Goursat's Lemma characterizes the subgroups of direct products. Is there a similar characterization for the subgroups of semidirect products? What about if I'm only interested in the normal subgroups?
The short answer is that there is nothing nearly so nice as Goursat's Lemma. You can certainly reduce easily to the case where $\pi_K(H)=K$, much like you can reduce Goursat's Lemma to the case of a subdirect product, but after that it gets complicated. To give you an idea, here are three references.
A theorem of Rosenbaum (Die Untergruppen von halbdirekten Produkten, Rostock. Math. Kolloq. No. 35 (1988), 21-30) gives (from the MathScieNet Review MR991728 (90c:20032)):
The criterion was then used by Gutiérrez-Barrios to develop a criterion for a set of elements to be a normal subgroup of the semidirect product (Die Normalteiler von halbdirekten Produkten. Wiss. Z. Pädagog. Hochsch. Erfurt/Mühlhausen Math.-Natur. Reihe 25 (1989), no. 2, 108-114. MR1044548 (91b:20029))
Usenko (Subgroups of semidirect products, english translation in Ukrainian Math. J. 43 (1991), no. 7-8, 982-988 (1992), MR1148867 (92k:20045)) uses crossed homomorphisms to study subgroups of semidirect products.
While searching on the net i found this article. See whether it helps or not: http://www.springerlink.com/content/l272110h87u05667/fulltext.pdf
Thanks to Jack.