What is the maximum value of Easting in UTM coordinates?
You are correct about the maximum value, you just have the easting and northing backwards.

In your first example, the easting is 414668 m and the northing is 6812844 m. The convention is eastings first. You can think of UTM coordinates as $(x, y)$ pairs, and they are sometimes labelled as such. Here's the conversion offered by one converter:



As you say, eastings are usually less than 834 000 m, and more than 160 000 m. 

Note than some UTM zones are non-standard widths. Also, you will sometimes see coordinates referenced to some neighbouring zone, which isn't a good idea but is sometimes convenient, so the numbers can be unexpectedly large. Lastly: be aware that in the US you may sometimes come across UTM coordinates in other units, such as feet. Horrible.