Subgroups — #13646

Order 256.

101 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₆₄	C₂	1	2	yes	maximal, complemented
2	D₄×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²⋊C₃₂	C₂	1	2	yes	maximal, complemented
5	C₄×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₁₆	C₂²	1	2	yes	—
7	C₈×D₄	C₄	1	1	yes	characteristic
8	M₆₄	—	2	4	no	—
9	C₂×C₃₂	C₂²	1	1	yes	characteristic
10	C₂×C₃₂	C₂²	1	2	yes	—
11	C₂×C₃₂	C₂²	1	1	yes	characteristic
12	C₄⋊C₁₆	C₄	1	1	yes	characteristic
13	C₂²⋊C₁₆	C₄	1	2	yes	—
14	C₄×C₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₈	C₂×C₄	1	2	yes	—
16	C₄×D₄	C₈	1	1	yes	characteristic
17	C₂×C₁₆	—	2	2	no	—
18	C₂×C₁₆	—	2	2	no	—
19	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₆	C₂×C₄	1	2	yes	—
21	C₂×C₁₆	C₂³	1	1	yes	characteristic
22	C₄⋊C₈	C₈	1	1	yes	characteristic
23	C₂²⋊C₈	C₈	1	2	yes	—
24	C₄×C₈	C₂×C₄	1	1	yes	characteristic
25	C₃₂	—	2	2	no	—
26	C₃₂	—	2	1	no	—
27	C₃₂	D₄	1	2	yes	complemented
28	C₂×D₄	C₁₆	1	1	yes	characteristic
29	C₂²×C₄	C₂×C₈	1	2	yes	—
30	C₂×C₈	—	2	2	no	—
31	C₂×C₈	—	2	2	no	—
32	C₂×C₈	C₂×C₈	1	1	yes	characteristic
33	C₂×C₈	C₂×C₈	1	2	yes	—
34	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
35	C₄⋊C₄	C₁₆	1	1	yes	characteristic
36	C₂²⋊C₄	C₁₆	1	2	yes	—
37	C₄²	C₂×C₈	1	1	yes	characteristic
38	C₁₆	—	2	2	no	—
39	C₁₆	—	2	1	no	—
40	C₁₆	C₂×D₄	1	1	yes	characteristic
41	C₁₆	#41	1	1	yes	characteristic
42	C₂³	C₂×C₁₆	1	2	yes	—
43	D₄	—	2	2	no	—
44	C₂×C₄	—	2	2	no	—
45	C₂×C₄	—	2	2	no	—
46	C₂×C₄	C₂×C₁₆	1	1	yes	characteristic
47	C₂×C₄	C₂×C₁₆	1	2	yes	—
48	C₂×C₄	C₂×C₁₆	1	1	yes	characteristic
49	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
50	C₈	—	2	2	no	—
51	C₈	—	2	1	no	—
52	C₈	C₄×D₄	1	1	yes	characteristic
53	C₈	#131	1	1	yes	characteristic
54	C₂²	—	2	2	no	—
55	C₂²	—	2	2	no	—
56	C₂²	—	2	2	no	—
57	C₂²	C₂²×C₁₆	1	1	yes	characteristic
58	C₄	—	2	2	no	—
59	C₄	—	2	1	no	—
60	C₄	M₆₄	1	2	yes	complemented
61	C₄	C₈×D₄	1	1	yes	characteristic
62	C₄	#504	1	1	yes	characteristic
63	C₂	—	4	2	no	—
64	C₂	D₄×C₁₆	1	1	yes	characteristic
65	C₂	C₂×M₆₄	1	1	yes	characteristic
66	C₂	#2258	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂×C₁₆	C₂³
commutator, socle	C₂²	C₂²×C₁₆
prime radical	C₂×D₄	C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×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×M₆₄	C₂	semidirect
3. C₄⋊C₃₂	C₂	semidirect
4. C₂²⋊C₃₂	C₂	semidirect
5. C₄×C₃₂	C₂	semidirect
27. C₃₂	D₄	semidirect
60. C₄	M₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₆₄	2
D₄×C₁₆	2
C₄⋊C₃₂	2
C₂²⋊C₃₂	2
C₄×C₃₂	2