Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
749 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 749 conjugacy classes of subgroups — too many to draw. Only its 199 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#525 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C22⋊C8 | C2 | semidirect |
| 3. C2×#381 | C2 | semidirect |
| 4. #1516 | C2 | direct (N × H) |
| 5. C2×#328 | C2 | semidirect |
| 6. C2×#327 | C2 | semidirect |
| 7. C22×#41 | C4 | semidirect |
| 12. #525 | C4 | semidirect |
| 15. C2×C22⋊Q8 | C4 | semidirect |
| 17. C2×#121 | C4 | semidirect |
| 22. C2×#117 | C4 | semidirect |
| 27. C2×C22⋊C8 | C22 | semidirect |
| 29. #381 | C22 | semidirect |
| 32. #328 | C22 | semidirect |
| 33. #327 | C22 | semidirect |
| 37. C2×#41 | C2×C4 | semidirect |
| 65. C22⋊Q8 | C2×C4 | semidirect |
| 71. #121 | C2×C4 | semidirect |
| 80. #117 | C2×C4 | semidirect |
| 288. C2 | #1516 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#525 | 2 |
| C22×C22⋊C8 | 2 |
| C2×#381 | 2 |
| #1516 | 2 |
| C2×#328 | 2 |
| C2×#327 | 2 |