i'm not familiar with the cough syrup but for the time being i'll assume it has either codeine in it or hydrocodone. The simplest most basic urine test does indeed look for opiates, but it only looks for heroin, morphine etc (and possibly codeine). In order to find a related drug like hydrocodone, which is the active drug in Lortab, then an expanded test is required.
However, in either setting, these drugs will not come up as oxycontin because (assuming the syrup doesn't have any percocet or oxycodone in it) these drugs are not in fact oxycodone (which is the active opiate in oxycontin, percocet percodan etc). Granted they're molecularly very similar but if a test is specifically looking for oxycodone it shouldn't be able to find it in someone taking hydrocodone products perse.
Another thing to keep in mind is that hydrocodone products have a very very short half-life- they're designed to be in your body for a brief time. So assuming you haven't been taking lots of Lortab for an extended period of time, you can expect to pass a urine test looking for hydrocodone within 2-4 days of taking the drug, even fewer if you drink lots of water and flush your system.
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