Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
510 conjugacy classes of subgroups; only the 61 normal subgroups are shown.
This group has 510 conjugacy classes of subgroups — too many to draw. Only its 61 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 | SL(2,𝔽3)×Dic10 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C4×C5⋊GL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C5×#5386 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #11751610 | C2 | 1 | 2 | yes | maximal |
| 5 | #11751608 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #10612770 | — | 3 | 1 | no | maximal, Hall |
| 7 | C2×D15⋊Q8 | — | 4 | 1 | no | maximal |
| 8 | C2×C5⋊GL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 9 | C4×C5×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 10 | Dic5×SL(2,𝔽3) | C22 | 1 | 2 | yes | — |
| 11 | #91184 | C22 | 1 | 1 | yes | characteristic |
| 12 | C5×#859 | C22 | 1 | 2 | yes | — |
| 13 | Q8×GL(2,𝔽3) | — | 5 | 1 | no | maximal, Hall |
| 14 | #66061 | — | 3 | 1 | no | — |
| 15 | Q8×Dic10 | D3 | 1 | 1 | yes | characteristic, complemented |
| 16 | C5×#475 | — | 3 | 1 | no | — |
| 17 | #65368 | — | 3 | 1 | no | — |
| 18 | C4×#688 | — | 3 | 1 | no | — |
| 19 | #65159 | — | 3 | 2 | no | — |
| 20 | #65144 | — | 3 | 2 | no | — |
| 21 | #65131 | — | 3 | 1 | no | — |
| 22 | #65119 | — | 3 | 1 | no | — |
| 23 | #65080 | — | 3 | 2 | no | — |
| 24 | #65077 | — | 3 | 2 | no | — |
| 25 | C2×C4×D15 | — | 4 | 1 | no | — |
| 26 | C2×C5×Dic6 | — | 4 | 1 | no | — |
| 27 | C2×C3×Dic10 | — | 4 | 1 | no | — |
| 28 | C2×C5×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 29 | C2×#1204 | — | 4 | 2 | no | — |
| 30 | C2×#1200 | — | 4 | 2 | no | — |
| 31 | D15⋊Q8 | — | 4 | 8 | no | — |
| 32 | C5⋊GL(2,𝔽3) | Q8 | 1 | 2 | yes | complemented |
| 33 | Q8×SL(2,𝔽3) | — | 5 | 1 | no | — |
| 34 | C4×GL(2,𝔽3) | — | 5 | 1 | no | — |
| 35 | C4×GL(2,𝔽3) | — | 5 | 2 | no | — |
| 36 | #5386 | — | 5 | 2 | no | — |
| 37 | #5386 | D5 | 1 | 1 | yes | characteristic, complemented |
| 38 | C2×Q8×D5 | — | 6 | 1 | no | — |
| 39 | C5×C4⋊Q8 | — | 3 | 1 | no | — |
| 40 | C4×C5×Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 41 | #4092 | — | 6 | 1 | no | — |
| 42 | Q8×Dic5 | C2×D3 | 1 | 2 | yes | — |
| 43 | C2×#688 | — | 3 | 1 | no | — |
| 44 | #4042 | — | 6 | 2 | no | — |
| 45 | #4039 | — | 3 | 2 | no | — |
| 46 | #4034 | — | 3 | 2 | no | — |
| 47 | #4034 | — | 3 | 2 | no | — |
| 48 | C4×D20 | — | 3 | 1 | no | — |
| 49 | #4015 | — | 3 | 1 | no | — |
