Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
248 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24 |
| prime radical | #525 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#137 | C2 | semidirect |
| 2. C22×M16 | C2 | semidirect |
| 4. C2×C22⋊Q8 | C2 | semidirect |
| 5. #470 | C2 | semidirect |
| 6. #468 | C2 | semidirect |
| 7. #467 | C2 | semidirect |
| 8. #465 | C2 | semidirect |
| 9. #428 | C2 | semidirect |
| 10. #419 | C2 | semidirect |
| 11. #417 | C2 | semidirect |
| 16. C2×Q16 | C22 | semidirect |
| 17. C2×SD16 | C22 | semidirect |
| 39. #107 | C22 | semidirect |
| 40. #106 | C22 | semidirect |
| 41. #103 | C22 | semidirect |
| 42. #103 | C22 | semidirect |
| 43. #103 | C22 | semidirect |
| 44. #102 | C22 | semidirect |
| 72. M16 | D4 | semidirect |
| 144. C22 | #137 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#137 | 2 |
| C22×M16 | 2 |
| #525 | 2 |
| C2×C22⋊Q8 | 2 |
| #470 | 2 |
| #468 | 2 |
| #467 | 2 |
| #465 | 2 |
| #428 | 2 |
| #419 | 2 |
| #417 | 2 |