Poset-blowdowns of generalized quaternion groups

Poset-blowdown of subgroup posets of groups is an analog of blowdown in algebraic geometry. It is a poset map obtained by contracting normal subgroups. For finite groups, this is considered as a map between the Hasse diagrams of the subgroup posets. Poset-blowdowns are classified into three types: \textit{tame, wild}, and \textit{hybrid} depending on the sizes of their fibers. In this paper we describe the poset-blowdowns for generalized quaternion groups $Q_{2^n}$ $(n \geq 3)$. They have distinguished nature in that all types (tame, wild, and hybrid) appear in the successive poset-blowdowns associated with the three chief series of $Q_{2^n}$.