Subgroups — #13721

Order 256.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#2253	C₂	1	1	yes	maximal, characteristic, complemented
2	#2249	C₂	1	1	yes	maximal, characteristic
3	#1416	C₂	1	2	yes	maximal, complemented
4	#1415	C₂	1	2	yes	maximal
5	C₄×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₃₂	C₂²	1	1	yes	characteristic
7	C₂×SD₃₂	—	2	1	no	—
8	C₂×D₁₆	C₂²	1	1	yes	characteristic
9	#499	—	2	1	no	—
10	#495	—	2	1	no	—
11	C₂×C₃₂	C₂²	1	2	yes	—
12	#367	—	2	1	no	—
13	#366	C₂²	1	2	yes	—
14	C₄×C₁₆	C₂²	1	1	yes	characteristic
15	C₂×Q₁₆	—	2	1	no	—
16	C₂×SD₁₆	—	4	1	no	—
17	C₂×D₈	—	2	1	no	—
18	#125	—	4	1	no	—
19	#124	—	4	1	no	—
20	Q₃₂	—	2	2	no	—
21	SD₃₂	—	4	2	no	—
22	D₁₆	—	2	2	no	—
23	C₂×C₁₆	D₄	1	1	yes	characteristic
24	C₂×C₁₆	C₂³	1	1	yes	characteristic
25	#107	—	2	2	no	—
26	#106	—	4	1	no	—
27	C₄×C₈	D₄	1	1	yes	characteristic
28	C₃₂	—	2	2	no	—
29	C₂×Q₈	—	4	1	no	—
30	C₂×D₄	—	4	1	no	—
31	Q₁₆	—	4	1	no	—
32	Q₁₆	—	4	1	no	—
33	SD₁₆	—	8	2	no	—
34	D₈	—	4	1	no	—
35	D₈	—	4	1	no	—
36	C₂×C₈	D₈	1	1	yes	characteristic
37	C₂×C₈	C₂×D₄	1	1	yes	characteristic
38	C₄⋊C₄	—	4	2	no	—
39	C₄⋊C₄	—	8	2	no	—
40	C₂²⋊C₄	—	8	2	no	—
41	C₄²	D₈	1	1	yes	characteristic
42	C₁₆	SD₁₆	1	2	yes	—
43	C₁₆	#41	1	2	yes	—
44	C₂³	—	8	1	no	—
45	Q₈	—	8	1	no	—
46	Q₈	—	8	1	no	—
47	D₄	—	8	1	no	—
48	D₄	—	8	1	no	—
49	C₂×C₄	—	8	1	no	—
50	C₂×C₄	—	8	2	no	—
51	C₂×C₄	D₁₆	1	2	yes	—
52	C₂×C₄	C₂×D₈	1	1	yes	characteristic
53	C₈	SD₃₂	1	2	yes	—
54	C₈	#124	1	1	yes	characteristic
55	C₈	C₂×SD₁₆	1	1	yes	characteristic
56	C₂²	—	16	1	no	—
57	C₂²	—	16	2	no	—
58	C₂²	C₂×D₁₆	1	1	yes	characteristic
59	C₄	—	2	2	no	—
60	C₄	—	16	1	no	—
61	C₄	—	16	2	no	—
62	C₄	#486	1	1	yes	characteristic
63	C₄	C₂×SD₃₂	1	1	yes	characteristic
64	C₂	—	32	1	no	—
65	C₂	#2240	1	1	yes	characteristic
66	C₂	#2262	1	2	yes	—

	subgroup	G/H
center, socle	C₂²	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³
prime radical	C₂×D₁₆	C₂²
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. #2253	C₂	semidirect
3. #1416	C₂	semidirect
5. 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
#2253	2
#2249	2
#1416	2
#1415	2
C₄×C₃₂	2