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This page shows the most symmetric solutions for 4x6 solutions with a stripe of squares along the diagonal.
There are a total of 806 solutions (344 and 462 for the two directions of stripe). For the first one, there are no solutions with solids in any corner, and for the second, solids never appear one away from a corner on the long side. (When the stripe is diamonds, there are NO solutions at all.)

The maximum symmetry is 50/58, or 86%.
SolidsMaximum
Symmetry/58
SolidsMaximum
Symmetry/58
0,2,11 38 1,3,6 37
0,2,16 30 1,3,13 39
0,2,21 43 1,3,20 41
0,11,12 45 1,6,10 35
0,12,16 35 1,6,17 45
0,12,21 39 1,10,20 36
2,9,11 37 1,13,17 40
2,9,16 39 1,17,20 39
2,9,21 50 6,8,10 39
2,11,14 37 6,8,17 50
2,14,16 40 6,10,15 39