Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
5899 conjugacy classes of subgroups; only the 3479 normal subgroups are shown.
This group has 5899 conjugacy classes of subgroups — too many to draw. Only its 3479 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C24 |
| commutator, Frattini, cosocle | C22 | C26 |
| prime radical | C23×D4 | C22 |
| 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. C24×Q8 | C2 | semidirect |
| 2. C2×D4×Q8 | C2 | direct (N × H) |
| 3. C22×C4⋊Q8 | C2 | semidirect |
| 4. C22×C22⋊Q8 | C2 | semidirect |
| 5. C22×C4×Q8 | C2 | semidirect |
| 10. C23×Q8 | C22 | semidirect |
| 13. D4×Q8 | C22 | direct (N × H) |
| 14. C2×C4⋊Q8 | C22 | semidirect |
| 15. C2×C22⋊Q8 | C22 | semidirect |
| 16. C2×C4×Q8 | C22 | semidirect |
| 26. C22×Q8 | D4 | direct (N × H) |
| 29. C22×Q8 | C23 | semidirect |
| 30. C22×D4 | Q8 | direct (N × H) |
| 38. C4⋊Q8 | C23 | semidirect |
| 39. C22⋊Q8 | C23 | semidirect |
| 40. C4×Q8 | C23 | semidirect |
| 49. C24 | C2×Q8 | semidirect |
| 54. C2×Q8 | C2×D4 | direct (N × H) |
| 57. C2×D4 | C2×Q8 | direct (N × H) |
| 63. C22×C4 | C2×Q8 | semidirect |
| 79. C23 | C22×Q8 | semidirect |
| 84. Q8 | C22×D4 | direct (N × H) |
| 85. D4 | C22×Q8 | direct (N × H) |
| 90. C2×C4 | C22×Q8 | semidirect |
| 98. C22 | D4×Q8 | direct (N × H) |
| 101. C22 | C23×Q8 | semidirect |
| 107. C4 | C23×Q8 | semidirect |
| 109. C2 | C2×D4×Q8 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×Q8 | 2 |
| C2×D4×Q8 | 2 |
| C22×C4⋊Q8 | 2 |
| C22×C22⋊Q8 | 2 |
| C22×C4×Q8 | 2 |
| C22×C4×D4 | 2 |