Maximus as praenomen

Hi,

I'm reading on behindthename.com that the name of Maximilian I, Holy Roman Emperor "was a blend of the names of the Roman generals Fabius Maximus and Cornelius Scipio Aemilianus."

I have a question about Maximus and would like a confirmation that Aemilianus was always used to mean "son of Aemilius."

Now, Maximus is always a cognomen? Or can it be a praenomen as well? And take for instance Fabius Maximus. If Maximus is here a cognomen, where's the family name??

Thank you!