Is there really a question?
A simple idea by example (from ornithology, passerines). crow --> an Order, Crow and his specie (corvus albicolis, corvus corax, etc).
Passerine birds with many suborders (למינהו) as suborder passery, infraorder corvida
An order that has not suborders --> למינו
An order that has suborders --> למינהו
Arbe is one order without suborders
Sal'am is an order with suborders.
Additionnal remark, the word סוג is introduced by Yehuda Ibn Tibbon and is not used as super species or super order in Tora and Chazal. It is in Chazal a big container (see Dmay 5, 6:פרק ה - משנה ו
הַלּוֹקֵחַ מִן הַסִּיטוֹן וְחָזַר וְלָקַח מִמֶּנּוּ שְׁנִיָּה, לֹא יְעַשֵּׂר מִזֶּה עַל זֶה, אֲפִלּוּ מֵאוֹתוֹ הַסּוּג, אֲפִלּוּ מֵאוֹתוֹ הַמִּין. נֶאֱמָן הַסִּיטוֹן לוֹמַר, מִשֶּׁל אֶחָד הֵם: & rlm;
See here and here
it is as leminam and leminehem. עושם באימה רצון קונם עושים באימה רצון קוניהם are 2 girsaot of kel adon and I think that who say konam is because of the difference between למינם למינהם and the cholek think that if there is elokim and baalim we need to save a plural i.e. konim instead of kone. anyway minehu = המינים שלו and למינם is כל אחד עם המין שלו.