Department reputation in the field is very important, particularly how they are seen by industry.
> well your research group is organized and how much the people there support you.
Controversial perhaps, but when academics are being dicks, nobody ever calls them out on it, academics close ranks and protect their own. Students come and go, professors are there for life.
If there are issues with ones research group, there is no support. Chose wisely, but it is hard to know in advance what sort of environment there really is and how things will really play out.
That's not my experience, actually. There is a lot of snobbery in some cases. Some places (UK) make a point of only hiring from certain universities. A friend of mine did his PhD at Birmingham - good university, and excellent at his field, but not a top-5 one. He was on the LIGO experiment, and on the first gravitational wave detection. Got grilled about "why on earth did you go to study at Birmingham???" at a job interview.
You might say, not a good place to work, and you'd be right, but the trend is real.
Really depends on the field. On the whole people who have a PhD in Not-Computer Engineering tend to end up working in fields connected to what their PhD was in.
I work in Civil Engineering and not only are there lots of PhDs around here most of them are working on things pretty close to what their research topic was, and where in fact specifically hired to do so. The PhDs who aren't working in their 'field' are generally more senior people who used to work or more relevant stuff but are now in more management roles.
I've noticed this as well and I'm not even a PhD student! For example, the Vrije Universiteit Amsterdam doesn't rank super high. Yet, when I look at their cyber security department, then they're top notch.
In most cases though, industry doesn't care about any of this. So only do it if you're actually interested in research. PhD to industry doesn't translate well (high variability in where people end up).