Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
264 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #99 |
| commutator | C23 | C22×C3×C4 |
| Frattini, cosocle | C2×D4 | C23×C3 |
| socle | C2×C3 | C2×#99 |
| prime radical | C3×#525 | C2 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C2×C3×C4 ◁ C2×C3×#41 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#413 | C2 | semidirect |
| 3. C3×#410 | C2 | semidirect |
| 4. C3×#352 | C2 | semidirect |
| 5. C3×AΣL(1,𝔽16) | C2 | semidirect |
| 6. #2120 | C3 | direct (N × H) |
| 8. C23×C3×C4 | C4 | semidirect |
| 10. C3×#123 | C4 | semidirect |
| 14. C3×#120 | C4 | semidirect |
| 19. C3×#117 | C4 | semidirect |
| 25. #413 | C2×C3 | semidirect |
| 26. #410 | C2×C3 | semidirect |
| 27. #352 | C2×C3 | semidirect |
| 28. AΣL(1,𝔽16) | C2×C3 | semidirect |
| 63. C23×C4 | C3×C4 | semidirect |
| 65. #123 | C3×C4 | semidirect |
| 69. #120 | C3×C4 | semidirect |
| 74. #117 | C3×C4 | semidirect |
| 222. C3 | #2120 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#525 | 2 |
| C3×#413 | 2 |
| C3×#410 | 2 |
| C3×#352 | 2 |
| C3×AΣL(1,𝔽16) | 2 |
| #2120 | 3 |