2	2	no	—
9	#357	—	2	8	no	—
10	C₄×C₁₆	C₂²	1	2	yes	—
11	C₈²	C₂²	1	1	yes	characteristic
12	C₂×D₈	—	2	4	no	—
13	C₂×D₈	—	4	4	no	—
14	C₄⋊Q₈	—	2	2	no	—
15	#127	—	2	2	no	—
16	C₂×C₁₆	D₄	1	4	yes	—
17	#107	—	2	4	no	—
18	#103	—	4	4	no	—
19	C₄×C₈	D₄	1	2	yes	—
20	C₄×C₈	C₂³	1	1	yes	characteristic
21	C₂×Q₈	—	4	2	no	—
22	C₂×D₄	—	4	2	no	—
23	C₂×D₄	—	4	4	no	—
24	D₈	—	4	8	no	—
25	D₈	—	8	8	no	—
26	C₂×C₈	D₈	1	4	yes	—
27	C₂×C₈	C₂×D₄	1	2	yes	—
28	C₄⋊C₄	—	4	2	no	—
29	C₄⋊C₄	—	4	2	no	—
30	C₄²	C₂×D₄	1	1	yes	characteristic
31	C₁₆	—	2	4	no	—
32	C₂³	—	8	2	no	—
33	Q₈	—	8	4	no	—
34	D₄	—	8	4	no	—
35	D₄	—	8	8	no	—
36	C₂×C₄	—	8	2	no	—
37	C₂×C₄	#127	1	1	yes	characteristic
38	C₂×C₄	C₂×D₈	1	2	yes	—
39	C₈	—	2	4	no	—
40	C₈	#135	1	4	yes	—
41	C₂²	—	16	2	no	—
42	C₂²	—	16	4	no	—
43	C₂²	#493	1	1	yes	characteristic
44	C₄	—	16	2	no	—
45	C₄	#495	1	2	yes	—
46	C₄	#509	1	4	yes	—
47	C₂	—	32	2	no	—
48	C₂	#1709	1	1	yes	characteristic
49	C₂	#2250	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³
prime radical	#1712	C₂
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.