I'm assuming you want to know more about group theory, so here is an answer to point you to some of the interesting things that you can learn if you are interested in computing with finite groups.
(i) If you have the conjugacy classes and there are not too many, then writing normal subgroups as unions of conjugacy classes is fast and easy. This is often the case when you only have access to a group through its character table, such as when the Suzuki and the sporadic simple groups were being discovered, and we wanted to understand some of their (non-simple) subgroups.
For a typical finite group given concretely as a permutation group, you use a special type of induction working along what is called a chief series, where you find a maximal chain of normal subgroups. It turns out there are some fairly easy ways to find these: for a solvable group, or any group G with an abelian quotient group, you can fairly easily and concretely find the derived subgroup, [G,G]. The quotient group is an abelian group, so every subgroup between the whole group and the derived subgroup is normal. You then take the derived subgroup of the derived subgroup, but now you only take subgroups that are normalized by G/[G,G]. The action of G/[G,G] is relatively plain and easy to understand, so these "invariant" subgroups are pretty easy to find. To finish the inductive step, you need to find the normal subgroups of G/[[G,G],[G,G]] that don't contain [G,G], and this can be done by a slightly easier version of the general subgroup lattice algorithm using what is called the first cohomology group, or "derivations" (basically derivatives for groups, and related in a way to the derived subgroup). When you reach the point where the group is its own derived subgroup, a perfect group, then a second algorithm begins, that identifies the simple groups involved in the top of the group, and then constructs the solvable group below them. Sometimes you even have to repeat those last two steps, but not for groups of order less than 6024 ≈ 4.7e42 or so.
(ii) Often in mathematics, you want to use "induction". A normal subgroup is one of the two main ways to do induction in group theory. Usually it is not necessary to find all normal subgroups, but rather a single (nice) chief series will do. Some groups only have a single chief series (a fair number of dihedral groups are like this), and so finding the chief series and finding all normal subgroups is the same question. You might try the symmetric group of degree 4 and order 24 as an example.