If he dies and does not leave behind anyone but a brother or sister through his mother

Kalaalah refers to a person who dies and has no father or
son. If he has a brother or a sister through his mother, each of them gets
one-sixth (of the estate), and if they are more than that then they share
one third, because Allah, may He be exalted, says (interpretation of the
meaning):

“If the man or woman whose
inheritance is in question has left neither ascendants nor descendants, but
has left a brother or a sister, each one of the two gets a sixth; but if
more than two, they share in a third”

[al-Nisa’ 4:12]. 

If he had a wife, she gets one half. 

If he had a full brother, then he gets the entire estate or
what is left, by virtue of being a male relative on the father’s side
(ta‘seeb), after those who are entitled to allotted shares have received
them, if there are any such heirs. 

If he has a full sister, then she gets half, and if there are
two sisters then they get two thirds, because Allah, may He be exalted, says
(interpretation of the meaning):

“They ask you for a legal
verdict. Say: “Allâh directs (thus) about Al-Kalâlah (those who leave
neither descendants nor ascendants as heirs). If it is a man that dies,
leaving a sister, but no child, she shall have half the inheritance. If
(such a deceased was) a woman, who left no child, her brother takes her
inheritance. If there are two sisters, they shall have two-thirds of the
inheritance; if there are brothers and sisters, the male will have twice the
share of the female.”

[al-Nisa’ 4:176]. 

If he only has a brother or a sister through the mother, then
he or she takes one-sixth as an allotted share and the rest of the estate
comes to him (or her) according to the Hanafis and Hanbalis. Thus he (or
she) takes the entire estate.  

Maalik and al-Shaafa‘i were of the view that the remainder of
the estate should go to the bayt al-maal if there are no male relatives on
the father’s side (‘asbah). 

Ibn Qudaamah (may Allah have mercy on him) said in
al-Mughni (6/186): To sum up, if the deceased did not leave behind any
heir except those who are entitled to an allotted share, such as daughters
or sisters or grandmothers, and there is some wealth left over, then what is
left over from the allotted shares should be given to them on the same basis
as the allotted shares, except in the case of a husband or wife. This was
narrated from ‘Umar, ‘Ali, Ibn Mas‘ood and Ibn ‘Abbaas (may Allah be pleased
with him). It was also narrated from al-Hasan, Ibn Sireen, Shurayh, ‘Ata’,
Mujaahid, al-Thawri, and Abu Haneefah and his companions. Ibn Suraaqah said:
Things are done on this basis now in the regions. 

Zayd ibn Thaabit was of the view that what is left over from
the allotted shares should go to the bayt al-maal and no one should be given
more than his allotted share. This was also the view of Maalik, al-Awzaa‘i
and al-Shaafa‘i (may Allah be pleased with them). End quote. 

And he said: With regard to the spouses, what is left over
(after giving the allotted shares) should not be given to them, according to
the consensus of the scholars, but it was narrated from ‘Uthmaan (may Allah
be pleased with him) that he did give the left-over wealth to the husband,
but perhaps he was a relative on the father’s side (‘asbah) or on the
mother’s side (dhu rahm), so he gave that to him or he gave it from the bayt
al-maal and not by way of inheritance. End quote. 

When it comes to matters of inheritance, it is important to
ask about every case according to what the deceased has left behind in his
estate, lest there be any error in applying the rulings in a given case. 

And Allah knows best.