Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
503 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 503 conjugacy classes of subgroups — too many to draw. Only its 116 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×A4 |
| commutator, cosocle | C2×Q8 | C23×C3 |
| Frattini | C22 | C23×A4 |
| Fitting | C2×D4×Q8 | C3 |
| hypercenter | C22×D4 | A4 |
| hypocenter | Q8 | C2×C3×D4 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C23 ◁ C22×D4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×SL(2,𝔽3) | C2 | semidirect |
| 2. D4×SL(2,𝔽3) | C2 | direct (N × H) |
| 3. C2×C4×SL(2,𝔽3) | C2 | semidirect |
| 4. C2×D4×Q8 | C3 | semidirect |
| 8. C22×SL(2,𝔽3) | C22 | semidirect |
| 9. C4×SL(2,𝔽3) | C22 | semidirect |
| 10. C23×Q8 | C2×C3 | semidirect |
| 12. D4×Q8 | C2×C3 | semidirect |
| 15. C2×C4×Q8 | C2×C3 | semidirect |
| 22. C2×SL(2,𝔽3) | D4 | direct (N × H) |
| 29. C22×Q8 | C22×C3 | semidirect |
| 37. C4×Q8 | C22×C3 | semidirect |
| 53. SL(2,𝔽3) | C2×D4 | direct (N × H) |
| 65. C2×Q8 | C3×D4 | semidirect |
| 67. C2×D4 | SL(2,𝔽3) | direct (N × H) |
| 95. C23 | C2×SL(2,𝔽3) | semidirect |
| 104. Q8 | C2×C3×D4 | semidirect |
| 105. D4 | C2×SL(2,𝔽3) | direct (N × H) |
| 117. C2×C4 | C2×SL(2,𝔽3) | semidirect |
| 129. C22 | C22×SL(2,𝔽3) | semidirect |
| 137. C4 | C22×SL(2,𝔽3) | semidirect |
| 140. C2 | D4×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 |
| D4×SL(2,𝔽3) | 2 |
| C2×C4×SL(2,𝔽3) | 2 |
| C2×D4×Q8 | 3 |
| C22×C3×D4 | 4 |