| 50 | C4×Dic10 | — | 3 | 1 | no | — |
| 51 | C5×#106 | — | 3 | 2 | no | — |
| 52 | C5×C4⋊C8 | — | 3 | 1 | no | — |
| 53 | C5×#103 | — | 3 | 2 | no | — |
| 54 | #3967 | — | 3 | 1 | no | — |
| 55 | #3953 | — | 3 | 2 | no | — |
| 56 | #3949 | — | 3 | 2 | no | — |
| 57 | #3946 | — | 3 | 2 | no | — |
| 58 | C8×Dic5 | — | 3 | 2 | no | — |
| 59 | #3941 | — | 3 | 1 | no | — |
| 60 | #3940 | — | 3 | 1 | no | — |
| 61 | #3940 | — | 3 | 2 | no | — |
| 62 | #3940 | — | 3 | 2 | no | — |
| 63 | C4×#168 | — | 3 | 1 | no | — |
| 64 | Q8×SD16 | — | 15 | 1 | no | Sylow |
| 65 | C22×D15 | — | 4 | 1 | no | — |
| 66 | C2×C3×C4×C5 | — | 4 | 1 | no | — |
| 67 | C2×Dic15 | — | 4 | 1 | no | — |
| 68 | C4×D15 | — | 4 | 4 | no | — |
| 69 | C2×C5×Dic3 | — | 4 | 2 | no | — |
| 70 | C5×Dic6 | — | 4 | 4 | no | — |
| 71 | C2×C3×Dic5 | — | 4 | 2 | no | — |
| 72 | C3×Dic10 | — | 4 | 4 | no | — |
| 73 | C5×SL(2,𝔽3) | C2×Q8 | 1 | 1 | yes | characteristic, complemented |
| 74 | #1204 | — | 4 | 8 | no | — |
| 75 | #1200 | — | 4 | 8 | no | — |
| 76 | C2×D3×Q8 | — | 20 | 1 | no | — |
| 77 | C2×GL(2,𝔽3) | — | 5 | 1 | no | — |
| 78 | C4×SL(2,𝔽3) | — | 5 | 2 | no | — |
| 79 | C4×SL(2,𝔽3) | C2×D5 | 1 | 1 | yes | characteristic |
| 80 | #859 | — | 5 | 1 | no | — |
| 81 | #859 | C2×D5 | 1 | 2 | yes | — |
| 82 | C2×C5×Q8 | — | 6 | 1 | no | — |
| 83 | C2×C5×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 84 | Q8×D5 | — | 12 | 4 | no | — |
| 85 | C2×D20 | — | 3 | 1 | no | — |
| 86 | C2×C4×D5 | — | 6 | 1 | no | — |
| 87 | C2×C4×D5 | — | 6 | 2 | no | — |
| 88 | C2×Dic10 | — | 6 | 2 | no | — |
| 89 | C2×Dic10 | — | 6 | 1 | no | — |
| 90 | C2×Dic10 | S4 | 1 | 1 | yes | characteristic |
| 91 | C2×C5×C8 | — | 3 | 2 | no | — |
| 92 | C5×C4⋊C4 | — | 3 | 1 | no | — |
| 93 | C5×C4⋊C4 | — | 3 | 2 | no | — |
| 94 | C5×C4⋊C4 | — | 6 | 1 | no | — |
| 95 | C42×C5 | — | 3 | 1 | no | — |
| 96 | #688 | — | 3 | 2 | no | — |
| 97 | #688 | — | 3 | 2 | no | — |
| 98 | #685 | — | 6 | 2 | no | — |
| 99 | #685 | — | 6 | 1 | no | — |
| 100 | C4⋊Dic5 | — | 3 | 1 | no | — |
| 101 | C4⋊Dic5 | — | 3 | 2 | no | — |
| 102 | C4⋊Dic5 | — | 3 | 1 | no | — |
| 103 | #683 | — | 3 | 2 | no | — |
| 104 | #683 | — | 6 | 2 | no | — |
| 105 | C4×Dic5 | — | 3 | 2 | no | — |
| 106 | C2×#168 | — | 3 | 1 | no | — |
| 107 | C2×#168 | — | 3 | 1 | no | — |
| 108 | Q82 | — | 5 | 1 | no | — |
| 109 | D4×Q8 | — | 15 | 1 | no | — |
| 110 | #498 | — | 15 | 2 | no | — |
| 111 | #498 | — | 15 | 1 | no | — |
