Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
174 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 174 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C23 |
| commutator | C22 | C2×C42 |
| socle | C23 | C22×C4 |
| prime radical | C24 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×#97 | C2 | semidirect |
| 4. C2×C4×C8 | C2 | semidirect |
| 6. #336 | C2 | semidirect |
| 7. #336 | C2 | semidirect |
| 11. C2×M16 | C4 | semidirect |
| 19. C22⋊C8 | C4 | semidirect |
| 70. C2×C4 | M16 | semidirect |
| 80. C8 | C22⋊C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×M16 | 2 |
| C2×C22⋊C8 | 2 |
| C2×#97 | 2 |
| C2×C4×C8 | 2 |
| C4×C22⋊C4 | 2 |
| #336 | 2 |