Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
551 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 551 conjugacy classes of subgroups — too many to draw. Only its 132 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C24 |
| prime radical | C2×#120 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×C4 | C2 | semidirect |
| 2. C2×#123 | C2 | semidirect |
| 4. #386 | C2 | semidirect |
| 5. #385 | C2 | semidirect |
| 6. #381 | C2 | semidirect |
| 7. #379 | C2 | semidirect |
| 17. C2×C4⋊C4 | C22 | semidirect |
| 22. C2×C22⋊C4 | C22 | semidirect |
| 23. #95 | C22 | semidirect |
| 39. C22×C4 | D4 | semidirect |
| 87. C22 | #123 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C4 | 2 |
| C2×#123 | 2 |
| C2×#120 | 2 |
| #386 | 2 |
| #385 | 2 |
| #381 | 2 |
| #379 | 2 |