Subgroups — #13727

Order 256.

125 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×SD₆₄	C₂	1	4	yes	maximal, complemented
2	#2247	C₂	1	1	yes	maximal, characteristic, complemented
3	#2246	C₂	1	1	yes	maximal, characteristic
4	C₄×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Q₃₂	—	2	1	no	—
6	C₂×Q₃₂	C₂²	1	2	yes	—
7	C₂×D₁₆	—	2	1	no	—
8	C₂×D₁₆	C₂²	1	2	yes	—
9	#494	—	2	1	no	—
10	#493	—	2	1	no	—
11	SD₆₄	—	2	8	no	—
12	C₂×C₃₂	C₂²	1	2	yes	—
13	C₄×C₁₆	C₂²	1	1	yes	characteristic
14	C₂×Q₁₆	—	2	2	no	—
15	C₂×Q₁₆	—	4	1	no	—
16	C₂×D₈	—	2	2	no	—
17	C₂×D₈	—	4	1	no	—
18	C₄⋊Q₈	—	4	1	no	—
19	#127	—	4	1	no	—
20	Q₃₂	—	2	4	no	—
21	Q₃₂	—	4	2	no	—
22	D₁₆	—	2	4	no	—
23	D₁₆	—	4	2	no	—
24	C₂×C₁₆	D₄	1	1	yes	characteristic
25	C₂×C₁₆	C₂³	1	1	yes	characteristic
26	C₄×C₈	D₄	1	1	yes	characteristic
27	C₃₂	D₄	1	4	yes	complemented
28	C₂×Q₈	—	4	2	no	—
29	C₂×D₄	—	4	2	no	—
30	C₂×D₄	—	8	2	no	—
31	Q₁₆	—	4	2	no	—
32	Q₁₆	—	4	2	no	—
33	Q₁₆	—	8	2	no	—
34	D₈	—	4	2	no	—
35	D₈	—	4	2	no	—
36	D₈	—	8	2	no	—
37	C₂×C₈	D₈	1	1	yes	characteristic
38	C₂×C₈	C₂×D₄	1	1	yes	characteristic
39	C₄⋊C₄	—	8	2	no	—
40	C₄²	D₈	1	1	yes	characteristic
41	C₁₆	D₈	1	2	yes	—
42	C₁₆	C₂×D₄	1	2	yes	—
43	C₂³	—	8	2	no	—
44	Q₈	—	8	2	no	—
45	Q₈	—	8	2	no	—
46	D₄	—	8	2	no	—
47	D₄	—	8	2	no	—
48	D₄	—	16	4	no	—
49	C₂×C₄	—	8	2	no	—
50	C₂×C₄	D₁₆	1	2	yes	—
51	C₂×C₄	C₂×D₈	1	1	yes	characteristic
52	C₈	D₁₆	1	2	yes	—
53	C₈	#127	1	1	yes	characteristic
54	C₈	C₂×D₈	1	1	yes	characteristic
55	C₂²	—	16	2	no	—
56	C₂²	—	16	4	no	—
57	C₂²	C₂×D₁₆	1	1	yes	characteristic
58	C₄	—	16	2	no	—
59	C₄	SD₆₄	1	4	yes	complemented
60	C₄	#493	1	1	yes	characteristic
61	C₄	C₂×D₁₆	1	1	yes	characteristic
62	C₂	—	32	2	no	—
63	C₂	#2246	1	1	yes	characteristic
64	C₂	C₂×SD₆₄	1	2	yes	—

	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. C₂×SD₆₄	C₂	semidirect
2. #2247	C₂	semidirect
4. C₄×C₃₂	C₂	semidirect
27. C₃₂	D₄	semidirect
59. C₄	SD₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×SD₆₄	2
#2247	2
#2246	2
C₄×C₃₂	2