What has the LNH done? It might be easier to answer what the LNH hasn't done.
They've travelled back in time to the creation of the Looniverse. They've repelled numerous alien invasions. They've foiled countless nefarious plots to take over the globe. They've been to other worlds, other times, and other planets. In short, there's very little they haven't done.
But you probably want specifics. There are a few sources for this kind of information.
This page last updated: 18 November 1997.