Wawan Purwanto, communication director at Indonesia's intelligence agency said ISIS-inspired group Jemaah Ansharut Daulah (JAD) was believed to be behind the bombings.
Further details on the terror suspect would be released after examination by police Disaster Victim Identification unit was done, Frans said.
The family's father exploded a vehicle bomb, while the mother was with their two daughters aged nine and 12 and two sons aged 18 and 16 used a motorbike in an attack on another church, police said.
In Surabaya, Indonesia, a family has involved an 8-year-old girl, in a suicide bomb attack in order to settle scores with an opposition.
National police chief Tito Karnavian said the girl, who was with two of the attackers on a motorcycle, survived being thrown by the blast at Surabaya's police headquarters.
Inspector general Machfud Arifin told CNN Indonesia that the suicide attacks were carried out using a motorbike at St Mary's church and a vehicle at another. He also said he will look into enforcing new anti-terrorism regulations if parliament fails to pass a bill in the next month.
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In the past, the European Union has also lodged complaints at the World Trade Organisation. But European diplomats were despondent. "We don't have much to threaten the Americans".
Security forces have arrested hundreds of militants during a sustained crackdown that smashed some networks.
A Surabaya police spokesman said police arrested 13 suspects over the two suicide bombings carried out by two families, after conducting raids in Surabaya and its neighbouring cities of Malang and Pasuruan. All six members of the family were killed.
The mother also participated in the attack alongside their daughters, aged 12 and 9.
Television footage appeared to show a person on a motorcycle driving into the grounds of a church before a bomb was detonated.
At least four people were killed and more than 40 wounded in the first attack at the Santa Maria Roman Catholic Church, officials said.
Earlier, media reports said a woman with a younger child and a teenager had just entered one church and was being questioned by security when the bomb exploded. It is also hard to detect it by X-ray, making it a popular type of explosives among terrorists.