Living beings, I believe, is a strictly broader group than conscious beings. I'm not claiming to be certain about this - it's a belief I've taken for granted most of my life - but it would take a good argument to convince me otherwise.
There are plenty of different ways to define life, and some of them include entities that, I believe, are clearly not conscious. For example, starting way at the bottom, I strongly believe these things are not conscious: genes, DNA, RNA, proteins, prions, viruses. I also assume that some other things are not conscious, such as single cells. Beyond that, things get more murky.
As far as I can tell, "being conscious" and "having a mind" are equivalent. I can't define what a mind is, but I can say it would be hard for me to accept that something could have a mind without anything resembling a brain. Plants seem not to have anything resembling a brain.
Of course, I could be wrong about that. Mechanically, a brain is nothing but a complex network of electrochemical signaling devices (plus a bunch of sensors and actuators, but I'll ignore those for now even though they appear to be crucial), and there are lots of other places to find electrochemical signals, including both within and between plants. There just aren't any that seem to resemble the structure, function, or complexity of a brain.