Finite Group DatabasebetaFinite Group DatabasebetaOrder 9504.
288 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 20 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 | D11×((C3⋊D3)⋊C3)⋊C2 | — | 4 | 1 | no | maximal |
| 2 | D11×AΓL(1,𝔽9) | — | 3 | 1 | no | maximal |
| 3 | #72259104 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C11×AGL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | D11×PSU(3,𝔽2)⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C11×PSU(3,𝔽2)⋊C3 | C22 | 1 | 1 | yes | characteristic, complemented |
| 7 | D11×PSU(3,𝔽2) | D3 | 1 | 1 | yes | characteristic, complemented |
| 8 | D11×GO+(4,𝔽2) | — | 3 | 1 | no | — |
| 9 | AGL(1,𝔽9)⋊D11 | — | 3 | 1 | no | — |
| 10 | PSU(3,𝔽2)⋊D11 | — | 3 | 1 | no | — |
| 11 | #13097490 | — | 3 | 1 | no | — |
| 12 | C11×AΓL(1,𝔽9) | — | 3 | 1 | no | — |
| 13 | D11×AGL(1,𝔽9) | — | 3 | 1 | no | — |
| 14 | #272⋊D11 | — | 4 | 1 | no | — |
| 15 | ((C3⋊D3)⋊C3)⋊D11 | — | 4 | 1 | no | — |
| 16 | ((C3⋊D3)⋊C3)⋊D11 | — | 4 | 1 | no | — |
| 17 | C11×((C3⋊D3)⋊C3)⋊C2 | — | 4 | 1 | no | — |
| 18 | D11×#272 | — | 4 | 1 | no | — |
| 19 | D11×(C3⋊D3)⋊C3 | — | 4 | 1 | no | — |
| 20 | D11×(C3⋊D3)⋊C3 | — | 4 | 1 | no | — |
| 21 | D11×GL(2,𝔽3) | — | 9 | 1 | no | maximal |
| 22 | C2×AGL(2,𝔽3) | — | 11 | 1 | no | maximal, Hall |
| 23 | D32×D11 | — | 6 | 1 | no | — |
| 24 | D32×D11 | — | 12 | 1 | no | — |
| 25 | C11⋊PSU(3,𝔽2) | — | 3 | 1 | no | — |
| 26 | C11×PSU(3,𝔽2) | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 27 | #157⋊D11 | — | 3 | 1 | no | — |
| 28 | D32⋊D11 | — | 6 | 1 | no | — |
| 29 | C11×GO+(4,𝔽2) | — | 3 | 1 | no | — |
| 30 | D11×#157 | — | 3 | 1 | no | — |
| 31 | C11⋊AGL(1,𝔽9) | — | 3 | 1 | no | — |
| 32 | C11×AGL(1,𝔽9) | — | 3 | 1 | no | — |
| 33 | C11⋊#272 | — | 4 | 1 | no | — |
| 34 | C11×#272 | — | 4 | 1 | no | — |
| 35 | C11⋊((C3⋊D3)⋊C3) | — | 4 | 1 | no | — |
| 36 | C11⋊((C3⋊D3)⋊C3) | — | 4 | 1 | no | — |
| 37 | C11×(C3⋊D3)⋊C3 | — | 4 | 1 | no | — |
| 38 | C11×(C3⋊D3)⋊C3 | — | 4 | 1 | no | — |
| 39 | D11×H(3,𝔽3) | — | 4 | 1 | no | — |
| 40 | D11×SL(2,𝔽3) | — | 9 | 1 | no | — |
| 41 | C11⋊GL(2,𝔽3) | — | 9 | 1 | no | — |
| 42 | C11×GL(2,𝔽3) | — | 9 | 1 | no | — |
| 43 | C2×PSU(3,𝔽2)⋊C3 | — | 11 | 1 | no | — |
| 44 | AGL(2,𝔽3) | — | 11 | 1 | no | — |
| 45 | AGL(2,𝔽3) | D11 | 1 | 1 | yes | characteristic, direct product factor |
| 46 | (C3⋊D3)⋊D11 | — | 6 | 1 | no | — |
| 47 | (C3⋊D3)⋊D11 | — | 12 | 1 | no | — |
| 48 | D3×D33 | — | 12 | 1 | no | — |
| 49 | D3×D33 | — | 12 | 1 | no | — |
| 50 | D3×D33 | — | 12 | 1 | no | — |
| 51 | D32×C11 | — | 6 | 1 | no | — |
| 52 | D32×C11 | — | 12 | 1 | no | — |
