Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
552 conjugacy classes of subgroups; only the 368 normal subgroups are shown.
This group has 552 conjugacy classes of subgroups — too many to draw. Only its 368 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 | C23×C97 | C24 |
| commutator | C22 | C23×C4×C97 |
| Frattini, cosocle | C23 | C24×C97 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C97 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 8. C4×Q8×C97 | C4 | semidirect |
| 31. Q8×C97 | C4⋊C4 | semidirect |
| 60. #2351 | C97 | direct (N × H) |
| 61. C97 | #2351 | direct (N × H) |
| 69. C4×Q8 | C4×C97 | semidirect |
| 92. Q8 | C4⋊C4×C97 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×Q8×C97 | 2 |
| #389×C97 | 2 |
| #382×C97 | 2 |
| C4×C4⋊C4×C97 | 2 |
| #2351 | 97 |