Subgroups — #12529

Order 256.

159 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2248	C₂	1	2	yes	maximal, complemented
2	#2240	C₂	1	2	yes	maximal, complemented
3	#1712	C₂	1	1	yes	maximal, characteristic
4	#1707	C₂	1	1	yes	maximal, characteristic, complemented
5	#1312	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×SD₃₂	—	2	4	no	—
7	#500	C₂²	1	1	yes	characteristic
8	#498	—	2	1	no	—
9	#494	C₂²	1	1	yes	characteristic
10	#493	—	2	1	no	—
11	#493	—	2	1	no	—
12	#493	C₂²	1	2	yes	—
13	#487	—	2	1	no	—
14	#357	—	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×D₈	—	2	2	no	—
19	C₂×D₈	—	2	2	no	—
20	C₂×D₈	—	4	2	no	—
21	C₂×D₈	—	4	2	no	—
22	C₄⋊Q₈	—	2	1	no	—
23	C₄⋊Q₈	—	2	1	no	—
24	#127	—	2	2	no	—
25	SD₃₂	—	4	8	no	—
26	C₂×C₁₆	D₄	1	4	yes	—
27	#107	—	2	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	1	no	—
35	C₂×D₄	—	4	2	no	—
36	C₂×D₄	—	4	2	no	—
37	C₂×D₄	—	4	2	no	—
38	Q₁₆	—	4	4	no	—
39	D₈	—	4	4	no	—
40	D₈	—	4	4	no	—
41	D₈	—	8	4	no	—
42	D₈	—	8	4	no	—
43	C₂×C₈	D₈	1	2	yes	—
44	C₂×C₈	D₈	1	2	yes	—
45	C₂×C₈	C₂×D₄	1	2	yes	—
46	C₄⋊C₄	—	4	1	no	—
47	C₄⋊C₄	—	4	1	no	—
48	C₄⋊C₄	—	4	1	no	—
49	C₄⋊C₄	—	4	1	no	—
50	C₄²	C₂×D₄	1	1	yes	characteristic
51	C₁₆	—	2	4	no	—
52	C₂³	—	8	2	no	—
53	Q₈	—	8	2	no	—
54	Q₈	—	8	2	no	—
55	D₄	—	8	2	no	—
56	D₄	—	8	2	no	—
57	D₄	—	8	4	no	—
58	D₄	—	8	4	no	—
59	C₂×C₄	—	8	1	no	—
60	C₂×C₄	—	8	1	no	—
61	C₂×C₄	#127	1	1	yes	characteristic
62	C₂×C₄	C₂×D₈	1	1	yes	characteristic
63	C₂×C₄	C₂×D₈	1	1	yes	characteristic
64	C₈	—	2	2	no	—
65	C₈	SD₃₂	1	4	yes	complemented
66	C₈	AGL(1,ℤ/8)	1	4	yes	—
67	C₂²	—	16	2	no	—
68	C₂²	—	16	2	no	—
69	C₂²	—	16	2	no	—
70	C₂²	#493	1	1	yes	characteristic
71	C₄	—	16	1	no	—
72	C₄	—	16	1	no	—
73	C₄	#496	1	2	yes	—
74	C₄	C₂×SD₃₂	1	2	yes	—
75	C₄	#509	1	2	yes	—
76	C₂	—	32	2	no	—
77	C₂	#1721	1	1	yes	characteristic
78	C₂	#2248	1	1	yes	characteristic
79	C₂	#2250	1	1	yes	characteristic

	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

Complements

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

N	H ≅ G/N	type
1. #2248	C₂	semidirect
2. #2240	C₂	semidirect
4. #1707	C₂	semidirect
5. #1312	C₂	semidirect
65. C₈	SD₃₂	semidirect

Maximal subgroups

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

subgroup	index
#2248	2
#2240	2
#1712	2
#1707	2
#1312	2