Given a finite dimensional local commutative algebra over a finite field $K$ and a finite dimensional module $M$. What is the fastest/best way to obtain all submodule from $M$ using a Computer algebra system like GAP/QPA? It might be also interesting to hear how you might do that by hand.
Here is how you can do it with QPA:
gap> A := NakayamaAlgebra(GF(3), [4]);
<GF(3)[<quiver with 1 vertices and 1 arrows>]/<twosided ideal in <GF(3)[<quiver with 1 vertices and 1 arrows>]>, (1 generators)>>
gap> M := IndecProjectiveModules(A)[1];
<[ 4 ]>
gap> subs := AllSubmodulesOfModule(M);
[ [ <<[ 0 ]> > <[ 4 ]>> ], [ <<[ 1 ]> > <[ 4 ]>> ], [ <<[ 2 ]> > <[ 4 ]>> ], [ <<[ 3 ]> > <[ 4 ]>> ],
[ <<[ 4 ]> > <[ 4 ]>> ] ]
I hope that this is helpful.
Best regards, the QPAteam.
The generic method for submodules would be to write down a matrix representation and to use MeatAxe tools  in GAP there is e.g. a function MTX.BasesSubmodules
. Since your algebra is commutative, there might be better adhoc methods, e.g. by looking at generalized eigenspaces of the generator matrices first, but I'm not aware of any turnkey implementation of such an approach.

$\begingroup$ Thanks. Can you give an example? For example using A:=NakayamaAlgebra([n],GF(3)) for n=3 or 4 from QPA. This gives $K[x]/(x^n)$ with K=GF(3) and as module the radical. $\endgroup$ – Mare Sep 4 '17 at 14:02
