Finite Group DatabasebetaFinite Group DatabasebetaOrder 5184.
1246 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 1246 conjugacy classes of subgroups — too many to draw. Only its 74 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×#13049456 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C2×#13049467 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #72213762 | C2 | 1 | 4 | yes | maximal, direct product factor |
| 4 | C22×#10620192 | C2 | 1 | 1 | yes | maximal, characteristic |
| 5 | C22×#66472 | — | 2 | 1 | no | — |
| 6 | C22×#66472 | C22 | 1 | 1 | yes | characteristic |
| 7 | C2×#10620192 | C4 | 1 | 4 | yes | complemented |
| 8 | C2×#10620192 | C22 | 1 | 2 | yes | — |
| 9 | C2×#10620150 | — | 2 | 1 | no | — |
| 10 | #13049467 | — | 2 | 1 | no | — |
| 11 | #13049467 | C22 | 1 | 2 | yes | — |
| 12 | #13049456 | — | 2 | 1 | no | — |
| 13 | #13049456 | C22 | 1 | 2 | yes | — |
| 14 | C22×C3⋊(C3⋊(C3⋊D3)) | D4 | 1 | 1 | yes | characteristic |
| 15 | C2×#66472 | — | 2 | 2 | no | — |
| 16 | C2×#66472 | — | 4 | 2 | no | — |
| 17 | C2×#66472 | C2×C4 | 1 | 2 | yes | — |
| 18 | C2×#66472 | D4 | 1 | 1 | yes | characteristic |
| 19 | C2×#66472 | Q8 | 1 | 2 | yes | — |
| 20 | C2×#66472 | C23 | 1 | 1 | yes | characteristic |
| 21 | #10620192 | C2×C4 | 1 | 4 | yes | complemented |
| 22 | C2×C32×C3⋊Dic3 | — | 4 | 1 | no | — |
| 23 | #10620150 | — | 2 | 2 | no | — |
| 24 | #10620150 | — | 4 | 1 | no | — |
| 25 | C2×AGL(1,𝔽9)⋊C4 | — | 9 | 4 | no | maximal |
| 26 | C2×#6289 | — | 12 | 4 | no | — |
| 27 | C22×C34 | SD16 | 1 | 1 | yes | characteristic |
| 28 | C2×C3⋊(C3⋊(C3⋊D3)) | C4⋊C4 | 1 | 2 | yes | — |
| 29 | C2×C3⋊(C3⋊(C3⋊D3)) | SD16 | 1 | 1 | yes | characteristic |
| 30 | C2×C3⋊(C3⋊(C3⋊D3)) | SD16 | 1 | 2 | yes | — |
| 31 | C2×C3⋊(C3⋊(C3⋊D3)) | C2×D4 | 1 | 1 | yes | characteristic |
| 32 | #66472 | — | 4 | 2 | no | — |
| 33 | #66472 | C4⋊C4 | 1 | 2 | yes | — |
| 34 | #66472 | C22×C4 | 1 | 1 | yes | characteristic |
| 35 | #66472 | C2×Q8 | 1 | 1 | yes | characteristic |
| 36 | C32×C3⋊Dic3 | — | 4 | 2 | no | — |
| 37 | C22×AGL(1,𝔽9) | — | 9 | 4 | no | — |
| 38 | C22×AGL(1,𝔽9) | — | 18 | 3 | no | — |
| 39 | C2×#3777 | — | 9 | 4 | no | — |
| 40 | C2×#3773 | — | 9 | 4 | no | — |
| 41 | AGL(1,𝔽9)⋊C4 | — | 9 | 16 | no | — |
| 42 | C22×C3⋊(C3⋊D3) | — | 12 | 4 | no | — |
| 43 | C22×C3⋊(C3⋊D3) | — | 24 | 3 | no | — |
| 44 | C2×C3×C3⋊Dic3 | — | 4 | 4 | no | — |
| 45 | C2×C32×Dic3 | — | 12 | 4 | no | — |
| 46 | #6289 | — | 12 | 8 | no | — |
| 47 | #6289 | — | 12 | 8 | no | — |
