Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1471 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C2×#401 | C2 | 1 | 24 | yes | maximal, direct product factor |
| 2 | C22×#95 | C2 | 1 | 7 | yes | maximal |
| 3 | C24×C4 | C22 | 1 | 7 | yes | — |
| 4 | #401 | C22 | 1 | 64 | yes | direct product factor |
| 5 | C2×#95 | C22 | 1 | 84 | yes | — |
| 6 | C25 | C23 | 1 | 1 | yes | characteristic |
| 7 | C23×C4 | — | 2 | 84 | no | — |
| 8 | C23×C4 | C23 | 1 | 42 | yes | — |
| 9 | #95 | C23 | 1 | 112 | yes | — |
| 10 | C24 | #41 | 1 | 28 | yes | — |
| 11 | C24 | C24 | 1 | 3 | yes | — |
| 12 | C22×C4 | — | 2 | 252 | no | — |
| 13 | C22×C4 | — | 4 | 112 | no | — |
| 14 | C22×C4 | C24 | 1 | 28 | yes | — |
| 15 | C23 | #126 | 1 | 112 | yes | — |
| 16 | C23 | C2×#41 | 1 | 42 | yes | — |
| 17 | C23 | C25 | 1 | 1 | yes | characteristic |
| 18 | C2×C4 | — | 2 | 84 | no | — |
| 19 | C2×C4 | — | 4 | 168 | no | — |
| 20 | C22 | #401 | 1 | 64 | yes | direct product factor |
| 21 | C22 | C2×#126 | 1 | 84 | yes | — |
| 22 | C22 | C22×#41 | 1 | 7 | yes | — |
| 23 | C4 | — | 4 | 28 | no | — |
| 24 | C2 | C2×#401 | 1 | 24 | yes | direct product factor |
| 25 | C2 | C22×#126 | 1 | 7 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C23 |
| commutator, Frattini, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#401 | C2 | direct (N × H) |
| 4. #401 | C22 | direct (N × H) |
| 20. C22 | #401 | direct (N × H) |
| 24. C2 | C2×#401 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#401 | 2 |
| C22×#95 | 2 |