Subgroups — #13627

Order 256.

109 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	C₄⋊C₃₂	C₂	1	4	yes	maximal, complemented
4	C₂²×C₁₆	C₄	1	1	yes	characteristic
5	C₂²×C₁₆	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₈	C₄	1	1	yes	characteristic
7	M₆₄	—	2	4	no	—
8	C₂×C₃₂	C₂²	1	4	yes	—
9	C₄×C₁₆	C₄	1	2	yes	—
10	C₄×C₁₆	C₂²	1	2	yes	—
11	C₂²×C₈	C₈	1	1	yes	characteristic
12	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
13	C₂×C₄²	C₈	1	1	yes	characteristic
14	C₂×C₁₆	—	2	2	no	—
15	C₂×C₁₆	C₂×C₄	1	2	yes	—
16	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
17	C₂×C₁₆	D₄	1	2	yes	—
18	C₂×C₁₆	Q₈	1	2	yes	—
19	C₂×C₁₆	C₂³	1	1	yes	characteristic
20	C₄×C₈	C₈	1	2	yes	—
21	C₄×C₈	C₂×C₄	1	2	yes	—
22	C₃₂	—	2	4	no	—
23	C₂²×C₄	C₁₆	1	2	yes	—
24	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
25	C₂×C₈	—	2	2	no	—
26	C₂×C₈	C₄⋊C₄	1	2	yes	—
27	C₂×C₈	C₂×C₈	1	2	yes	—
28	C₂×C₈	C₂×C₈	1	1	yes	characteristic
29	C₂×C₈	M₁₆	1	2	yes	—
30	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
31	C₄²	C₁₆	1	1	yes	characteristic
32	C₄²	C₁₆	1	1	yes	characteristic
33	C₄²	C₂×C₈	1	2	yes	—
34	C₁₆	—	2	2	no	—
35	C₁₆	C₄⋊C₄	1	2	yes	—
36	C₁₆	C₂×D₄	1	1	yes	characteristic
37	C₁₆	C₂×Q₈	1	1	yes	characteristic
38	C₂³	C₂×C₁₆	1	1	yes	characteristic
39	C₂×C₄	—	2	4	no	—
40	C₂×C₄	C₄⋊C₈	1	2	yes	—
41	C₂×C₄	C₂×C₁₆	1	2	yes	—
42	C₂×C₄	C₂×C₁₆	1	1	yes	characteristic
43	C₂×C₄	C₂×C₁₆	1	2	yes	—
44	C₂×C₄	M₃₂	1	2	yes	—
45	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
46	C₈	—	2	2	no	—
47	C₈	C₄⋊C₈	1	2	yes	—
48	C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₈	C₂×M₁₆	1	1	yes	characteristic
50	C₂²	—	2	2	no	—
51	C₂²	C₄⋊C₁₆	1	2	yes	—
52	C₂²	C₂²×C₁₆	1	1	yes	characteristic
53	C₄	—	2	2	no	—
54	C₄	C₄⋊C₁₆	1	2	yes	—
55	C₄	M₆₄	1	4	yes	complemented
56	C₄	C₂×C₄⋊C₈	1	1	yes	characteristic
57	C₄	C₂×M₃₂	1	1	yes	characteristic
58	C₂	—	2	2	no	—
59	C₂	C₂×C₄⋊C₁₆	1	1	yes	characteristic
60	C₂	C₂×M₆₄	1	2	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₂×C₁₆	C₂³
commutator, socle	C₂²	C₂²×C₁₆
prime radical	C₂³	C₂×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
3. C₄⋊C₃₂	C₂	semidirect
55. 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
C₂×C₄×C₁₆	2
C₄⋊C₃₂	2