Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
516 conjugacy classes of subgroups; only the 296 normal subgroups are shown.
This group has 516 conjugacy classes of subgroups — too many to draw. Only its 296 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×C97 | C23 |
| commutator | C22 | C23×C4×C97 |
| Frattini, cosocle | C23 | C24×C97 |
| socle | C23×C97 | C24 |
| prime radical | C24×C97 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C97 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #387×C97 | C2 | semidirect |
| 4. #383×C97 | C2 | semidirect |
| 5. C4×C4⋊C4×C97 | C2 | semidirect |
| 7. C43×C97 | C2 | semidirect |
| 9. #126×C97 | C4 | direct (N × H) |
| 31. C4⋊C4×C97 | C2×C4 | semidirect |
| 37. C42×C97 | C2×C4 | semidirect |
| 68. C4×C97 | #126 | direct (N × H) |
| 73. C4×#126 | C97 | direct (N × H) |
| 74. C97 | C4×#126 | direct (N × H) |
| 77. #387 | C2×C97 | semidirect |
| 78. #383 | C2×C97 | semidirect |
| 79. C4×C4⋊C4 | C2×C97 | semidirect |
| 81. C43 | C2×C97 | semidirect |
| 83. #126 | C4×C97 | direct (N × H) |
| 105. C4⋊C4 | C2×C4×C97 | semidirect |
| 111. C42 | C2×C4×C97 | semidirect |
| 142. C4 | #126×C97 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#126×C97 | 2 |
| #388×C97 | 2 |
| #387×C97 | 2 |
| #383×C97 | 2 |
| C4×C4⋊C4×C97 | 2 |
| C4×C22⋊C4×C97 | 2 |
| C43×C97 | 2 |
| C4×#126 | 97 |