Finite Group DatabasebetaFinite Group DatabasebetaOrder 28224.
287 conjugacy classes of subgroups; only the 18 normal subgroups are shown.
This group has 287 conjugacy classes of subgroups — too many to draw. Only its 18 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 | #72244497 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | AGL(1,𝔽8)×AΓL(1,𝔽8) | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | AGL(1,𝔽8)×AΓL(1,𝔽8) | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C2×A4×AΓL(1,𝔽8) | — | 7 | 1 | no | maximal |
| 5 | C2×A4×AΓL(1,𝔽8) | — | 7 | 1 | no | maximal |
| 6 | A4×AΓL(1,𝔽8) | — | 7 | 1 | no | — |
| 7 | A4×AΓL(1,𝔽8) | — | 7 | 1 | no | — |
| 8 | ΓL(1,𝔽8)×AΓL(1,𝔽8) | — | 8 | 1 | no | maximal |
| 9 | ΓL(1,𝔽8)×AΓL(1,𝔽8) | — | 8 | 1 | no | maximal |
| 10 | #72214179 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 11 | AGL(1,𝔽8)2 | C32 | 1 | 1 | yes | characteristic, complemented, Hall |
| 12 | C2×A4×AGL(1,𝔽8) | — | 7 | 1 | no | — |
| 13 | C2×A4×AGL(1,𝔽8) | — | 7 | 1 | no | — |
| 14 | #90632⋊C3 | — | 21 | 1 | no | — |
| 15 | #13062401 | — | 21 | 1 | no | — |
| 16 | C2×C22⋊AΓL(1,𝔽8) | — | 7 | 1 | no | — |
| 17 | C2×C22⋊AΓL(1,𝔽8) | — | 7 | 1 | no | — |
| 18 | C2×C22⋊AΓL(1,𝔽8) | — | 7 | 1 | no | — |
| 19 | C2×C22⋊AΓL(1,𝔽8) | — | 7 | 1 | no | — |
| 20 | C23×AΓL(1,𝔽8) | ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, complemented |
| 21 | C23×AΓL(1,𝔽8) | ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, complemented |
| 22 | ΓL(1,𝔽8)×AGL(1,𝔽8) | — | 8 | 1 | no | — |
| 23 | ΓL(1,𝔽8)×AGL(1,𝔽8) | — | 8 | 1 | no | — |
| 24 | #11924306 | — | 8 | 1 | no | — |
| 25 | #11924306 | — | 8 | 1 | no | — |
| 26 | #11924305 | — | 8 | 1 | no | — |
| 27 | #11924305 | — | 8 | 1 | no | — |
| 28 | C7×AΓL(1,𝔽8) | — | 24 | 1 | no | — |
| 29 | C7×AΓL(1,𝔽8) | — | 24 | 1 | no | — |
| 30 | C2×C3×AΓL(1,𝔽8) | — | 28 | 1 | no | — |
| 31 | C2×C3×AΓL(1,𝔽8) | — | 28 | 1 | no | — |
| 32 | A4×AGL(1,𝔽8) | — | 7 | 1 | no | — |
| 33 | A4×AGL(1,𝔽8) | — | 7 | 1 | no | — |
| 34 | C22⋊AΓL(1,𝔽8) | — | 7 | 1 | no | — |
| 35 | C22⋊AΓL(1,𝔽8) | — | 7 | 1 | no | — |
| 36 | C22⋊AΓL(1,𝔽8) | — | 7 | 1 | no | — |
| 37 | C22⋊AΓL(1,𝔽8) | — | 7 | 1 | no | — |
| 38 | C22×AΓL(1,𝔽8) | — | 7 | 1 | no | — |
| 39 | C22×AΓL(1,𝔽8) | — | 21 | 1 | no | — |
