Would a cocaroch survive a nuclear explosion?

not at ground zero

Cockroaches (and most insects) do have a higher chance of surviving nuclear radiation (e.g. from fallout) than do humans (and most mammals) because the insects have a higher LD50 than mammals do. LD50 stands for Lethal Dose 50% - the dose that will kill 50% of a group receiving that dose without medical care.

Here are approximate LD50s for gamma radiation for several species:

1. Cockroach : 5000 Rads
2. Rabbit : 800 Rads
3. Goldfish : 750 Rads
4. Rat : 600 Rads
5. Mouse : 450 Rads
6. Monkey : 450 Rads
7. Humans : 250-450 Rads