Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
2154 conjugacy classes of subgroups.
The subgroup lattice has 288 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C7 | C24 |
| commutator | C22 | C24×C4×C7 |
| Frattini, cosocle | C2×C4 | C25×C7 |
| socle | C23×C7 | C23×C4 |
| prime radical | C24×C7 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C7×Q8×M16 | C2 | direct (N × H) |
| 3. C2×C7×#446 | C2 | semidirect |
| 4. C2×C7×C8×Q8 | C2 | semidirect |
| 18. C7×#446 | C22 | semidirect |
| 19. C7×C8×Q8 | C22 | semidirect |
| 26. C2×Q8×M16 | C7 | direct (N × H) |
| 35. C2×C7×M16 | Q8 | direct (N × H) |
| 61. Q8×M16 | C2×C7 | direct (N × H) |
| 62. C2×#446 | C2×C7 | semidirect |
| 63. C2×C8×Q8 | C2×C7 | semidirect |
| 69. C2×C7×Q8 | M16 | direct (N × H) |
| 87. C7×M16 | C2×Q8 | direct (N × H) |
| 89. C2×C7×C8 | C2×Q8 | semidirect |
| 112. #446 | C22×C7 | semidirect |
| 113. C8×Q8 | C22×C7 | semidirect |
| 126. C7×Q8 | C2×M16 | direct (N × H) |
| 147. C7×C8 | C22×Q8 | semidirect |
| 156. C2×M16 | C7×Q8 | direct (N × H) |
| 200. C2×Q8 | C7×M16 | direct (N × H) |
| 218. M16 | C2×C7×Q8 | direct (N × H) |
| 220. C2×C8 | C2×C7×Q8 | semidirect |
| 233. C2×C7 | Q8×M16 | direct (N × H) |
| 243. Q8 | C2×C7×M16 | direct (N × H) |
| 264. C8 | C22×C7×Q8 | semidirect |
| 265. C7 | C2×Q8×M16 | direct (N × H) |
| 283. C2 | C7×Q8×M16 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C4×C7×Q8 | 2 |
| C7×Q8×M16 | 2 |
| C2×C7×#446 | 2 |
| C2×C7×C8×Q8 | 2 |
| C2×C7×#423 | 2 |
| C2×C4×C7×M16 | 2 |
| C2×Q8×M16 | 7 |