  In this paper we consider the solvability of pseudodifferential operators in
the case when the principal symbol vanishes of order $k \ge 2 $ at a nonradial
involutive manifold $\Sigma_2$. We shall assume that the operator is of
subprincipal type, which means that the $ k$:th inhomogeneous blowup at
$\Sigma_2$ of the refined principal symbol is of principal type with Hamilton
vector field parallel to the base $\Sigma_2$, but transversal to the symplectic
leaves of $\Sigma_2$ at the characteristics. When $k = \infty $ this blowup
reduces to the subprincipal symbol. We also assume that the blowup is
essentially constant on the leaves of $\Sigma_2$, and does not satisfying the
Nirenberg-Treves condition (${\Psi}$). We also have conditions on the vanishing
of the normal gradient and the Hessian of the blowup at the characteristics.
Under these conditions, we show that $P$ is not solvable.
