Simplify the given expression by factorization gives:

$$\frac{6^{r+s}\times 12^{r-s}}{8^r\times 9^{r+2s}}=\frac{2^{(r+s)+2(r-s)}\times 3^{(r+s)+(r-s)}}{2^{3r}\times 3^{2(r+2s)}}=\frac{2^{3r-s}\times 3^{2r}}{2^{3r}\times 3^{2r+4s}}=\frac{1}{2^s\times 3^{4s}}$$

In order for this number to be an integer, it must hold that $\boxed{s\le 0}$.