Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
670 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 670 conjugacy classes of subgroups — too many to draw. Only its 136 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 | #120 |
| commutator | C23 | C22×C3×C4 |
| Frattini, cosocle | C24 | C23×C3 |
| prime radical | C2×C3×H(5,𝔽2) | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C24×C3 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C3×#99 | C2 | semidirect |
| 3. C3×#386 | C2 | semidirect |
| 4. #1790 | C3 | direct (N × H) |
| 6. C3×H(5,𝔽2) | C4 | semidirect |
| 10. C23×C3×C4 | C22 | semidirect |
| 12. C2×C3×C22⋊C4 | C22 | semidirect |
| 16. C2×#99 | C2×C3 | semidirect |
| 17. #386 | C2×C3 | semidirect |
| 39. H(5,𝔽2) | C3×C4 | semidirect |
| 43. C23×C4 | C22×C3 | semidirect |
| 45. C2×C22⋊C4 | C22×C3 | semidirect |
| 57. C23×C3 | C22⋊C4 | semidirect |
| 74. C2×C3×C4 | C22⋊C4 | semidirect |
| 120. C23 | C3×C22⋊C4 | semidirect |
| 137. C2×C4 | C3×C22⋊C4 | semidirect |
| 160. C3 | #1790 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×H(5,𝔽2) | 2 |
| C2×C3×#99 | 2 |
| C3×#386 | 2 |
| #1790 | 3 |