how to change position to sorter.phtml block?

i’m trying to move this UI element
vendor/magento/module-catalog/view/frontend/templates/product/list/toolbar/sorter.phtml from its original position to the right side of the screen inside another container.

The problem is that i couldn’t for the life of me find its name to select it so instead i tried selecting the entire toolbar.phtml block and move it like this:

<move element="product_list_toolbar" destination="sidebar.additional" />

but nothing changes on the front end.

i found the name product_list_toolbar in vendor/magento/module-catalog/view/frontend/layout/catalog_category_view.xml specifically here:

<block class="MagentoCatalogBlockProductProductListToolbar" name="product_list_toolbar" template="Magento_Catalog::product/list/toolbar.phtml">
     <block class="MagentoThemeBlockHtmlPager" name="product_list_toolbar_pager"/>
</block>

Can anyone help?

The ideal would be to move only the sorter element that is found inside toolbar but even if you can help me select the entire toolbar it would be of huge help!