  We deal with the Sobolev space theory for the stochastic partial differential
equation (SPDE) driven by Wiener processes $$ \partial_{t}^{\alpha}u=\left(
\phi(\Delta) u +f(u) \right) + \partial_t^\beta \sum_{k=1}^\infty \int_0^t
g^k(u)\,dw_s^k, \quad t>0, x\in \mathbb{R}^d; \,\,\, u(0,\cdot)=u_0 $$ as well
as the SPDE driven by space-time white noise $$
\partial^{\alpha}_{t}u=\phi(\Delta)u + f(u) + \partial^{\beta-1}_{t}h(u)
\dot{W}, \quad t>0,x\in \mathbb{R}^d; \quad u(0,\cdot)=u_{0}. $$
  Here, $\alpha\in (0,1), \beta\in (-\infty, \alpha+1/2)$, $\{w_t^k :
k=1,2,\cdots\}$ is a family of independent one-dimensional Wiener processes,
and $\dot{W}$ is a space-time white noise defined on $[0,\infty)\times
\mathbb{R}^d$. The time non-local operator $\partial_{t}^{\gamma}$ denotes the
Caputo fractional derivative if $\gamma>0$ and the Riemann-Liouville fractional
integral if $\gamma\leq0$. The the spatial non-local operator $\phi(\Delta)$ is
a type of integro-differential operator whose symbol is $-\phi(|\xi|^2)$, where
$\phi$ is a Bernstein function satisfying \begin{equation*}
\kappa_0\left(\frac{R}{r}\right)^{\delta_{0}} \leq \frac{\phi(R)}{\phi(r)},
\qquad \forall\,\, 0<r<R<\infty \end{equation*} with some constants
$\kappa_0>0$ and $\delta_0\in (0,1]$.
  We prove the uniqueness and existence results in Sobolev spaces, and obtain
the maximal regularity results of solutions.
