Consider the planes (100), (001), and (111) of an fcc crystal. The indices refer to the conventional cubic unit cell.
What are the indices of these planes when referred to the primitive unit cell of the fcc lattice? The primitive lattice vectors are,
$$ \vec{a}_1=\frac{a}{2}(\hat{x}+\hat{z}),\quad \vec{a}_2=\frac{a}{2}(\hat{x}+\hat{y}),\quad\vec{a}_3=\frac{a}{2}(\hat{y}+\hat{z}),$$where $a$ is the lattice constant.