Finite Group DatabasebetaFinite Group DatabasebetaOrder 1248.
260 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D3×D13 |
| commutator, socle, cosocle | C22×C3×C13 | C23 |
| Fitting | C23×C3×C13 | C22 |
| hypercenter | C23 | D3×D13 |
| hypocenter | C3×C13 | #120 |
| LCS | G ▷ C22×C3×C13 ▷ C3×C13 | |
| UCS | C1 ◁ C22 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×C3×D13 | C2 | semidirect |
| 2. C2×#64625 | C2 | semidirect |
| 3. C2×#64603 | C2 | semidirect |
| 4. C13×#217 | C2 | semidirect |
| 5. #10597608 | C2 | semidirect |
| 8. C22×D39 | C22 | semidirect |
| 13. C2×Dic39 | C22 | semidirect |
| 14. C2×Dic3×C13 | C22 | semidirect |
| 16. C23×D13 | D3 | semidirect |
| 22. C22×C3×C13 | D4 | semidirect |
| 29. C2×C3×D13 | D4 | semidirect |
| 48. C22×C13 | #67 | semidirect |
| 55. C2×D13 | #67 | semidirect |
| 62. #217 | D13 | semidirect |
| 63. C3×C13 | #120 | semidirect |
| 78. C2×Dic3 | C2×D13 | semidirect |
| 84. C13 | #953 | semidirect |
| 90. C22×C3 | D52 | semidirect |
| 117. C22 | #64603 | semidirect |
| 121. C3 | #88136 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×D13 | 2 |
| C2×#64625 | 2 |
| C2×#64603 | 2 |
| C13×#217 | 2 |
| #10597608 | 2 |
| #88136 | 3 |
| #953 | 13 |