Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
2847 conjugacy classes of subgroups; only the 1695 normal subgroups are shown.
This group has 2847 conjugacy classes of subgroups — too many to draw. Only its 1695 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C42 | C22 |
| commutator | C2 | C23×C42 |
| Frattini, cosocle | C23 | C25 |
| socle | C24 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C23×C42 | C2 | semidirect |
| 3. C42×D4 | C2 | direct (N × H) |
| 4. C2×C4×C4⋊C4 | C2 | semidirect |
| 5. C2×C4×C22⋊C4 | C2 | semidirect |
| 6. C2×C43 | C2 | semidirect |
| 9. C2×C4×D4 | C4 | direct (N × H) |
| 18. C22×C42 | C22 | semidirect |
| 19. C4×C4⋊C4 | C22 | semidirect |
| 20. C4×C22⋊C4 | C22 | semidirect |
| 22. C43 | C22 | semidirect |
| 28. C23×C4 | C2×C4 | semidirect |
| 34. C4×D4 | C2×C4 | direct (N × H) |
| 35. C2×C4⋊C4 | C2×C4 | semidirect |
| 37. C2×C22⋊C4 | C2×C4 | semidirect |
| 42. C2×C42 | C2×C4 | semidirect |
| 43. C2×C42 | D4 | direct (N × H) |
| 54. C2×D4 | C42 | direct (N × H) |
| 66. C22×C4 | C22×C4 | semidirect |
| 73. C4⋊C4 | C22×C4 | semidirect |
| 74. C22⋊C4 | C22×C4 | semidirect |
| 77. C42 | C22×C4 | semidirect |
| 78. C42 | C2×D4 | direct (N × H) |
| 83. C23 | C2×C42 | semidirect |
| 89. D4 | C2×C42 | direct (N × H) |
| 96. C2×C4 | C2×C42 | semidirect |
| 97. C2×C4 | C4×D4 | direct (N × H) |
| 108. C22 | C22×C42 | semidirect |
| 116. C4 | C22×C42 | semidirect |
| 117. C4 | C2×C4×D4 | direct (N × H) |
| 119. C2 | C42×D4 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4×D4 | 2 |
| C23×C42 | 2 |
| C42×D4 | 2 |
| C2×C4×C4⋊C4 | 2 |
| C2×C4×C22⋊C4 | 2 |
| C2×C43 | 2 |