Subgroups — #12528

Order 256.

137 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2247	C₂	1	2	yes	maximal
2	#2241	C₂	1	2	yes	maximal
3	#1713	C₂	1	1	yes	maximal, characteristic
4	#1707	C₂	1	1	yes	maximal, characteristic
5	#1312	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₃₂	—	2	4	no	—
7	#500	—	2	1	no	—
8	#500	—	2	1	no	—
9	#500	C₂²	1	1	yes	characteristic
10	#498	—	2	1	no	—
11	#494	C₂²	1	1	yes	characteristic
12	#494	C₂²	1	2	yes	—
13	#487	—	2	1	no	—
14	#358	—	2	4	no	—
15	C₄×C₁₆	C₂²	1	2	yes	—
16	C₈²	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆	—	2	2	no	—
18	C₂×Q₁₆	—	2	2	no	—
19	C₂×Q₁₆	—	2	2	no	—
20	C₄⋊Q₈	—	2	1	no	—
21	C₄⋊Q₈	—	2	1	no	—
22	C₄⋊Q₈	—	2	2	no	—
23	Q₃₂	—	4	8	no	—
24	C₂×C₁₆	D₄	1	4	yes	—
25	#107	—	2	2	no	—
26	#107	—	4	2	no	—
27	#107	—	4	2	no	—
28	#106	—	4	2	no	—
29	#103	—	4	2	no	—
30	C₄×C₈	D₄	1	1	yes	characteristic
31	C₄×C₈	D₄	1	1	yes	characteristic
32	C₄×C₈	C₂³	1	1	yes	characteristic
33	C₂×Q₈	—	4	1	no	—
34	C₂×Q₈	—	4	2	no	—
35	C₂×Q₈	—	4	1	no	—
36	Q₁₆	—	4	4	no	—
37	Q₁₆	—	4	4	no	—
38	Q₁₆	—	4	4	no	—
39	C₂×C₈	D₈	1	2	yes	—
40	C₂×C₈	D₈	1	2	yes	—
41	C₂×C₈	C₂×D₄	1	2	yes	—
42	C₄⋊C₄	—	4	1	no	—
43	C₄⋊C₄	—	4	1	no	—
44	C₄⋊C₄	—	4	1	no	—
45	C₄⋊C₄	—	4	1	no	—
46	C₄⋊C₄	—	4	2	no	—
47	C₄⋊C₄	—	4	2	no	—
48	C₄²	C₂×D₄	1	1	yes	characteristic
49	C₁₆	—	2	4	no	—
50	Q₈	—	8	2	no	—
51	Q₈	—	8	2	no	—
52	Q₈	—	8	2	no	—
53	Q₈	—	8	2	no	—
54	C₂×C₄	—	8	1	no	—
55	C₂×C₄	—	8	2	no	—
56	C₂×C₄	—	8	1	no	—
57	C₂×C₄	#127	1	1	yes	characteristic
58	C₂×C₄	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄	C₂×D₈	1	1	yes	characteristic
60	C₈	—	2	2	no	—
61	C₈	Q₃₂	1	4	yes	complemented
62	C₈	AGL(1,ℤ/8)	1	4	yes	—
63	C₂²	#493	1	1	yes	characteristic
64	C₄	—	16	1	no	—
65	C₄	—	16	2	no	—
66	C₄	—	16	1	no	—
67	C₄	#496	1	2	yes	—
68	C₄	C₂×Q₃₂	1	2	yes	—
69	C₄	#510	1	2	yes	—
70	C₂	#1721	1	1	yes	characteristic
71	C₂	#2247	1	1	yes	characteristic
72	C₂	#2251	1	1	yes	characteristic

	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

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
61. C₈	Q₃₂	semidirect

Maximal subgroups

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

subgroup	index
#2247	2
#2241	2
#1713	2
#1707	2
#1312	2