This mixer does not seem to have recording outputs at a glance. You can use the insert plugs instead but they will be unbalanced and thus prone to hum and other noise insertion.
If you had recording outputs, something like an RME Multiface (an age-old high-end soundcard needing either a PCI, Pccard, or Expresscard interface) will take 8 inputs and deliver 8 outputs. That still puts you at your current limit. You can extend this with additional external A/D-converters that have syncable clock (the Multiface can work as clock master or clock slave) and use ADAT.
Without using all this external stuff, something like a Mackie Onyx 1640 with the optional Firewire card installed or the Mackie Onyx 1640i will deliver 16 channels on Firewire. However, driver situation may look dire by now on Windows or MacOSX (with official support ended). I use a Mackie Onyx 1620 myself (only 8 mic inputs and 4 stereo line inputs) and it works fine under Linux and Ffado with the Firewire card (which sends 18 channels: the 16 input channels and the finished main mix). The Firewire card only permits 2 channels back into the mixer; in contrast, the native Mackie Onyx 1640i allows 16 channels back in. I can't vouch for the latter's function, though.
This mixer+interface combination is actually only strictly necessary for live performances (it records all inputs before faders and EQ, so the live mix does not affect the recording). If you only want to record, an interface alone will do the trick.