Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
204 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 204 conjugacy classes of subgroups — too many to draw. Only its 68 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, socle | C23 | C22×C4 |
| prime radical | C22×D4 | C4 |
| LCS | G ▷ C23 ▷ 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 |
|---|---|---|
| 1. C22×M16 | C2 | semidirect |
| 3. C2×C22⋊C8 | C2 | semidirect |
| 4. C2×C22⋊C8 | C2 | semidirect |
| 5. #336 | C2 | semidirect |
| 45. C2×C8 | D4 | semidirect |
| 60. C23 | M16 | semidirect |
| 77. C2×C4 | M16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×M16 | 2 |
| C2×C4×D4 | 2 |
| C2×C22⋊C8 | 2 |
| #336 | 2 |