Given the group SU(N) of NxN unitary matrices, does there exist a subgroup S
with a manifold dimension larger than the SU(N-1) manifold dimension and
smaller than the SU(N) one? S should not necessarily have SU(N-1) as subgroup.



subgroup of SU(N) with maximal manifold dimension 
Your question is equivalent to the question about maximal dimension of a proper Lie subalgebra
of $su(N)$. Clearly such subalgebra is reductive since it has a positive definite invariant form.
Thus you are looking for a reductive Lie algebra of maximal possible dimension strictly smaller
than $N^2-1$ with
faithful representation $V$ of dimension $N$. We claim that this dimension is $(N-1)^2$ for $N>4$.
Here is a sketch of argument:


1) representation $V$ is irreducible: if it splits into direct sum of subrepresentations of dimensions $a$ and $b$, then dimension of our Lie algebra is less than $a^2+b^2\le (N-1)^2$ (one needs to
consider the special case when $a=1$ separately).


2) representation $V$ is not a tensor product of irreducible representations: if it is a tensor
product of representations of dimensions $a$ and $b$, then our Lie algebra has dimension less than $a^2+b^2\le (N/2)^2+(N/2)^2=N^2/2\le (N-1)^2$.


3) it follows from 2) above that our Lie algebra is in fact simple Lie algebra. The smallest possible dimension of an irreducible representation of any simple Lie algebra is well known 
(this is always one of the fundamental representations). Now quick search gives you the result
(and the counterexample pointed out by Somnath). 


