Finite Group DatabasebetaFinite Group DatabasebetaOrder 5184.
1159 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 1159 conjugacy classes of subgroups — too many to draw. Only its 42 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 | #72220241 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#13049467 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #72219709 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #72219702 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | #72213762 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C22×#66472 | — | 2 | 1 | no | — |
| 7 | C22×#66472 | C22 | 1 | 1 | yes | characteristic |
| 8 | C2×#10620206 | C22 | 1 | 1 | yes | characteristic |
| 9 | C2×#10620192 | C22 | 1 | 2 | yes | — |
| 10 | #13049703 | — | 2 | 1 | no | — |
| 11 | #13049467 | — | 2 | 1 | no | — |
| 12 | #13049467 | C22 | 1 | 2 | yes | — |
| 13 | #13049456 | C22 | 1 | 1 | yes | characteristic |
| 14 | #13049455 | — | 2 | 1 | no | — |
| 15 | C22×C3⋊(C3⋊(C3⋊D3)) | D4 | 1 | 1 | yes | characteristic |
| 16 | C2×C3⋊D32 | — | 2 | 1 | no | — |
| 17 | C2×#66472 | — | 2 | 2 | no | — |
| 18 | C2×#66472 | — | 2 | 2 | no | — |
| 19 | C2×#66472 | — | 4 | 2 | no | — |
| 20 | C2×#66472 | D4 | 1 | 1 | yes | characteristic |
| 21 | C2×#66472 | C23 | 1 | 1 | yes | characteristic |
| 22 | #10620206 | — | 2 | 2 | no | — |
| 23 | #10620192 | — | 2 | 2 | no | — |
| 24 | C32×#638 | — | 4 | 1 | no | — |
| 25 | #10620152 | — | 4 | 1 | no | — |
| 26 | #10620150 | — | 2 | 1 | no | — |
| 27 | #10596412 | — | 9 | 4 | no | maximal |
| 28 | #89039 | — | 12 | 4 | no | — |
| 29 | C22×C34 | SD16 | 1 | 1 | yes | characteristic |
| 30 | C2×C3⋊(C3⋊(C3⋊D3)) | — | 2 | 1 | no | — |
| 31 | C2×C3⋊(C3⋊(C3⋊D3)) | — | 2 | 1 | no | — |
| 32 | C2×C3⋊(C3⋊(C3⋊D3)) | SD16 | 1 | 1 | yes | characteristic |
| 33 | C2×C3⋊(C3⋊(C3⋊D3)) | C2×D4 | 1 | 1 | yes | characteristic |
| 34 | C2×C32×C3⋊D3 | — | 4 | 1 | no | — |
| 35 | C3⋊D32 | — | 4 | 1 | no | — |
| 36 | C3⋊D32 | — | 4 | 1 | no | — |
| 37 | #66472 | — | 2 | 1 | no | — |
| 38 | #66472 | — | 4 | 2 | no | — |
| 39 | #66472 | #41 | 1 | 2 | yes | — |
| 40 | C32×C3⋊Dic3 | — | 4 | 1 | no | — |
| 41 | C2×#3777 | — | 9 | 4 | no | — |
| 42 | #64242 | — | 9 | 4 | no | — |
| 43 | C22⋊AGL(1,𝔽9) | — | 9 | 4 | no | — |
| 44 | C22⋊AGL(1,𝔽9) | — | 18 | 3 | no | — |
