Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
440 conjugacy classes of subgroups; only the 392 normal subgroups are shown.
This group has 440 conjugacy classes of subgroups — too many to draw. Only its 392 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C8 | C22 |
| commutator | C2 | C23×C3×C8 |
| Frattini, cosocle | C2×C4 | C24×C3 |
| socle, prime radical | C23×C3 | C22×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C8 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×C8×Q8 | C2 | direct (N × H) |
| 5. C2×C8×Q8 | C3 | direct (N × H) |
| 16. C8×Q8 | C2×C3 | direct (N × H) |
| 19. C2×C3×Q8 | C8 | direct (N × H) |
| 25. C2×C3×C8 | Q8 | direct (N × H) |
| 41. C3×Q8 | C2×C8 | direct (N × H) |
| 51. C3×C8 | C2×Q8 | direct (N × H) |
| 52. C2×Q8 | C3×C8 | direct (N × H) |
| 58. C2×C8 | C3×Q8 | direct (N × H) |
| 74. Q8 | C2×C3×C8 | direct (N × H) |
| 84. C8 | C2×C3×Q8 | direct (N × H) |
| 85. C2×C3 | C8×Q8 | direct (N × H) |
| 98. C3 | C2×C8×Q8 | direct (N × H) |
| 99. C2 | C3×C8×Q8 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×C4×Q8 | 2 |
| C3×C8×Q8 | 2 |
| C2×C3×C4⋊C8 | 2 |
| C2×C3×C4×C8 | 2 |
| C2×C8×Q8 | 3 |