Can I use div as a direct child of UL?

No. The only element that may be a child of <ul> is <li>.

HTML 4:

<!ELEMENT UL - - (LI)+                 -- unordered list -->

(See also how to read a content model definition in a DTD)

HTML 5:

Content model: Zero or more li elements.

No the div cannot be nested inside the element, only < li> can be used as child element. instead of wrapping div inside ul element. you can do something like this

<ul class="class1">
 <li class="child1">
   <ul>
     <li>item1</li>
     <li>item2</li>
     <li>item3</li>
   </ul>
 </li>
<ul> 

For HTML 5 :

http://www.w3.org/TR/html-markup/ul.html

Permitted contents

Zero or more li elements

For HTML 4 :

http://www.w3.org/TR/html401/struct/lists.html#h-10.2

<!ELEMENT UL - - (LI)+

EDIT :

I forget the other HTML5 :D (which have the same specification on this than the W3C's one)

http://www.whatwg.org/specs/web-apps/current-work/multipage/grouping-content.html#the-ul-element