Finite Group DatabasebetaFinite Group DatabasebetaOrder 16256.
872 conjugacy classes of subgroups; only the 368 normal subgroups are shown.
This group has 872 conjugacy classes of subgroups — too many to draw. Only its 368 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C127 | C24 |
| commutator | C22 | C23×C4×C127 |
| Frattini, cosocle | C23 | C24×C127 |
| prime radical | C22×#41×C127 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C127 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C4×Q8×C127 | C2 | semidirect |
| 3. #390×C127 | C2 | semidirect |
| 4. #380×C127 | C2 | semidirect |
| 5. C4×C22⋊C4×C127 | C2 | semidirect |
| 7. C2×#41×C127 | C4 | semidirect |
| 40. Q8×C127 | C22⋊C4 | semidirect |
| 52. C2×C4×C127 | #41 | semidirect |
| 73. #2341 | C127 | direct (N × H) |
| 74. C127 | #2341 | direct (N × H) |
| 76. C2×C4×Q8 | C2×C127 | semidirect |
| 77. #390 | C2×C127 | semidirect |
| 78. #380 | C2×C127 | semidirect |
| 79. C4×C22⋊C4 | C2×C127 | semidirect |
| 81. C2×#41 | C4×C127 | semidirect |
| 114. Q8 | C22⋊C4×C127 | semidirect |
| 126. C2×C4 | #41×C127 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#41×C127 | 2 |
| C2×C4×Q8×C127 | 2 |
| #390×C127 | 2 |
| #380×C127 | 2 |
| C4×C22⋊C4×C127 | 2 |
| #2341 | 127 |