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The topic of this thesis is the theory of holomorphic extensions of orbifolds of holomorphic vertex operator algebras. We develop an orbifold theory for non-abelian groups of the form $\mathbb{Z}_q \rtimes \mathbb{Z}_p$ and give an explicit formula for the characters of the holomorphic extensions. Additionally, we show how groups of the form $\mathbb{Z}_q \rtimes \mathbb{Z}_p$ can be realised as automorphism groups of lattice vertex operator algebras. Furthermore, we construct the action of automorphisms on twisted modules for lattice vertex operator algebras and give explicit expressions for the corresponding graded trace functions. Using these results, we conduct a survey of orbifolds by cyclic groups and groups of the form $\mathbb{Z}_q \rtimes \mathbb{Z}_p$ of lattice vertex operator algebras associated to extremal lattices in dimensions $48$ and $72$. This is the first systematic survey of holomorphic vertex operator algebras at higher central charge, as well as the first systematic survey of non-abelian orbifold theories. We construct around $200$ new holomorphic vertex operator algebras at central charge $c=48$ and $c=72$. These include those with the lowest number of low-weight states currently known among holomorphic vertex operator algebras of the respective central charges. These allow us to prove that there exist holomorphic vertex operator algebras that cannot be constructed as abelian orbifolds of lattice vertex operator algebras. Finally, we investigate the notion of `large-central charge limit' of a family of vertex operator algebras. We define a vertex algebra that can be considered such a large-central charge limit for a family of permutation orbifolds and give a sufficient condition for its existence.