Bounds for moments of quadratic Dirichlet $L$-functions of prime-related moduli

We study the $k$th moment of central values of the family of quadratic Dirichlet $L$-functions of moduli $8p$, with $p$ ranging over odd primes. Assuming the truth of the generalized Riemann hypothesis, we establish sharp upper and lower bounds for the $k$th power moment of these $L$-values for all real $k \geq 0$.