Non-induced subgraphs

I would like to construct graphs that are generated by subsets of the graph edges but that are non necessarily induced subgraphs. I thought this could be done with the Subgraph function, but the example below shows that this is not the case.

G = Graph[{1 <-> 2, 1 <-> 3, 2 <-> 3, 2 <-> 4, 3 <-> 4}];  H = Subgraph[G, {1 <-> 2, 1 <-> 3, 2 <-> 4, 3 <-> 4}];  H == G (* True *) 

(However in this example I would like to get the subgraph without the edge 2<->3)

How can I proceed? Do I need to implement the whole construction, or are there pre-implemented functions that I could use?