TITLE: What are a couple of examples of finite sized but interesting categories?
QUESTION [6 upvotes]: I'm studying category theory and, given that I don't have a background in topology, I'm struggling to think of some finite categories that interesting.
The main one I know of is finite preorders -- I find this category interesting because it has products, sums, etc. I'd love a couple of other categories like that that I could use to better understand (by graphing out the objects and morphisms) things like equalizers, pullbacks, etc.
I could of course do this by construction, but I'd much prefer some finite categories that include these ideas in a useful way.
Thank you!

REPLY [2 votes]: A well-known example is the equivalence between groups and
and one-object categories with all morphisms invertible
(i.e. all morphisms are isomorphisms).  The group is finite
iff the category is.
A more exotic example of a finite category with all morphisms invertible is
Conway's $M_{13}$ groupoid
(see also this
blog entry by John Baez, 
who is fond of most things categorical).
Conway calls it "$M_{13}$" because there are $13$ objects each of whose
automorphism groups is isomorphic with the Mathieu group $M_{12}$ 
(and any two objects are connected by $|M_{12}|$ isomorphisms).