Subgroups — 3₋¹⁺²×M₆₄ #13136113

Order 1728.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₆×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	3₋¹⁺²×C₃₂	C₂	1	2	yes	maximal, complemented
3	C₃²×M₆₄	C₃	1	1	yes	maximal, characteristic
4	C₉×M₆₄	C₃	1	3	yes	maximal, complemented
5	C₂×C₈×3₋¹⁺²	C₄	1	1	yes	characteristic
6	C₁₆×3₋¹⁺²	C₄	1	1	yes	characteristic
7	C₁₆×3₋¹⁺²	C₂²	1	1	yes	characteristic
8	C₂×C₃²×C₁₆	C₂×C₃	1	1	yes	characteristic
9	C₃²×C₃₂	C₂×C₃	1	2	yes	—
10	C₂×C₉×C₁₆	C₂×C₃	1	3	yes	—
11	C₉×C₃₂	C₂×C₃	1	6	yes	complemented
12	C₂×C₄×3₋¹⁺²	C₈	1	1	yes	characteristic
13	C₈×3₋¹⁺²	C₈	1	1	yes	characteristic
14	C₈×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
15	C₃×M₆₄	—	3	1	no	—
16	C₃×M₆₄	C₃²	1	1	yes	characteristic
17	C₂×C₃²×C₈	C₃×C₄	1	1	yes	characteristic
18	C₃²×C₁₆	C₃×C₄	1	1	yes	characteristic
19	C₃²×C₁₆	C₂²×C₃	1	1	yes	characteristic
20	C₂×C₈×C₉	C₃×C₄	1	3	yes	—
21	C₉×C₁₆	C₃×C₄	1	3	yes	—
22	C₉×C₁₆	C₂²×C₃	1	3	yes	—
23	C₂²×3₋¹⁺²	C₁₆	1	1	yes	characteristic
24	C₄×3₋¹⁺²	C₁₆	1	1	yes	characteristic
25	C₄×3₋¹⁺²	C₂×C₈	1	1	yes	characteristic
26	C₂×C₃×C₁₆	—	3	1	no	—
27	C₂×C₃×C₁₆	C₂×C₃²	1	1	yes	characteristic
28	C₃×C₃₂	—	3	2	no	—
29	C₃×C₃₂	C₂×C₃²	1	2	yes	—
30	C₂×C₃²×C₄	C₃×C₈	1	1	yes	characteristic
31	C₃²×C₈	C₃×C₈	1	1	yes	characteristic
32	C₃²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
33	C₂×C₄×C₉	C₃×C₈	1	3	yes	—
34	C₈×C₉	C₃×C₈	1	3	yes	—
35	C₈×C₉	C₂×C₃×C₄	1	3	yes	—
36	M₆₄	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
37	C₂×3₋¹⁺²	—	2	1	no	—
38	C₂×3₋¹⁺²	C₂×C₁₆	1	1	yes	characteristic
39	C₂×C₃×C₈	—	3	1	no	—
40	C₂×C₃×C₈	C₃²×C₄	1	1	yes	characteristic
41	C₃×C₁₆	—	3	1	no	—
42	C₃×C₁₆	—	3	1	no	—
43	C₃×C₁₆	C₃²×C₄	1	1	yes	characteristic
44	C₃×C₁₆	C₂²×C₃²	1	1	yes	characteristic
45	C₂²×C₃²	C₃×C₁₆	1	1	yes	characteristic
46	C₃²×C₄	C₃×C₁₆	1	1	yes	characteristic
47	C₃²×C₄	C₂×C₃×C₈	1	1	yes	characteristic
48	C₂²×C₉	C₃×C₁₆	1	3	yes	—
49	C₄×C₉	C₃×C₁₆	1	3	yes	—
50	C₄×C₉	C₂×C₃×C₈	1	3	yes	—
51	C₂×C₁₆	C₂×3₋¹⁺²	1	1	yes	characteristic
52	C₃₂	C₂×3₋¹⁺²	1	2	yes	complemented
53	3₋¹⁺²	M₆₄	1	1	yes	characteristic, direct product factor, Sylow
54	C₂×C₃×C₄	—	3	1	no	—
55	C₂×C₃×C₄	C₃²×C₈	1	1	yes	characteristic
56	C₃×C₈	—	3	1	no	—
57	C₃×C₈	—	3	1	no	—
58	C₃×C₈	C₃²×C₈	1	1	yes	characteristic
59	C₃×C₈	C₂×C₃²×C₄	1	1	yes	characteristic
60	C₂×C₃²	—	2	1	no	—
61	C₂×C₃²	C₂×C₃×C₁₆	1	1	yes	characteristic
62	C₂×C₉	—	2	3	no	—
63	C₂×C₉	C₂×C₃×C₁₆	1	3	yes	—
64	C₂×C₈	C₄×3₋¹⁺²	1	1	yes	characteristic
65	C₁₆	C₄×3₋¹⁺²	1	1	yes	characteristic
66	C₁₆	C₂²×3₋¹⁺²	1	1	yes	characteristic
67	C₂²×C₃	—	3	1	no	—
68	C₂²×C₃	C₃²×C₁₆	1	1	yes	characteristic
69	C₃×C₄	—	3	1	no	—
70	C₃×C₄	—	3	1	no	—
71	C₃×C₄	C₃²×C₁₆	1	1	yes	characteristic
72	C₃×C₄	C₂×C₃²×C₈	1	1	yes	characteristic
73	C₃²	C₃×M₆₄	1	1	yes	characteristic
74	C₉	C₃×M₆₄	1	3	yes	complemented
75	C₂×C₄	C₈×3₋¹⁺²	1	1	yes	characteristic
76	C₈	C₈×3₋¹⁺²	1	1	yes	characteristic
77	C₈	C₂×C₄×3₋¹⁺²	1	1	yes	characteristic
78	C₂×C₃	—	2	1	no	—
79	C₂×C₃	—	3	1	no	—
80	C₂×C₃	—	6	1	no	—
81	C₂×C₃	C₂×C₃²×C₁₆	1	1	yes	characteristic
82	C₂²	C₁₆×3₋¹⁺²	1	1	yes	characteristic
83	C₄	C₁₆×3₋¹⁺²	1	1	yes	characteristic
84	C₄	C₂×C₈×3₋¹⁺²	1	1	yes	characteristic
85	C₃	—	3	1	no	—
86	C₃	C₃²×M₆₄	1	1	yes	characteristic
87	C₂	—	2	1	no	—
88	C₂	C₂×C₁₆×3₋¹⁺²	1	1	yes	characteristic

	subgroup	G/H
center, Frattini, cosocle	C₃×C₁₆	C₂²×C₃²
commutator, socle	C₂×C₃	C₂×C₃²×C₁₆
prime radical	C₂²×C₃²	C₃×C₁₆
LCS	G ▷ C₂×C₃ ▷ C₁
UCS	C₁ ◁ C₃×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
2. 3₋¹⁺²×C₃₂	C₂	semidirect
4. C₉×M₆₄	C₃	semidirect
11. C₉×C₃₂	C₂×C₃	semidirect
36. M₆₄	3₋¹⁺²	direct (N × H)
52. C₃₂	C₂×3₋¹⁺²	semidirect
53. 3₋¹⁺²	M₆₄	direct (N × H)
74. C₉	C₃×M₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₆×3₋¹⁺²	2
3₋¹⁺²×C₃₂	2
C₃²×M₆₄	3
C₉×M₆₄	3

Subgroups — 3−1+2×M64 #13136113

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×M₆₄ #13136113