Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
395 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 395 conjugacy classes of subgroups — too many to draw. Only its 92 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, hypocenter | Q8 | C22×C3×C4 |
| Frattini | C22 | C23×A4 |
| Fitting | C2×#520 | C3 |
| prime radical | C22×#41⋊C3 | C2 |
| cosocle | C2×Q8 | C23×C3 |
| hypercenter | C2×C4⋊C4 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C23 ◁ C2×C4⋊C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #520⋊C3 | C2 | direct (N × H) |
| 3. C2×C4×SL(2,𝔽3) | C2 | semidirect |
| 4. C2×#520 | C3 | semidirect |
| 5. C2×#41⋊C3 | C4 | semidirect |
| 9. C4×SL(2,𝔽3) | C22 | semidirect |
| 12. #520 | C2×C3 | semidirect |
| 19. #41⋊C3 | C2×C4 | semidirect |
| 20. C2×SL(2,𝔽3) | C2×C4 | semidirect |
| 25. C2×#41 | C3×C4 | semidirect |
| 49. SL(2,𝔽3) | C22×C4 | semidirect |
| 54. #41 | C2×C3×C4 | semidirect |
| 132. C2 | #520⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#41⋊C3 | 2 |
| #520⋊C3 | 2 |
| C2×C4×SL(2,𝔽3) | 2 |
| C2×#520 | 3 |
| C2×C3×C4⋊C4 | 4 |