| 112 | #476 | — | 15 | 1 | no | — |
| 113 | #476 | — | 15 | 2 | no | — |
| 114 | #475 | — | 3 | 1 | no | — |
| 115 | #475 | — | 15 | 2 | no | — |
| 116 | C8×Q8 | — | 15 | 1 | no | — |
| 117 | C4×SD16 | — | 15 | 2 | no | — |
| 118 | C4×SD16 | — | 15 | 1 | no | — |
| 119 | C22×C3×C5 | — | 4 | 1 | no | — |
| 120 | C2×D15 | — | 4 | 2 | no | — |
| 121 | C2×D15 | — | 4 | 2 | no | — |
| 122 | C2×D15 | — | 4 | 2 | no | — |
| 123 | C3×C4×C5 | — | 4 | 2 | no | — |
| 124 | Dic15 | — | 4 | 2 | no | — |
| 125 | C3×Dic5 | — | 4 | 4 | no | — |
| 126 | C5×Dic3 | — | 4 | 4 | no | — |
| 127 | C2×C3×Q8 | — | 20 | 1 | no | — |
| 128 | D3×Q8 | — | 20 | 8 | no | — |
| 129 | C2×C4×D3 | — | 20 | 2 | no | — |
| 130 | C2×C4×D3 | — | 20 | 1 | no | — |
| 131 | C2×Dic6 | — | 4 | 1 | no | — |
| 132 | C2×Dic6 | — | 20 | 2 | no | — |
| 133 | C2×SL(2,𝔽3) | C22×D5 | 1 | 1 | yes | characteristic |
| 134 | GL(2,𝔽3) | — | 5 | 2 | no | — |
| 135 | C22×D5 | — | 6 | 1 | no | — |
| 136 | C5×Q8 | — | 3 | 1 | no | — |
| 137 | C5×Q8 | — | 12 | 2 | no | — |
| 138 | C5×Q8 | D3×Q8 | 1 | 1 | yes | characteristic, complemented |
| 139 | C2×C4×C5 | — | 3 | 1 | no | — |
| 140 | C2×C4×C5 | — | 3 | 1 | no | — |
| 141 | C2×C4×C5 | — | 6 | 2 | no | — |
| 142 | C2×C4×C5 | C2×S4 | 1 | 1 | yes | characteristic |
| 143 | C2×Dic5 | — | 3 | 2 | no | — |
| 144 | C2×Dic5 | — | 6 | 1 | no | — |
| 145 | C2×Dic5 | C2×S4 | 1 | 2 | yes | — |
| 146 | D20 | — | 3 | 2 | no | — |
| 147 | D20 | — | 6 | 1 | no | — |
| 148 | C4×D5 | — | 12 | 2 | no | — |
| 149 | C4×D5 | — | 12 | 4 | no | — |
| 150 | Dic10 | — | 12 | 4 | no | — |
| 151 | Dic10 | — | 24 | 1 | no | — |
| 152 | Dic10 | GL(2,𝔽3) | 1 | 4 | yes | complemented |
| 153 | C5×C8 | — | 6 | 2 | no | — |
| 154 | #168 | — | 3 | 2 | no | — |
| 155 | #168 | — | 6 | 1 | no | — |
| 156 | C22×Q8 | — | 30 | 1 | no | — |
| 157 | C2×SD16 | — | 15 | 1 | no | — |
| 158 | C4⋊Q8 | — | 3 | 1 | no | — |
| 159 | C4⋊Q8 | — | 15 | 1 | no | — |
| 160 | C4⋊Q8 | — | 15 | 2 | no | — |
| 161 | C4⋊Q8 | — | 15 | 2 | no | — |
| 162 | C22⋊Q8 | — | 30 | 2 | no | — |
| 163 | C22⋊Q8 | — | 30 | 1 | no | — |
| 164 | C4×Q8 | — | 5 | 2 | no | — |
| 165 | C4×Q8 | — | 15 | 1 | no | — |
| 166 | C4×Q8 | D3×D5 | 1 | 1 | yes | characteristic |
| 167 | C4×D4 | — | 15 | 1 | no | — |
| 168 | C4×D4 | — | 15 | 2 | no | — |
| 169 | #106 | — | 3 | 2 | no | — |
| 170 | #106 | — | 15 | 1 | no | — |
| 171 | #106 | — | 15 | 2 | no | — |
| 172 | #106 | — | 15 | 2 | no | — |
| 173 | #106 | — | 15 | 2 | no | — |
