Subgroups — #12504

Order 256.

131 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2252	C₂	1	1	yes	maximal, characteristic
2	#2247	C₂	1	1	yes	maximal, characteristic
3	#2241	C₂	1	2	yes	maximal
4	#1707	C₂	1	2	yes	maximal
5	C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₃₂	—	2	2	no	—
7	#500	C₂²	1	2	yes	—
8	#498	—	2	2	no	—
9	#494	C₂²	1	2	yes	—
10	#487	—	2	2	no	—
11	#366	—	2	2	no	—
12	#358	—	2	4	no	—
13	C₄×C₁₆	C₂²	1	1	yes	characteristic
14	C₄×C₁₆	C₂²	1	1	yes	characteristic
15	C₈²	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆	—	2	2	no	—
17	C₂×Q₁₆	—	2	2	no	—
18	C₄⋊Q₈	—	2	2	no	—
19	C₄⋊Q₈	—	2	2	no	—
20	Q₃₂	—	4	4	no	—
21	C₂×C₁₆	D₄	1	2	yes	—
22	C₂×C₁₆	D₄	1	2	yes	—
23	#107	—	2	2	no	—
24	#107	—	2	2	no	—
25	#106	—	4	4	no	—
26	#103	—	4	4	no	—
27	C₄×C₈	D₄	1	1	yes	characteristic
28	C₄×C₈	D₄	1	1	yes	characteristic
29	C₄×C₈	C₂³	1	1	yes	characteristic
30	C₂×Q₈	—	4	2	no	—
31	C₂×Q₈	—	4	2	no	—
32	Q₁₆	—	4	4	no	—
33	Q₁₆	—	4	4	no	—
34	C₂×C₈	D₈	1	2	yes	—
35	C₂×C₈	D₈	1	2	yes	—
36	C₂×C₈	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈	C₂×D₄	1	1	yes	characteristic
38	C₄⋊C₄	—	4	2	no	—
39	C₄⋊C₄	—	4	2	no	—
40	C₄⋊C₄	—	4	2	no	—
41	C₄⋊C₄	—	4	2	no	—
42	C₄²	C₂×D₄	1	1	yes	characteristic
43	C₁₆	—	2	2	no	—
44	C₁₆	SD₁₆	1	4	yes	—
45	Q₈	—	8	2	no	—
46	Q₈	—	8	2	no	—
47	Q₈	—	8	4	no	—
48	C₂×C₄	—	8	2	no	—
49	C₂×C₄	—	8	2	no	—
50	C₂×C₄	#127	1	1	yes	characteristic
51	C₂×C₄	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄	C₂×D₈	1	1	yes	characteristic
53	C₈	—	2	2	no	—
54	C₈	Q₃₂	1	4	yes	complemented
55	C₈	C₂×SD₁₆	1	2	yes	—
56	C₈	#135	1	2	yes	—
57	C₂²	#493	1	1	yes	characteristic
58	C₄	—	16	2	no	—
59	C₄	—	16	2	no	—
60	C₄	#492	1	1	yes	characteristic
61	C₄	#495	1	1	yes	characteristic
62	C₄	C₂×Q₃₂	1	2	yes	—
63	C₄	#508	1	2	yes	—
64	C₂	#1706	1	1	yes	characteristic
65	C₂	#2247	1	1	yes	characteristic
66	C₂	#2249	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
54. C₈	Q₃₂	semidirect

Maximal subgroups

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

subgroup	index
#2252	2
#2247	2
#2241	2
#1707	2
C₈×C₁₆	2