| 48 | C2×C34 | #106 | 1 | 2 | yes | complemented |
| 49 | C2×C34 | C2×SD16 | 1 | 1 | yes | characteristic |
| 50 | C3⋊(C3⋊(C3⋊D3)) | #106 | 1 | 2 | yes | complemented |
| 51 | C3⋊(C3⋊(C3⋊D3)) | C2×C4⋊C4 | 1 | 1 | yes | characteristic |
| 52 | C3⋊(C3⋊(C3⋊D3)) | C2×SD16 | 1 | 1 | yes | characteristic |
| 53 | C22×#157 | — | 9 | 4 | no | — |
| 54 | C22×#157 | — | 18 | 3 | no | — |
| 55 | C22×#157 | — | 18 | 4 | no | — |
| 56 | C22×#157 | — | 36 | 3 | no | — |
| 57 | C2×AGL(1,𝔽9) | — | 9 | 8 | no | — |
| 58 | C2×AGL(1,𝔽9) | — | 9 | 16 | no | — |
| 59 | C2×AGL(1,𝔽9) | — | 18 | 12 | no | — |
| 60 | C2×AGL(1,𝔽9) | — | 18 | 6 | no | — |
| 61 | C2×C4×C3⋊D3 | — | 36 | 1 | no | — |
| 62 | C2×#624 | — | 18 | 4 | no | — |
| 63 | C2×#624 | — | 36 | 6 | no | — |
| 64 | #3777 | — | 9 | 8 | no | — |
| 65 | #3777 | — | 18 | 4 | no | — |
| 66 | #3773 | — | 9 | 8 | no | — |
| 67 | #3773 | — | 18 | 4 | no | — |
| 68 | C22×C33 | — | 4 | 4 | no | — |
| 69 | C22×C33 | — | 8 | 3 | no | — |
| 70 | C2×C3⋊(C3⋊D3) | — | 12 | 4 | no | — |
| 71 | C2×C3⋊(C3⋊D3) | — | 12 | 4 | no | — |
| 72 | C2×C3⋊(C3⋊D3) | — | 12 | 8 | no | — |
| 73 | C2×C3⋊(C3⋊D3) | — | 12 | 8 | no | — |
| 74 | C2×C3⋊(C3⋊D3) | — | 24 | 6 | no | — |
| 75 | C2×C3⋊(C3⋊D3) | — | 24 | 3 | no | — |
| 76 | C2×C3⋊(C3⋊D3) | — | 24 | 6 | no | — |
| 77 | C2×C3⋊(C3⋊D3) | — | 24 | 3 | no | — |
| 78 | C3×C3⋊Dic3 | — | 4 | 8 | no | — |
| 79 | C32×Dic3 | — | 12 | 8 | no | — |
| 80 | C34 | C2×#106 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 81 | C22×C3⋊D3 | — | 9 | 4 | no | — |
| 82 | C22×C3⋊D3 | — | 18 | 3 | no | — |
| 83 | C22×C3⋊D3 | — | 36 | 1 | no | — |
| 84 | C22×C3⋊D3 | — | 36 | 6 | no | — |
| 85 | C22×C3⋊D3 | — | 36 | 3 | no | — |
| 86 | C22×C3⋊D3 | — | 72 | 6 | no | — |
| 87 | C22×C3⋊D3 | — | 72 | 4 | no | — |
| 88 | C2×#157 | — | 9 | 8 | no | — |
| 89 | C2×#157 | — | 9 | 4 | no | — |
| 90 | C2×#157 | — | 9 | 4 | no | — |
| 91 | C2×#157 | — | 9 | 8 | no | — |
| 92 | C2×#157 | — | 18 | 6 | no | — |
| 93 | C2×#157 | — | 18 | 3 | no | — |
| 94 | C2×#157 | — | 18 | 3 | no | — |
| 95 | C2×#157 | — | 18 | 6 | no | — |
| 96 | C2×#157 | — | 18 | 8 | no | — |
| 97 | C2×#157 | — | 36 | 8 | no | — |
| 98 | C2×#157 | — | 36 | 12 | no | — |
| 99 | C2×#157 | — | 36 | 6 | no | — |
| 100 | AGL(1,𝔽9) | — | 9 | 16 | no | — |
| 101 | AGL(1,𝔽9) | — | 18 | 12 | no | — |
| 102 | C2×C32×C4 | — | 36 | 1 | no | — |
| 103 | C2×C3⋊Dic3 | — | 4 | 1 | no | — |
| 104 | C4×C3⋊D3 | — | 36 | 2 | no | — |
| 105 | C4×C3⋊D3 | — | 36 | 2 | no | — |
