Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
268 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 268 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×A4 |
| commutator, cosocle | C2×Q8 | C23×C3 |
| Frattini, socle | C22 | C23×A4 |
| Fitting | Q8×#41 | C3 |
| prime radical | C2×#41⋊C3 | C22 |
| hypercenter | C4×Q8 | A4 |
| hypocenter | Q8 | C2×C3×Q8 |
| LCS | G ▷ C2×Q8 ▷ Q8 | |
| UCS | C1 ◁ C2×C4 ◁ C4×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q8×SL(2,𝔽3) | C2 | semidirect |
| 2. #557⋊C3 | C2 | semidirect |
| 4. Q8×#41 | C3 | semidirect |
| 17. #41⋊C3 | Q8 | direct (N × H) |
| 44. SL(2,𝔽3) | C2×Q8 | semidirect |
| 46. #41 | C3×Q8 | semidirect |
| 75. Q8 | #41⋊C3 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×SL(2,𝔽3) | 2 |
| #557⋊C3 | 2 |
| C4×#41⋊C3 | 2 |
| Q8×#41 | 3 |
| C3×C4×Q8 | 4 |