Subgroups — #94755

Order 512.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₆₄	C₂	1	2	yes	maximal
2	C₂×C₄×C₃₂	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₃₂	C₄	1	1	yes	characteristic
4	C₂²×C₃₂	C₂²	1	1	yes	characteristic
5	C₂×C₄×C₁₆	C₄	1	1	yes	characteristic
6	C₂×C₆₄	—	2	2	no	—
7	C₂×C₆₄	—	2	2	no	—
8	C₂×C₆₄	C₄	1	4	yes	complemented
9	C₄×C₃₂	—	2	2	no	—
10	C₂²×C₁₆	C₈	1	1	yes	characteristic
11	C₂²×C₁₆	C₂×C₄	1	1	yes	characteristic
12	C₂×C₄×C₈	C₈	1	1	yes	characteristic
13	C₂×C₃₂	—	2	1	no	—
14	C₂×C₃₂	—	2	1	no	—
15	C₂×C₃₂	C₈	1	2	yes	—
16	C₂×C₃₂	C₂×C₄	1	2	yes	—
17	C₂×C₃₂	D₄	1	1	yes	characteristic
18	C₂×C₃₂	D₄	1	2	yes	—
19	C₂×C₃₂	Q₈	1	1	yes	characteristic
20	C₄×C₁₆	—	2	1	no	—
21	C₄×C₁₆	—	2	1	no	—
22	C₆₄	—	2	4	no	—
23	C₂²×C₈	C₁₆	1	1	yes	characteristic
24	C₂²×C₈	C₂×C₈	1	1	yes	characteristic
25	C₂×C₄²	C₁₆	1	1	yes	characteristic
26	C₂×C₁₆	—	2	1	no	—
27	C₂×C₁₆	—	2	1	no	—
28	C₂×C₁₆	C₁₆	1	2	yes	—
29	C₂×C₁₆	C₄²	1	1	yes	characteristic
30	C₂×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₁₆	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₁₆	C₂×C₈	1	1	yes	characteristic
33	C₂×C₁₆	M₁₆	1	1	yes	characteristic
34	C₂×C₁₆	M₁₆	1	1	yes	characteristic
35	C₄×C₈	—	2	1	no	—
36	C₄×C₈	—	2	1	no	—
37	C₃₂	—	2	2	no	—
38	C₃₂	C₂²⋊C₄	1	2	yes	—
39	C₃₂	C₄⋊C₄	1	2	yes	—
40	C₂²×C₄	C₃₂	1	2	yes	—
41	C₂²×C₄	C₂×C₁₆	1	1	yes	characteristic
42	C₂×C₈	—	2	1	no	—
43	C₂×C₈	—	2	1	no	—
44	C₂×C₈	C₃₂	1	2	yes	—
45	C₂×C₈	C₄×C₈	1	1	yes	characteristic
46	C₂×C₈	C₂²⋊C₈	1	1	yes	characteristic
47	C₂×C₈	C₄⋊C₈	1	1	yes	characteristic
48	C₂×C₈	C₂×C₁₆	1	1	yes	characteristic
49	C₂×C₈	M₃₂	1	1	yes	characteristic
50	C₂×C₈	M₃₂	1	1	yes	characteristic
51	C₄²	—	2	1	no	—
52	C₄²	—	2	1	no	—
53	C₁₆	—	2	2	no	—
54	C₁₆	#95	1	1	yes	characteristic
55	C₁₆	#97	1	1	yes	characteristic
56	C₁₆	C₂²⋊C₈	1	1	yes	characteristic
57	C₁₆	C₄⋊C₈	1	1	yes	characteristic
58	C₂³	C₂×C₃₂	1	1	yes	characteristic
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	2	2	no	—
61	C₂×C₄	C₆₄	1	4	yes	complemented
62	C₂×C₄	C₄×C₁₆	1	1	yes	characteristic
63	C₂×C₄	C₂²⋊C₁₆	1	1	yes	characteristic
64	C₂×C₄	C₄⋊C₁₆	1	1	yes	characteristic
65	C₂×C₄	C₂×C₃₂	1	1	yes	characteristic
66	C₂×C₄	M₆₄	1	1	yes	characteristic
67	C₂×C₄	M₆₄	1	1	yes	characteristic
68	C₈	—	2	2	no	—
69	C₈	#336	1	1	yes	characteristic
70	C₈	#346	1	1	yes	characteristic
71	C₈	C₂²⋊C₁₆	1	1	yes	characteristic
72	C₈	C₄⋊C₁₆	1	1	yes	characteristic
73	C₂²	C₄×C₃₂	1	1	yes	characteristic
74	C₂²	C₂²⋊C₃₂	1	1	yes	characteristic
75	C₂²	C₄⋊C₃₂	1	1	yes	characteristic
76	C₂²	C₂×C₆₄	1	2	yes	—
77	C₂²	M₁₂₈	1	2	yes	—
78	C₄	—	2	4	no	—
79	C₄	#1374	1	1	yes	characteristic
80	C₄	#1397	1	1	yes	characteristic
81	C₄	C₂²⋊C₃₂	1	1	yes	characteristic
82	C₄	C₄⋊C₃₂	1	1	yes	characteristic
83	C₂	#7465	1	1	yes	characteristic
84	C₂	C₄×C₆₄	1	1	yes	characteristic
85	C₂	#7510	1	1	yes	characteristic
86	C₂	C₂²⋊C₆₄	1	2	yes	—
87	C₂	C₄⋊C₆₄	1	2	yes	—

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₃₂	C₂²
commutator	C₂	C₄×C₆₄
socle, prime radical	C₂³	C₂×C₃₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₂ ◁ G
LECS	G ▷ C₂²×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
8. C₂×C₆₄	C₄	semidirect
61. C₂×C₄	C₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₆₄	2
C₂×C₄×C₃₂	2