If $G$ is a finite subgroup of $G!L(3,{mathbb C})$, then $G$ acts on ${mathbb C}^3$, and it is known that ${mathbb C}^3/G$ is Gorenstein if and only if $G$ is a subgroup of $S!L(3,{mathbb C})$. In this work, the authors begin with a classification of finite subgroups of $S!L(3,{mathbb C})$, including two types, (J) and (K), which have often been overlooked. They go on to present a general method for finding invariant polynomials and their relations to finite subgroups of $G!L(3,{mathbb C})$. The method is, in practice, substantially better than the classical method due to Noether. Some properties of quotient varieties are presented, along with a proof that ${mathbb C}^3/G$ has isolated singularities if and only if $G$ is abelian and 1 is not an eigenvalue of $g$ for every nontrivial $g in G$. The authors also find minimal quotient generators of the ring of invariant polynomials and relations among them.