A terrific recent addition to the literature I reviewed for the MAA Online is *"Finite Group Theory*by I.Martin Theory" by I. Martin Issacs. It contains all the standard material one would expect in a graduate group theory text as well as a number of topics you don't normally see in such texts,like texts, like subnormality and the Chernoff measure.It's measure. It's quite a bit more advanced then than the usual group theory texts,as texts as well. All of it is presented beautifully with Issacs' usual authority and scholarship. A GREAT book for anyone interested in group theory with a basic knowledge of algebra.
For older and more standard texts,there's texts, there's always the old classic by Philip Hall.One Hall. One of the first of the post 1960's post-1960's texts and STILL one of the best.
And, of course, there's always John S.Rose's "A Course In Group TheoryTheory", available to all in Dover,thankfully.A Dover, thankfully. A classic with one of the most complete presentations of th the theory of group actions there is.