Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
816 conjugacy classes of subgroups; only the 440 normal subgroups are shown.
This group has 816 conjugacy classes of subgroups — too many to draw. Only its 440 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 | C24×C3 | D4 |
| commutator | C4 | C23×C3×C4 |
| Frattini, cosocle | C2×C4 | C24×C3 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ C23×C3×C4 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C2×C3×#103 | C2 | direct (N × H) |
| 5. C22×#103 | C3 | direct (N × H) |
| 8. C22×C3×Q8 | C4 | semidirect |
| 17. C3×#103 | C22 | direct (N × H) |
| 21. C2×#103 | C2×C3 | direct (N × H) |
| 26. C2×C3×Q8 | C2×C4 | semidirect |
| 42. C22×Q8 | C3×C4 | semidirect |
| 51. #103 | C22×C3 | direct (N × H) |
| 58. C3×Q8 | C22×C4 | semidirect |
| 73. C2×Q8 | C2×C3×C4 | semidirect |
| 87. C22×C3 | #103 | direct (N × H) |
| 103. Q8 | C22×C3×C4 | semidirect |
| 114. C2×C3 | C2×#103 | direct (N × H) |
| 118. C22 | C3×#103 | direct (N × H) |
| 128. C3 | C22×#103 | direct (N × H) |
| 129. C2 | C2×C3×#103 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×Q8 | 2 |
| C23×C3×C8 | 2 |
| C22×C3×C4⋊C4 | 2 |
| C2×C3×#103 | 2 |
| C22×#103 | 3 |