Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
372 conjugacy classes of subgroups; only the 164 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 164 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3×C7 | C22⋊C4 |
| commutator | C22 | C22×C3×C4×C7 |
| Frattini, cosocle | C2×C4 | C23×C3×C7 |
| prime radical | C22×C3×C7×D4 | C2 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C7 ◁ C22×C3×C4×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C3×C7×M16 | C2 | semidirect |
| 3. C3×C7×#100 | C2 | direct (N × H) |
| 4. C2×C7×#100 | C3 | direct (N × H) |
| 12. C3×C7×M16 | C22 | semidirect |
| 15. C2×C7×M16 | C2×C3 | semidirect |
| 16. C7×#100 | C2×C3 | direct (N × H) |
| 17. C2×C3×#100 | C7 | direct (N × H) |
| 37. C7×M16 | C22×C3 | semidirect |
| 40. C2×C3×M16 | C2×C7 | semidirect |
| 41. C3×#100 | C2×C7 | direct (N × H) |
| 52. C2×#100 | C3×C7 | direct (N × H) |
| 72. C3×M16 | C22×C7 | semidirect |
| 77. C2×C3×C7 | #100 | direct (N × H) |
| 80. C2×M16 | C2×C3×C7 | semidirect |
| 81. #100 | C2×C3×C7 | direct (N × H) |
| 104. C3×C7 | C2×#100 | direct (N × H) |
| 112. M16 | C22×C3×C7 | semidirect |
| 117. C2×C7 | C3×#100 | direct (N × H) |
| 141. C7 | C2×C3×#100 | direct (N × H) |
| 145. C2×C3 | C7×#100 | direct (N × H) |
| 157. C3 | C2×C7×#100 | direct (N × H) |
| 161. C2 | C3×C7×#100 | 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×D4 | 2 |
| C2×C3×C7×M16 | 2 |
| C3×C7×#100 | 2 |
| C2×C7×#100 | 3 |
| C2×C3×#100 | 7 |