Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
388 conjugacy classes of subgroups; only the 196 normal subgroups are shown.
This group has 388 conjugacy classes of subgroups — too many to draw. Only its 196 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C24 |
| commutator | C22 | C23×C4 |
| prime radical | C22×D4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C22×C4⋊C4 | C2 | semidirect |
| 4. #389 | C2 | semidirect |
| 5. #385 | C2 | semidirect |
| 6. #384 | C2 | semidirect |
| 7. #380 | C2 | semidirect |
| 8. C4×C4⋊C4 | C2 | semidirect |
| 13. C4×D4 | C4 | semidirect |
| 37. C22×C4 | C2×C4 | semidirect |
| 50. C4⋊C4 | C2×C4 | semidirect |
| 51. C4⋊C4 | D4 | direct (N × H) |
| 53. C22⋊C4 | C2×C4 | semidirect |
| 55. C42 | C2×C4 | semidirect |
| 64. D4 | C4⋊C4 | direct (N × H) |
| 94. C22 | C2×C4⋊C4 | semidirect |
| 107. C4 | C2×C4⋊C4 | semidirect |
| 108. C4 | C4×D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×D4 | 2 |
| C22×C4⋊C4 | 2 |
| #389 | 2 |
| #385 | 2 |
| #384 | 2 |
| #380 | 2 |
| C4×C4⋊C4 | 2 |