Automorphism groups of smooth quartic threefolds

We classify finite groups of automorphisms of smooth quartic threefolds over $\mathbb{C}$. Every such group embeds into one of $17$ maximal groups, which we realize by explicit equations and matrix generators. Among finite subgroups of $\mathrm{PGL}(5,\mathbb{C})$ preserving a smooth quartic threefold, there are exactly $24$ maximal ones up to conjugacy, and we use them to determine the $194$ groups that occur as full automorphism groups. As an application, we classify the finite groups of projective automorphisms of smooth quartic surfaces and show that there are exactly $12$ maximal ones up to conjugacy in $\mathrm{PGL}(4,\mathbb{C})$.