Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
266 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 266 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, socle | C2×C3 | C2×#99 |
| commutator | C23 | C22×C3×C4 |
| Frattini, cosocle | C2×D4 | C23×C3 |
| prime radical | C3×#535 | C2 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ 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×AΣL(1,𝔽16) | C2 | semidirect |
| 5. #2122 | C3 | direct (N × H) |
| 7. C22×C3×D4 | C4 | semidirect |
| 13. C3×#120 | C4 | semidirect |
| 16. C3×#117 | C4 | semidirect |
| 22. #413 | C2×C3 | semidirect |
| 23. #410 | C2×C3 | semidirect |
| 24. AΣL(1,𝔽16) | C2×C3 | semidirect |
| 53. C22×D4 | C3×C4 | semidirect |
| 59. #120 | C3×C4 | semidirect |
| 62. #117 | C3×C4 | semidirect |
| 192. C3 | #2122 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#535 | 2 |
| C3×#413 | 2 |
| C3×#410 | 2 |
| C3×AΣL(1,𝔽16) | 2 |
| #2122 | 3 |