Finite Group DatabasebetaFinite Group DatabasebetaOrder 8576.
392 conjugacy classes of subgroups; only the 192 normal subgroups are shown.
This group has 392 conjugacy classes of subgroups — too many to draw. Only its 192 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×C67 | C22×D4 |
| commutator, Frattini, cosocle | C2×C4 | C24×C67 |
| prime radical | #550×C67 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C67 ◁ C4×Q8×C67 ◁ G | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q82×C67 | C2 | semidirect |
| 3. #486×C67 | C2 | semidirect |
| 4. #476×C67 | C2 | semidirect |
| 5. #460×C67 | C2 | semidirect |
| 6. C8×Q8×C67 | C2 | semidirect |
| 7. C4×SD16×C67 | C2 | semidirect |
| 46. Q8×C67 | SD16 | semidirect |
| 47. Q8×C67 | #41 | semidirect |
| 76. #3362 | C67 | direct (N × H) |
| 77. C67 | #3362 | direct (N × H) |
| 78. Q82 | C2×C67 | semidirect |
| 80. #486 | C2×C67 | semidirect |
| 81. #476 | C2×C67 | semidirect |
| 82. #460 | C2×C67 | semidirect |
| 83. C8×Q8 | C2×C67 | semidirect |
| 84. C4×SD16 | C2×C67 | semidirect |
| 123. Q8 | SD16×C67 | semidirect |
| 124. Q8 | #41×C67 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q82×C67 | 2 |
| #550×C67 | 2 |
| #486×C67 | 2 |
| #476×C67 | 2 |
| #460×C67 | 2 |
| C8×Q8×C67 | 2 |
| C4×SD16×C67 | 2 |
| #3362 | 67 |