Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
336 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 336 conjugacy classes of subgroups — too many to draw. Only its 168 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | C23 |
| commutator | C22 | C2×C3×C42 |
| Frattini, cosocle | C22×C4 | C23×C3 |
| socle | C23×C3 | C22×C4 |
| prime radical | C24×C3 | C2×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×C3×#97 | C2 | semidirect |
| 4. C2×C3×C4×C8 | C2 | semidirect |
| 5. C3×#336 | C2 | semidirect |
| 6. #1750 | C3 | direct (N × H) |
| 16. C3×C22⋊C8 | C4 | semidirect |
| 21. C2×#97 | C2×C3 | semidirect |
| 22. C2×C4×C8 | C2×C3 | semidirect |
| 23. #336 | C2×C3 | semidirect |
| 54. C22⋊C8 | C3×C4 | semidirect |
| 149. C3 | #1750 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#117 | 2 |
| C2×C3×C22⋊C8 | 2 |
| C2×C3×#97 | 2 |
| C2×C3×C4×C8 | 2 |
| C3×#336 | 2 |
| #1750 | 3 |