Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
760 conjugacy classes of subgroups; only the 498 normal subgroups are shown.
This group has 760 conjugacy classes of subgroups — too many to draw. Only its 498 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C4 | #87604 |
| commutator, hypocenter | C97 | C22×C4×C8 |
| Frattini | C22 | C23×#87604 |
| Fitting | C23×C4×C97 | C4 |
| socle, prime radical | C24×C97 | C2×C4 |
| cosocle | C2×Dic97 | C24 |
| LCS | G ▷ C97 | |
| UCS | C1 ◁ C23×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C4×#11715897 | C2 | direct (N × H) |
| 4. C22×#11715897 | C4 | direct (N × H) |
| 6. C4×#11715897 | C22 | direct (N × H) |
| 12. C2×#11715897 | C2×C4 | direct (N × H) |
| 14. C22×Dic97 | C8 | semidirect |
| 22. C22×C4×C97 | C8 | semidirect |
| 28. C2×C4×C97 | C2×C8 | semidirect |
| 31. C2×Dic97 | C2×C8 | semidirect |
| 36. #11715897 | C22×C4 | direct (N × H) |
| 37. C22×C97 | C4×C8 | semidirect |
| 42. C4×C97 | C22×C8 | semidirect |
| 44. Dic97 | C22×C8 | semidirect |
| 46. C2×C97 | C2×C4×C8 | semidirect |
| 51. C97 | C22×C4×C8 | semidirect |
| 69. C22×C4 | #11715897 | direct (N × H) |
| 85. C2×C4 | C2×#11715897 | direct (N × H) |
| 88. C22 | C4×#11715897 | direct (N × H) |
| 96. C4 | C22×#11715897 | direct (N × H) |
| 97. C2 | C2×C4×#11715897 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#11715897 | 2 |
| C2×C4×#11715897 | 2 |
| C22×C4×Dic97 | 2 |
| C22×C4×C8 | 97 |