Subgroups — #13734

Order 256.

75 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2253	C₂	1	1	yes	maximal, characteristic
2	#2252	C₂	1	1	yes	maximal, characteristic
3	#1424	C₂	1	2	yes	maximal
4	#1423	C₂	1	2	yes	maximal
5	#1397	C₂	1	1	yes	maximal, characteristic
6	#500	—	2	1	no	—
7	#499	—	2	1	no	—
8	C₂×C₃₂	C₂²	1	2	yes	—
9	#367	—	2	1	no	—
10	#366	—	2	1	no	—
11	#366	C₂²	1	1	yes	characteristic
12	#366	C₂²	1	2	yes	—
13	#366	C₂²	1	1	yes	characteristic
14	C₄×C₁₆	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈	—	4	1	no	—
16	#125	—	4	1	no	—
17	C₂×C₁₆	D₄	1	1	yes	characteristic
18	C₂×C₁₆	C₂³	1	1	yes	characteristic
19	#107	—	2	2	no	—
20	#107	—	2	1	no	—
21	#107	—	2	1	no	—
22	#107	—	4	1	no	—
23	#106	—	4	1	no	—
24	C₄×C₈	D₄	1	1	yes	characteristic
25	C₃₂	Q₈	1	4	yes	complemented
26	C₂×Q₈	—	8	1	no	—
27	C₂×C₈	D₈	1	1	yes	characteristic
28	C₂×C₈	C₂×D₄	1	1	yes	characteristic
29	C₄⋊C₄	—	4	1	no	—
30	C₄⋊C₄	—	4	1	no	—
31	C₄⋊C₄	—	4	2	no	—
32	C₄⋊C₄	—	8	1	no	—
33	C₄⋊C₄	—	8	1	no	—
34	C₄⋊C₄	—	8	1	no	—
35	C₄²	D₈	1	1	yes	characteristic
36	C₁₆	Q₁₆	1	2	yes	—
37	C₁₆	C₂×Q₈	1	2	yes	—
38	Q₈	—	16	2	no	—
39	C₂×C₄	—	8	1	no	—
40	C₂×C₄	—	8	1	no	—
41	C₂×C₄	—	8	2	no	—
42	C₂×C₄	D₁₆	1	1	yes	characteristic
43	C₂×C₄	D₁₆	1	1	yes	characteristic
44	C₂×C₄	C₂×D₈	1	1	yes	characteristic
45	C₈	Q₃₂	1	2	yes	—
46	C₈	C₄⋊Q₈	1	1	yes	characteristic
47	C₈	C₂×Q₁₆	1	1	yes	characteristic
48	C₂²	C₂×D₁₆	1	1	yes	characteristic
49	C₄	—	2	1	no	—
50	C₄	—	2	1	no	—
51	C₄	—	16	1	no	—
52	C₄	—	16	2	no	—
53	C₄	—	16	1	no	—
54	C₄	#500	1	1	yes	characteristic
55	C₄	C₂×Q₃₂	1	1	yes	characteristic
56	C₂	#2252	1	1	yes	characteristic
57	C₂	#2263	1	1	yes	characteristic
58	C₂	#2264	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
25. C₃₂	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#2253	2
#2252	2
#1424	2
#1423	2
#1397	2