| 40 | C22×A42 | — | 49 | 1 | no | Hall |
| 41 | C2×A4×ΓL(1,𝔽8) | — | 56 | 1 | no | — |
| 42 | C2×A4×ΓL(1,𝔽8) | — | 56 | 1 | no | — |
| 43 | C3×AΓL(1,𝔽8) | — | 28 | 1 | no | — |
| 44 | C3×AΓL(1,𝔽8) | — | 28 | 1 | no | — |
| 45 | #90632 | — | 3 | 1 | no | — |
| 46 | #90631 | — | 3 | 1 | no | — |
| 47 | C23×AGL(1,𝔽8) | C3×ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, complemented |
| 48 | C23×AGL(1,𝔽8) | C3×ΓL(1,𝔽8) | 1 | 1 | yes | characteristic, complemented |
| 49 | ΓL(1,𝔽8)2 | — | 64 | 1 | no | Hall |
| 50 | C7×AGL(1,𝔽8) | — | 24 | 1 | no | — |
| 51 | C7×AGL(1,𝔽8) | — | 24 | 1 | no | — |
| 52 | C2×C3×AGL(1,𝔽8) | — | 28 | 1 | no | — |
| 53 | C2×C3×AGL(1,𝔽8) | — | 28 | 1 | no | — |
| 54 | C2×AΓL(1,𝔽8) | — | 7 | 1 | no | — |
| 55 | C2×AΓL(1,𝔽8) | — | 21 | 1 | no | — |
| 56 | C2×AΓL(1,𝔽8) | — | 28 | 1 | no | — |
| 57 | C2×AΓL(1,𝔽8) | — | 28 | 1 | no | — |
| 58 | C2×AΓL(1,𝔽8) | — | 28 | 1 | no | — |
| 59 | C2×AΓL(1,𝔽8) | — | 28 | 1 | no | — |
| 60 | C2×A42 | — | 49 | 1 | no | — |
| 61 | C2×A42 | — | 49 | 1 | no | — |
| 62 | C2×A42 | — | 49 | 1 | no | — |
| 63 | A4×ΓL(1,𝔽8) | — | 56 | 1 | no | — |
| 64 | A4×ΓL(1,𝔽8) | — | 56 | 1 | no | — |
| 65 | C22×AGL(1,𝔽8) | — | 7 | 1 | no | — |
| 66 | C22×AGL(1,𝔽8) | — | 21 | 1 | no | — |
| 67 | C22×C22⋊A4 | — | 49 | 1 | no | — |
| 68 | C22×C22⋊A4 | — | 49 | 1 | no | — |
| 69 | C24×A4 | — | 7 | 1 | no | — |
| 70 | C24×A4 | — | 7 | 1 | no | — |
| 71 | C2×C22⋊ΓL(1,𝔽8) | — | 56 | 1 | no | — |
| 72 | C2×C22⋊ΓL(1,𝔽8) | — | 56 | 1 | no | — |
| 73 | C2×C22⋊ΓL(1,𝔽8) | — | 56 | 1 | no | — |
| 74 | C2×C22⋊ΓL(1,𝔽8) | — | 56 | 1 | no | — |
| 75 | C2×C7×A4 | — | 168 | 1 | no | — |
| 76 | C2×C7×A4 | — | 168 | 1 | no | — |
| 77 | C23×ΓL(1,𝔽8) | — | 8 | 1 | no | — |
| 78 | C23×ΓL(1,𝔽8) | — | 8 | 1 | no | — |
| 79 | C3×AGL(1,𝔽8) | — | 28 | 1 | no | — |
| 80 | C3×AGL(1,𝔽8) | — | 28 | 1 | no | — |
| 81 | AΓL(1,𝔽8) | — | 28 | 1 | no | — |
| 82 | AΓL(1,𝔽8) | — | 28 | 1 | no | — |
| 83 | AΓL(1,𝔽8) | — | 28 | 1 | no | — |
| 84 | AΓL(1,𝔽8) | — | 28 | 1 | no | — |
| 85 | AΓL(1,𝔽8) | — | 168 | 1 | no | — |
| 86 | AΓL(1,𝔽8) | — | 168 | 1 | no | — |
| 87 | AΓL(1,𝔽8) | — | 168 | 1 | no | — |
| 88 | AΓL(1,𝔽8) | — | 168 | 1 | no | — |
| 89 | AΓL(1,𝔽8) | — | 168 | 1 | no | — |
