  The intrinsic symmetries of physical systems have been employed to reduce the
number of degrees of freedom of systems, thereby simplifying computations. In
this work, we investigate the properties of $\mathcal{M}SU(2^N)$,
$\mathcal{M}$-invariant subspaces of the special unitary Lie group $SU(2^N)$
acting on $N$ qubits, for some $\mathcal{M}\subseteq M_{2^N}(\mathbb{C})$. We
demonstrate that for certain choices of $\mathcal{M}$, the subset
$\mathcal{M}SU(2^N)$ inherits many topological and group properties from
$SU(2^N)$. We then present a combinatorial method for computing the dimension
of such subspaces when $\mathcal{M}$ is a representation of a permutation group
acting on qubits $(GSU(2^N))$, or a Hamiltonian $(H^{(N)}SU(2^N))$. The
Kronecker product of $\mathfrak{su}(2)$ matrices is employed to construct the
Lie algebras associated with different permutation-invariant groups $GSU(2^N)$.
Numerical results on the number of dimensions support the the developed theory.
