Finite Group DatabasebetaFinite Group DatabasebetaOrder 2592.
2218 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 2218 conjugacy classes of subgroups — too many to draw. Only its 187 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 | C2×#10620150 | C2 | 1 | 3 | yes | maximal |
| 2 | C3×#88813 | C2 | 1 | 4 | yes | maximal, complemented |
| 3 | #11722900 | — | 3 | 4 | no | maximal |
| 4 | C22×C3⋊(C3⋊(C3⋊D3)) | C22 | 1 | 1 | yes | characteristic |
| 5 | C2×C32×C3⋊Dic3 | C22 | 1 | 6 | yes | — |
| 6 | #10620150 | C4 | 1 | 12 | yes | complemented |
| 7 | C2×#6289 | — | 3 | 12 | no | — |
| 8 | C2×#6289 | — | 6 | 6 | no | — |
| 9 | #88813 | D3 | 1 | 4 | yes | complemented |
| 10 | C3×Dic32 | — | 3 | 12 | no | — |
| 11 | C22×C34 | C23 | 1 | 1 | yes | characteristic |
| 12 | C2×C3⋊(C3⋊(C3⋊D3)) | C2×C4 | 1 | 6 | yes | — |
| 13 | C32×C3⋊Dic3 | C2×C4 | 1 | 12 | yes | complemented |
| 14 | C4×#624 | — | 9 | 6 | no | — |
| 15 | C22×C3⋊(C3⋊D3) | — | 3 | 4 | no | — |
| 16 | C22×C3⋊(C3⋊D3) | — | 6 | 6 | no | — |
| 17 | C22×C3⋊(C3⋊D3) | — | 12 | 4 | no | — |
| 18 | C22×C3⋊(C3⋊D3) | — | 12 | 2 | no | — |
| 19 | C2×C3×C3⋊Dic3 | — | 2 | 6 | no | — |
| 20 | C2×C3×C3⋊Dic3 | C2×D3 | 1 | 12 | yes | — |
| 21 | C2×C32×Dic3 | — | 3 | 12 | no | — |
| 22 | C2×C32×Dic3 | — | 6 | 6 | no | — |
| 23 | #6289 | — | 3 | 48 | no | — |
| 24 | #6289 | — | 6 | 24 | no | — |
| 25 | C2×C34 | C22×C4 | 1 | 3 | yes | — |
| 26 | C3⋊(C3⋊(C3⋊D3)) | C42 | 1 | 4 | yes | complemented |
| 27 | C2×C4×C3⋊D3 | — | 9 | 6 | no | — |
| 28 | C2×#624 | — | 9 | 12 | no | — |
| 29 | C2×#624 | — | 18 | 12 | no | — |
| 30 | C2×#624 | — | 18 | 6 | no | — |
| 31 | C3×C4×Dic3 | — | 9 | 12 | no | — |
| 32 | Dic32 | — | 3 | 12 | no | — |
| 33 | C22×C33 | — | 2 | 6 | no | — |
| 34 | C22×C33 | — | 4 | 2 | no | — |
| 35 | C22×C33 | — | 4 | 4 | no | — |
| 36 | C22×C33 | C22×D3 | 1 | 4 | yes | — |
| 37 | C2×C3⋊(C3⋊D3) | — | 3 | 24 | no | — |
| 38 | C2×C3⋊(C3⋊D3) | — | 6 | 12 | no | — |
| 39 | C2×C3⋊(C3⋊D3) | — | 6 | 24 | no | — |
| 40 | C2×C3⋊(C3⋊D3) | — | 12 | 24 | no | — |
| 41 | C2×C3⋊(C3⋊D3) | — | 12 | 12 | no | — |
| 42 | C3×C3⋊Dic3 | — | 2 | 12 | no | — |
| 43 | C3×C3⋊Dic3 | C4×D3 | 1 | 24 | yes | complemented |
| 44 | C32×Dic3 | — | 3 | 24 | no | — |
| 45 | C32×Dic3 | — | 6 | 12 | no | — |
| 46 | C42×D3 | — | 27 | 4 | no | — |
| 47 | C34 | C2×C42 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 48 | C22×C3⋊D3 | — | 9 | 6 | no | — |
| 49 | C22×C3⋊D3 | — | 18 | 12 | no | — |
| 50 | C22×C3⋊D3 | — | 18 | 6 | no | — |
| 51 | C22×C3⋊D3 | — | 36 | 4 | no | — |
| 52 | C22×C3⋊D3 | — | 36 | 4 | no | — |
| 53 | C22×C3⋊D3 | — | 36 | 12 | no | — |
| 54 | C22×C3⋊D3 | — | 36 | 2 | no | — |
| 55 | C2×C32×C4 | — | 9 | 6 | no | — |
| 56 | C2×C3⋊Dic3 | D32 | 1 | 6 | yes | — |
