  In this work, we consider an initial-boundary value problem for a
time-fractional biharmonic equation in a bounded polygonal domain with a
Lipschitz continuous boundary in $\mathbb{R}^2$ with clamped boundary
conditions. After establishing the well-posedness, we focus on some regularity
results of the solution with respect to the regularity of the problem data. The
spatially semidiscrete scheme covers several popular lowest-order
piecewise-quadratic finite element schemes, namely, Morley, discontinuous
Galerkin, and $C^0$ interior penalty methods, and includes both smooth and
nonsmooth initial data. Optimal order error bounds with respect to the
regularity assumptions on the data are proved for both homogeneous and
nonhomogeneous problems. The numerical experiments validate the theoretical
convergence rate results.
