Subgroups — #13551

Order 256.

107 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	#2108	C₂	1	1	yes	maximal, characteristic
3	#1398	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂	C₄	1	1	yes	characteristic
5	C₂²×C₁₆	C₂²	1	1	yes	characteristic
6	#405	C₄	1	1	yes	characteristic
7	M₆₄	C₄	1	4	yes	complemented
8	M₆₄	C₂²	1	4	yes	—
9	C₂×C₃₂	C₄	1	4	yes	complemented
10	#346	C₂²	1	2	yes	—
11	C₄×C₁₆	C₄	1	2	yes	—
12	C₂×M₁₆	C₈	1	1	yes	characteristic
13	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
14	#117	C₈	1	1	yes	characteristic
15	M₃₂	C₈	1	4	yes	—
16	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
17	C₂×C₁₆	C₂×C₄	1	2	yes	—
18	C₂×C₁₆	C₂×C₄	1	2	yes	—
19	C₂×C₁₆	C₂×C₄	1	2	yes	—
20	C₂×C₁₆	C₂³	1	1	yes	characteristic
21	#97	C₈	1	2	yes	—
22	C₄×C₈	C₂×C₄	1	2	yes	—
23	C₃₂	C₂×C₄	1	8	yes	—
24	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
25	M₁₆	—	2	2	no	—
26	C₂×C₈	C₄²	1	2	yes	—
27	C₂×C₈	C₂×C₈	1	2	yes	—
28	C₂×C₈	C₂×C₈	1	1	yes	characteristic
29	C₂×C₈	C₂×C₈	1	2	yes	—
30	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
31	C₄⋊C₄	—	2	1	no	—
32	C₂²⋊C₄	—	2	1	no	—
33	C₄²	C₂×C₈	1	2	yes	—
34	C₁₆	C₄²	1	1	yes	characteristic
35	C₁₆	C₄²	1	1	yes	characteristic
36	C₁₆	C₂×C₈	1	4	yes	—
37	C₁₆	C₂²×C₄	1	2	yes	—
38	C₂³	M₃₂	1	1	yes	characteristic
39	C₂×C₄	—	2	2	no	—
40	C₂×C₄	C₄×C₈	1	2	yes	—
41	C₂×C₄	M₃₂	1	1	yes	characteristic
42	C₂×C₄	M₃₂	1	2	yes	—
43	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
44	C₈	—	2	2	no	—
45	C₈	C₄×C₈	1	2	yes	—
46	C₈	C₂×C₄²	1	1	yes	characteristic
47	C₈	C₂²×C₈	1	1	yes	characteristic
48	C₂²	—	4	1	no	—
49	C₂²	#346	1	2	yes	—
50	C₂²	C₂×M₃₂	1	1	yes	characteristic
51	C₄	—	4	2	no	—
52	C₄	#346	1	2	yes	—
53	C₄	C₂×C₄×C₈	1	1	yes	characteristic
54	C₄	C₂×M₃₂	1	1	yes	characteristic
55	C₂	—	2	1	no	—
56	C₂	—	2	2	no	—
57	C₂	C₂×#346	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	C₄²
commutator	C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₁₆	C₂³
socle	C₂	C₂×#346
prime radical	C₂³	M₃₂
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ 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. #1398	C₂	semidirect
7. M₆₄	C₄	semidirect
9. C₂×C₃₂	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
#2108	2
#1398	2