  We study maximal subalgebras of an arbitrary finite dimensional algebra over
a field, and obtain full classification/description results of such algebras.
This is done by first obtaining a complete classification in the semisimple
case, and then lifting to non-semisimple algebras. The results are sharpest in
the case of algebraically closed fields, and take special forms for algebras
presented by quivers with relations. We also relate representation theoretic
properties of the algebra and its maximal and other subalgebras, and provide a
series of embeddings between quivers, incidence algebras and other structures
which relate indecomposable representations of algebras and some subalgebras
via induction/restriction functors. Some results in literature are also
re-derived as a particular case, and other applications are given.
