Functional testing comes under system testing.
Functional Testing : To test the software according to the 
prgrammable knowledge of the software. To test the output 
according to the requirements. To tset something what it 
supposed to do.
System Testing : To test the whole system by integrating it.
It is of two types. 1- Functional 2- Non functional
User acceptance testing, usability testing, localization 
testing etc. are coming under functional testing; but 
security testing, penetration testing, load, stress, volume 
testing etc. are coming under non functional testing.
So functional testing is a subset of system testing.

Functional testing? It is a test to check whether each and
every function of that application is working as per the
requirement (remember this work “as per requirement
document”-you must say this in the interview). It is a major
test where 80% of the tests are done. In this test, the Test
Cases are executed (or run).

What is System testing? When testers complete testing (The
testers test the application in the test environments,
meaning they test with the test data only, NOT with the real
data), the application (software) has to be tested in the
real environment. What it means is, since the testers test
it in the test environment with the test data, we have to
make sure that the application works well in the real
environment with the real data. In test environment, some of
the things cannot be simulated or tested. Al though the test
environment is very similar to the production (real)
environment, we need to make sure that we get a smooth
delivery in the real system as well (As servers are
different and database is different, things may not work as
expected when the application is moved from test environment
to production environment)