I've been able to come up with a number of categories. Here are three of them as a partial answer (ASCII art because I'm having trouble figuring out the current state of LaTeX tables on SE):

>! <pre>
>!                               Cu Ga Ru Lu Tl Po Pa
>! named after a country            X  X        X
>! prime atomic number           X  X     X
>! post-transition metal            X        X  X</pre>