Finite Group DatabasebetaFinite Group DatabasebetaValid at every prime. The subgroup lattice of this shape is one object: each count below is a polynomial in p, and substituting a prime gives the lattice of that group.
| Order | p6 |
| Subgroups | 4p6 + 4p5 + 11p4 + 9p3 + 8p2 + 5p + 7 |
| Classes of subgroups | 13 |
| Automorphisms | p26 − 2p25 + p23 + 2p21 − 2p20 − p18 + 2p16 − p15 |
| Classes with a complement | 8 |
Each circle is one class of subgroups, labeled by the exponents of its shape — 3,2 is Cp3 × Cp2, and 2²,1 is (Cp2)2 × Cp. The badge is how many subgroups the class holds. Hover a row of the table to light its circle, or click a circle to jump to its row.
| Subgroup | Quotient | Subgroups |
|---|---|---|
| Cp (direct factor) | Cp2 × (Cp)3 | p4 + p3 + p2 + p |
| Cp (characteristic) | (Cp)5 | 1 |
| Cp2 (direct factor) | (Cp)4 | p4 |
| (Cp)2 (direct factor) | Cp2 × (Cp)2 | p6 + p5 + 2p4 + p3 + p2 |
| (Cp)2 | (Cp)4 | p3 + p2 + p + 1 |
| Cp2 × Cp (direct factor) | (Cp)3 | p6 + p5 + p4 + p3 |
| (Cp)3 (direct factor) | Cp2 × Cp | p6 + p5 + p4 + p3 |
| (Cp)3 | (Cp)3 | p4 + p3 + 2p2 + p + 1 |
| Cp2 × (Cp)2 (direct factor) | (Cp)2 | p6 + p5 + 2p4 + p3 + p2 |
| (Cp)4 (direct factor) | Cp2 | p4 |
| (Cp)4 | (Cp)2 | p3 + p2 + p + 1 |
| Cp2 × (Cp)3 (direct factor) | Cp | p4 + p3 + p2 + p |
| (Cp)5 (characteristic) | Cp | 1 |