| 106 | C2×C3×Dic3 | — | 12 | 12 | no | — |
| 107 | C2×C3×Dic3 | — | 12 | 4 | no | — |
| 108 | #624 | — | 18 | 8 | no | — |
| 109 | #624 | — | 36 | 4 | no | — |
| 110 | #624 | — | 36 | 12 | no | — |
| 111 | #624 | — | 36 | 12 | no | — |
| 112 | C2×#106 | — | 81 | 1 | no | Sylow |
| 113 | C2×C33 | — | 4 | 4 | no | — |
| 114 | C2×C33 | — | 4 | 8 | no | — |
| 115 | C2×C33 | — | 8 | 6 | no | — |
| 116 | C2×C33 | — | 8 | 3 | no | — |
| 117 | C3⋊(C3⋊D3) | — | 12 | 8 | no | — |
| 118 | C3⋊(C3⋊D3) | — | 12 | 4 | no | — |
| 119 | C3⋊(C3⋊D3) | — | 12 | 4 | no | — |
| 120 | C3⋊(C3⋊D3) | — | 24 | 6 | no | — |
| 121 | C3⋊(C3⋊D3) | — | 24 | 3 | no | — |
| 122 | C3⋊(C3⋊D3) | — | 24 | 3 | no | — |
| 123 | C2×C4×D3 | — | 108 | 4 | no | — |
| 124 | C22×C32 | — | 2 | 3 | no | — |
| 125 | C22×C32 | — | 4 | 1 | no | — |
| 126 | C22×C32 | — | 4 | 6 | no | — |
| 127 | C22×C32 | — | 4 | 3 | no | — |
| 128 | C22×C32 | — | 8 | 6 | no | — |
| 129 | C22×C32 | — | 8 | 4 | no | — |
| 130 | C22×C32 | AΓL(1,𝔽9) | 1 | 4 | yes | — |
| 131 | C2×C3⋊D3 | — | 9 | 4 | no | — |
| 132 | C2×C3⋊D3 | — | 9 | 4 | no | — |
| 133 | C2×C3⋊D3 | — | 9 | 8 | no | — |
| 134 | C2×C3⋊D3 | — | 9 | 8 | no | — |
| 135 | C2×C3⋊D3 | — | 18 | 6 | no | — |
| 136 | C2×C3⋊D3 | — | 18 | 3 | no | — |
| 137 | C2×C3⋊D3 | — | 18 | 6 | no | — |
| 138 | C2×C3⋊D3 | — | 18 | 3 | no | — |
| 139 | C2×C3⋊D3 | — | 36 | 1 | no | — |
| 140 | C2×C3⋊D3 | — | 36 | 1 | no | — |
| 141 | C2×C3⋊D3 | — | 36 | 2 | no | — |
| 142 | C2×C3⋊D3 | — | 36 | 2 | no | — |
| 143 | C2×C3⋊D3 | — | 36 | 6 | no | — |
| 144 | C2×C3⋊D3 | — | 36 | 6 | no | — |
| 145 | C2×C3⋊D3 | — | 36 | 12 | no | — |
| 146 | C2×C3⋊D3 | — | 36 | 12 | no | — |
| 147 | C2×C3⋊D3 | — | 36 | 6 | no | — |
| 148 | C2×C3⋊D3 | — | 36 | 3 | no | — |
| 149 | C2×C3⋊D3 | — | 36 | 6 | no | — |
| 150 | C2×C3⋊D3 | — | 36 | 3 | no | — |
| 151 | C2×C3⋊D3 | — | 72 | 6 | no | — |
| 152 | C2×C3⋊D3 | — | 72 | 6 | no | — |
| 153 | C2×C3⋊D3 | — | 72 | 12 | no | — |
| 154 | C2×C3⋊D3 | — | 72 | 12 | no | — |
| 155 | C2×C3⋊D3 | — | 72 | 4 | no | — |
| 156 | C2×C3⋊D3 | — | 72 | 4 | no | — |
| 157 | C2×C3⋊D3 | — | 72 | 8 | no | — |
| 158 | C2×C3⋊D3 | — | 72 | 8 | no | — |
| 159 | #157 | — | 9 | 4 | no | — |
| 160 | #157 | — | 9 | 8 | no | — |
| 161 | #157 | — | 9 | 4 | no | — |
| 162 | #157 | — | 18 | 3 | no | — |
| 163 | #157 | — | 18 | 6 | no | — |
| 164 | #157 | — | 18 | 3 | no | — |
| 165 | #157 | — | 36 | 8 | no | — |
| 166 | #157 | — | 36 | 12 | no | — |
| 167 | C32×C4 | — | 36 | 2 | no | — |
