Order 24576.
17180 conjugacy classes of subgroups.
The subgroup lattice has 1228 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2×C8×C32 | C24×C3 |
| socle, prime radical | C24×C3 | C2×C8×C32 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C3×C4×C16×C64 | C2 | direct (N × H) |
| 5. C2×C4×C16×C64 | C3 | direct (N × H) |
| 18. C2×C3×C16×C64 | C4 | direct (N × H) |
| 23. C4×C16×C64 | C2×C3 | direct (N × H) |
| 56. C3×C16×C64 | C2×C4 | direct (N × H) |
| 69. C2×C16×C64 | C3×C4 | direct (N × H) |
| 122. C2×C3×C4×C64 | C16 | direct (N × H) |
| 164. C16×C64 | C2×C3×C4 | direct (N × H) |
| 235. C3×C4×C64 | C2×C16 | direct (N × H) |
| 269. C2×C4×C64 | C3×C16 | direct (N × H) |
| 337. C2×C3×C4×C16 | C64 | direct (N × H) |
| 387. C2×C3×C64 | C4×C16 | direct (N × H) |
| 490. C4×C64 | C2×C3×C16 | direct (N × H) |
| 582. C3×C4×C16 | C2×C64 | direct (N × H) |
| 613. C3×C64 | C2×C4×C16 | direct (N × H) |
| 637. C2×C4×C16 | C3×C64 | direct (N × H) |
| 687. C2×C64 | C3×C4×C16 | direct (N × H) |
| 759. C2×C3×C16 | C4×C64 | direct (N × H) |
| 882. C4×C16 | C2×C3×C64 | direct (N × H) |
| 913. C64 | C2×C3×C4×C16 | direct (N × H) |
| 971. C3×C16 | C2×C4×C64 | direct (N × H) |
| 1014. C2×C16 | C3×C4×C64 | direct (N × H) |
| 1067. C2×C3×C4 | C16×C64 | direct (N × H) |
| 1149. C16 | C2×C3×C4×C64 | direct (N × H) |
| 1163. C3×C4 | C2×C16×C64 | direct (N × H) |
| 1175. C2×C4 | C3×C16×C64 | direct (N × H) |
| 1204. C2×C3 | C4×C16×C64 | direct (N × H) |
| 1214. C4 | C2×C3×C16×C64 | direct (N × H) |
| 1222. C3 | C2×C4×C16×C64 | direct (N × H) |
| 1223. C2 | C3×C4×C16×C64 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×C4×C16×C32 | 2 |
| C2×C3×C4×C8×C64 | 2 |
| C22×C3×C16×C64 | 2 |
| C3×C4×C16×C64 | 2 |
| C2×C4×C16×C64 | 3 |