Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
132 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 42 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #99 |
| commutator | C23 | C22×C4 |
| Frattini, cosocle | C2×D4 | C23 |
| socle | C2 | C2×#99 |
| prime radical | #525 | C2 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #413 | C2 | semidirect |
| 3. #410 | C2 | semidirect |
| 4. #352 | C2 | semidirect |
| 5. AΣL(1,𝔽16) | C2 | semidirect |
| 7. C23×C4 | C4 | semidirect |
| 9. #123 | C4 | semidirect |
| 13. #120 | C4 | semidirect |
| 18. #117 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #525 | 2 |
| #413 | 2 |
| #410 | 2 |
| #352 | 2 |
| AΣL(1,𝔽16) | 2 |