Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
417199 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C27 | C2 | 1 | 255 | yes | maximal, direct product factor |
| 2 | C26 | C22 | 1 | 10795 | yes | direct product factor |
| 3 | C25 | C23 | 1 | 97155 | yes | direct product factor |
| 4 | C24 | C24 | 1 | 200787 | yes | direct product factor |
| 5 | C23 | C25 | 1 | 97155 | yes | direct product factor |
| 6 | C22 | C26 | 1 | 10795 | yes | direct product factor |
| 7 | C2 | C27 | 1 | 255 | yes | direct product factor |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator, Frattini | C1 | G |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
| LECS | G ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C27 | C2 | direct (N × H) |
| 2. C26 | C22 | direct (N × H) |
| 3. C25 | C23 | direct (N × H) |
| 4. C24 | C24 | direct (N × H) |
| 5. C23 | C25 | direct (N × H) |
| 6. C22 | C26 | direct (N × H) |
| 7. C2 | C27 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C27 | 2 |