| 57 | C4×C3⋊D3 | — | 9 | 24 | no | — |
| 58 | C2×C3×Dic3 | — | 3 | 24 | no | — |
| 59 | C2×C3×Dic3 | — | 6 | 12 | no | — |
| 60 | C2×C3×Dic3 | — | 6 | 12 | no | — |
| 61 | C2×C3×Dic3 | — | 6 | 12 | no | — |
| 62 | #624 | — | 9 | 48 | no | — |
| 63 | #624 | — | 18 | 48 | no | — |
| 64 | #624 | — | 18 | 24 | no | — |
| 65 | C2×C33 | — | 2 | 12 | no | — |
| 66 | C2×C33 | — | 2 | 6 | no | — |
| 67 | C2×C33 | — | 4 | 6 | no | — |
| 68 | C2×C33 | — | 4 | 12 | no | — |
| 69 | C2×C33 | C2×C4×D3 | 1 | 12 | yes | — |
| 70 | C3⋊(C3⋊D3) | — | 3 | 16 | no | — |
| 71 | C3⋊(C3⋊D3) | — | 6 | 24 | no | — |
| 72 | C3⋊(C3⋊D3) | — | 12 | 16 | no | — |
| 73 | C3⋊(C3⋊D3) | — | 12 | 8 | no | — |
| 74 | C2×C4×D3 | — | 27 | 12 | no | — |
| 75 | C2×C4×D3 | — | 54 | 6 | no | — |
| 76 | C3×C42 | — | 27 | 4 | no | — |
| 77 | C4×Dic3 | — | 9 | 12 | no | — |
| 78 | C22×C32 | — | 2 | 6 | no | — |
| 79 | C22×C32 | — | 2 | 12 | no | — |
| 80 | C22×C32 | — | 4 | 4 | no | — |
| 81 | C22×C32 | — | 4 | 12 | no | — |
| 82 | C22×C32 | — | 4 | 2 | no | — |
| 83 | C22×C32 | — | 4 | 4 | no | — |
| 84 | C22×C32 | C2×D32 | 1 | 6 | yes | — |
| 85 | C2×C3⋊D3 | — | 9 | 24 | no | — |
| 86 | C2×C3⋊D3 | — | 9 | 12 | no | — |
| 87 | C2×C3⋊D3 | — | 18 | 24 | no | — |
| 88 | C2×C3⋊D3 | — | 18 | 48 | no | — |
| 89 | C2×C3⋊D3 | — | 18 | 12 | no | — |
| 90 | C2×C3⋊D3 | — | 18 | 24 | no | — |
| 91 | C2×C3⋊D3 | — | 36 | 24 | no | — |
| 92 | C2×C3⋊D3 | — | 36 | 24 | no | — |
| 93 | C2×C3⋊D3 | — | 36 | 24 | no | — |
| 94 | C2×C3⋊D3 | — | 36 | 48 | no | — |
| 95 | C2×C3⋊D3 | — | 36 | 12 | no | — |
| 96 | C32×C4 | — | 9 | 12 | no | — |
| 97 | C3⋊Dic3 | #624 | 1 | 12 | yes | complemented |
| 98 | C3×Dic3 | — | 3 | 48 | no | — |
| 99 | C3×Dic3 | — | 6 | 24 | no | — |
| 100 | C3×Dic3 | — | 6 | 24 | no | — |
| 101 | C3×Dic3 | — | 6 | 24 | no | — |
| 102 | C2×C42 | — | 81 | 1 | no | Sylow |
| 103 | C33 | — | 2 | 6 | no | — |
| 104 | C33 | — | 4 | 2 | no | — |
| 105 | C33 | — | 4 | 4 | no | — |
| 106 | C33 | C42×D3 | 1 | 4 | yes | complemented |
| 107 | C22×D3 | — | 27 | 4 | no | — |
| 108 | C22×D3 | — | 54 | 6 | no | — |
| 109 | C22×D3 | — | 108 | 4 | no | — |
| 110 | C22×D3 | — | 108 | 2 | no | — |
| 111 | C2×C3×C4 | — | 9 | 12 | no | — |
| 112 | C2×C3×C4 | — | 18 | 6 | no | — |
| 113 | C2×Dic3 | — | 3 | 12 | no | — |
| 114 | C2×Dic3 | — | 6 | 6 | no | — |
| 115 | C4×D3 | — | 27 | 48 | no | — |
| 116 | C4×D3 | — | 54 | 24 | no | — |
| 117 | C2×C32 | — | 2 | 6 | no | — |
| 118 | C2×C32 | — | 2 | 12 | no | — |
| 119 | C2×C32 | — | 2 | 12 | no | — |
| 120 | C2×C32 | — | 2 | 24 | no | — |
| 121 | C2×C32 | — | 4 | 12 | no | — |
| 122 | C2×C32 | — | 4 | 12 | no | — |
| 123 | C2×C32 | — | 4 | 24 | no | — |
| 124 | C2×C32 | — | 4 | 6 | no | — |
| 125 | C2×C32 | — | 4 | 12 | no | — |
