Ah, interesting screenshot! However it does actually tell us the GPU... it's an 865G IGP. Intel's very old integrated graphics. (I'm not kidding, my very first self-built computer used the 865P chipset

) Unfortunately I think that also solves the cause of the issue, I can't find the right page but I'm pretty sure this IGP does not support the requirements needed to offer standby mode.
To be absolutely sure you can try installing this:
865G chipset drivers
LINK
Install, reboot, then install this:
865G Graphics driver
LINK
After another reboot, if Windows 7 doesn't allow you to use Standby then it's a safe bet the integrated graphics doesn't support it.