In your particular case the fastest sort is probably the one described in this answer. It is exactly optimized for an array of 6 ints and uses sorting networks. It is 20 times (measured on x86) faster than library qsort. Sorting networks are optimal for sort of fixed length arrays. As they are a fixed sequence of instructions they can even be implemented easily by hardware.
Generally speaking there is many sorting algorithms optimized for some specialized case. The general purpose algorithms like heap sort or quick sort are optimized for in place sorting of an array of items. They yield a complexity of O(n.log(n)), n being the number of items to sort.
The library function qsort() is very well coded and efficient in terms of complexity, but uses a call to some comparizon function provided by user, and this call has a quite high cost.
For sorting very large amount of datas algorithms have also to take care of swapping of data to and from disk, this is the kind of sorts implemented in databases and your best bet if you have such needs is to put datas in some database and use the built in sort.