Finite Group DatabasebetaFinite Group DatabasebetaOrder 1248.
188 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 56 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 | C2×C3×C4×C13 | C22 |
| hypercenter | C2×C4 | D3×D13 |
| hypocenter | C3×C13 | #121 |
| LCS | G ▷ C22×C3×C13 ▷ C3×C13 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×D156 | C2 | semidirect |
| 2. C2×C4×D3×C13 | C2 | semidirect |
| 3. C2×#64604 | C2 | semidirect |
| 4. C3×C4⋊Dic13 | C2 | semidirect |
| 5. #10597610 | C2 | semidirect |
| 19. C4⋊Dic13 | D3 | semidirect |
| 26. C2×D3×C13 | D4 | semidirect |
| 27. C3×C4×C13 | D4 | semidirect |
| 52. C4×C13 | D12 | semidirect |
| 56. C2×C4×D3 | D13 | semidirect |
| 59. C3×C13 | #121 | semidirect |
| 79. C13 | #895 | semidirect |
| 86. C2×D3 | D52 | semidirect |
| 88. C3×C4 | C22⋊D13 | semidirect |
| 112. C4 | #64604 | semidirect |
| 113. C3 | #88184 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×D156 | 2 |
| C2×C4×D3×C13 | 2 |
| C2×#64604 | 2 |
| C3×C4⋊Dic13 | 2 |
| #10597610 | 2 |
| #88184 | 3 |
| #895 | 13 |