How to make symbols get vertically aligned as elements of a set?

I need to vertically align each element in the set instead of floating above the commas.

\documentclass[border=3pt]{standalone}\usepackage{amsmath,mathtools,bclogo}\begin{document}\begin{minipage}{7cm}\abovedisplayskip=0pt\relax\begin{align*}\Omega &=\{\bccube,\bctetraedre,\bcoctaedre\}\\2^\Omega &= \begin{multlined}[t]\{\{\},\\\{\bccube\},\{\bctetraedre\},\{\bcoctaedre\}, \\\{\bccube,\bctetraedre\},\{\bccube,\bcoctaedre\},\{\bctetraedre,\bcoctaedre\},\\\{\bccube,\bctetraedre,\bcoctaedre\}\}\end{multlined}\end{align*}\end{minipage}\end{document}

Extra problems: I also need to split the power set into 4 rows and each row is left aligned.