| 53 | C3⋊D3×D11 | — | 4 | 1 | no | — |
| 54 | C3⋊D3×D11 | S4 | 1 | 1 | yes | characteristic |
| 55 | C3×D3×D11 | — | 12 | 1 | no | — |
| 56 | C3×D3×D11 | — | 12 | 1 | no | — |
| 57 | C3×D3×D11 | — | 12 | 1 | no | — |
| 58 | C3×D3×D11 | — | 24 | 1 | no | — |
| 59 | C11⋊#157 | — | 3 | 1 | no | — |
| 60 | C11×#157 | — | 3 | 1 | no | — |
| 61 | SD16×D11 | — | 27 | 1 | no | Hall |
| 62 | C11×H(3,𝔽3) | — | 4 | 1 | no | Hall |
| 63 | C2×AΓL(1,𝔽9) | — | 33 | 1 | no | — |
| 64 | C2×D3×D11 | — | 36 | 1 | no | — |
| 65 | C2×D3×D11 | — | 36 | 1 | no | — |
| 66 | C11×SL(2,𝔽3) | — | 9 | 1 | no | — |
| 67 | PSU(3,𝔽2)⋊C3 | C2×D11 | 1 | 1 | yes | characteristic, complemented |
| 68 | C2×((C3⋊D3)⋊C3)⋊C2 | — | 44 | 1 | no | — |
| 69 | C3⋊D33 | — | 4 | 1 | no | — |
| 70 | C3⋊D33 | GL(2,𝔽3) | 1 | 1 | yes | characteristic, complemented |
| 71 | C11×C3⋊D3 | — | 4 | 1 | no | — |
| 72 | C11×C3⋊D3 | C2×S4 | 1 | 1 | yes | characteristic |
| 73 | C3×D33 | — | 12 | 1 | no | — |
| 74 | C3×D33 | — | 12 | 1 | no | — |
| 75 | C3×D33 | — | 12 | 1 | no | — |
| 76 | C3×D33 | — | 24 | 1 | no | — |
| 77 | C3×D3×C11 | — | 12 | 1 | no | — |
| 78 | C3×D3×C11 | — | 12 | 1 | no | — |
| 79 | C3×D3×C11 | — | 12 | 1 | no | — |
| 80 | C3×D3×C11 | — | 24 | 1 | no | — |
| 81 | C32×D11 | — | 4 | 1 | no | — |
| 82 | C32×D11 | — | 8 | 1 | no | — |
| 83 | C32×D11 | GL(2,𝔽3) | 1 | 1 | yes | characteristic, complemented |
| 84 | Q8×D11 | — | 9 | 1 | no | — |
| 85 | D4×D11 | — | 27 | 1 | no | — |
| 86 | C11×SD16 | — | 27 | 1 | no | — |
| 87 | #4325 | — | 27 | 1 | no | — |
| 88 | #4324 | — | 27 | 1 | no | — |
| 89 | #4314 | — | 27 | 1 | no | — |
| 90 | C8×D11 | — | 27 | 1 | no | — |
| 91 | C2×PSU(3,𝔽2) | — | 11 | 1 | no | — |
| 92 | C2×GO+(4,𝔽2) | — | 33 | 1 | no | — |
| 93 | C2×AGL(1,𝔽9) | — | 33 | 1 | no | — |
| 94 | AΓL(1,𝔽9) | — | 3 | 1 | no | — |
| 95 | AΓL(1,𝔽9) | — | 33 | 1 | no | — |
| 96 | AΓL(1,𝔽9) | — | 33 | 1 | no | — |
| 97 | AΓL(1,𝔽9) | — | 33 | 1 | no | — |
| 98 | C2×D33 | — | 36 | 1 | no | — |
| 99 | C2×D33 | — | 36 | 1 | no | — |
| 100 | C2×D3×C11 | — | 36 | 1 | no | — |
| 101 | C2×D3×C11 | — | 36 | 1 | no | — |
| 102 | C2×C3×D11 | — | 36 | 1 | no | — |
| 103 | C2×C3×D11 | — | 36 | 1 | no | — |
| 104 | D3×D11 | — | 12 | 1 | no | — |
| 105 | D3×D11 | — | 12 | 1 | no | — |
| 106 | D3×D11 | — | 12 | 1 | no | — |
| 107 | D3×D11 | — | 24 | 1 | no | — |
| 108 | D3×D11 | — | 36 | 1 | no | — |
| 109 | D3×D11 | — | 36 | 1 | no | — |
| 110 | D3×D11 | — | 36 | 1 | no | — |
| 111 | D3×D11 | — | 36 | 1 | no | — |
| 112 | D3×D11 | — | 36 | 1 | no | — |
| 113 | C2×#272 | — | 44 | 1 | no | — |
| 114 | C2×(C3⋊D3)⋊C3 | — | 44 | 1 | no | — |
