Sai Ram A - this will not give what is required. it will return the company name.

Richard Brescia 1 - This code gives so much mis-information in 1 answer, not acceptable at all, overly heavy and and just generally weird, and does not answer the question, or provide a correct result.

So if this was my "My Domain" : "myCo--myBox.my.salesforce.com" then the above code would return "myCo", which is not what the question asked for. And what a very elaborate way to get the production  sub domain.

It should return myBox

If you want to know if it is a Sandbox, use "SELECT IsSandbox FROM Organization"

myHostStr.indexOf(myOrg.Name.toLowerCase() + '--'); will almost always return -1, because the org name is unlikely to be in the baseUrl
also immediately following you try to write th index straight back to 0 with if (sandboxTag == 0) sandboxTag = myOrg.Name.length() + 2; else sandboxTag = 0;

then a while loop to finally find the sandbox name OMG, it works but seriously.

Here is my rewrite of your code, the assumption is no "--" or "." in your my domain:

and that org info is helpful in otherways.

string hostStr='Production';
Organization myOrg = [Select Id, Name, instanceName, isSandbox From Organization];
system.debug(myOrg);
if(myOrg.isSandbox){
    String myHostStr = System.Url.getSalesforceBaseUrl().getHost();
    Integer startIndex = myHostStr.indexOf('--')+2;
    Integer endIndex = myHostStr.indexOf('.');
    hostStr = myHostStr.mid(startIndex,endIndex-startIndex);
}
return hostStr;
Which by the way is quite readable.

but even better because we don't need to query the org & utilising the getter:

String myHostStr = System.Url.getSalesforceBaseUrl().getHost();
Integer startIndex = myHostStr.indexOf('--')+2;
if(startIndex != 1){ //because if prod then -1 is returned  +2
    Integer endIndex = myHostStr.indexOf('.');
    return myHostStr.mid(startIndex,endIndex-startIndex);
}else
    return 'Production';
}

global class SandboxPostRefresh_AC implements SandboxPostCopy { code is nice option if you are that well organised.
use this option only if those values get stored in custom hierarchy settings, and if on every refresh you remember to run the apex at refresh time, because that apex cannot be instantiated during normal runtime.

the simpliest fastest answer is:

UserInfo.getUserName().substringAfterLast('.');
which works if you don't change the username. Best practice assumes you don't change the username.

so I like Yuchen's answer the best.

For others trying to organise community urls use a network query:

list<Network> communities = [SELECT Id, Name FROM Network WHERE status = 'Live'];
for(Network comm : communities){
    string community = comm.Name.toLowerCase();
    string loginURL = Network.getLoginUrl(comm.id);
    integer pointer = loginURL.indexOf(community);
    string domain = loginURL.left(pointer-1);
    string baseUrl = domain+'/'+community+'/';
    system.debug(community);
    system.debug(pointer);
    system.debug(LoginUrl);
    system.debug(domain);
    system.debug(baseUrl);
}