In general 1024x768 should be enough to do dragging and dropping of device to new group.
What are the version and build numbers of your MobiControl server?
What browser and which version are you using to open the web-console?
How many level do you have in your device group tree?
What zoom level did you use in your browser session?
Doesn't appear to matter. Got back into the office today on my normal monitor and it appears that if your browser window is less than 1281 px wide you can't, at least I can't display both the devices frame and the group frame at the same time. I tried both firefox and chrome. If I resize the horizontal width of the window to 1280 or less, the devices frame goes dark.