  In this paper, 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 generlized Riemann hypothesis, we establish
sharp upper and lower bounds for the $k$-th power moment of these $L$-values
for all real $k \geq 0$.
