Finite Group DatabasebetaFinite Group DatabasebetaOrder 4096.
Only the 1123 normal subgroups are shown; non-normal subgroups are omitted.
Only the group's 1123 normal subgroups are shown; non-normal subgroups are omitted.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #72216432 |
| commutator, Frattini, cosocle | C27 | C25 |
| LCS | G ▷ C27 ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ C28 ◁ G | |
| LECS | G ▷ C27 ▷ C24 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#72216427 | C2 | semidirect |
| 2. C2×#72216292 | C2 | semidirect |
| 3. C2×#72209716 | C2 | semidirect |
| 4. C2×#115690 | C22 | semidirect |
| 6. #72216433 | C2 | direct (N × H) |
| 7. #72216427 | C22 | semidirect |
| 8. #72216292 | C22 | semidirect |
| 9. C2×#6342371 | C22 | semidirect |
| 10. #72209716 | C22 | semidirect |
| 11. C29 | C23 | semidirect |
| 12. C2×#62817 | D4 | semidirect |
| 14. #6499867 | D4 | semidirect |
| 16. C2×#21746 | D4 | semidirect |
| 17. #6342371 | C23 | semidirect |
| 18. C2×#14679 | C23 | semidirect |
| 19. #115690 | C23 | semidirect |
| 21. C28 | C24 | semidirect |
| 22. #62817 | C2×D4 | semidirect |
| 27. #21746 | C2×D4 | semidirect |
| 28. #14679 | C24 | semidirect |
| 38. C26 | B(4,𝔽2) | semidirect |
| 42. C23×D4 | B(4,𝔽2) | semidirect |
| 44. C24×C4 | B(4,𝔽2) | semidirect |
| 50. C25 | C2×B(4,𝔽2) | semidirect |
| 54. C23×C4 | C2×B(4,𝔽2) | semidirect |
| 68. C2 | #72216433 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#72216427 | 2 |
| C2×#72216292 | 2 |
| C2×#72209716 | 2 |
| #72216433 | 2 |