| 45 | AGL(1,𝔽9)⋊C4 | — | 9 | 8 | no | — |
| 46 | GO+(4,𝔽2)⋊C4 | — | 9 | 8 | no | — |
| 47 | C22×C3⋊(C3⋊D3) | — | 12 | 4 | no | — |
| 48 | C22×C3⋊(C3⋊D3) | — | 24 | 3 | no | — |
| 49 | C2×D3×C3⋊D3 | — | 12 | 4 | no | — |
| 50 | C3×#638 | — | 4 | 4 | no | — |
| 51 | C32×#67 | — | 12 | 4 | no | — |
| 52 | #6292 | — | 12 | 4 | no | — |
| 53 | #6291 | — | 12 | 4 | no | — |
| 54 | #6289 | — | 12 | 4 | no | — |
| 55 | C2×C34 | — | 2 | 1 | no | — |
| 56 | C2×C34 | C2×SD16 | 1 | 1 | yes | characteristic |
| 57 | C3⋊(C3⋊(C3⋊D3)) | — | 2 | 1 | no | — |
| 58 | C3⋊(C3⋊(C3⋊D3)) | #123 | 1 | 1 | yes | characteristic |
| 59 | C3⋊(C3⋊(C3⋊D3)) | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 60 | C32×C3⋊D3 | — | 8 | 1 | no | — |
| 61 | C22×#157 | — | 9 | 4 | no | — |
| 62 | C22×#157 | — | 18 | 3 | no | — |
| 63 | C22×#157 | — | 18 | 4 | no | — |
| 64 | C22×#157 | — | 36 | 3 | no | — |
| 65 | C2×GO+(4,𝔽2) | — | 9 | 4 | no | — |
| 66 | C2×AGL(1,𝔽9) | — | 9 | 8 | no | — |
| 67 | C2×AGL(1,𝔽9) | — | 18 | 6 | no | — |
| 68 | D4×C3⋊D3 | — | 36 | 1 | no | — |
| 69 | #3811 | — | 18 | 4 | no | — |
| 70 | #3811 | — | 36 | 6 | no | — |
| 71 | #3777 | — | 9 | 8 | no | — |
| 72 | #3777 | — | 18 | 4 | no | — |
| 73 | #3773 | — | 9 | 4 | no | — |
| 74 | D32⋊C4 | — | 18 | 4 | no | — |
| 75 | C22×C33 | — | 4 | 4 | no | — |
| 76 | C22×C33 | — | 8 | 3 | no | — |
| 77 | C2×C3⋊(C3⋊D3) | — | 12 | 4 | no | — |
| 78 | C2×C3⋊(C3⋊D3) | — | 12 | 4 | no | — |
| 79 | C2×C3⋊(C3⋊D3) | — | 24 | 4 | no | — |
| 80 | C2×C3⋊(C3⋊D3) | — | 24 | 4 | no | — |
| 81 | C2×C3⋊(C3⋊D3) | — | 24 | 3 | no | — |
| 82 | C2×C3⋊(C3⋊D3) | — | 24 | 3 | no | — |
| 83 | C2×C3⋊(C3⋊D3) | — | 24 | 6 | no | — |
| 84 | C2×C3⋊(C3⋊D3) | — | 24 | 6 | no | — |
| 85 | C2×C3×C3⋊D3 | — | 4 | 4 | no | — |
| 86 | C2×C32×D3 | — | 12 | 4 | no | — |
| 87 | D3×C3⋊D3 | — | 24 | 4 | no | — |
| 88 | D3×C3⋊D3 | — | 24 | 4 | no | — |
| 89 | C3×C3⋊Dic3 | — | 4 | 4 | no | — |
| 90 | C32×Dic3 | — | 12 | 4 | no | — |
| 91 | C34 | #481 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 92 | C22×C3⋊D3 | — | 9 | 4 | no | — |
| 93 | C22×C3⋊D3 | — | 18 | 3 | no | — |
| 94 | C22×C3⋊D3 | — | 36 | 1 | no | — |
| 95 | C22×C3⋊D3 | — | 36 | 1 | no | — |
| 96 | C22×C3⋊D3 | — | 36 | 6 | no | — |
| 97 | C22×C3⋊D3 | — | 36 | 3 | no | — |
