  In present paper, we study the following nonlinear Schr\"{o}dinger equation
with combined power nonlinearities
  \begin{align*}
  - \Delta u+V(x)u+\lambda u=|u|^{2^*-2}u+\mu |u|^{q-2}u \quad \quad \text{in}
\ \mathbb{ R}^N, \ N\geq 3
  \end{align*}
  having prescribed mass
  \begin{align*}
  \int_{ \mathbb{ R}^N}u^2dx=a^2,
  \end{align*} where $\mu, a>0$, $q\in(2, 2^*)$, $2^*=\frac{2N}{N-2}$ is the
critical Sobolev exponent, $V$ is an external potential vanishing at infinity,
and the parameter $\lambda\in \mathbb{R}$ appears as a Lagrange multiplier.
Under some mild assumptions on $V$, for the $L^2$-subcritical perturbation
$q\in(2, 2+\frac{4}{N})$, we prove that there exists $a_0>0$ such that the
normalized solution with negative energy to the above problem with $\mu>0$ can
be obtained for $a\in (0, a_0)$; for the $L^2$-critical perturbation
$q=2+\frac{4}{N}$, by limiting the range of $\mu$, the positive ground state
normalized solution to the above problem for any $a>0$ is also found with the
aid of the Poho\v{z}aev constraint; moreover, for the $L^2$-supercritical
perturbation $q\in( 2+\frac{4}{N}, 2^*)$, we get a positive ground state
normalized solution for the above problem with $a>0$ and $\mu>0$ by using the
Poho\v{z}aev constraint.
  At the same time, the exponential decay property of the positive normalized
solution is established, which is important for the instability analysis of the
standing waves. Furthermore, we give a description of the ground state set and
obtain the strong instability of the standing waves for $q\in[2+\frac{4}{N},
2^*)$. This paper can be regarded as a generalization of Soave [J. Funct. Anal.
(2020)] in a sense.
