Subgroups — #13710

Order 256.

80 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2254	C₂	1	1	yes	maximal, characteristic, complemented
2	#2178	C₂	1	1	yes	maximal, characteristic
3	#1426	C₂	1	2	yes	maximal, complemented
4	#1422	C₂	1	1	yes	maximal, characteristic, complemented
5	#1418	C₂	1	2	yes	maximal
6	#508	C₂²	1	1	yes	characteristic
7	#500	—	2	1	no	—
8	#443	—	2	1	no	—
9	M₆₄	C₂²	1	2	yes	—
10	#368	C₂²	1	1	yes	characteristic
11	#367	—	2	1	no	—
12	#367	C₂²	1	2	yes	—
13	#359	—	2	1	no	—
14	C₄×C₁₆	C₂²	1	1	yes	characteristic
15	#135	—	2	1	no	—
16	#131	—	4	1	no	—
17	C₄⋊Q₈	—	4	1	no	—
18	Q₃₂	Q₈	1	1	yes	characteristic, complemented
19	SD₃₂	—	2	1	no	—
20	D₁₆	Q₈	1	1	yes	characteristic, complemented
21	C₂×C₁₆	D₄	1	1	yes	characteristic
22	C₂×C₁₆	C₂³	1	1	yes	characteristic
23	#108	—	2	1	no	—
24	#107	—	2	2	no	—
25	#107	—	4	1	no	—
26	#104	—	4	1	no	—
27	C₄×C₈	D₄	1	1	yes	characteristic
28	C₃₂	—	2	2	no	—
29	#41	—	4	1	no	—
30	C₂×Q₈	—	8	1	no	—
31	Q₁₆	—	2	1	no	—
32	SD₁₆	—	4	1	no	—
33	D₈	—	2	1	no	—
34	M₁₆	—	4	1	no	—
35	M₁₆	—	8	1	no	—
36	C₂×C₈	—	8	1	no	—
37	C₂×C₈	D₈	1	1	yes	characteristic
38	C₂×C₈	C₂×D₄	1	1	yes	characteristic
39	C₄⋊C₄	—	4	2	no	—
40	C₄⋊C₄	—	8	1	no	—
41	C₄²	D₈	1	1	yes	characteristic
42	C₁₆	—	2	1	no	—
43	C₁₆	C₂×Q₈	1	1	yes	characteristic
44	C₁₆	#41	1	1	yes	characteristic
45	Q₈	—	4	1	no	—
46	Q₈	—	16	2	no	—
47	D₄	—	4	1	no	—
48	D₄	—	8	1	no	—
49	C₂×C₄	—	2	1	no	—
50	C₂×C₄	—	8	1	no	—
51	C₂×C₄	—	8	2	no	—
52	C₂×C₄	C₂×D₈	1	1	yes	characteristic
53	C₈	—	8	1	no	—
54	C₈	SD₃₂	1	1	yes	characteristic
55	C₈	SD₃₂	1	1	yes	characteristic
56	C₈	C₂²⋊Q₈	1	1	yes	characteristic
57	C₈	#137	1	1	yes	characteristic
58	C₂²	—	8	1	no	—
59	C₂²	#509	1	1	yes	characteristic
60	C₄	—	2	2	no	—
61	C₄	—	8	1	no	—
62	C₄	—	16	2	no	—
63	C₄	#474	1	1	yes	characteristic
64	C₄	C₂×SD₃₂	1	1	yes	characteristic
65	C₂	—	2	1	no	—
66	C₂	—	16	1	no	—
67	C₂	#2226	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#2226
commutator, Frattini, cosocle	C₂×C₁₆	C₂³
prime radical	#508	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ 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
1. #2254	C₂	semidirect
3. #1426	C₂	semidirect
4. #1422	C₂	semidirect
18. Q₃₂	Q₈	semidirect
20. D₁₆	Q₈	semidirect

Maximal subgroups

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

subgroup	index
#2254	2
#2178	2
#1426	2
#1422	2
#1418	2