Finite Group DatabasebetaFinite Group DatabasebetaOrder 1792.
2830 conjugacy classes of subgroups.
The subgroup lattice has 232 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4×C7 | D4 |
| commutator | C4 | C24×C4×C7 |
| Frattini, cosocle | C2×C4 | C25×C7 |
| socle, prime radical | C24×C7 | C2×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4×C7 ◁ C22×C42×C7 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×C4×C7×Q16 | C2 | direct (N × H) |
| 8. C22×C7×Q16 | C4 | direct (N × H) |
| 18. C4×C7×Q16 | C22 | direct (N × H) |
| 22. C22×C4×Q16 | C7 | direct (N × H) |
| 32. C2×C7×Q16 | C2×C4 | direct (N × H) |
| 51. C2×C4×Q16 | C2×C7 | direct (N × H) |
| 66. C22×C4×C7 | Q16 | direct (N × H) |
| 73. C7×Q16 | C22×C4 | direct (N × H) |
| 85. C22×Q16 | C4×C7 | direct (N × H) |
| 95. C4×Q16 | C22×C7 | direct (N × H) |
| 112. C2×C4×C7 | C2×Q16 | direct (N × H) |
| 130. C2×Q16 | C2×C4×C7 | direct (N × H) |
| 147. C22×C7 | C4×Q16 | direct (N × H) |
| 156. C4×C7 | C22×Q16 | direct (N × H) |
| 170. C22×C4 | C7×Q16 | direct (N × H) |
| 177. Q16 | C22×C4×C7 | direct (N × H) |
| 188. C2×C7 | C2×C4×Q16 | direct (N × H) |
| 205. C2×C4 | C2×C7×Q16 | direct (N × H) |
| 214. C7 | C22×C4×Q16 | direct (N × H) |
| 215. C22 | C4×C7×Q16 | direct (N × H) |
| 224. C4 | C22×C7×Q16 | direct (N × H) |
| 227. C2 | C2×C4×C7×Q16 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C7×Q16 | 2 |
| C22×C4×C7×Q8 | 2 |
| C2×C4×C7×Q16 | 2 |
| C22×C7×#107 | 2 |
| C22×C7×#103 | 2 |
| C22×C4×C7×C8 | 2 |
| C22×C4×Q16 | 7 |