  Making the first steps towards a classification of simple partial comodules,
we give a general construction for partial comodules of a Hopf algebra \(H\)
using central idempotents in right coideal subalgebras and show that any
\(1\)-dimensional partial comodule is of that form. We conjecture that in fact
all finite-dimensional simple partial \(H\)-comodules arise this way. For \(H =
kG\) for some finite group \(G\), we give conditions for the constructed
partial comodule to be simple, and we determine when two of them are
isomorphic. If \(H = kG^*,\) then our construction recovers the work of M.
Dokuchaev and N. Zhukavets. We also study the partial modules and comodules of
the non-commutative non-cocommutative Kac-Paljutkin algebra \(\mathcal{A}\).
