Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
260 conjugacy classes of subgroups; only the 196 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 196 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3×C7 | D4 |
| commutator | C4 | C22×C3×C4×C7 |
| Frattini, cosocle | C2×C4 | C23×C3×C7 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3×C7 ◁ C22×C3×C4×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C3×C7×#106 | C2 | direct (N × H) |
| 4. C2×C7×#106 | C3 | direct (N × H) |
| 7. C2×C3×C7×C8 | C4 | semidirect |
| 13. C7×#106 | C2×C3 | direct (N × H) |
| 14. C2×C3×#106 | C7 | direct (N × H) |
| 22. C3×C7×C8 | C2×C4 | semidirect |
| 25. C2×C7×C8 | C3×C4 | semidirect |
| 31. C3×#106 | C2×C7 | direct (N × H) |
| 39. C2×#106 | C3×C7 | direct (N × H) |
| 47. C7×C8 | C2×C3×C4 | semidirect |
| 50. C2×C3×C8 | C4×C7 | semidirect |
| 54. C2×C3×C7 | #106 | direct (N × H) |
| 59. #106 | C2×C3×C7 | direct (N × H) |
| 74. C3×C8 | C2×C4×C7 | semidirect |
| 75. C3×C7 | C2×#106 | direct (N × H) |
| 78. C2×C8 | C3×C4×C7 | semidirect |
| 82. C2×C7 | C3×#106 | direct (N × H) |
| 99. C8 | C2×C3×C4×C7 | semidirect |
| 100. C7 | C2×C3×#106 | direct (N × H) |
| 101. C2×C3 | C7×#106 | direct (N × H) |
| 111. C3 | C2×C7×#106 | direct (N × H) |
| 112. C2 | C3×C7×#106 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×C7×C8 | 2 |
| C2×C3×C7×C4⋊C4 | 2 |
| C3×C7×#106 | 2 |
| C2×C7×#106 | 3 |
| C2×C3×#106 | 7 |