  A notion of intermediate vertex subalgebras of lattice vertex operator
algebras is introduced, as a generalization of the notion of principal
subspaces. Bases and the graded dimensions of such subalgebras are given.As an
application, it is shown that the characters of some modules of an intermediate
vertex subalgebra between $E_7$ and $E_8$ lattice vertex operator algebras
satisfy some modular differential equations.This result is an analogue of the
result concerning the "hole" of the Deligne dimension formulas and the
intermediate Lie algebra between the simple Lie algebras $E_7$ and $E_8$.
