The symmetry for the green pixels is fairly simple. The box below denotes the unit cell of the pattern. Translating it in two dimensions creates the entire pattern. Within this unit cell, the red dots make up an asymmetric unit. Combined with a four-fold rotation axis perpendicular to the plane of the pattern, located in the center of the unit cell (blue dot), this asymmetric unit can recreate the entire pattern.
It's not so simple for the red and blue pixels. As far as I can see, the unit cell of that pattern is indicated below, but it's early in the morning still... There is no simple asymmetric unit within this unit cell.
That unit cell seems to be generated by an asymmetric unit in combination with rotational and translational symmetry. For example, the two smaller boxes in purple combined are sufficient to create the entire pattern by translating them in one direction (left and right), but down and up, it is shifted by half a unit cell length. The two boxes themselves are related by a 90-degree rotation around the center of the box and a translation of half a unit cell length.
Now, why am I rambling on about these things? Because they are at the heart of efficient computational routines that deal with such patterns, for example, demosaicing algorithms. When the underlying symmetry of such patterns can be defined in its simplest terms, the algorithms will generally be fast and efficient.
Carry on!: -D