| 90 | AΓL(1,𝔽8) | AΓL(1,𝔽8) | 1 | 1 | yes | characteristic, direct product factor |
| 91 | AΓL(1,𝔽8) | AΓL(1,𝔽8) | 1 | 1 | yes | characteristic, direct product factor |
| 92 | #3862 | — | 64 | 1 | no | — |
| 93 | #3861 | — | 64 | 1 | no | — |
| 94 | C7×ΓL(1,𝔽8) | — | 192 | 1 | no | — |
| 95 | C7×ΓL(1,𝔽8) | — | 192 | 1 | no | — |
| 96 | C22×C3×A4 | — | 196 | 1 | no | — |
| 97 | C22×C3×A4 | — | 196 | 1 | no | — |
| 98 | A42 | — | 49 | 1 | no | — |
| 99 | C2×C3×ΓL(1,𝔽8) | — | 224 | 1 | no | — |
| 100 | C2×C3×ΓL(1,𝔽8) | — | 224 | 1 | no | — |
| 101 | C2×AGL(1,𝔽8) | — | 7 | 1 | no | — |
| 102 | C2×AGL(1,𝔽8) | — | 21 | 1 | no | — |
| 103 | C2×C22⋊A4 | — | 49 | 1 | no | — |
| 104 | C2×C22⋊A4 | — | 49 | 1 | no | — |
| 105 | C2×C22⋊A4 | — | 49 | 1 | no | — |
| 106 | C2×C22⋊A4 | — | 49 | 1 | no | — |
| 107 | C2×C22⋊A4 | — | 49 | 1 | no | — |
| 108 | C2×C22⋊A4 | — | 49 | 1 | no | — |
| 109 | C23×A4 | — | 7 | 1 | no | — |
| 110 | C23×A4 | — | 7 | 1 | no | — |
| 111 | C23×A4 | — | 49 | 1 | no | — |
| 112 | C23×A4 | — | 49 | 1 | no | — |
| 113 | C23×A4 | — | 147 | 1 | no | — |
| 114 | C23×A4 | — | 147 | 1 | no | — |
| 115 | C22⋊ΓL(1,𝔽8) | — | 56 | 1 | no | — |
| 116 | C22⋊ΓL(1,𝔽8) | — | 56 | 1 | no | — |
| 117 | C22⋊ΓL(1,𝔽8) | — | 56 | 1 | no | — |
| 118 | C22⋊ΓL(1,𝔽8) | — | 56 | 1 | no | — |
| 119 | C7×A4 | — | 168 | 1 | no | — |
| 120 | C7×A4 | — | 168 | 1 | no | — |
| 121 | C22×ΓL(1,𝔽8) | — | 168 | 1 | no | — |
| 122 | C22×ΓL(1,𝔽8) | — | 168 | 1 | no | — |
| 123 | C2×C3×A4 | — | 196 | 1 | no | — |
| 124 | C2×C3×A4 | — | 196 | 1 | no | — |
| 125 | C2×C3×A4 | — | 196 | 1 | no | — |
| 126 | C2×C3×A4 | — | 196 | 1 | no | — |
| 127 | C2×C3×A4 | — | 196 | 1 | no | — |
| 128 | C2×C3×A4 | — | 196 | 1 | no | — |
| 129 | C26 | ΓL(1,𝔽8)2 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 130 | C3×ΓL(1,𝔽8) | — | 224 | 1 | no | — |
| 131 | C3×ΓL(1,𝔽8) | — | 224 | 1 | no | — |
| 132 | C23×C7 | — | 24 | 1 | no | — |
| 133 | C23×C7 | — | 24 | 1 | no | — |
| 134 | AGL(1,𝔽8) | — | 72 | 1 | no | — |
| 135 | AGL(1,𝔽8) | — | 72 | 1 | no | — |
| 136 | AGL(1,𝔽8) | — | 72 | 1 | no | — |
| 137 | AGL(1,𝔽8) | — | 72 | 1 | no | — |
| 138 | AGL(1,𝔽8) | — | 504 | 1 | no | — |
| 139 | AGL(1,𝔽8) | C3×AΓL(1,𝔽8) | 1 | 1 | yes | characteristic, complemented |
