Subgroups — #13549

Order 256.

119 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	C₂²×C₃₂	C₂	1	1	yes	maximal, characteristic
3	#2108	C₂	1	1	yes	maximal, characteristic
4	#1397	C₂	1	2	yes	maximal, complemented
5	C₄×C₃₂	C₂	1	2	yes	maximal, complemented
6	C₂×M₃₂	C₄	1	1	yes	characteristic
7	C₂²×C₁₆	C₂²	1	1	yes	characteristic
8	#405	C₄	1	1	yes	characteristic
9	M₆₄	C₄	1	4	yes	complemented
10	C₂×C₃₂	C₄	1	4	yes	complemented
11	C₂×C₃₂	C₂²	1	2	yes	—
12	C₂×C₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₃₂	C₂²	1	1	yes	characteristic
14	#346	C₄	1	2	yes	—
15	C₄×C₁₆	C₂²	1	2	yes	—
16	C₂×M₁₆	C₈	1	1	yes	characteristic
17	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
18	#117	C₈	1	1	yes	characteristic
19	M₃₂	C₈	1	4	yes	—
20	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆	C₂×C₄	1	2	yes	—
22	C₂×C₁₆	C₂×C₄	1	2	yes	—
23	C₂×C₁₆	C₂×C₄	1	2	yes	—
24	C₂×C₁₆	C₂³	1	1	yes	characteristic
25	#97	C₈	1	2	yes	—
26	C₄×C₈	C₂×C₄	1	2	yes	—
27	C₃₂	C₂×C₄	1	2	yes	—
28	C₃₂	C₂×C₄	1	2	yes	—
29	C₃₂	C₂×C₄	1	4	yes	—
30	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
31	M₁₆	C₁₆	1	4	yes	—
32	C₂×C₈	C₄²	1	2	yes	—
33	C₂×C₈	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈	C₂×C₈	1	2	yes	—
35	C₂×C₈	C₂×C₈	1	2	yes	—
36	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
37	C₄⋊C₄	C₁₆	1	2	yes	—
38	C₂²⋊C₄	C₁₆	1	2	yes	—
39	C₄²	C₂×C₈	1	2	yes	—
40	C₁₆	C₄²	1	2	yes	—
41	C₁₆	C₂×C₈	1	4	yes	—
42	C₁₆	C₂²×C₄	1	1	yes	characteristic
43	C₁₆	C₂²×C₄	1	1	yes	characteristic
44	C₂³	C₂×C₁₆	1	1	yes	characteristic
45	C₂×C₄	C₄×C₈	1	2	yes	—
46	C₂×C₄	C₂×C₁₆	1	4	yes	—
47	C₂×C₄	C₂×C₁₆	1	1	yes	characteristic
48	C₂×C₄	C₂×C₁₆	1	2	yes	—
49	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
50	C₈	C₄×C₈	1	2	yes	—
51	C₈	C₂×C₁₆	1	4	yes	—
52	C₈	C₂×C₄²	1	1	yes	characteristic
53	C₈	C₂²×C₈	1	1	yes	characteristic
54	C₂²	—	2	2	no	—
55	C₂²	C₄×C₁₆	1	2	yes	—
56	C₂²	C₂²×C₁₆	1	1	yes	characteristic
57	C₄	—	2	4	no	—
58	C₄	C₄×C₁₆	1	2	yes	—
59	C₄	C₂×C₄×C₈	1	1	yes	characteristic
60	C₄	C₂²×C₁₆	1	1	yes	characteristic
61	C₂	—	2	2	no	—
62	C₂	C₂×C₄×C₁₆	1	1	yes	characteristic
63	C₂	#2258	1	2	yes	—

	subgroup	G/H
center	C₂×C₃₂	C₂²
commutator	C₂	C₂×C₄×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂³
socle	C₂²	C₂²×C₁₆
prime radical	C₂³	C₂×C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₂ ◁ 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
4. #1397	C₂	semidirect
5. C₄×C₃₂	C₂	semidirect
9. M₆₄	C₄	semidirect
10. 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
C₂²×C₃₂	2
#2108	2
#1397	2
C₄×C₃₂	2