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A finite $p$-group $G$ is called \textit{powerful} if either $p$ is odd and$[G,G]\subseteq G^p$ or $p=2$ and $[G,G]\subseteq G^4$. A {\em{cover}} for agroup is a collection of subgroups whose union is equal to the entire group. Wewill discuss covers of $p$-groups by powerful $p$-subgroups. The size of thesmallest cover of a $p$-group by powerful $p$-subgroups is called the\textit{powerful covering number}. In this paper we determine the powerfulcovering number of the dihedral 2-groups.

On Covers of Dihedral 2-Groups by Powerful Subgroups