Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
219 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 219 conjugacy classes of subgroups — too many to draw. Only its 44 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 | #99 |
| commutator | C23 | C22×C4 |
| Frattini, cosocle | C2×D4 | C23 |
| prime radical | C2×#120 | C2 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#100 | C2 | semidirect |
| 3. C2×#99 | C2 | semidirect |
| 4. AΣL(1,𝔽16) | C2 | direct (N × H) |
| 5. C25 | C4 | semidirect |
| 11. #120 | C4 | semidirect |
| 15. C2×C22⋊C4 | C4 | semidirect |
| 17. #100 | C22 | semidirect |
| 19. #99 | C22 | semidirect |
| 26. C24 | C2×C4 | semidirect |
| 48. C22⋊C4 | C2×C4 | semidirect |
| 122. C2 | AΣL(1,𝔽16) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#120 | 2 |
| C2×#100 | 2 |
| C2×#99 | 2 |
| AΣL(1,𝔽16) | 2 |