| 98 | C22×C3⋊D3 | — | 72 | 6 | no | — |
| 99 | C22×C3⋊D3 | — | 72 | 4 | no | — |
| 100 | C2×D32 | — | 18 | 4 | no | — |
| 101 | C2×D32 | — | 36 | 6 | no | — |
| 102 | C2×#157 | — | 9 | 4 | no | — |
| 103 | C2×#157 | — | 9 | 4 | no | — |
| 104 | C2×#157 | — | 18 | 3 | no | — |
| 105 | C2×#157 | — | 18 | 3 | no | — |
| 106 | C2×#157 | — | 18 | 8 | no | — |
| 107 | C2×#157 | — | 18 | 8 | no | — |
| 108 | C2×#157 | — | 36 | 8 | no | — |
| 109 | C2×#157 | — | 36 | 12 | no | — |
| 110 | C2×#157 | — | 36 | 6 | no | — |
| 111 | C2×#157 | — | 36 | 6 | no | — |
| 112 | GO+(4,𝔽2) | — | 18 | 8 | no | — |
| 113 | AGL(1,𝔽9) | — | 18 | 8 | no | — |
| 114 | AGL(1,𝔽9) | — | 36 | 6 | no | — |
| 115 | C32×D4 | — | 36 | 1 | no | — |
| 116 | #638 | — | 4 | 1 | no | — |
| 117 | #638 | — | 36 | 1 | no | — |
| 118 | C4⋊(C3⋊D3) | — | 36 | 1 | no | — |
| 119 | C4×C3⋊D3 | — | 36 | 1 | no | — |
| 120 | C3×#67 | — | 12 | 12 | no | — |
| 121 | C3×#67 | — | 12 | 4 | no | — |
| 122 | #626 | — | 36 | 4 | no | — |
| 123 | #626 | — | 36 | 12 | no | — |
| 124 | #624 | — | 18 | 4 | no | — |
| 125 | #624 | — | 36 | 6 | no | — |
| 126 | #481 | — | 81 | 1 | no | Sylow |
| 127 | C2×C33 | — | 4 | 4 | no | — |
| 128 | C2×C33 | — | 8 | 3 | no | — |
| 129 | C2×C33 | — | 8 | 4 | no | — |
| 130 | C2×C33 | — | 8 | 6 | no | — |
| 131 | C3⋊(C3⋊D3) | — | 12 | 4 | no | — |
| 132 | C3⋊(C3⋊D3) | — | 12 | 4 | no | — |
| 133 | C3⋊(C3⋊D3) | — | 24 | 3 | no | — |
| 134 | C3⋊(C3⋊D3) | — | 24 | 3 | no | — |
| 135 | C3⋊(C3⋊D3) | — | 24 | 4 | no | — |
| 136 | C3⋊(C3⋊D3) | — | 24 | 6 | no | — |
| 137 | C3×C3⋊D3 | — | 8 | 4 | no | — |
| 138 | C32×D3 | — | 24 | 4 | no | — |
| 139 | D3×D4 | — | 108 | 4 | no | — |
| 140 | C22×C32 | — | 2 | 3 | no | — |
| 141 | C22×C32 | — | 4 | 1 | no | — |
| 142 | C22×C32 | — | 4 | 6 | no | — |
| 143 | C22×C32 | — | 4 | 3 | no | — |
| 144 | C22×C32 | — | 8 | 6 | no | — |
| 145 | C22×C32 | — | 8 | 4 | no | — |
| 146 | C22×C32 | — | 36 | 1 | no | — |
| 147 | C22×C32 | AΓL(1,𝔽9) | 1 | 4 | yes | — |
| 148 | C2×C3⋊D3 | — | 4 | 1 | no | — |
| 149 | C2×C3⋊D3 | — | 9 | 4 | no | — |
| 150 | C2×C3⋊D3 | — | 9 | 4 | no | — |
| 151 | C2×C3⋊D3 | — | 18 | 4 | no | — |
| 152 | C2×C3⋊D3 | — | 18 | 4 | no | — |
| 153 | C2×C3⋊D3 | — | 18 | 3 | no | — |
| 154 | C2×C3⋊D3 | — | 18 | 3 | no | — |
