[javascript] JavaScript get child element

I'd suggest doing something similar to:

function show_sub(cat) {
    if (!cat) {
        return false;
    }
    else if (document.getElementById(cat)) {
        var parent = document.getElementById(cat),
            sub = parent.getElementsByClassName('sub');
        if (sub[0].style.display == 'inline'){
            sub[0].style.display = 'none';
        }
        else {
            sub[0].style.display = 'inline';
        }
    }
}

document.getElementById('cat').onclick = function(){
    show_sub(this.id);
};????

JS Fiddle demo.

Though the above relies on the use of a class rather than a name attribute equal to sub.

As to why your original version "didn't work" (not, I must add, a particularly useful description of the problem), all I can suggest is that, in Chromium, the JavaScript console reported that:

Uncaught TypeError: Object # has no method 'getElementsByName'.

One approach to working around the older-IE family's limitations is to use a custom function to emulate getElementsByClassName(), albeit crudely:

function eBCN(elem,classN){
    if (!elem || !classN){
        return false;
    }
    else {
        var children = elem.childNodes;
        for (var i=0,len=children.length;i<len;i++){
            if (children[i].nodeType == 1
                &&
                children[i].className == classN){
                    var sub = children[i];
            }
        }
        return sub;
    }
}

function show_sub(cat) {
    if (!cat) {
        return false;
    }
    else if (document.getElementById(cat)) {
        var parent = document.getElementById(cat),
            sub = eBCN(parent,'sub');
        if (sub.style.display == 'inline'){
            sub.style.display = 'none';
        }
        else {
            sub.style.display = 'inline';
        }
    }
}

var D = document,
    listElems = D.getElementsByTagName('li');
for (var i=0,len=listElems.length;i<len;i++){
    listElems[i].onclick = function(){
        show_sub(this.id);
    };
}?

JS Fiddle demo.