| 168 | C3⋊Dic3 | — | 4 | 2 | no | — |
| 169 | C3×Dic3 | — | 12 | 24 | no | — |
| 170 | C3×Dic3 | — | 12 | 8 | no | — |
| 171 | C22×C8 | — | 81 | 1 | no | — |
| 172 | C2×C4⋊C4 | — | 81 | 1 | no | — |
| 173 | C2×C4⋊C4 | — | 81 | 1 | no | — |
| 174 | #106 | — | 81 | 4 | no | — |
| 175 | C33 | — | 4 | 4 | no | — |
| 176 | C33 | — | 8 | 3 | no | — |
| 177 | C22×D3 | — | 108 | 4 | no | — |
| 178 | C22×D3 | — | 216 | 3 | no | — |
| 179 | C2×C3×C4 | — | 36 | 4 | no | — |
| 180 | C2×Dic3 | — | 12 | 4 | no | — |
| 181 | C4×D3 | — | 108 | 8 | no | — |
| 182 | C4×D3 | — | 108 | 8 | no | — |
| 183 | C2×C32 | — | 2 | 6 | no | — |
| 184 | C2×C32 | — | 2 | 3 | no | — |
| 185 | C2×C32 | — | 4 | 1 | no | — |
| 186 | C2×C32 | — | 4 | 2 | no | — |
| 187 | C2×C32 | — | 4 | 6 | no | — |
| 188 | C2×C32 | — | 4 | 12 | no | — |
| 189 | C2×C32 | — | 4 | 6 | no | — |
| 190 | C2×C32 | — | 4 | 3 | no | — |
| 191 | C2×C32 | — | 8 | 12 | no | — |
| 192 | C2×C32 | — | 8 | 6 | no | — |
| 193 | C2×C32 | — | 8 | 8 | no | — |
| 194 | C2×C32 | — | 8 | 4 | no | — |
| 195 | C2×C32 | AGL(1,𝔽9)⋊C4 | 1 | 8 | yes | complemented |
| 196 | C2×C32 | C2×AΓL(1,𝔽9) | 1 | 4 | yes | — |
| 197 | C3⋊D3 | — | 9 | 4 | no | — |
| 198 | C3⋊D3 | — | 9 | 4 | no | — |
| 199 | C3⋊D3 | — | 9 | 8 | no | — |
| 200 | C3⋊D3 | — | 18 | 3 | no | — |
| 201 | C3⋊D3 | — | 18 | 6 | no | — |
| 202 | C3⋊D3 | — | 18 | 3 | no | — |
| 203 | C3⋊D3 | — | 36 | 2 | no | — |
| 204 | C3⋊D3 | — | 36 | 1 | no | — |
| 205 | C3⋊D3 | — | 36 | 1 | no | — |
| 206 | C3⋊D3 | — | 36 | 12 | no | — |
| 207 | C3⋊D3 | — | 36 | 6 | no | — |
| 208 | C3⋊D3 | — | 36 | 6 | no | — |
| 209 | C3⋊D3 | — | 36 | 3 | no | — |
| 210 | C3⋊D3 | — | 36 | 6 | no | — |
| 211 | C3⋊D3 | — | 36 | 3 | no | — |
| 212 | C3⋊D3 | — | 72 | 12 | no | — |
| 213 | C3⋊D3 | — | 72 | 6 | no | — |
| 214 | C3⋊D3 | — | 72 | 6 | no | — |
| 215 | C3⋊D3 | — | 72 | 8 | no | — |
| 216 | C3⋊D3 | — | 72 | 4 | no | — |
| 217 | C3⋊D3 | — | 72 | 4 | no | — |
| 218 | C22×C4 | — | 81 | 1 | no | — |
| 219 | C22×C4 | — | 162 | 1 | no | — |
| 220 | C22×C4 | — | 162 | 1 | no | — |
| 221 | C2×C8 | — | 81 | 2 | no | — |
| 222 | C2×C8 | — | 81 | 4 | no | — |
| 223 | C4⋊C4 | — | 81 | 2 | no | — |
| 224 | C4⋊C4 | — | 81 | 2 | no | — |
| 225 | C4⋊C4 | — | 162 | 1 | no | — |
| 226 | C4⋊C4 | — | 162 | 1 | no | — |
| 227 | C22×C3 | — | 4 | 4 | no | — |
| 228 | C22×C3 | — | 8 | 3 | no | — |
| 229 | C2×D3 | — | 108 | 4 | no | — |
| 230 | C2×D3 | — | 108 | 4 | no | — |
| 231 | C2×D3 | — | 108 | 8 | no | — |
| 232 | C2×D3 | — | 108 | 8 | no | — |