| 155 | C2×C3⋊D3 | — | 36 | 1 | no | — |
| 156 | C2×C3⋊D3 | — | 36 | 1 | no | — |
| 157 | C2×C3⋊D3 | — | 36 | 6 | no | — |
| 158 | C2×C3⋊D3 | — | 36 | 6 | no | — |
| 159 | C2×C3⋊D3 | — | 36 | 3 | no | — |
| 160 | C2×C3⋊D3 | — | 36 | 3 | no | — |
| 161 | C2×C3⋊D3 | — | 36 | 3 | no | — |
| 162 | C2×C3⋊D3 | — | 36 | 6 | no | — |
| 163 | C2×C3⋊D3 | — | 36 | 3 | no | — |
| 164 | C2×C3⋊D3 | — | 36 | 6 | no | — |
| 165 | C2×C3⋊D3 | — | 72 | 6 | no | — |
| 166 | C2×C3⋊D3 | — | 72 | 6 | no | — |
| 167 | C2×C3⋊D3 | — | 72 | 4 | no | — |
| 168 | C2×C3⋊D3 | — | 72 | 4 | no | — |
| 169 | C2×C3⋊D3 | — | 72 | 1 | no | — |
| 170 | C2×C3⋊D3 | — | 72 | 1 | no | — |
| 171 | C2×C3⋊D3 | — | 72 | 1 | no | — |
| 172 | C2×C3⋊D3 | — | 72 | 1 | no | — |
| 173 | C2×C3⋊D3 | — | 72 | 6 | no | — |
| 174 | C2×C3⋊D3 | — | 72 | 6 | no | — |
| 175 | C2×C3⋊D3 | — | 72 | 12 | no | — |
| 176 | C2×C3⋊D3 | — | 72 | 12 | no | — |
| 177 | C2×C3⋊D3 | — | 72 | 8 | no | — |
| 178 | C2×C3⋊D3 | — | 72 | 8 | no | — |
| 179 | C2×C3×D3 | — | 12 | 12 | no | — |
| 180 | C2×C3×D3 | — | 12 | 4 | no | — |
| 181 | D32 | — | 36 | 4 | no | — |
| 182 | D32 | — | 36 | 4 | no | — |
| 183 | D32 | — | 72 | 6 | no | — |
| 184 | D32 | — | 72 | 6 | no | — |
| 185 | #157 | — | 9 | 8 | no | — |
| 186 | #157 | — | 18 | 6 | no | — |
| 187 | #157 | — | 18 | 4 | no | — |
| 188 | #157 | — | 36 | 8 | no | — |
| 189 | #157 | — | 36 | 12 | no | — |
| 190 | #157 | — | 36 | 3 | no | — |
| 191 | C32×C4 | — | 36 | 1 | no | — |
| 192 | C3⋊Dic3 | — | 4 | 1 | no | — |
| 193 | C3×Dic3 | — | 12 | 12 | no | — |
| 194 | C3×Dic3 | — | 12 | 4 | no | — |
| 195 | #121 | — | 81 | 1 | no | — |
| 196 | C2×C4⋊C4 | — | 81 | 1 | no | — |
| 197 | #106 | — | 81 | 2 | no | — |
| 198 | #102 | — | 81 | 2 | no | — |
| 199 | C22⋊C8 | — | 81 | 1 | no | — |
| 200 | C33 | — | 4 | 4 | no | — |
| 201 | C33 | — | 8 | 3 | no | — |
| 202 | C22×D3 | — | 108 | 4 | no | — |
| 203 | C22×D3 | — | 108 | 4 | no | — |
| 204 | C22×D3 | — | 216 | 3 | no | — |
| 205 | C3×D4 | — | 36 | 4 | no | — |
| 206 | #67 | — | 12 | 4 | no | — |
| 207 | #67 | — | 108 | 4 | no | — |
| 208 | D12 | — | 108 | 4 | no | — |
| 209 | C4×D3 | — | 108 | 4 | no | — |
| 210 | C2×C32 | — | 2 | 3 | no | — |
| 211 | C2×C32 | — | 2 | 4 | no | — |
| 212 | C2×C32 | — | 4 | 1 | no | — |
| 213 | C2×C32 | — | 4 | 6 | no | — |
