Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
3187 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#564 | C2 | 1 | 48 | yes | maximal, direct product factor |
| 2 | C22×#125 | C2 | 1 | 15 | yes | maximal |
| 3 | #564 | C22 | 1 | 256 | yes | direct product factor |
| 4 | C2×#125 | C22 | 1 | 360 | yes | — |
| 5 | C22×C4⋊C4 | C22 | 1 | 30 | yes | — |
| 6 | C22×C42 | C22 | 1 | 5 | yes | — |
| 7 | C23×C4 | C23 | 1 | 15 | yes | — |
| 8 | #125 | C23 | 1 | 960 | yes | — |
| 9 | C2×C4⋊C4 | C23 | 1 | 360 | yes | — |
| 10 | C2×C42 | C23 | 1 | 60 | yes | — |
| 11 | C24 | C24 | 1 | 1 | yes | characteristic |
| 12 | C22×C4 | — | 2 | 60 | no | — |
| 13 | C22×C4 | C24 | 1 | 90 | yes | — |
| 14 | C4⋊C4 | C24 | 1 | 480 | yes | — |
| 15 | C42 | C24 | 1 | 80 | yes | — |
| 16 | C23 | 2−1+4 | 1 | 12 | yes | — |
| 17 | C23 | C25 | 1 | 3 | yes | — |
| 18 | C2×C4 | — | 2 | 180 | no | — |
| 19 | C2×C4 | C25 | 1 | 60 | yes | — |
| 20 | C22 | #564 | 1 | 16 | yes | direct product factor |
| 21 | C22 | C2×2−1+4 | 1 | 18 | yes | — |
| 22 | C22 | C26 | 1 | 1 | yes | characteristic |
| 23 | C4 | — | 2 | 60 | no | — |
| 24 | C2 | C2×#564 | 1 | 12 | yes | direct product factor |
| 25 | C2 | C22×2−1+4 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C24 |
| commutator, Frattini, cosocle | C22 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#564 | C2 | direct (N × H) |
| 3. #564 | C22 | direct (N × H) |
| 20. C22 | #564 | direct (N × H) |
| 24. C2 | C2×#564 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#564 | 2 |
| C22×#125 | 2 |