| 115 | C2×(C3⋊D3)⋊C3 | — | 44 | 1 | no | — |
| 116 | ((C3⋊D3)⋊C3)⋊C2 | — | 4 | 1 | no | — |
| 117 | ((C3⋊D3)⋊C3)⋊C2 | — | 44 | 1 | no | — |
| 118 | ((C3⋊D3)⋊C3)⋊C2 | — | 44 | 1 | no | — |
| 119 | ((C3⋊D3)⋊C3)⋊C2 | — | 44 | 1 | no | — |
| 120 | C32×C11 | — | 4 | 1 | no | — |
| 121 | C32×C11 | — | 8 | 1 | no | — |
| 122 | C32×C11 | C2×GL(2,𝔽3) | 1 | 1 | yes | characteristic, complemented |
| 123 | C2×GL(2,𝔽3) | — | 99 | 1 | no | — |
| 124 | C22×D11 | — | 54 | 1 | no | — |
| 125 | Q8×C11 | — | 9 | 1 | no | — |
| 126 | D4×C11 | — | 27 | 1 | no | — |
| 127 | C22⋊D11 | — | 54 | 1 | no | — |
| 128 | D44 | — | 27 | 1 | no | — |
| 129 | C4×D11 | — | 27 | 1 | no | — |
| 130 | Dic22 | — | 27 | 1 | no | — |
| 131 | C8×C11 | — | 27 | 1 | no | — |
| 132 | C11⋊C8 | — | 27 | 1 | no | — |
| 133 | C2×D32 | — | 66 | 1 | no | — |
| 134 | C2×D32 | — | 132 | 1 | no | — |
| 135 | C2×#157 | — | 33 | 1 | no | — |
| 136 | PSU(3,𝔽2) | — | 33 | 1 | no | — |
| 137 | PSU(3,𝔽2) | D3×D11 | 1 | 1 | yes | characteristic, complemented |
| 138 | GO+(4,𝔽2) | — | 3 | 1 | no | — |
| 139 | GO+(4,𝔽2) | — | 33 | 1 | no | — |
| 140 | GO+(4,𝔽2) | — | 66 | 1 | no | — |
| 141 | AGL(1,𝔽9) | — | 3 | 1 | no | — |
| 142 | AGL(1,𝔽9) | — | 33 | 1 | no | — |
| 143 | C2×C3×C11 | — | 36 | 1 | no | — |
| 144 | C2×C3×C11 | — | 36 | 1 | no | — |
| 145 | D33 | — | 12 | 1 | no | — |
| 146 | D33 | — | 12 | 1 | no | — |
| 147 | D33 | — | 12 | 1 | no | — |
| 148 | D33 | — | 24 | 1 | no | — |
| 149 | D33 | — | 36 | 1 | no | — |
| 150 | C3×D11 | — | 4 | 1 | no | — |
| 151 | C3×D11 | — | 12 | 1 | no | — |
| 152 | C3×D11 | — | 24 | 1 | no | — |
| 153 | C3×D11 | — | 36 | 1 | no | — |
| 154 | C3×D11 | — | 36 | 1 | no | — |
| 155 | D3×C11 | — | 12 | 1 | no | — |
| 156 | D3×C11 | — | 12 | 1 | no | — |
| 157 | D3×C11 | — | 12 | 1 | no | — |
| 158 | D3×C11 | — | 24 | 1 | no | — |
| 159 | D3×C11 | — | 36 | 1 | no | — |
| 160 | C2×H(3,𝔽3) | — | 44 | 1 | no | — |
| 161 | #272 | — | 4 | 1 | no | — |
| 162 | #272 | — | 44 | 1 | no | — |
| 163 | (C3⋊D3)⋊C3 | — | 4 | 1 | no | — |
| 164 | (C3⋊D3)⋊C3 | — | 4 | 1 | no | — |
| 165 | (C3⋊D3)⋊C3 | — | 44 | 1 | no | — |
| 166 | (C3⋊D3)⋊C3 | — | 44 | 1 | no | — |
| 167 | C2×SL(2,𝔽3) | — | 99 | 1 | no | — |
| 168 | GL(2,𝔽3) | — | 9 | 1 | no | — |
| 169 | GL(2,𝔽3) | — | 99 | 1 | no | — |
| 170 | C22×C11 | — | 54 | 1 | no | — |
| 171 | C2×D11 | — | 9 | 1 | no | — |
| 172 | C2×D11 | — | 36 | 1 | no | — |
| 173 | C2×D11 | — | 54 | 1 | no | — |
| 174 | C2×D11 | — | 108 | 1 | no | — |
| 175 | C4×C11 | — | 27 | 1 | no | — |
| 176 | Dic11 | — | 27 | 1 | no | — |
| 177 | C2×C3⋊D3 | — | 11 | 1 | no | — |
