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 | p9 + 3p8 + 4p7 + 7p6 + 9p5 + 11p4 + 9p3 + 8p2 + 5p + 7 |
| Classes of subgroups | 5 |
| Automorphisms | p36 − p35 − p34 + p31 + 2p29 − p27 − p26 − p25 − p24 + 2p22 + p20 − p17 − p16 + p15 |
| Classes with a complement | 5 |
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) | (Cp)5 | p5 + p4 + p3 + p2 + p + 1 |
| (Cp)2 (direct factor) | (Cp)4 | p8 + p7 + 2p6 + 2p5 + 3p4 + 2p3 + 2p2 + p + 1 |
| (Cp)3 (direct factor) | (Cp)3 | p9 + p8 + 2p7 + 3p6 + 3p5 + 3p4 + 3p3 + 2p2 + p + 1 |
| (Cp)4 (direct factor) | (Cp)2 | p8 + p7 + 2p6 + 2p5 + 3p4 + 2p3 + 2p2 + p + 1 |
| (Cp)5 (direct factor) | Cp | p5 + p4 + p3 + p2 + p + 1 |