Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
242 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 242 conjugacy classes of subgroups — too many to draw. Only its 142 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D4 |
| commutator | C4 | C23×C4 |
| Frattini, cosocle | C2×C4 | C24 |
| socle | C22 | C22×D4 |
| prime radical | C2×#41 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×C4×Q8 | C2 | semidirect |
| 4. #441 | C2 | semidirect |
| 5. #440 | C2 | semidirect |
| 6. C4×Q16 | C2 | semidirect |
| 7. C4×SD16 | C2 | semidirect |
| 8. #428 | C2 | semidirect |
| 9. #419 | C2 | semidirect |
| 11. C4×M16 | C2 | semidirect |
| 14. #137 | C4 | direct (N × H) |
| 48. C2×Q8 | C2×C4 | semidirect |
| 61. Q16 | C2×C4 | semidirect |
| 62. SD16 | C2×C4 | semidirect |
| 64. M16 | C2×C4 | semidirect |
| 135. C4 | #137 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#137 | 2 |
| C4×#41 | 2 |
| C2×C4×Q8 | 2 |
| #441 | 2 |
| #440 | 2 |
| C4×Q16 | 2 |
| C4×SD16 | 2 |
| #428 | 2 |
| #419 | 2 |
| #417 | 2 |
| C4×M16 | 2 |