| 178 | C2×C3⋊D3 | — | 44 | 1 | no | — |
| 179 | C2×C3×D3 | — | 264 | 1 | no | — |
| 180 | C2×C3×D3 | — | 132 | 1 | no | — |
| 181 | C2×C3×D3 | — | 132 | 1 | no | — |
| 182 | C2×C3×D3 | — | 132 | 1 | no | — |
| 183 | D32 | — | 6 | 1 | no | — |
| 184 | D32 | — | 12 | 1 | no | — |
| 185 | D32 | — | 66 | 1 | no | — |
| 186 | D32 | — | 132 | 1 | no | — |
| 187 | D32 | — | 132 | 1 | no | — |
| 188 | D32 | — | 132 | 1 | no | — |
| 189 | D32 | — | 132 | 1 | no | — |
| 190 | #157 | — | 3 | 1 | no | — |
| 191 | #157 | — | 33 | 1 | no | — |
| 192 | C3×C11 | — | 4 | 1 | no | — |
| 193 | C3×C11 | — | 12 | 1 | no | — |
| 194 | C3×C11 | — | 24 | 1 | no | — |
| 195 | C2×SD16 | — | 297 | 1 | no | Sylow |
| 196 | H(3,𝔽3) | — | 4 | 1 | no | Sylow |
| 197 | C22×D3 | — | 396 | 1 | no | — |
| 198 | C22×D3 | — | 396 | 1 | no | — |
| 199 | SL(2,𝔽3) | — | 9 | 1 | no | — |
| 200 | C2×C11 | — | 9 | 1 | no | — |
| 201 | C2×C11 | — | 36 | 1 | no | — |
| 202 | D11 | — | 9 | 1 | no | — |
| 203 | D11 | — | 36 | 1 | no | — |
| 204 | D11 | AGL(2,𝔽3) | 1 | 1 | yes | characteristic, direct product factor |
| 205 | C2×C32 | — | 11 | 1 | no | — |
| 206 | C2×C32 | — | 44 | 1 | no | — |
| 207 | C2×C32 | — | 88 | 1 | no | — |
| 208 | C3⋊D3 | — | 4 | 1 | no | — |
| 209 | C3⋊D3 | — | 11 | 1 | no | — |
| 210 | C3⋊D3 | — | 44 | 1 | no | — |
| 211 | C3⋊D3 | D11×S4 | 1 | 1 | yes | characteristic |
| 212 | C3×D3 | — | 264 | 1 | no | — |
| 213 | C3×D3 | — | 12 | 1 | no | — |
| 214 | C3×D3 | — | 12 | 1 | no | — |
| 215 | C3×D3 | — | 12 | 1 | no | — |
| 216 | C3×D3 | — | 24 | 1 | no | — |
| 217 | C3×D3 | — | 132 | 1 | no | — |
| 218 | C3×D3 | — | 132 | 1 | no | — |
| 219 | C3×D3 | — | 132 | 1 | no | — |
| 220 | C2×Q8 | — | 99 | 1 | no | — |
| 221 | C2×D4 | — | 297 | 1 | no | — |
| 222 | SD16 | — | 27 | 1 | no | — |
| 223 | SD16 | — | 297 | 1 | no | — |
| 224 | SD16 | — | 297 | 1 | no | — |
| 225 | SD16 | — | 297 | 1 | no | — |
| 226 | C2×C8 | — | 297 | 1 | no | — |
| 227 | C22×C3 | — | 396 | 1 | no | — |
| 228 | C22×C3 | — | 396 | 1 | no | — |
| 229 | C2×D3 | — | 264 | 1 | no | — |
| 230 | C2×D3 | — | 36 | 1 | no | — |
| 231 | C2×D3 | — | 36 | 1 | no | — |
| 232 | C2×D3 | — | 132 | 1 | no | — |
| 233 | C2×D3 | — | 132 | 1 | no | — |
| 234 | C2×D3 | — | 132 | 1 | no | — |
| 235 | C2×D3 | — | 396 | 1 | no | — |
| 236 | C2×D3 | — | 396 | 1 | no | — |
| 237 | C2×D3 | — | 396 | 1 | no | — |
| 238 | C2×D3 | — | 396 | 1 | no | — |
| 239 | C2×D3 | — | 396 | 1 | no | — |
| 240 | C2×D3 | — | 396 | 1 | no | — |
| 241 | C2×D3 | — | 396 | 1 | no | — |
| 242 | C11 | C2×AGL(2,𝔽3) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 243 | C32 | — | 4 | 1 | no | — |
