<div class="input-group">
  <div class="input-group-btn">
    <!-- Button and dropdown menu -->
  </div>
  <input type="text" class="form-control" aria-label="...">
</div>

<div class="input-group">
  <input type="text" class="form-control" aria-label="...">
  <div class="input-group-btn">
    <!-- Button and dropdown menu -->
  </div>
</div>