Finite Group DatabasebetaFinite Group DatabasebetaOrder 8576.
740 conjugacy classes of subgroups; only the 248 normal subgroups are shown.
This group has 740 conjugacy classes of subgroups — too many to draw. Only its 248 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C67 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C67 |
| prime radical | C24×C67 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C67 ◁ C23×C4×C67 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×Q16×C67 | C2 | semidirect |
| 4. #451×C67 | C2 | direct (N × H) |
| 5. C2×#103×C67 | C2 | semidirect |
| 16. C2×Q16×C67 | C22 | semidirect |
| 23. #103×C67 | C22 | semidirect |
| 56. C23×C67 | Q16 | semidirect |
| 85. C22×C67 | C2×Q16 | semidirect |
| 95. C2×C67 | #451 | direct (N × H) |
| 99. C2×#451 | C67 | direct (N × H) |
| 100. C67 | C2×#451 | direct (N × H) |
| 102. C22×Q16 | C2×C67 | semidirect |
| 104. #451 | C2×C67 | direct (N × H) |
| 105. C2×#103 | C2×C67 | semidirect |
| 116. C2×Q16 | C22×C67 | semidirect |
| 123. #103 | C22×C67 | semidirect |
| 156. C23 | Q16×C67 | semidirect |
| 185. C22 | C2×Q16×C67 | semidirect |
| 195. C2 | #451×C67 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×Q8×C67 | 2 |
| C22×Q16×C67 | 2 |
| C2×C22⋊Q8×C67 | 2 |
| #451×C67 | 2 |
| C2×#103×C67 | 2 |
| C2×C22⋊C8×C67 | 2 |
| C2×#451 | 67 |