Lattice Points

Probability

A lattice point is defined as a point in the two-dimensional plane with integral coordinates. We define the centroid of four points $(x_i,y_i), i=1,2,3,4$ as point $\left(\frac{x_1+x_2+x_3+x_4}4,\frac{y_1+y_2+y_3+y_4}4\right).$

What is the largest number of distinct lattice points in the plane such that the centroid of any four of them is not a lattice point?