Subgroups — #7467

Order 256.

95 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₆₄	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₆	C₄	1	1	yes	characteristic
4	C₂²×C₁₆	C₂²	1	1	yes	characteristic
5	C₂×C₄×C₈	C₄	1	1	yes	characteristic
6	M₆₄	—	2	2	no	—
7	M₆₄	C₄	1	4	yes	complemented
8	C₂×C₃₂	—	2	2	no	—
9	C₄×C₁₆	—	2	2	no	—
10	C₂²×C₈	C₈	1	1	yes	characteristic
11	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄²	C₈	1	1	yes	characteristic
13	C₂×C₁₆	—	2	1	no	—
14	C₂×C₁₆	—	2	1	no	—
15	C₂×C₁₆	C₈	1	2	yes	—
16	C₂×C₁₆	C₂×C₄	1	2	yes	—
17	C₂×C₁₆	D₄	1	2	yes	—
18	C₂×C₁₆	D₄	1	1	yes	characteristic
19	C₂×C₁₆	Q₈	1	1	yes	characteristic
20	C₄×C₈	—	2	1	no	—
21	C₄×C₈	—	2	1	no	—
22	C₃₂	—	2	4	no	—
23	C₂²×C₄	—	2	1	no	—
24	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
25	C₂×C₈	—	2	1	no	—
26	C₂×C₈	—	2	1	no	—
27	C₂×C₈	C₁₆	1	2	yes	—
28	C₂×C₈	C₄²	1	1	yes	characteristic
29	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₈	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈	C₂×C₈	1	1	yes	characteristic
32	C₂×C₈	M₁₆	1	1	yes	characteristic
33	C₂×C₈	M₁₆	1	1	yes	characteristic
34	C₄²	—	2	1	no	—
35	C₄²	C₁₆	1	2	yes	—
36	C₁₆	—	2	2	no	—
37	C₁₆	C₂²⋊C₄	1	2	yes	—
38	C₁₆	C₄⋊C₄	1	2	yes	—
39	C₂³	M₃₂	1	1	yes	characteristic
40	C₂×C₄	—	2	4	no	—
41	C₂×C₄	—	2	2	no	—
42	C₂×C₄	C₄×C₈	1	1	yes	characteristic
43	C₂×C₄	C₂²⋊C₈	1	1	yes	characteristic
44	C₂×C₄	C₄⋊C₈	1	1	yes	characteristic
45	C₂×C₄	C₂×C₁₆	1	1	yes	characteristic
46	C₂×C₄	C₂×C₁₆	1	1	yes	characteristic
47	C₂×C₄	M₃₂	1	1	yes	characteristic
48	C₈	—	2	2	no	—
49	C₈	#95	1	1	yes	characteristic
50	C₈	#97	1	1	yes	characteristic
51	C₈	C₂²⋊C₈	1	1	yes	characteristic
52	C₈	C₄⋊C₈	1	1	yes	characteristic
53	C₂²	—	2	2	no	—
54	C₂²	#346	1	1	yes	characteristic
55	C₂²	C₂²⋊C₁₆	1	1	yes	characteristic
56	C₂²	C₄⋊C₁₆	1	1	yes	characteristic
57	C₄	—	2	4	no	—
58	C₄	#336	1	1	yes	characteristic
59	C₄	C₄×C₁₆	1	1	yes	characteristic
60	C₄	C₂²⋊C₁₆	1	1	yes	characteristic
61	C₄	C₄⋊C₁₆	1	1	yes	characteristic
62	C₂	—	2	1	no	—
63	C₂	—	2	1	no	—
64	C₂	#1374	1	1	yes	characteristic
65	C₂	#1422	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	D₄
commutator	C₄	C₄×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂²
socle	C₂²	C₄⋊C₁₆
prime radical	C₂³	M₃₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
7. M₆₄	C₄	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
C₂×C₄×C₁₆	2