Best design pattern/approach for a long list of if/else/execute brances of code

I have a "legacy" code that I want to refactor.

The code basically does a remote call to a server and gets back a reply. Then according to the reply executes accordingly.

Example of skeleton of the code:

public Object processResponse(String responseType, Object response) {
    if(responseType.equals(CLIENT_REGISTERED)) {  
       //code  
       //code ...
    }  
    else if (responseType.equals(CLIENT_ABORTED)) {
       //code  
       //code....
    }    
    else if (responseType.equals(DATA_SPLIT)) {
      //code
      //code... 
   }  
   etc  

The problem is that there are many-many if/else branches and the code inside each if is not trivial.

So it becomes hard to maintain.

I was wondering what is that best pattern for this?

One thought I had was to create a single object with method names the same as the responseType and then inside processResponse just using reflection call the method with the same name as the responseType.

This would clean up processResponse but it moves the code to a single object with many/many methods and I think reflection would cause performance issues.

Is there a nice design approach/pattern to clean this up?