Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
1812 conjugacy classes of subgroups; only the 360 normal subgroups are shown.
This group has 1812 conjugacy classes of subgroups — too many to draw. Only its 360 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24×C97 | C23 |
| commutator, Frattini, cosocle | C23 | C24×C97 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24×C97 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×D4×C97 | C2 | semidirect |
| 2. C22×C22⋊C4×C97 | C2 | semidirect |
| 3. #392×C97 | C2 | direct (N × H) |
| 4. C2×#95×C97 | C2 | semidirect |
| 8. C22×D4×C97 | C22 | semidirect |
| 12. C2×C22⋊C4×C97 | C22 | semidirect |
| 13. #95×C97 | C22 | semidirect |
| 18. C24×C97 | D4 | semidirect |
| 26. C22×C4×C97 | D4 | semidirect |
| 41. C23×C97 | C2×D4 | semidirect |
| 49. C2×C4×C97 | C2×D4 | semidirect |
| 65. C2×C97 | #392 | direct (N × H) |
| 69. C2×#392 | C97 | direct (N × H) |
| 70. C97 | C2×#392 | direct (N × H) |
| 71. C23×D4 | C2×C97 | semidirect |
| 72. C22×C22⋊C4 | C2×C97 | semidirect |
| 73. #392 | C2×C97 | direct (N × H) |
| 74. C2×#95 | C2×C97 | semidirect |
| 78. C22×D4 | C22×C97 | semidirect |
| 82. C2×C22⋊C4 | C22×C97 | semidirect |
| 83. #95 | C22×C97 | semidirect |
| 88. C24 | D4×C97 | semidirect |
| 96. C22×C4 | D4×C97 | semidirect |
| 111. C23 | C2×D4×C97 | semidirect |
| 119. C2×C4 | C2×D4×C97 | semidirect |
| 135. C2 | #392×C97 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×D4×C97 | 2 |
| C22×C22⋊C4×C97 | 2 |
| #392×C97 | 2 |
| C2×#95×C97 | 2 |
| C2×#392 | 97 |