jmc

There are 8 Republicans and 3 spots for them, so there are $\left(\genfrac{}{}{0ex}{}{8}{3}\right)=56$ ways to choose the Republicans. There are 6 Democrats and 2 spots for them, so there are $\left(\genfrac{}{}{0ex}{}{6}{2}\right)=15$ ways to choose the Democrats. So there are $56\times 15=\boxed{840}$ ways to choose the subcommittee.