Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
155 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 155 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C8 | A4 |
| commutator, hypocenter | Q8 | C2×C3×C8 |
| Frattini | C2×C4 | C22×A4 |
| Fitting | C2×C8×Q8 | C3 |
| socle | C23 | C4×A4 |
| prime radical | C22×SL(2,𝔽3) | C4 |
| cosocle | C4×Q8 | C22×C3 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C22×C8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C8×SL(2,𝔽3) | C2 | direct (N × H) |
| 3. C2×C8×Q8 | C3 | semidirect |
| 10. C8×Q8 | C2×C3 | semidirect |
| 14. C2×SL(2,𝔽3) | C8 | direct (N × H) |
| 36. SL(2,𝔽3) | C2×C8 | direct (N × H) |
| 40. C2×Q8 | C3×C8 | semidirect |
| 47. C2×C8 | SL(2,𝔽3) | direct (N × H) |
| 67. Q8 | C2×C3×C8 | semidirect |
| 80. C8 | C2×SL(2,𝔽3) | direct (N × H) |
| 98. C2 | C8×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 |
|---|---|
| C2×C4×SL(2,𝔽3) | 2 |
| C8×SL(2,𝔽3) | 2 |
| C2×C8×Q8 | 3 |
| C22×C3×C8 | 4 |