You have a couple of ways you could do this, depending on how you'll present this diagram (or have users interact with it themselves). Also, the type of data relationships you have in your diagram will dictate what is visually most effective for the purpose of your overall diagram.
Layers:
Use the layers at the building blocks of your objects and connections. If you want to have certain groups shown/not shown independent from other content - put those in their own layer. Personally, I would put all connections on their own layer(s), to give you maximum flexibility.
Hotspots
After you've set up your layers, create hotspots (i.e. interactions) within your diagram to show or hide your layers and connections as needed. This is shown as the green lightning bolt in the Lucidchart UI.
This case study on the NJ Criminal Justice system has a very complex Lucidchart with layers and actions to create an interactive mapping. Hopefully it will be a useful example and create some ideas for you!