Consider the matrix

$$ P = \begin{pmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix}. $$

Let the transpose of a matrix $X$ be denoted by $X^T$. Then the number of $3 \times 3$ invertible matrices $Q$ with integer entries, such that

$$ Q^{-1} = Q^T \quad \text{and} \quad PQ = QP, $$

is