Subgroups — #13553

Order 256.

165 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×M₆₄	C₂	1	2	yes	maximal, complemented
3	C₂²⋊C₃₂	C₂	1	4	yes	maximal, complemented
4	C₂³×C₈	C₄	1	1	yes	characteristic
5	C₂²×C₁₆	—	2	4	no	—
6	C₂²×C₁₆	C₄	1	1	yes	characteristic
7	C₂²×C₁₆	C₄	1	2	yes	—
8	C₂²×C₁₆	C₂²	1	2	yes	—
9	C₂²×C₁₆	C₂²	1	1	yes	characteristic
10	M₆₄	—	2	4	no	—
11	C₂×C₃₂	C₂²	1	4	yes	complemented
12	C₂³×C₄	C₈	1	1	yes	characteristic
13	C₂²×C₈	—	2	2	no	—
14	C₂²×C₈	—	2	2	no	—
15	C₂²×C₈	C₈	1	1	yes	characteristic
16	C₂²×C₈	C₈	1	2	yes	—
17	C₂²×C₈	C₂×C₄	1	2	yes	—
18	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₆	—	2	2	no	—
20	C₂×C₁₆	—	2	4	no	—
21	C₂×C₁₆	—	2	4	no	—
22	C₂×C₁₆	C₂×C₄	1	2	yes	—
23	C₂×C₁₆	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₆	D₄	1	4	yes	—
25	C₂×C₁₆	C₂³	1	1	yes	characteristic
26	C₃₂	—	2	4	no	—
27	C₂⁴	C₁₆	1	1	yes	characteristic
28	C₂²×C₄	—	2	2	no	—
29	C₂²×C₄	—	2	2	no	—
30	C₂²×C₄	C₁₆	1	1	yes	characteristic
31	C₂²×C₄	C₁₆	1	2	yes	—
32	C₂²×C₄	C₂×C₈	1	2	yes	—
33	C₂²×C₄	C₂×C₈	1	1	yes	characteristic
34	C₂×C₈	—	2	2	no	—
35	C₂×C₈	—	2	4	no	—
36	C₂×C₈	—	2	2	no	—
37	C₂×C₈	—	2	2	no	—
38	C₂×C₈	C₂²⋊C₄	1	2	yes	—
39	C₂×C₈	C₂×C₈	1	2	yes	—
40	C₂×C₈	C₂×C₈	1	1	yes	characteristic
41	C₂×C₈	M₁₆	1	2	yes	—
42	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
43	C₁₆	—	2	2	no	—
44	C₁₆	C₂²⋊C₄	1	2	yes	—
45	C₁₆	C₂×D₄	1	2	yes	—
46	C₂³	—	2	2	no	—
47	C₂³	—	2	4	no	—
48	C₂³	C₂×C₁₆	1	1	yes	characteristic
49	C₂³	C₂×C₁₆	1	2	yes	—
50	C₂×C₄	—	2	2	no	—
51	C₂×C₄	—	2	4	no	—
52	C₂×C₄	—	2	2	no	—
53	C₂×C₄	—	2	2	no	—
54	C₂×C₄	C₂²⋊C₈	1	2	yes	—
55	C₂×C₄	C₂×C₁₆	1	2	yes	—
56	C₂×C₄	C₂×C₁₆	1	1	yes	characteristic
57	C₂×C₄	M₃₂	1	2	yes	—
58	C₂×C₄	C₂²×C₈	1	1	yes	characteristic
59	C₈	—	2	2	no	—
60	C₈	C₂²⋊C₈	1	2	yes	—
61	C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
62	C₈	C₂×M₁₆	1	1	yes	characteristic
63	C₂²	—	2	8	no	—
64	C₂²	—	2	2	no	—
65	C₂²	—	2	2	no	—
66	C₂²	—	2	2	no	—
67	C₂²	C₂²⋊C₁₆	1	2	yes	—
68	C₂²	M₆₄	1	4	yes	complemented
69	C₂²	C₂²×C₁₆	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	C₂²⋊C₁₆	1	2	yes	—
72	C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
73	C₄	C₂×M₃₂	1	1	yes	characteristic
74	C₂	—	2	2	no	—
75	C₂	—	2	4	no	—
76	C₂	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
77	C₂	C₂×M₆₄	1	2	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₂×C₁₆	C₂³
commutator, socle	C₂²	C₂²×C₁₆
prime radical	C₂⁴	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
2. C₂×M₆₄	C₂	semidirect
3. C₂²⋊C₃₂	C₂	semidirect
11. C₂×C₃₂	C₂²	semidirect
68. C₂²	M₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₁₆	2
C₂×M₆₄	2
C₂²⋊C₃₂	2