| 214 | C2×C32 | — | 4 | 3 | no | — |
| 215 | C2×C32 | — | 4 | 6 | no | — |
| 216 | C2×C32 | — | 4 | 3 | no | — |
| 217 | C2×C32 | — | 8 | 6 | no | — |
| 218 | C2×C32 | — | 8 | 4 | no | — |
| 219 | C2×C32 | — | 8 | 1 | no | — |
| 220 | C2×C32 | — | 8 | 6 | no | — |
| 221 | C2×C32 | — | 8 | 12 | no | — |
| 222 | C2×C32 | — | 8 | 8 | no | — |
| 223 | C2×C32 | — | 72 | 1 | no | — |
| 224 | C2×C32 | C2×AΓL(1,𝔽9) | 1 | 4 | yes | — |
| 225 | C3⋊D3 | — | 8 | 1 | no | — |
| 226 | C3⋊D3 | — | 9 | 4 | no | — |
| 227 | C3⋊D3 | — | 9 | 4 | no | — |
| 228 | C3⋊D3 | — | 18 | 3 | no | — |
| 229 | C3⋊D3 | — | 18 | 3 | no | — |
| 230 | C3⋊D3 | — | 18 | 4 | no | — |
| 231 | C3⋊D3 | — | 36 | 1 | no | — |
| 232 | C3⋊D3 | — | 36 | 1 | no | — |
| 233 | C3⋊D3 | — | 36 | 6 | no | — |
| 234 | C3⋊D3 | — | 36 | 6 | no | — |
| 235 | C3⋊D3 | — | 36 | 3 | no | — |
| 236 | C3⋊D3 | — | 36 | 3 | no | — |
| 237 | C3⋊D3 | — | 36 | 3 | no | — |
| 238 | C3⋊D3 | — | 36 | 6 | no | — |
| 239 | C3⋊D3 | — | 72 | 6 | no | — |
| 240 | C3⋊D3 | — | 72 | 6 | no | — |
| 241 | C3⋊D3 | — | 72 | 4 | no | — |
| 242 | C3⋊D3 | — | 72 | 4 | no | — |
| 243 | C3⋊D3 | — | 72 | 1 | no | — |
| 244 | C3⋊D3 | — | 72 | 6 | no | — |
| 245 | C3⋊D3 | — | 72 | 12 | no | — |
| 246 | C3⋊D3 | — | 72 | 8 | no | — |
| 247 | C3×D3 | — | 24 | 12 | no | — |
| 248 | C3×D3 | — | 24 | 4 | no | — |
| 249 | C2×D4 | — | 81 | 1 | no | — |
| 250 | C2×D4 | — | 162 | 1 | no | — |
| 251 | C22×C4 | — | 81 | 1 | no | — |
| 252 | C22×C4 | — | 162 | 1 | no | — |
| 253 | C2×C8 | — | 81 | 2 | no | — |
| 254 | C4⋊C4 | — | 81 | 2 | no | — |
| 255 | C4⋊C4 | — | 81 | 1 | no | — |
| 256 | C4⋊C4 | — | 162 | 1 | no | — |
| 257 | C22⋊C4 | — | 162 | 1 | no | — |
| 258 | C22×C3 | — | 4 | 4 | no | — |
| 259 | C22×C3 | — | 8 | 3 | no | — |
| 260 | C22×C3 | — | 36 | 4 | no | — |
| 261 | C2×D3 | — | 12 | 4 | no | — |
| 262 | C2×D3 | — | 108 | 4 | no | — |
| 263 | C2×D3 | — | 108 | 4 | no | — |
| 264 | C2×D3 | — | 216 | 4 | no | — |
| 265 | C2×D3 | — | 216 | 4 | no | — |
| 266 | C2×D3 | — | 216 | 3 | no | — |
| 267 | C2×D3 | — | 216 | 3 | no | — |
| 268 | C2×D3 | — | 216 | 4 | no | — |
| 269 | C2×D3 | — | 216 | 4 | no | — |
| 270 | C2×D3 | — | 216 | 6 | no | — |
| 271 | C2×D3 | — | 216 | 6 | no | — |
| 272 | C3×C4 | — | 36 | 4 | no | — |
| 273 | Dic3 | — | 12 | 4 | no | — |
