If your Hardware ID shows VEN_10EC&DEV_8168 , you need a Realtek PCIe GBE Family Controller driver.
To get your Esonic H61 motherboard's LAN working, you primarily need the Realtek Fast Ethernet . Most H61 models from Esonic use the Realtek 8105E or similar 10/100 Mbps chipset. Download Options Official Resource esonic h61 motherboard lan driver