Subgroups — #7513

Order 256.

45 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₆₄	C₂	1	1	yes	maximal, characteristic
2	M₁₂₈	C₂	1	2	yes	maximal, complemented
3	C₂²×C₁₆	C₄	1	1	yes	characteristic
4	M₆₄	—	2	2	no	—
5	C₂×C₃₂	C₄	1	1	yes	characteristic
6	C₂×C₃₂	C₂²	1	1	yes	characteristic
7	C₆₄	—	2	2	no	—
8	C₂²×C₈	C₈	1	1	yes	characteristic
9	C₂×C₁₆	—	2	1	no	—
10	C₂×C₁₆	—	2	1	no	—
11	C₂×C₁₆	C₈	1	1	yes	characteristic
12	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
13	C₃₂	—	2	1	no	—
14	C₃₂	D₄	1	2	yes	—
15	C₂²×C₄	C₁₆	1	1	yes	characteristic
16	C₂×C₈	—	2	1	no	—
17	C₂×C₈	—	2	1	no	—
18	C₂×C₈	C₁₆	1	1	yes	characteristic
19	C₂×C₈	C₂×C₈	1	1	yes	characteristic
20	C₁₆	—	2	1	no	—
21	C₁₆	C₂²⋊C₄	1	1	yes	characteristic
22	C₁₆	M₁₆	1	1	yes	characteristic
23	C₂³	C₃₂	1	1	yes	characteristic
24	C₂×C₄	—	2	1	no	—
25	C₂×C₄	—	2	1	no	—
26	C₂×C₄	C₃₂	1	1	yes	characteristic
27	C₂×C₄	C₂×C₁₆	1	1	yes	characteristic
28	C₈	—	2	1	no	—
29	C₈	C₂²⋊C₈	1	1	yes	characteristic
30	C₈	M₃₂	1	1	yes	characteristic
31	C₂²	—	2	1	no	—
32	C₂²	—	4	1	no	—
33	C₂²	C₂×C₃₂	1	1	yes	characteristic
34	C₄	—	2	1	no	—
35	C₄	C₂²⋊C₁₆	1	1	yes	characteristic
36	C₄	M₆₄	1	1	yes	characteristic
37	C₂	—	2	1	no	—
38	C₂	—	4	1	no	—
39	C₂	C₂²⋊C₃₂	1	1	yes	characteristic

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

N	H ≅ G/N	type
2. M₁₂₈	C₂	semidirect

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

subgroup	index
C₂×M₆₄	2
M₁₂₈	2