Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
348 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 348 conjugacy classes of subgroups — too many to draw. Only its 216 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C3 |
| prime radical | C23×C3 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C3×#501 | C2 | semidirect |
| 4. C3×#499 | C2 | semidirect |
| 7. #3166 | C3 | direct (N × H) |
| 26. #501 | C2×C3 | semidirect |
| 27. #499 | C2×C3 | semidirect |
| 34. C3×M16 | Q8 | semidirect |
| 80. M16 | C3×Q8 | semidirect |
| 127. C3 | #3166 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#533 | 2 |
| C2×C3×#125 | 2 |
| C3×#501 | 2 |
| C3×#499 | 2 |
| C3×#428 | 2 |
| C3×C4×M16 | 2 |
| #3166 | 3 |