Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
519 conjugacy classes of subgroups; only the 172 normal subgroups are shown.
This group has 519 conjugacy classes of subgroups — too many to draw. Only its 172 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C4 | A4 |
| commutator, hypocenter | Q8 | C22×C3×C4 |
| Frattini | C22 | C23×A4 |
| Fitting | C22×C4×Q8 | C3 |
| socle | C24 | C2×A4 |
| prime radical | C23×SL(2,𝔽3) | C2 |
| cosocle | C2×Q8 | C23×C3 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C23×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C4×SL(2,𝔽3) | C2 | direct (N × H) |
| 3. C22×C4×Q8 | C3 | semidirect |
| 5. C22×SL(2,𝔽3) | C4 | direct (N × H) |
| 7. C4×SL(2,𝔽3) | C22 | direct (N × H) |
| 10. C2×C4×Q8 | C2×C3 | semidirect |
| 16. C2×SL(2,𝔽3) | C2×C4 | direct (N × H) |
| 20. C22×Q8 | C3×C4 | semidirect |
| 27. C4×Q8 | C22×C3 | semidirect |
| 38. SL(2,𝔽3) | C22×C4 | direct (N × H) |
| 45. C2×Q8 | C2×C3×C4 | semidirect |
| 53. C22×C4 | SL(2,𝔽3) | direct (N × H) |
| 73. Q8 | C22×C3×C4 | semidirect |
| 81. C2×C4 | C2×SL(2,𝔽3) | direct (N × H) |
| 87. C22 | C4×SL(2,𝔽3) | direct (N × H) |
| 96. C4 | C22×SL(2,𝔽3) | direct (N × H) |
| 98. C2 | C2×C4×SL(2,𝔽3) | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×SL(2,𝔽3) | 2 |
| C2×C4×SL(2,𝔽3) | 2 |
| C22×C4×Q8 | 3 |
| C23×C3×C4 | 4 |