Finite Group DatabasebetaFinite Group DatabasebetaOrder 2592.
3284 conjugacy classes of subgroups; only the 325 normal subgroups are shown.
This group has 3284 conjugacy classes of subgroups — too many to draw. Only its 325 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | C33×C22⋊A4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3⋊(C3⋊#1020) | — | 3 | 39 | no | maximal |
| 3 | #11723994 | — | 3 | 1 | no | maximal |
| 4 | C3⋊(C3⋊(C3⋊S4)) | — | 4 | 3 | no | maximal |
| 5 | C24×C33 | D3 | 1 | 1 | yes | characteristic, complemented |
| 6 | C32×C22⋊A4 | D3 | 1 | 39 | yes | complemented |
| 7 | C2×#6312 | — | 3 | 3 | no | — |
| 8 | #88864 | — | 3 | 3 | no | — |
| 9 | C33×A4 | — | 4 | 3 | no | — |
| 10 | C33×A4 | — | 8 | 1 | no | — |
| 11 | C3⋊#1020 | — | 9 | 117 | no | — |
| 12 | #64162 | — | 9 | 13 | no | — |
| 13 | C23×C33 | — | 3 | 3 | no | — |
| 14 | C23×C33 | — | 6 | 1 | no | — |
| 15 | C22×C3⋊(C3⋊D3) | — | 3 | 1 | no | — |
| 16 | C3⋊(C3⋊S4) | — | 12 | 117 | no | — |
| 17 | #6312 | — | 6 | 3 | no | — |
| 18 | #6312 | — | 6 | 3 | no | — |
| 19 | C2×C3⋊(C3⋊Dic3) | — | 3 | 3 | no | — |
| 20 | C3⋊(C3⋊(C3⋊D3)) | — | 16 | 1 | no | — |
| 21 | C24×C32 | C3⋊D3 | 1 | 13 | yes | complemented |
| 22 | C3×C22⋊A4 | C3⋊D3 | 1 | 117 | yes | complemented |
| 23 | C2×#638 | — | 9 | 39 | no | — |
| 24 | #3757 | — | 9 | 39 | no | — |
| 25 | C22×C33 | — | 2 | 1 | no | — |
| 26 | C22×C33 | — | 3 | 1 | no | — |
| 27 | C22×C33 | — | 3 | 3 | no | — |
| 28 | C22×C33 | — | 6 | 3 | no | — |
| 29 | C22×C33 | S4 | 1 | 3 | yes | complemented |
| 30 | C2×C3⋊(C3⋊D3) | — | 6 | 3 | no | — |
| 31 | C32×A4 | — | 4 | 117 | no | — |
| 32 | C32×A4 | — | 8 | 39 | no | — |
| 33 | C3⋊(C3⋊Dic3) | — | 6 | 3 | no | — |
| 34 | #1020 | — | 27 | 27 | no | — |
| 35 | #953 | — | 27 | 13 | no | — |
| 36 | C34 | — | 16 | 1 | no | Sylow |
| 37 | C23×C32 | — | 3 | 39 | no | — |
| 38 | C23×C32 | — | 6 | 13 | no | — |
| 39 | C22×C3⋊D3 | — | 9 | 13 | no | — |
| 40 | C3⋊S4 | — | 36 | 351 | no | — |
| 41 | #638 | — | 18 | 39 | no | — |
| 42 | #638 | — | 18 | 39 | no | — |
| 43 | C2×C3⋊Dic3 | — | 9 | 39 | no | — |
| 44 | C2×C33 | — | 3 | 3 | no | — |
| 45 | C2×C33 | — | 6 | 1 | no | — |
| 46 | C3⋊(C3⋊D3) | — | 12 | 1 | no | — |
| 47 | C3⋊(C3⋊D3) | — | 48 | 39 | no | — |
| 48 | C24×C3 | C3⋊(C3⋊D3) | 1 | 13 | yes | complemented |
| 49 | C22⋊A4 | C3⋊(C3⋊D3) | 1 | 27 | yes | complemented |
| 50 | C2×#67 | — | 27 | 39 | no | — |
| 51 | #217 | — | 27 | 39 | no | — |
| 52 | C22×C32 | — | 2 | 13 | no | — |
| 53 | C22×C32 | — | 3 | 13 | no | — |
| 54 | C22×C32 | — | 3 | 39 | no | — |
| 55 | C22×C32 | — | 6 | 39 | no | — |
| 56 | C22×C32 | C3⋊S4 | 1 | 39 | yes | complemented |
| 57 | C2×C3⋊D3 | — | 18 | 39 | no | — |
| 58 | C3×A4 | — | 4 | 351 | no | — |
| 59 | C3×A4 | — | 8 | 117 | no | — |
| 60 | C3⋊Dic3 | — | 18 | 39 | no | — |
