Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
456 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 456 conjugacy classes of subgroups — too many to draw. Only its 144 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×C3 | C2×D4 |
| commutator | C2×C4 | C22×C3×C4 |
| Frattini, cosocle | C22×C4 | C23×C3 |
| prime radical | C2×C3×#121 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C23×C3×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×C3×C4⋊C4 | C2 | semidirect |
| 3. C2×C3×#102 | C2 | semidirect |
| 4. C2×C3×C22⋊C8 | C2 | semidirect |
| 5. C3×#340 | C2 | semidirect |
| 6. #1874 | C3 | direct (N × H) |
| 13. C3×#121 | C4 | semidirect |
| 22. C22×C4⋊C4 | C2×C3 | semidirect |
| 23. C2×#102 | C2×C3 | semidirect |
| 24. C2×C22⋊C8 | C2×C3 | semidirect |
| 25. #340 | C2×C3 | semidirect |
| 34. C2×C3×D4 | C2×C4 | semidirect |
| 52. C3×C4⋊C4 | C2×C4 | semidirect |
| 53. C3×C4⋊C4 | D4 | semidirect |
| 61. #121 | C3×C4 | semidirect |
| 112. C2×D4 | C2×C3×C4 | semidirect |
| 130. C4⋊C4 | C2×C3×C4 | semidirect |
| 131. C4⋊C4 | C3×D4 | semidirect |
| 146. C22×C3 | #102 | semidirect |
| 216. C22 | C3×#102 | semidirect |
| 229. C3 | #1874 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×#121 | 2 |
| C22×C3×C4⋊C4 | 2 |
| C2×C3×#102 | 2 |
| C2×C3×C22⋊C8 | 2 |
| C3×#340 | 2 |
| #1874 | 3 |