Consider the matrix
$$
P=\left(\begin{array}{lll}
2 & 0 & 0 \\
0 & 2 & 0 \\
0 & 0 & 3
\end{array}\right)
$$
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 $Q^{-1}=Q^T$ and $P Q=Q P$ is
