jmc

For every divisor $d$ of $12$, the number $12/d$ is also a divisor of $12$. Their product is $d\cdot (12/d)=12$. It follows that every divisor can be paired with another divisor of $12$ such that their product is $12=2^2\cdot 3$. There are $(2+1)(1+1)=6$ divisors of $12$, namely $1,2,3,4,6,12$. Thus, the product of the divisors is given by $12^{6/2}=12^3.$ Since we need to exclude $12$ itself, the answer is $\frac{12^3}{12}=\boxed{144}$.