| 244 | C32 | — | 8 | 1 | no | — |
| 245 | C32 | D11×GL(2,𝔽3) | 1 | 1 | yes | characteristic, complemented |
| 246 | C23 | — | 594 | 1 | no | — |
| 247 | Q8 | — | 9 | 1 | no | — |
| 248 | Q8 | — | 297 | 1 | no | — |
| 249 | D4 | — | 27 | 1 | no | — |
| 250 | D4 | — | 297 | 1 | no | — |
| 251 | D4 | — | 594 | 1 | no | — |
| 252 | C2×C4 | — | 297 | 1 | no | — |
| 253 | C8 | — | 27 | 1 | no | — |
| 254 | C8 | — | 297 | 1 | no | — |
| 255 | C2×C3 | — | 264 | 1 | no | — |
| 256 | C2×C3 | — | 36 | 1 | no | — |
| 257 | C2×C3 | — | 36 | 1 | no | — |
| 258 | C2×C3 | — | 44 | 1 | no | — |
| 259 | C2×C3 | — | 132 | 1 | no | — |
| 260 | C2×C3 | — | 396 | 1 | no | — |
| 261 | C2×C3 | — | 396 | 1 | no | — |
| 262 | D3 | — | 264 | 1 | no | — |
| 263 | D3 | — | 12 | 1 | no | — |
| 264 | D3 | — | 12 | 1 | no | — |
| 265 | D3 | — | 12 | 1 | no | — |
| 266 | D3 | — | 24 | 1 | no | — |
| 267 | D3 | — | 36 | 1 | no | — |
| 268 | D3 | — | 132 | 1 | no | — |
| 269 | D3 | — | 132 | 1 | no | — |
| 270 | D3 | — | 132 | 1 | no | — |
| 271 | D3 | — | 396 | 1 | no | — |
| 272 | C22 | — | 54 | 1 | no | — |
| 273 | C22 | — | 594 | 1 | no | — |
| 274 | C22 | — | 99 | 1 | no | — |
| 275 | C22 | — | 396 | 1 | no | — |
| 276 | C22 | — | 1188 | 1 | no | — |
| 277 | C4 | — | 27 | 1 | no | — |
| 278 | C4 | — | 297 | 1 | no | — |
| 279 | C3 | — | 4 | 1 | no | — |
| 280 | C3 | — | 12 | 1 | no | — |
| 281 | C3 | — | 24 | 1 | no | — |
| 282 | C2 | — | 9 | 1 | no | — |
| 283 | C2 | — | 11 | 1 | no | — |
| 284 | C2 | — | 36 | 1 | no | — |
| 285 | C2 | — | 99 | 1 | no | — |
| 286 | C2 | — | 396 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C11×PSU(3,𝔽2)⋊C3 | C22 |
| Fitting, socle | C32×C11 | C2×GL(2,𝔽3) |
| LCS | G ▷ C11×PSU(3,𝔽2)⋊C3 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #72259104 | C2 | semidirect |
| 4. C11×AGL(2,𝔽3) | C2 | semidirect |
| 5. D11×PSU(3,𝔽2)⋊C3 | C2 | semidirect |
| 6. C11×PSU(3,𝔽2)⋊C3 | C22 | semidirect |
| 7. D11×PSU(3,𝔽2) | D3 | semidirect |
| 26. C11×PSU(3,𝔽2) | C2×D3 | semidirect |
| 45. AGL(2,𝔽3) | D11 | direct (N × H) |
| 67. PSU(3,𝔽2)⋊C3 | C2×D11 | semidirect |
| 70. C3⋊D33 | GL(2,𝔽3) | semidirect |
| 83. C32×D11 | GL(2,𝔽3) | semidirect |
| 122. C32×C11 | C2×GL(2,𝔽3) | semidirect |
| 137. PSU(3,𝔽2) | D3×D11 | semidirect |
| 204. D11 | AGL(2,𝔽3) | direct (N × H) |
| 242. C11 | C2×AGL(2,𝔽3) | semidirect |
| 245. C32 | D11×GL(2,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D11×((C3⋊D3)⋊C3)⋊C2 | 4 |
| D11×AΓL(1,𝔽9) | 3 |
| #72259104 | 2 |
| C11×AGL(2,𝔽3) | 2 |
| D11×PSU(3,𝔽2)⋊C3 | 2 |
| D11×GL(2,𝔽3) | 9 |
| C2×AGL(2,𝔽3) | 11 |