Subgroups — #13639

Order 256.

101 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₃₂	C₂	1	2	yes	maximal, complemented
3	C₂²⋊C₃₂	C₂	1	2	yes	maximal
4	C₄×C₃₂	C₂	1	2	yes	maximal, complemented
5	C₂²×C₁₆	C₄	1	1	yes	characteristic
6	C₂²×C₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₄×C₈	C₄	1	1	yes	characteristic
8	C₂×C₃₂	C₂²	1	4	yes	—
9	C₄×C₁₆	C₄	1	1	yes	characteristic
10	C₄×C₁₆	C₄	1	1	yes	characteristic
11	C₄×C₁₆	C₂²	1	1	yes	characteristic
12	C₄×C₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₈	C₈	1	1	yes	characteristic
14	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²	C₈	1	1	yes	characteristic
16	C₂×C₁₆	—	2	2	no	—
17	C₂×C₁₆	—	2	2	no	—
18	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆	C₂³	1	1	yes	characteristic
22	C₄×C₈	C₈	1	1	yes	characteristic
23	C₄×C₈	C₈	1	1	yes	characteristic
24	C₄×C₈	C₂×C₄	1	1	yes	characteristic
25	C₄×C₈	C₂×C₄	1	1	yes	characteristic
26	C₃₂	—	2	4	no	—
27	C₂²×C₄	C₁₆	1	1	yes	characteristic
28	C₂²×C₄	C₁₆	1	1	yes	characteristic
29	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
30	C₂×C₈	—	2	2	no	—
31	C₂×C₈	—	2	1	no	—
32	C₂×C₈	—	2	1	no	—
33	C₂×C₈	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈	C₂×C₈	1	1	yes	characteristic
35	C₂×C₈	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
37	C₄²	C₁₆	1	1	yes	characteristic
38	C₄²	C₁₆	1	1	yes	characteristic
39	C₄²	C₂×C₈	1	1	yes	characteristic
40	C₄²	C₂×C₈	1	1	yes	characteristic
41	C₁₆	—	2	1	no	—
42	C₁₆	—	2	1	no	—
43	C₁₆	M₁₆	1	2	yes	—
44	C₁₆	#41	1	2	yes	—
45	C₂³	C₂×C₁₆	1	1	yes	characteristic
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	2	1	no	—
48	C₂×C₄	—	2	1	no	—
49	C₂×C₄	C₃₂	1	4	yes	complemented
50	C₂×C₄	C₂×C₁₆	1	1	yes	characteristic
51	C₂×C₄	C₂×C₁₆	1	1	yes	characteristic
52	C₂×C₄	C₂×C₁₆	1	1	yes	characteristic
53	C₂×C₄	C₂×C₁₆	1	1	yes	characteristic
54	C₂×C₄	C₂×C₁₆	1	1	yes	characteristic
55	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
56	C₈	—	2	1	no	—
57	C₈	—	2	1	no	—
58	C₈	M₃₂	1	2	yes	—
59	C₈	#117	1	1	yes	characteristic
60	C₈	C₂×M₁₆	1	1	yes	characteristic
61	C₂²	—	2	1	no	—
62	C₂²	—	2	1	no	—
63	C₂²	C₂×C₃₂	1	2	yes	—
64	C₂²	C₂²×C₁₆	1	1	yes	characteristic
65	C₄	—	2	1	no	—
66	C₄	—	2	1	no	—
67	C₄	C₂×C₃₂	1	2	yes	—
68	C₄	C₂×C₃₂	1	2	yes	—
69	C₄	M₆₄	1	2	yes	—
70	C₄	#431	1	1	yes	characteristic
71	C₄	C₂×M₃₂	1	1	yes	characteristic
72	C₂	—	2	2	no	—
73	C₂	#2162	1	1	yes	characteristic
74	C₂	C₂²×C₃₂	1	1	yes	characteristic
75	C₂	C₂×M₆₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₆	C₂²
commutator	C₂	C₂²×C₃₂
Frattini, cosocle	C₂×C₁₆	C₂³
socle	C₂²	C₂²×C₁₆
prime radical	C₂³	C₂×C₁₆
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×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
2. C₄⋊C₃₂	C₂	semidirect
4. C₄×C₃₂	C₂	semidirect
49. C₂×C₄	C₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₁₆	2
C₄⋊C₃₂	2
C₂²⋊C₃₂	2
C₄×C₃₂	2