Finite Group DatabasebetaFinite Group DatabasebetaOrder 16256.
332 conjugacy classes of subgroups; only the 184 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 184 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C127 | D4 |
| commutator | C4 | C2×C42×C127 |
| Frattini, cosocle | C22×C4 | C23×C127 |
| socle, prime radical | C23×C127 | C2×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C127 ◁ C2×C42×C127 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 9. C4×Q8×C127 | C4 | semidirect |
| 16. #103×C127 | C4 | direct (N × H) |
| 44. Q8×C127 | C42 | semidirect |
| 71. C4×C127 | #103 | direct (N × H) |
| 80. C4×#103 | C127 | direct (N × H) |
| 81. C127 | C4×#103 | direct (N × H) |
| 90. C4×Q8 | C4×C127 | semidirect |
| 97. #103 | C4×C127 | direct (N × H) |
| 125. Q8 | C42×C127 | semidirect |
| 152. C4 | #103×C127 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×Q8×C127 | 2 |
| C2×#103×C127 | 2 |
| C2×C4×C8×C127 | 2 |
| C4×C4⋊C4×C127 | 2 |
| #340×C127 | 2 |
| C4×#103 | 127 |