Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.
This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C16 | D21 |
| commutator, hypocenter | C3×C7 | C22×C16 |
| Frattini | C8 | C22×D21 |
| Fitting | C2×C3×C7×C16 | C2 |
| socle | C22×C3×C7 | C2×C8 |
| prime radical | C22×D21 | C8 |
| cosocle | C3×C7×C8 | C23 |
| LCS | G ▷ C3×C7 | |
| UCS | C1 ◁ C2×C16 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C3×C7×C16 | C2 | semidirect |
| 3. C2×C3⋊(C7⋊C16) | C2 | semidirect |
| 4. C16×D21 | C2 | direct (N × H) |
| 11. C3×C7×C16 | C22 | semidirect |
| 12. C3⋊(C7⋊C16) | C22 | semidirect |
| 14. C2×C7×C16 | D3 | semidirect |
| 31. C7×C16 | C2×D3 | semidirect |
| 34. C2×C3×C16 | D7 | semidirect |
| 38. C2×D21 | C16 | direct (N × H) |
| 57. C3×C16 | C2×D7 | semidirect |
| 59. C2×C3×C7 | C2×C16 | semidirect |
| 61. D21 | C2×C16 | direct (N × H) |
| 65. C2×C16 | D21 | direct (N × H) |
| 80. C3×C7 | C22×C16 | semidirect |
| 87. C16 | C2×D21 | direct (N × H) |
| 88. C2×C7 | D3×C16 | semidirect |
| 105. C7 | C2×D3×C16 | semidirect |
| 106. C2×C3 | D7×C16 | semidirect |
| 115. C3 | C2×D7×C16 | semidirect |
| 117. C2 | C16×D21 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C8×D21 | 2 |
| C2×C3×C7×C16 | 2 |
| C2×C3⋊(C7⋊C16) | 2 |
| C16×D21 | 2 |
| C2×D7×C16 | 3 |
| C2×D3×C16 | 7 |