Finite Group DatabasebetaFinite Group DatabasebetaOrder 2048.
1386 conjugacy classes of subgroups.
The subgroup lattice has 569 nodes — too large to display as a diagram.
| subgroup | G/H | |
|---|---|---|
| center | C8 | C42×D8 |
| commutator | C2×C8 | C22×C4×C8 |
| Frattini, cosocle | C2×C4×C16 | C24 |
| socle | C2 | D8×#346 |
| prime radical | C2×#508 | C2×C8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C2×#1398 ◁ C4×#1398 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #72225381 | C2 | semidirect |
| 2. #72225380 | C2 | semidirect |
| 3. #72225379 | C2 | semidirect |
| 4. #72225363 | C2 | semidirect |
| 6. #72216243 | C2 | semidirect |
| 7. C8×#1398 | C2 | semidirect |
| 21. #151073 | C4 | semidirect |
| 22. #151072 | C4 | semidirect |
| 45. #13647 | C2×C4 | semidirect |
| 87. #7459 | C2×C4 | semidirect |
| 88. #7459 | C2×C4 | semidirect |
| 89. #7381 | C2×C4 | semidirect |
| 91. #7329 | C2×C4 | semidirect |
| 92. C8×C32 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72225381 | 2 |
| #72225380 | 2 |
| #72225379 | 2 |
| #72225363 | 2 |
| #72225353 | 2 |
| #72216243 | 2 |
| C8×#1398 | 2 |
| #72216211 | 2 |