Calculating number of subnets question

I get most of the subnetting things, I think.
How to calculate how many subnets / hosts / Network ID / broadcast ...
ACCEPT !

I don't understand how to calculated subnets based on subnets like

10.1.0.0
255.255.255.0

I'm NOT confused (I don't think) about class or unclassfull / cidr stuff.

I'm just use to doing supernetting (I think its call) but confused when you
have an IP that is a full Octet.

for example
Here is my math on the above address.
10.1.0.1
0000001010.00000001.00000000.00000001 >IP 10.1.0.1
1111111111.11111111.11111111.00000000 >Subnet Mask 255.255.255.0
0000001010.00000001.00000000.00000000 Network IP 10.1.0.0 using Anding.
.11111111 I just add 1's to host bits to get broadcast
address. 10.1.0.255

I think that there are 8 subnet bits in the third octet and 8 host bits in
the 4th octet, that should give me 256 subnets (minus 2) and 256 hosts
(minus 2)

HOWEVER when I look at an ip subnet calculator I see that there are 65536
subnets available and 256 hosts. I'm confused where the 65536 subnets comes
from (other than its 2^16=65536) Where did the 16 bits come from for the
subnet.

Thanks.