Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
216 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 140 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 | C22×C3×C7 | C2×D4 |
| commutator | C4 | C22×C3×C4×C7 |
| Frattini, cosocle | C2×C4 | C23×C3×C7 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C7 ◁ C3×C42×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 6. C7×#441 | C3 | direct (N × H) |
| 8. C3×C7×Q16 | C4 | semidirect |
| 20. C3×#441 | C7 | direct (N × H) |
| 30. C7×Q16 | C3×C4 | semidirect |
| 48. #441 | C3×C7 | direct (N × H) |
| 58. C3×Q16 | C4×C7 | semidirect |
| 86. C3×C7 | #441 | direct (N × H) |
| 88. Q16 | C3×C4×C7 | semidirect |
| 111. C7 | C3×#441 | direct (N × H) |
| 120. C3 | C7×#441 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×C7×Q16 | 2 |
| C3×C4×C7×Q8 | 2 |
| C3×C7×#106 | 2 |
| C3×C7×#103 | 2 |
| C3×C7×#97 | 2 |
| C7×#441 | 3 |
| C3×#441 | 7 |