Finite Group DatabasebetaFinite Group DatabasebetaOrder 16256.
1036 conjugacy classes of subgroups; only the 328 normal subgroups are shown.
This group has 1036 conjugacy classes of subgroups — too many to draw. Only its 328 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×C127 | C23 |
| commutator, Frattini, cosocle | C23 | C24×C127 |
| prime radical | C25×C127 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C127 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×Q8×C127 | C2 | semidirect |
| 3. #393×C127 | C2 | direct (N × H) |
| 4. C2×#95×C127 | C2 | semidirect |
| 7. C22×Q8×C127 | C22 | semidirect |
| 12. #95×C127 | C22 | semidirect |
| 15. C24×C127 | Q8 | semidirect |
| 34. C23×C127 | C2×Q8 | semidirect |
| 57. C2×C127 | #393 | direct (N × H) |
| 61. C2×#393 | C127 | direct (N × H) |
| 62. C127 | C2×#393 | direct (N × H) |
| 63. C23×Q8 | C2×C127 | semidirect |
| 65. #393 | C2×C127 | direct (N × H) |
| 66. C2×#95 | C2×C127 | semidirect |
| 69. C22×Q8 | C22×C127 | semidirect |
| 74. #95 | C22×C127 | semidirect |
| 77. C24 | Q8×C127 | semidirect |
| 96. C23 | C2×Q8×C127 | semidirect |
| 119. C2 | #393×C127 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×Q8×C127 | 2 |
| C22×C22⋊C4×C127 | 2 |
| #393×C127 | 2 |
| C2×#95×C127 | 2 |
| C2×#393 | 127 |