| 61 | #120 | — | 81 | 1 | no | Sylow |
| 62 | C33 | — | 16 | 39 | no | — |
| 63 | C33 | #1020 | 1 | 1 | yes | characteristic, complemented |
| 64 | C23×C3 | — | 3 | 39 | no | — |
| 65 | C23×C3 | — | 6 | 13 | no | — |
| 66 | C22×D3 | — | 27 | 13 | no | — |
| 67 | S4 | — | 108 | 81 | no | — |
| 68 | #67 | — | 54 | 39 | no | — |
| 69 | #67 | — | 54 | 39 | no | — |
| 70 | C2×Dic3 | — | 27 | 39 | no | — |
| 71 | C2×C32 | — | 3 | 39 | no | — |
| 72 | C2×C32 | — | 6 | 13 | no | — |
| 73 | C3⋊D3 | — | 36 | 13 | no | — |
| 74 | C3⋊D3 | — | 144 | 117 | no | — |
| 75 | C24 | C3⋊(C3⋊(C3⋊D3)) | 1 | 1 | yes | characteristic, complemented |
| 76 | C2×D4 | — | 81 | 3 | no | — |
| 77 | C22⋊C4 | — | 81 | 3 | no | — |
| 78 | C22×C3 | — | 2 | 13 | no | — |
| 79 | C22×C3 | — | 3 | 13 | no | — |
| 80 | C22×C3 | — | 3 | 39 | no | — |
| 81 | C22×C3 | — | 6 | 39 | no | — |
| 82 | C22×C3 | C3⋊(C3⋊S4) | 1 | 39 | yes | complemented |
| 83 | C2×D3 | — | 54 | 39 | no | — |
| 84 | A4 | — | 4 | 81 | no | — |
| 85 | A4 | — | 8 | 27 | no | — |
| 86 | Dic3 | — | 54 | 39 | no | — |
| 87 | C32 | — | 16 | 117 | no | — |
| 88 | C32 | C3⋊#1020 | 1 | 13 | yes | complemented |
| 89 | C23 | — | 3 | 3 | no | — |
| 90 | C23 | — | 6 | 1 | no | — |
| 91 | C23 | — | 81 | 1 | no | — |
| 92 | D4 | — | 162 | 3 | no | — |
| 93 | D4 | — | 162 | 3 | no | — |
| 94 | C2×C4 | — | 81 | 3 | no | — |
| 95 | C2×C3 | — | 3 | 39 | no | — |
| 96 | C2×C3 | — | 6 | 13 | no | — |
| 97 | D3 | — | 108 | 13 | no | — |
| 98 | D3 | — | 432 | 27 | no | — |
| 99 | C22 | — | 2 | 1 | no | — |
| 100 | C22 | — | 3 | 1 | no | — |
| 101 | C22 | — | 3 | 3 | no | — |
| 102 | C22 | — | 6 | 3 | no | — |
| 103 | C22 | — | 162 | 3 | no | — |
| 104 | C22 | C3⋊(C3⋊(C3⋊S4)) | 1 | 3 | yes | complemented |
| 105 | C4 | — | 162 | 3 | no | — |
| 106 | C3 | — | 16 | 27 | no | — |
| 107 | C3 | C3⋊(C3⋊#1020) | 1 | 13 | yes | complemented |
| 108 | C2 | — | 3 | 3 | no | — |
| 109 | C2 | — | 6 | 1 | no | — |
| 110 | C2 | — | 324 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C33×C22⋊A4 | C2 |
| Fitting, socle | C24×C33 | D3 |
| LCS | G ▷ C33×C22⋊A4 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C33×C22⋊A4 | C2 | semidirect |
| 5. C24×C33 | D3 | semidirect |
| 6. C32×C22⋊A4 | D3 | semidirect |
| 21. C24×C32 | C3⋊D3 | semidirect |
| 22. C3×C22⋊A4 | C3⋊D3 | semidirect |
| 29. C22×C33 | S4 | semidirect |
| 48. C24×C3 | C3⋊(C3⋊D3) | semidirect |
| 49. C22⋊A4 | C3⋊(C3⋊D3) | semidirect |
| 56. C22×C32 | C3⋊S4 | semidirect |
| 63. C33 | #1020 | semidirect |
| 75. C24 | C3⋊(C3⋊(C3⋊D3)) | semidirect |
| 82. C22×C3 | C3⋊(C3⋊S4) | semidirect |
| 88. C32 | C3⋊#1020 | semidirect |
| 104. C22 | C3⋊(C3⋊(C3⋊S4)) | semidirect |
| 107. C3 | C3⋊(C3⋊#1020) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C33×C22⋊A4 | 2 |
| C3⋊(C3⋊#1020) | 3 |
| #11723994 | 3 |
| C3⋊(C3⋊(C3⋊S4)) | 4 |