Finite Group DatabasebetaFinite Group DatabasebetaOrder 1248.
332 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×D39 |
| commutator, socle, cosocle | C2×C3×C13 | C24 |
| Fitting | C2×C3×D4×C13 | C2 |
| hypercenter | C2×D4 | D39 |
| hypocenter | C3×C13 | H(5,𝔽2) |
| LCS | G ▷ C2×C3×C13 ▷ C3×C13 | |
| UCS | C1 ◁ C2 ◁ C2×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×D4×C13 | C2 | semidirect |
| 2. C2×#64625 | C2 | semidirect |
| 3. #10597792 | C2 | semidirect |
| 4. D4×D39 | C2 | semidirect |
| 5. #10597790 | C2 | semidirect |
| 7. C23×C3×C13 | C22 | semidirect |
| 8. C22×D39 | C22 | semidirect |
| 9. C3×D4×C13 | C22 | semidirect |
| 10. C2×C3×C4×C13 | C22 | semidirect |
| 11. #64625 | C22 | semidirect |
| 12. #64625 | C22 | semidirect |
| 13. C2×Dic39 | C22 | semidirect |
| 14. D156 | C22 | semidirect |
| 15. C4×D39 | C22 | semidirect |
| 16. Dic78 | C22 | semidirect |
| 17. C2×D4×C13 | D3 | semidirect |
| 29. C23×C13 | C2×D3 | semidirect |
| 31. D4×C13 | C2×D3 | semidirect |
| 32. C2×C4×C13 | C2×D3 | semidirect |
| 51. C2×C3×D4 | D13 | semidirect |
| 56. C3×C13 | H(5,𝔽2) | semidirect |
| 62. C23×C3 | C2×D13 | semidirect |
| 64. C3×D4 | C2×D13 | semidirect |
| 65. C2×C3×C4 | C2×D13 | semidirect |
| 76. C2×D4 | D39 | semidirect |
| 77. C13 | #1004 | semidirect |
| 86. C23 | C2×D39 | semidirect |
| 91. D4 | C2×D39 | semidirect |
| 94. C2×C4 | C2×D39 | semidirect |
| 106. C3 | #88251 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×D4×C13 | 2 |
| C2×#64625 | 2 |
| #10597792 | 2 |
| D4×D39 | 2 |
| #10597790 | 2 |
| #88251 | 3 |
| #1004 | 13 |