| 274 | C32 | — | 2 | 3 | no | — |
| 275 | C32 | — | 4 | 1 | no | — |
| 276 | C32 | — | 4 | 6 | no | — |
| 277 | C32 | — | 4 | 3 | no | — |
| 278 | C32 | — | 8 | 6 | no | — |
| 279 | C32 | — | 8 | 4 | no | — |
| 280 | C32 | #10596412 | 1 | 4 | yes | complemented |
| 281 | C23 | — | 81 | 1 | no | — |
| 282 | C23 | — | 162 | 1 | no | — |
| 283 | D4 | — | 36 | 1 | no | — |
| 284 | D4 | — | 324 | 1 | no | — |
| 285 | D4 | — | 162 | 2 | no | — |
| 286 | C2×C4 | — | 324 | 2 | no | — |
| 287 | C2×C4 | — | 81 | 1 | no | — |
| 288 | C2×C4 | — | 81 | 1 | no | — |
| 289 | C2×C4 | — | 162 | 1 | no | — |
| 290 | C2×C4 | — | 162 | 2 | no | — |
| 291 | C2×C4 | — | 162 | 2 | no | — |
| 292 | C8 | — | 162 | 2 | no | — |
| 293 | C2×C3 | — | 4 | 4 | no | — |
| 294 | C2×C3 | — | 8 | 4 | no | — |
| 295 | C2×C3 | — | 8 | 6 | no | — |
| 296 | C2×C3 | — | 8 | 3 | no | — |
| 297 | C2×C3 | — | 72 | 4 | no | — |
| 298 | D3 | — | 24 | 4 | no | — |
| 299 | D3 | — | 108 | 4 | no | — |
| 300 | D3 | — | 108 | 4 | no | — |
| 301 | D3 | — | 216 | 3 | no | — |
| 302 | D3 | — | 216 | 3 | no | — |
| 303 | D3 | — | 216 | 4 | no | — |
| 304 | D3 | — | 216 | 6 | no | — |
| 305 | C22 | — | 36 | 1 | no | — |
| 306 | C22 | — | 324 | 1 | no | — |
| 307 | C22 | — | 324 | 1 | no | — |
| 308 | C22 | — | 81 | 1 | no | — |
| 309 | C22 | — | 81 | 1 | no | — |
| 310 | C22 | — | 162 | 1 | no | — |
| 311 | C22 | — | 162 | 1 | no | — |
| 312 | C22 | #13049803 | 1 | 1 | yes | characteristic |
| 313 | C4 | — | 36 | 1 | no | — |
| 314 | C4 | — | 324 | 2 | no | — |
| 315 | C4 | — | 81 | 2 | no | — |
| 316 | C4 | — | 162 | 1 | no | — |
| 317 | C3 | — | 4 | 4 | no | — |
| 318 | C3 | — | 8 | 3 | no | — |
| 319 | C2 | — | 2 | 1 | no | — |
| 320 | C2 | — | 72 | 1 | no | — |
| 321 | C2 | — | 81 | 1 | no | — |
| 322 | C2 | — | 81 | 1 | no | — |
| 323 | C2 | — | 162 | 1 | no | — |
| 324 | C2 | C2×#13049803 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #72220241 | C2 | semidirect |
| 2. C2×#13049467 | C2 | semidirect |
| 4. #72219702 | C2 | semidirect |
| 5. #72213762 | C2 | semidirect |
| 91. C34 | #481 | semidirect |
| 280. C32 | #10596412 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72220241 | 2 |
| C2×#13049467 | 2 |
| #72219709 | 2 |
| #72219702 | 2 |
| #72213762 | 2 |
| #10596412 | 9 |