Finite Group DatabasebetaFinite Group DatabasebetaOrder 3072.
7914 conjugacy classes of subgroups.
The subgroup lattice has 246 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2×C16 | C25×C3 |
| socle, prime radical | C25×C3 | C2×C16 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C3×C4×C32 | C2 | direct (N × H) |
| 4. C23×C4×C32 | C3 | direct (N × H) |
| 8. C23×C3×C32 | C4 | direct (N × H) |
| 12. C2×C3×C4×C32 | C22 | direct (N × H) |
| 15. C22×C4×C32 | C2×C3 | direct (N × H) |
| 26. C22×C3×C32 | C2×C4 | direct (N × H) |
| 30. C3×C4×C32 | C23 | direct (N × H) |
| 34. C23×C32 | C3×C4 | direct (N × H) |
| 38. C2×C4×C32 | C22×C3 | direct (N × H) |
| 57. C2×C3×C32 | C22×C4 | direct (N × H) |
| 71. C22×C32 | C2×C3×C4 | direct (N × H) |
| 75. C4×C32 | C23×C3 | direct (N × H) |
| 77. C23×C3×C4 | C32 | direct (N × H) |
| 100. C3×C32 | C23×C4 | direct (N × H) |
| 119. C2×C32 | C22×C3×C4 | direct (N × H) |
| 127. C22×C3×C4 | C2×C32 | direct (N × H) |
| 146. C23×C4 | C3×C32 | direct (N × H) |
| 169. C32 | C23×C3×C4 | direct (N × H) |
| 170. C23×C3 | C4×C32 | direct (N × H) |
| 175. C2×C3×C4 | C22×C32 | direct (N × H) |
| 189. C22×C4 | C2×C3×C32 | direct (N × H) |
| 207. C22×C3 | C2×C4×C32 | direct (N × H) |
| 212. C3×C4 | C23×C32 | direct (N × H) |
| 215. C23 | C3×C4×C32 | direct (N × H) |
| 220. C2×C4 | C22×C3×C32 | direct (N × H) |
| 230. C2×C3 | C22×C4×C32 | direct (N × H) |
| 233. C22 | C2×C3×C4×C32 | direct (N × H) |
| 238. C4 | C23×C3×C32 | direct (N × H) |
| 241. C3 | C23×C4×C32 | direct (N × H) |
| 243. C2 | C22×C3×C4×C32 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×C4×C16 | 2 |
| C22×C3×C4×C32 | 2 |
| C24×C3×C32 | 2 |
| C23×C4×C32 | 3 |