Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
1018 conjugacy classes of subgroups; only the 216 normal subgroups are shown.
This group has 1018 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 | #120 |
| commutator, Frattini, cosocle | C23 | C24×C3 |
| 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 |
|---|---|---|
| 1. C2×C3×H(5,𝔽2) | C2 | semidirect |
| 2. C2×C3×#120 | C2 | semidirect |
| 3. C3×B(4,𝔽2) | C2 | direct (N × H) |
| 4. C2×C3×#99 | C2 | semidirect |
| 5. C2×B(4,𝔽2) | C3 | direct (N × H) |
| 6. C25×C3 | C22 | semidirect |
| 8. C3×H(5,𝔽2) | C22 | semidirect |
| 14. C3×#120 | C22 | semidirect |
| 16. C2×C3×C22⋊C4 | C22 | semidirect |
| 17. C3×#99 | C22 | semidirect |
| 18. C2×H(5,𝔽2) | C2×C3 | semidirect |
| 19. C2×#120 | C2×C3 | semidirect |
| 20. B(4,𝔽2) | C2×C3 | direct (N × H) |
| 21. C2×#99 | C2×C3 | semidirect |
| 26. C24×C3 | D4 | semidirect |
| 28. C24×C3 | C23 | semidirect |
| 39. C2×C3×D4 | D4 | semidirect |
| 43. C22×C3×C4 | D4 | semidirect |
| 46. C3×C22⋊C4 | C23 | semidirect |
| 47. C25 | C22×C3 | semidirect |
| 49. H(5,𝔽2) | C22×C3 | semidirect |
| 55. #120 | C22×C3 | semidirect |
| 57. C2×C22⋊C4 | C22×C3 | semidirect |
| 58. #99 | C22×C3 | semidirect |
| 74. C23×C3 | C2×D4 | semidirect |
| 86. C2×C3×C4 | C2×D4 | semidirect |
| 91. C24 | C3×D4 | semidirect |
| 93. C24 | C23×C3 | semidirect |
| 104. C2×D4 | C3×D4 | semidirect |
| 108. C22×C4 | C3×D4 | semidirect |
| 111. C22⋊C4 | C23×C3 | semidirect |
| 145. C23 | C2×C3×D4 | semidirect |
| 157. C2×C4 | C2×C3×D4 | semidirect |
| 162. C2×C3 | B(4,𝔽2) | direct (N × H) |
| 182. C3 | C2×B(4,𝔽2) | direct (N × H) |
| 187. C2 | C3×B(4,𝔽2) | 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×#120 | 2 |
| C3×B(4,𝔽2) | 2 |
| C2×C3×#99 | 2 |
| C2×B(4,𝔽2) | 3 |