I remember one of my school textbooks referring to the 741 as the "Model T Ford" of op-amps.
At the time it was written, I think this comparison was meant as a compliment - the standard, generic op-amp that anyone can afford and lots of people use. Nowadays it remains just as true now as it ever was; the 741 is utterly obsolete, and whilst there may still be quite a few around, they really have historical value only. There's no useful comparison with modern alternatives.
Both the input and output voltages are limited compared to the power supply - so, for example, if your're using it as a non-inverting amplifier with a gain of 3 or thereabouts, the 5.6V minimum output you're seeing corresponds to a minimum input voltage of about 1.8V above GND. That looks prettty normal for a 741, the data sheet is rather vague.
What you need is an op-amp with input and output voltages that include GND, and there's plenty of them out there. Search for ones that have rail-to-rail input and output capabilities if you want to avoid this problem in future.