Subgroups — #13719

Order 256.

101 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2252	C₂	1	1	yes	maximal, characteristic, complemented
2	#2246	C₂	1	1	yes	maximal, characteristic
3	#1415	C₂	1	4	yes	maximal, complemented
4	C₄×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₆	—	2	1	no	—
6	C₂×D₁₆	C₂²	1	2	yes	—
7	#500	—	2	1	no	—
8	#493	—	2	1	no	—
9	C₂×C₃₂	C₂²	1	2	yes	—
10	#366	—	2	1	no	—
11	#366	C₂²	1	2	yes	—
12	C₄×C₁₆	C₂²	1	1	yes	characteristic
13	C₂×D₈	—	2	2	no	—
14	C₂×D₈	—	4	1	no	—
15	C₄⋊Q₈	—	4	1	no	—
16	#127	—	4	1	no	—
17	D₁₆	—	2	4	no	—
18	D₁₆	—	4	2	no	—
19	C₂×C₁₆	D₄	1	1	yes	characteristic
20	C₂×C₁₆	C₂³	1	1	yes	characteristic
21	#107	—	2	2	no	—
22	#107	—	4	1	no	—
23	C₄×C₈	D₄	1	1	yes	characteristic
24	C₃₂	—	2	2	no	—
25	C₂×Q₈	—	8	1	no	—
26	C₂×D₄	—	4	2	no	—
27	C₂×D₄	—	8	1	no	—
28	C₂×D₄	—	8	1	no	—
29	D₈	—	4	2	no	—
30	D₈	—	4	2	no	—
31	D₈	—	8	2	no	—
32	C₂×C₈	D₈	1	1	yes	characteristic
33	C₂×C₈	C₂×D₄	1	1	yes	characteristic
34	C₄⋊C₄	—	4	2	no	—
35	C₄⋊C₄	—	8	1	no	—
36	C₄²	D₈	1	1	yes	characteristic
37	C₁₆	SD₁₆	1	2	yes	—
38	C₁₆	#41	1	2	yes	—
39	C₂³	—	8	2	no	—
40	Q₈	—	16	2	no	—
41	D₄	—	8	2	no	—
42	D₄	—	8	2	no	—
43	D₄	—	16	2	no	—
44	D₄	—	16	2	no	—
45	C₂×C₄	—	8	2	no	—
46	C₂×C₄	D₁₆	1	1	yes	characteristic
47	C₂×C₄	D₁₆	1	1	yes	characteristic
48	C₂×C₄	C₂×D₈	1	1	yes	characteristic
49	C₈	SD₃₂	1	2	yes	—
50	C₈	#124	1	1	yes	characteristic
51	C₈	C₂×SD₁₆	1	1	yes	characteristic
52	C₂²	—	16	2	no	—
53	C₂²	—	16	2	no	—
54	C₂²	—	16	2	no	—
55	C₂²	C₂×D₁₆	1	1	yes	characteristic
56	C₄	—	16	2	no	—
57	C₄	D₃₂	1	2	yes	—
58	C₄	SD₆₄	1	2	yes	—
59	C₄	#486	1	1	yes	characteristic
60	C₄	C₂×SD₃₂	1	1	yes	characteristic
61	C₂	—	32	2	no	—
62	C₂	#2240	1	1	yes	characteristic
63	C₂	C₂×D₃₂	1	1	yes	characteristic
64	C₂	C₂×SD₆₄	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³
prime radical	#2246	C₂
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
1. #2252	C₂	semidirect
3. #1415	C₂	semidirect
4. C₄×C₃₂	C₂	semidirect

Maximal subgroups

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

subgroup	index
#2252	2
#2246	2
#1415	2
C₄×C₃₂	2