Book on finite group theory, containing a sufficient number of examples

Why not try one of the following:

John S. Rose, A Course on Group Theory

Derek J.S. Robinson, A Course in The Theory of Groups

Dummit and Foote, Abstract Algebra.

Each of these books has a lot of good examples and exercises!

Groups: A Path to Geometry by R.P. Burn is an introduction to group theory that consists entirely of examples, problems and solutions.

Schaum's Outline of Group Theory by B. Baumslag contains lots of examples and problems with solutions.

Adventures in Group Theory: Rubik's Cube, Merlin's Machine, and Other Mathematical Toys by David Joyner is built around a series of concrete examples and applications of groups.

A very literal answer: Michael Weinstein has a book called Examples of groups.