| 140 | AGL(1,𝔽8) | C3×AΓL(1,𝔽8) | 1 | 1 | yes | characteristic, complemented |
| 141 | C72 | — | 576 | 1 | no | Sylow |
| 142 | C24×C3 | — | 28 | 1 | no | — |
| 143 | C24×C3 | — | 28 | 1 | no | — |
| 144 | C22⋊A4 | — | 49 | 1 | no | — |
| 145 | C22⋊A4 | — | 49 | 1 | no | — |
| 146 | C22×A4 | — | 588 | 1 | no | — |
| 147 | C22×A4 | — | 588 | 1 | no | — |
| 148 | C22×A4 | — | 147 | 1 | no | — |
| 149 | C22×A4 | — | 147 | 1 | no | — |
| 150 | C22×A4 | — | 147 | 1 | no | — |
| 151 | C22×A4 | — | 147 | 1 | no | — |
| 152 | C22×A4 | — | 147 | 1 | no | — |
| 153 | C22×A4 | — | 147 | 1 | no | — |
| 154 | C22×A4 | — | 196 | 1 | no | — |
| 155 | C22×A4 | — | 196 | 1 | no | — |
| 156 | C22×A4 | — | 196 | 1 | no | — |
| 157 | C22×A4 | — | 196 | 1 | no | — |
| 158 | C2×C3×C7 | — | 672 | 1 | no | — |
| 159 | C2×C3×C7 | — | 672 | 1 | no | — |
| 160 | C2×ΓL(1,𝔽8) | — | 168 | 1 | no | — |
| 161 | C2×ΓL(1,𝔽8) | — | 168 | 1 | no | — |
| 162 | C2×ΓL(1,𝔽8) | — | 224 | 1 | no | — |
| 163 | C2×ΓL(1,𝔽8) | — | 224 | 1 | no | — |
| 164 | C2×ΓL(1,𝔽8) | — | 224 | 1 | no | — |
| 165 | C2×ΓL(1,𝔽8) | — | 224 | 1 | no | — |
| 166 | C22×C32 | — | 784 | 1 | no | — |
| 167 | C3×A4 | — | 196 | 1 | no | — |
| 168 | C3×A4 | — | 196 | 1 | no | — |
| 169 | C25 | — | 7 | 1 | no | — |
| 170 | C25 | — | 7 | 1 | no | — |
| 171 | C25 | — | 441 | 1 | no | — |
| 172 | C22×C7 | — | 504 | 1 | no | — |
| 173 | C22×C7 | — | 504 | 1 | no | — |
| 174 | C23×C3 | — | 28 | 1 | no | — |
| 175 | C23×C3 | — | 28 | 1 | no | — |
| 176 | C23×C3 | — | 588 | 1 | no | — |
| 177 | C23×C3 | — | 588 | 1 | no | — |
| 178 | C23×C3 | — | 588 | 1 | no | — |
| 179 | C23×C3 | — | 588 | 1 | no | — |
| 180 | C2×A4 | — | 7 | 1 | no | — |
| 181 | C2×A4 | — | 7 | 1 | no | — |
| 182 | C2×A4 | — | 49 | 1 | no | — |
| 183 | C2×A4 | — | 588 | 1 | no | — |
| 184 | C2×A4 | — | 588 | 1 | no | — |
| 185 | C2×A4 | — | 588 | 1 | no | — |
| 186 | C2×A4 | — | 588 | 1 | no | — |
| 187 | C2×A4 | — | 588 | 1 | no | — |
| 188 | C2×A4 | — | 588 | 1 | no | — |
| 189 | C2×A4 | — | 147 | 1 | no | — |
| 190 | C2×A4 | — | 147 | 1 | no | — |
| 191 | C2×A4 | — | 147 | 1 | no | — |
| 192 | C2×A4 | — | 196 | 1 | no | — |
| 193 | C2×A4 | — | 196 | 1 | no | — |
| 194 | C2×A4 | — | 196 | 1 | no | — |
| 195 | C2×A4 | — | 196 | 1 | no | — |
