  In this paper, we study the Hasse norm principle for some non-Galois
extensions of number fields. Our main theorem is that for any square-free
composite number $d$ which is divisible by at least one of $3$, $55$, $91$ or
$95$, there exists a finite extension of degree $d$ for which the Hasse norm
principle fails. To accomplish it, we determine the structure of the
Tate--Shafarevich groups of norm one tori for finite extensions of degree $d$
under the normality of $p$-Sylow subgroups of the Galois groups of their Galois
closures for a square-free prime factor $p$ of $d$. Moreover, we reduce the
assertion to an investigation of $2$-dimensional $\mathbb{F}_p$-representations
of some groups of order coprime to $p$.