| 233 | C2×D3 | — | 216 | 6 | no | — |
| 234 | C2×D3 | — | 216 | 6 | no | — |
| 235 | C2×D3 | — | 216 | 3 | no | — |
| 236 | C2×D3 | — | 216 | 3 | no | — |
| 237 | C3×C4 | — | 36 | 8 | no | — |
| 238 | Dic3 | — | 12 | 8 | no | — |
| 239 | C32 | — | 2 | 3 | no | — |
| 240 | C32 | — | 4 | 1 | no | — |
| 241 | C32 | — | 4 | 6 | no | — |
| 242 | C32 | — | 4 | 3 | no | — |
| 243 | C32 | — | 8 | 6 | no | — |
| 244 | C32 | — | 8 | 4 | no | — |
| 245 | C32 | C2×AGL(1,𝔽9)⋊C4 | 1 | 4 | yes | complemented |
| 246 | C23 | — | 81 | 1 | no | — |
| 247 | C2×C4 | — | 36 | 1 | no | — |
| 248 | C2×C4 | — | 324 | 1 | no | — |
| 249 | C2×C4 | — | 324 | 2 | no | — |
| 250 | C2×C4 | — | 81 | 1 | no | — |
| 251 | C2×C4 | — | 81 | 1 | no | — |
| 252 | C2×C4 | — | 81 | 2 | no | — |
| 253 | C2×C4 | — | 81 | 2 | no | — |
| 254 | C2×C4 | — | 162 | 2 | no | — |
| 255 | C2×C4 | — | 162 | 2 | no | — |
| 256 | C8 | — | 81 | 4 | no | — |
| 257 | C2×C3 | — | 4 | 8 | no | — |
| 258 | C2×C3 | — | 4 | 4 | no | — |
| 259 | C2×C3 | — | 8 | 6 | no | — |
| 260 | C2×C3 | — | 8 | 3 | no | — |
| 261 | D3 | — | 108 | 4 | no | — |
| 262 | D3 | — | 108 | 4 | no | — |
| 263 | D3 | — | 108 | 8 | no | — |
| 264 | D3 | — | 216 | 3 | no | — |
| 265 | D3 | — | 216 | 3 | no | — |
| 266 | D3 | — | 216 | 6 | no | — |
| 267 | C22 | — | 81 | 2 | no | — |
| 268 | C22 | — | 81 | 2 | no | — |
| 269 | C22 | — | 81 | 1 | no | — |
| 270 | C22 | — | 81 | 1 | no | — |
| 271 | C22 | #13049803 | 1 | 1 | yes | characteristic |
| 272 | C4 | — | 36 | 2 | no | — |
| 273 | C4 | — | 324 | 2 | no | — |
| 274 | C4 | — | 81 | 1 | no | — |
| 275 | C4 | — | 81 | 1 | no | — |
| 276 | C4 | — | 81 | 2 | no | — |
| 277 | C3 | — | 4 | 4 | no | — |
| 278 | C3 | — | 8 | 3 | no | — |
| 279 | C2 | — | 81 | 1 | no | — |
| 280 | C2 | — | 81 | 1 | no | — |
| 281 | C2 | — | 81 | 2 | no | — |
| 282 | C2 | C2×#13049803 | 1 | 1 | yes | characteristic |
| 283 | C2 | #72213762 | 1 | 2 | yes | direct product factor |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #72213762 | C2 | direct (N × H) |
| 7. C2×#10620192 | C4 | semidirect |
| 21. #10620192 | C2×C4 | semidirect |
| 48. C2×C34 | #106 | semidirect |
| 50. C3⋊(C3⋊(C3⋊D3)) | #106 | semidirect |
| 80. C34 | C2×#106 | semidirect |
| 195. C2×C32 | AGL(1,𝔽9)⋊C4 | semidirect |
| 245. C32 | C2×AGL(1,𝔽9)⋊C4 | semidirect |
| 283. C2 | #72213762 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#13049456 | 2 |
| C2×#13049467 | 2 |
| #72213762 | 2 |
| C22×#10620192 | 2 |
| C2×AGL(1,𝔽9)⋊C4 | 9 |