Finite Group DatabasebetaFinite Group DatabasebetaOrder 1440.
704 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 704 conjugacy classes of subgroups — too many to draw. Only its 154 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D3×D15 |
| commutator, socle, cosocle | C2×C32×C5 | C24 |
| Fitting | C2×C32×C4×C5 | C22 |
| hypercenter | C2×C4 | D3×D15 |
| hypocenter | C32×C5 | H(5,𝔽2) |
| LCS | G ▷ C2×C32×C5 ▷ C32×C5 | |
| UCS | C1 ◁ C2 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10624283 | C2 | semidirect |
| 2. C3×#6817 | C2 | semidirect |
| 3. C2×C3×C5×D12 | C2 | semidirect |
| 4. D3×#1220 | C2 | semidirect |
| 5. #10624205 | C2 | semidirect |
| 6. D12×D15 | C2 | semidirect |
| 7. #10624200 | C2 | semidirect |
| 8. #10624196 | C2 | semidirect |
| 11. C22×C3×C5×D3 | C22 | semidirect |
| 12. C2×D3×D15 | C22 | semidirect |
| 13. C2×C32×C4×C5 | C22 | semidirect |
| 14. #67184 | C22 | semidirect |
| 15. C4⋊(C3⋊D15) | C22 | semidirect |
| 16. C4×C3⋊D15 | C22 | semidirect |
| 17. #67180 | C22 | semidirect |
| 18. C3×#1220 | C22 | semidirect |
| 19. C3×D60 | C22 | semidirect |
| 20. C3×C4×D15 | C22 | semidirect |
| 21. C3×Dic30 | C22 | semidirect |
| 22. C3×C5×D12 | C22 | semidirect |
| 23. C3×C5×D12 | C22 | semidirect |
| 24. #67152 | C22 | semidirect |
| 25. #67150 | C22 | semidirect |
| 26. D3×Dic15 | C22 | semidirect |
| 37. #6817 | D3 | semidirect |
| 38. C2×C5×D12 | D3 | semidirect |
| 58. C2×C3×D12 | D5 | semidirect |
| 66. C22×C5×D3 | C2×D3 | semidirect |
| 71. C2×C3×C4×C5 | C2×D3 | semidirect |
| 72. C2×C3×C4×C5 | C2×D3 | semidirect |
| 78. #1220 | C2×D3 | semidirect |
| 83. D60 | C2×D3 | semidirect |
| 87. C4×D15 | C2×D3 | semidirect |
| 91. Dic30 | C2×D3 | semidirect |
| 92. C5×D12 | C2×D3 | semidirect |
| 93. C5×D12 | C2×D3 | semidirect |
| 116. C22×C3×D3 | C2×D5 | semidirect |
| 118. C2×C32×C4 | C2×D5 | semidirect |
| 125. C3×D12 | C2×D5 | semidirect |
| 126. C3×D12 | C2×D5 | semidirect |
| 173. C2×D12 | D15 | semidirect |
| 174. C32×C5 | H(5,𝔽2) | semidirect |
| 179. C2×C4×C5 | D32 | semidirect |
| 218. C22×D3 | C2×D15 | semidirect |
| 226. C2×C3×C4 | D3×D5 | semidirect |
| 227. C2×C3×C4 | C2×D15 | semidirect |
| 241. D12 | C2×D15 | semidirect |
| 242. D12 | C2×D15 | semidirect |
| 276. C3×C5 | #1004 | semidirect |
| 277. C3×C5 | #1009 | semidirect |
| 308. C32 | #4144 | semidirect |
| 324. C2×C4 | D3×D15 | semidirect |
| 338. C5 | #64307 | semidirect |
| 350. C3 | #92018 | semidirect |
| 351. C3 | #92095 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10624283 | 2 |
| C3×#6817 | 2 |
| C2×C3×C5×D12 | 2 |
| D3×#1220 | 2 |
| #10624205 | 2 |
| D12×D15 | 2 |
| #10624200 | 2 |
| #10624196 | 2 |
| #92095 | 3 |
| #92018 | 3 |
| #64307 | 5 |