| 196 | C2×A4 | — | 196 | 1 | no | — |
| 197 | C2×A4 | — | 196 | 1 | no | — |
| 198 | C2×A4 | — | 196 | 1 | no | — |
| 199 | C2×A4 | — | 196 | 1 | no | — |
| 200 | C2×A4 | — | 196 | 1 | no | — |
| 201 | C2×A4 | — | 196 | 1 | no | — |
| 202 | C2×A4 | — | 196 | 1 | no | — |
| 203 | C2×A4 | — | 196 | 1 | no | — |
| 204 | C3×C7 | — | 672 | 1 | no | — |
| 205 | C3×C7 | — | 672 | 1 | no | — |
| 206 | ΓL(1,𝔽8) | — | 8 | 1 | no | — |
| 207 | ΓL(1,𝔽8) | — | 8 | 1 | no | — |
| 208 | ΓL(1,𝔽8) | — | 1344 | 1 | no | — |
| 209 | ΓL(1,𝔽8) | — | 1344 | 1 | no | — |
| 210 | ΓL(1,𝔽8) | — | 224 | 1 | no | — |
| 211 | ΓL(1,𝔽8) | — | 224 | 1 | no | — |
| 212 | ΓL(1,𝔽8) | — | 224 | 1 | no | — |
| 213 | ΓL(1,𝔽8) | — | 224 | 1 | no | — |
| 214 | C2×C32 | — | 784 | 1 | no | — |
| 215 | C2×C32 | — | 784 | 1 | no | — |
| 216 | C2×C32 | — | 784 | 1 | no | — |
| 217 | C24 | — | 21 | 1 | no | — |
| 218 | C24 | — | 21 | 1 | no | — |
| 219 | C24 | — | 441 | 1 | no | — |
| 220 | C24 | — | 441 | 1 | no | — |
| 221 | C24 | — | 441 | 1 | no | — |
| 222 | C24 | — | 441 | 1 | no | — |
| 223 | C24 | — | 441 | 1 | no | — |
| 224 | C2×C7 | — | 168 | 1 | no | — |
| 225 | C2×C7 | — | 504 | 1 | no | — |
| 226 | C22×C3 | — | 784 | 1 | no | — |
| 227 | C22×C3 | — | 784 | 1 | no | — |
| 228 | C22×C3 | — | 588 | 1 | no | — |
| 229 | C22×C3 | — | 588 | 1 | no | — |
| 230 | C22×C3 | — | 588 | 1 | no | — |
| 231 | C22×C3 | — | 588 | 1 | no | — |
| 232 | C22×C3 | — | 588 | 1 | no | — |
| 233 | C22×C3 | — | 196 | 1 | no | — |
| 234 | A4 | — | 7 | 1 | no | — |
| 235 | A4 | — | 7 | 1 | no | — |
| 236 | A4 | — | 588 | 1 | no | — |
| 237 | A4 | — | 588 | 1 | no | — |
| 238 | A4 | — | 196 | 1 | no | — |
| 239 | A4 | — | 196 | 1 | no | — |
| 240 | A4 | — | 196 | 1 | no | — |
| 241 | A4 | — | 196 | 1 | no | — |
| 242 | C32 | — | 784 | 1 | no | Sylow |
| 243 | C23 | — | 63 | 1 | no | — |
| 244 | C23 | — | 147 | 1 | no | — |
| 245 | C23 | — | 147 | 1 | no | — |
| 246 | C23 | — | 441 | 1 | no | — |
| 247 | C23 | — | 441 | 1 | no | — |
| 248 | C23 | — | 441 | 1 | no | — |
| 249 | C23 | — | 441 | 1 | no | — |
| 250 | C23 | — | 441 | 1 | no | — |
| 251 | C23 | — | 441 | 1 | no | — |
| 252 | C23 | — | 441 | 1 | no | — |
| 253 | C23 | — | 441 | 1 | no | — |
