Finite Group DatabasebetaFinite Group DatabasebetaOrder 16256.
908 conjugacy classes of subgroups; only the 360 normal subgroups are shown.
This group has 908 conjugacy classes of subgroups — too many to draw. Only its 360 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C127 | C23 |
| commutator, Frattini, cosocle | C23 | C24×C127 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C127 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #395×C127 | C2 | direct (N × H) |
| 21. C22×C4×C127 | Q8 | semidirect |
| 39. C2×C4×C127 | C2×Q8 | semidirect |
| 49. C2×C127 | #395 | direct (N × H) |
| 53. C2×#395 | C127 | direct (N × H) |
| 54. C127 | C2×#395 | direct (N × H) |
| 57. #395 | C2×C127 | direct (N × H) |
| 75. C22×C4 | Q8×C127 | semidirect |
| 93. C2×C4 | C2×Q8×C127 | semidirect |
| 103. C2 | #395×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×C4⋊C4×C127 | 2 |
| #395×C127 | 2 |
| C2×#95×C127 | 2 |
| C2×#395 | 127 |