How to determine if VPC subnet is public or private, EC2 instance cannot access Internet

Question

Hey all, I am sorry if this is a silly question but I cannot tell if the VPC subnet I created is public or private.  I am not seeing anything in the subnet details to tell me.  Also, my EC2 instance cannot access the Internet.
I setup a subnet for 10.0.40.0/24.  Assuming the subnet was private, I created a NAT gateway attached to VPC and created a route from the NAT gateway to the Internet.  The EC2 instance in that VPC still cannot get online.
So next I think figured it may have been a public subnet.  I created an Internet gateway in that VPC and created a route from the Internet gateway to the Internet.  Still no luck, instance still cannot get online.
Everything is attached to the same VPC.  I do not know what I am missing.  It does not seem like this should be this hard but for whatever reason, I cannot figure it out.  I read through the documentation at least five times and it still is not working.
The NAT gateway is using a primary private IP of 10.0.40.254.  The EC2 instance has IP 10.0.40.199, subnet 255.255.255.0, gateway 10.0.40.1, DNS 10.0.41.2.
I do not know where it got that ethernet gateway IP from.  10.0.40.1 is reachable but 10.0.40.254 is not reachable.  It would not let me set the primary private IP the same as the ethernet gateway.  I also do not know where it got the DNS IP 10.0.40.2 from but that too is not reachable.
So neither the NAT gateway or the Internet gateway work despite having it in the routing table per the instructions.

Accepted Answer

An EC2 instance cannot access the Internet if it does not have a public IP address, even if it is running on a public subnet with a route to the Internet Gateway.  
Configure Elastic IP or enable automatic public IP assignment for subnets.  
This document describes how to troubleshoot when you cannot connect to the Internet.  
https://repost.aws/knowledge-center/ec2-connect-internet-gateway  
For private subnets, the following documents can be used for troubleshooting.  
https://repost.aws/knowledge-center/ec2-internet-connectivity

I think the private subnet route table and the public subnet route table need to be separated.

Answer

**Step 1**: I recommend you check out the VPC (resource map) , which visualizes your network architecture.
meaning:
  - Public Route table: 
     * (1) VPC(eg: 10.0.0.0/16) -> local  
     * (2)0.0.0.0/0 -> igw-xxxx
  - Private Route table: 
     * (1) VPC(eg: 10.0.0.0/16) -> local  
     * (2)0.0.0.0/0 -> nat-xxxxx

**Step 2**: If you need to create Linux server and access the Internet, please enable Auto-assign public IP in the network settings. 
                  If you need a static IP, please apply for an Elastic IP and attach it to the Linux server.

**Step 3**:  Check whether the SSH port and access IP segment are enabled in the security group
Finally, if Linux server is placed in the private subnet, please use bastion  server to connect it . The Nat gateway can only allow the server to actively initiate requests to the Internet instead of Internet traffic can directly request private subnet servers, such as yum install ...

How to determine if VPC subnet is public or private, EC2 instance cannot access Internet

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