It is used extensively in Quantum Physics. For example, both Michael Tinkham and Volker Heine have written books titled "Group Theory and Quantum Physics".
Also group theory can be used to determine which polynomial equations can be solved in terms of radicals. (That was, in fact, the main reason for the development of group theory.) That is still mathematics but not what I would consider "pure" mathematics.
You use group theory to solve Rubik's cube! I was going to post a link but there are so many I'll just give the Google search and you can pick and choose. There are books on the subject and even college courses.
it is also used in chemistry to study molecular symmetry (many chemicals have chemical properties that derive partially or wholly from the types of symmetry they possess). this is especialy important in crystallography and spectroscopy.
group theory is an important part of cryptography (the study of codes), and plays a prominent role in understanding such techniques as RSA encryption.
cyclic groups play a role in musicology, and form a mathematical basis for the 12-tone system.
group theory is important in algebraic topology, which finds application, among other things, in design of computer networks and communication systems.
finally, there is group theory in symmetric artistic design elements (the so-called "frieze groups" and "ornament groups") used in textiles, printing, wall-papers, tiling (especially in arabic mosques), and such common-place items as kaleidoscopes (reflections form a group).
in short, groups are everywhere! if you've ever "counted by 2's or 3's" you were using intuitively a natural group property of the integers. the shuffling of a deck of cards, rotating a sphere in space, even our way of using a circle to measure time, these are all groups at work in our everyday lives.