Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
150 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 150 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C3 | #121 |
| commutator, Frattini, cosocle | C42 | C23×C3 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 8. #1672 | C3 | direct (N × H) |
| 79. C3×C8 | Q16 | semidirect |
| 123. C8 | C3×Q16 | semidirect |
| 137. C3 | #1672 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#498 | 2 |
| C3×#477 | 2 |
| C3×#462 | 2 |
| C3×C8×Q8 | 2 |
| C3×#334 | 2 |
| C3×#333 | 2 |
| C3×#332 | 2 |
| #1672 | 3 |