There is a nice semiregular tessellation with sixfold rotational symmetry made of hexagons, squares and triangles. Its grid is easy to find. From two of these grids we get a rather complicated tiling with twelvefold rotational symmetry:
Note that we have now some single hexagons. At the lower right we have a large nearly symmetric structure. Especially if one thinks of units of one rhomb together with a triangle or two hexagons together with two rhombs. In these units there are always two possible positions for the triangle or the hexagon. If these are not distinguished, then there would be much more symmetry.
An interesting structure arises in the lower left, up to a ring with connected squares. Alternating, there are single squares and four squares making a larger square. As the double grid becomes very complicated I do not further look at this.