| 254 | C23 | ΓL(1,𝔽8)×AΓL(1,𝔽8) | 1 | 1 | yes | characteristic, complemented |
| 255 | C23 | ΓL(1,𝔽8)×AΓL(1,𝔽8) | 1 | 1 | yes | characteristic, complemented |
| 256 | C7 | — | 24 | 1 | no | — |
| 257 | C7 | — | 24 | 1 | no | — |
| 258 | C7 | — | 576 | 1 | no | — |
| 259 | C7 | — | 576 | 1 | no | — |
| 260 | C2×C3 | — | 784 | 1 | no | — |
| 261 | C2×C3 | — | 784 | 1 | no | — |
| 262 | C2×C3 | — | 784 | 1 | no | — |
| 263 | C2×C3 | — | 784 | 1 | no | — |
| 264 | C2×C3 | — | 784 | 1 | no | — |
| 265 | C2×C3 | — | 784 | 1 | no | — |
| 266 | C2×C3 | — | 28 | 1 | no | — |
| 267 | C2×C3 | — | 28 | 1 | no | — |
| 268 | C2×C3 | — | 588 | 1 | no | — |
| 269 | C2×C3 | — | 588 | 1 | no | — |
| 270 | C2×C3 | — | 588 | 1 | no | — |
| 271 | C2×C3 | — | 588 | 1 | no | — |
| 272 | C22 | — | 21 | 1 | no | — |
| 273 | C22 | — | 21 | 1 | no | — |
| 274 | C22 | — | 441 | 1 | no | — |
| 275 | C22 | — | 441 | 1 | no | — |
| 276 | C22 | — | 441 | 1 | no | — |
| 277 | C22 | — | 441 | 1 | no | — |
| 278 | C22 | — | 441 | 1 | no | — |
| 279 | C3 | — | 784 | 1 | no | — |
| 280 | C3 | — | 784 | 1 | no | — |
| 281 | C3 | — | 28 | 1 | no | — |
| 282 | C3 | — | 28 | 1 | no | — |
| 283 | C2 | — | 21 | 1 | no | — |
| 284 | C2 | — | 21 | 1 | no | — |
| 285 | C2 | — | 441 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | AGL(1,𝔽8)2 | C32 |
| Fitting, socle | C26 | ΓL(1,𝔽8)2 |
| LCS | G ▷ AGL(1,𝔽8)2 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #72244497 | C3 | semidirect |
| 2. AGL(1,𝔽8)×AΓL(1,𝔽8) | C3 | semidirect |
| 3. AGL(1,𝔽8)×AΓL(1,𝔽8) | C3 | semidirect |
| 10. #72214179 | C3 | semidirect |
| 11. AGL(1,𝔽8)2 | C32 | semidirect |
| 20. C23×AΓL(1,𝔽8) | ΓL(1,𝔽8) | semidirect |
| 21. C23×AΓL(1,𝔽8) | ΓL(1,𝔽8) | semidirect |
| 47. C23×AGL(1,𝔽8) | C3×ΓL(1,𝔽8) | semidirect |
| 48. C23×AGL(1,𝔽8) | C3×ΓL(1,𝔽8) | semidirect |
| 90. AΓL(1,𝔽8) | AΓL(1,𝔽8) | direct (N × H) |
| 91. AΓL(1,𝔽8) | AΓL(1,𝔽8) | direct (N × H) |
| 129. C26 | ΓL(1,𝔽8)2 | semidirect |
| 139. AGL(1,𝔽8) | C3×AΓL(1,𝔽8) | semidirect |
| 140. AGL(1,𝔽8) | C3×AΓL(1,𝔽8) | semidirect |
| 254. C23 | ΓL(1,𝔽8)×AΓL(1,𝔽8) | semidirect |
| 255. C23 | ΓL(1,𝔽8)×AΓL(1,𝔽8) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72244497 | 3 |
| AGL(1,𝔽8)×AΓL(1,𝔽8) | 3 |
| C2×A4×AΓL(1,𝔽8) | 7 |
| ΓL(1,𝔽8)×AΓL(1,𝔽8) | 8 |
| #72214179 | 3 |