| 126 | C2×C32 | C4×D32 | 1 | 12 | yes | — |
| 127 | C2×C32 | C2×#624 | 1 | 6 | yes | — |
| 128 | C3⋊D3 | — | 9 | 24 | no | — |
| 129 | C3⋊D3 | — | 18 | 48 | no | — |
| 130 | C3⋊D3 | — | 18 | 24 | no | — |
| 131 | C3⋊D3 | — | 36 | 16 | no | — |
| 132 | C3⋊D3 | — | 36 | 16 | no | — |
| 133 | C3⋊D3 | — | 36 | 48 | no | — |
| 134 | C3⋊D3 | — | 36 | 8 | no | — |
| 135 | C22×C4 | — | 81 | 3 | no | — |
| 136 | C42 | — | 27 | 4 | no | — |
| 137 | C22×C3 | — | 2 | 6 | no | — |
| 138 | C22×C3 | — | 4 | 2 | no | — |
| 139 | C22×C3 | — | 4 | 4 | no | — |
| 140 | C22×C3 | D33 | 1 | 4 | yes | — |
| 141 | C2×D3 | — | 27 | 24 | no | — |
| 142 | C2×D3 | — | 54 | 12 | no | — |
| 143 | C2×D3 | — | 54 | 24 | no | — |
| 144 | C2×D3 | — | 108 | 24 | no | — |
| 145 | C2×D3 | — | 108 | 12 | no | — |
| 146 | C3×C4 | — | 9 | 24 | no | — |
| 147 | C3×C4 | — | 18 | 12 | no | — |
| 148 | Dic3 | — | 3 | 24 | no | — |
| 149 | Dic3 | — | 6 | 12 | no | — |
| 150 | C32 | — | 2 | 6 | no | — |
| 151 | C32 | — | 2 | 12 | no | — |
| 152 | C32 | — | 4 | 4 | no | — |
| 153 | C32 | — | 4 | 12 | no | — |
| 154 | C32 | — | 4 | 2 | no | — |
| 155 | C32 | — | 4 | 4 | no | — |
| 156 | C32 | C4×#624 | 1 | 6 | yes | complemented |
| 157 | C23 | — | 81 | 1 | no | — |
| 158 | C2×C4 | — | 9 | 6 | no | — |
| 159 | C2×C4 | — | 81 | 12 | no | — |
| 160 | C2×C3 | — | 2 | 6 | no | — |
| 161 | C2×C3 | — | 2 | 12 | no | — |
| 162 | C2×C3 | — | 4 | 6 | no | — |
| 163 | C2×C3 | — | 4 | 12 | no | — |
| 164 | C2×C3 | D3×#624 | 1 | 12 | yes | — |
| 165 | D3 | — | 27 | 16 | no | — |
| 166 | D3 | — | 54 | 24 | no | — |
| 167 | D3 | — | 108 | 16 | no | — |
| 168 | D3 | — | 108 | 8 | no | — |
| 169 | C22 | — | 81 | 6 | no | — |
| 170 | C22 | #10620215 | 1 | 1 | yes | characteristic |
| 171 | C4 | — | 9 | 12 | no | — |
| 172 | C3 | — | 2 | 6 | no | — |
| 173 | C3 | — | 4 | 2 | no | — |
| 174 | C3 | — | 4 | 4 | no | — |
| 175 | C3 | #11722900 | 1 | 4 | yes | complemented |
| 176 | C2 | — | 81 | 4 | no | — |
| 177 | C2 | #13049546 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | #10620215 |
| commutator, hypocenter | C34 | C2×C42 |
| Fitting, socle, cosocle | C22×C34 | C23 |
| prime radical | C22×C3⋊(C3⋊(C3⋊D3)) | C22 |
| LCS | G ▷ C34 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C3×#88813 | C2 | semidirect |
| 6. #10620150 | C4 | semidirect |
| 9. #88813 | D3 | semidirect |
| 13. C32×C3⋊Dic3 | C2×C4 | semidirect |
| 26. C3⋊(C3⋊(C3⋊D3)) | C42 | semidirect |
| 43. C3×C3⋊Dic3 | C4×D3 | semidirect |
| 47. C34 | C2×C42 | semidirect |
| 97. C3⋊Dic3 | #624 | semidirect |
| 106. C33 | C42×D3 | semidirect |
| 156. C32 | C4×#624 | semidirect |
| 175. C3 | #11722900 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#10620150 | 2 |
| C3×#88813 | 2 |
| #11722900 | 3 |