| 174 | C4⋊C8 | — | 3 | 1 | no | — |
| 175 | C4⋊C8 | — | 15 | 2 | no | — |
| 176 | #103 | — | 3 | 2 | no | — |
| 177 | #103 | — | 15 | 1 | no | — |
| 178 | #102 | — | 15 | 1 | no | — |
| 179 | #102 | — | 15 | 2 | no | — |
| 180 | C4×C8 | — | 15 | 1 | no | — |
| 181 | C4×C8 | — | 15 | 2 | no | — |
| 182 | C2×C3×C5 | — | 4 | 1 | no | — |
| 183 | C2×C3×C5 | — | 4 | 1 | no | — |
| 184 | C2×C3×C5 | — | 4 | 1 | no | — |
| 185 | D15 | — | 4 | 4 | no | — |
| 186 | C22×D3 | — | 20 | 1 | no | — |
| 187 | C3×Q8 | — | 20 | 4 | no | — |
| 188 | C2×C3×C4 | — | 4 | 1 | no | — |
| 189 | C2×C3×C4 | — | 20 | 2 | no | — |
| 190 | C2×Dic3 | — | 4 | 2 | no | — |
| 191 | C2×Dic3 | — | 20 | 1 | no | — |
| 192 | C4×D3 | — | 20 | 8 | no | — |
| 193 | C4×D3 | — | 20 | 4 | no | — |
| 194 | Dic6 | — | 4 | 4 | no | — |
| 195 | Dic6 | — | 20 | 8 | no | — |
| 196 | SL(2,𝔽3) | Q8×D5 | 1 | 1 | yes | characteristic, complemented |
| 197 | C22×C5 | C22×S4 | 1 | 1 | yes | characteristic |
| 198 | C2×D5 | — | 6 | 2 | no | — |
| 199 | C2×D5 | — | 12 | 1 | no | — |
| 200 | C2×D5 | — | 12 | 1 | no | — |
| 201 | C4×C5 | — | 3 | 1 | no | — |
| 202 | C4×C5 | — | 3 | 1 | no | — |
| 203 | C4×C5 | — | 6 | 1 | no | — |
| 204 | C4×C5 | — | 12 | 2 | no | — |
| 205 | C4×C5 | C2×GL(2,𝔽3) | 1 | 2 | yes | — |
| 206 | Dic5 | — | 6 | 2 | no | — |
| 207 | Dic5 | — | 12 | 1 | no | — |
| 208 | Dic5 | C2×GL(2,𝔽3) | 1 | 4 | yes | — |
| 209 | C2×Q8 | — | 5 | 1 | no | — |
| 210 | C2×Q8 | — | 6 | 1 | no | — |
| 211 | C2×Q8 | — | 30 | 1 | no | — |
| 212 | C2×Q8 | — | 30 | 2 | no | — |
| 213 | C2×Q8 | — | 60 | 4 | no | — |
| 214 | C2×Q8 | C2×D3×D5 | 1 | 1 | yes | characteristic |
| 215 | C2×D4 | — | 15 | 1 | no | — |
| 216 | C22×C4 | — | 30 | 2 | no | — |
| 217 | C22×C4 | — | 30 | 1 | no | — |
| 218 | SD16 | — | 15 | 2 | no | — |
| 219 | SD16 | — | 15 | 2 | no | — |
| 220 | C2×C8 | — | 3 | 2 | no | — |
| 221 | C2×C8 | — | 15 | 1 | no | — |
| 222 | C2×C8 | — | 15 | 1 | no | — |
| 223 | C4⋊C4 | — | 3 | 1 | no | — |
| 224 | C4⋊C4 | — | 3 | 2 | no | — |
| 225 | C4⋊C4 | — | 6 | 1 | no | — |
| 226 | C4⋊C4 | — | 15 | 2 | no | — |
| 227 | C4⋊C4 | — | 15 | 2 | no | — |
| 228 | C4⋊C4 | — | 15 | 1 | no | — |
| 229 | C4⋊C4 | — | 15 | 1 | no | — |
| 230 | C4⋊C4 | — | 30 | 2 | no | — |
| 231 | C22⋊C4 | — | 30 | 1 | no | — |
| 232 | C22⋊C4 | — | 30 | 2 | no | — |
| 233 | C42 | — | 3 | 1 | no | — |
| 234 | C42 | — | 15 | 2 | no | — |
| 235 | C3×C5 | — | 4 | 1 | no | Hall |
| 236 | C22×C3 | — | 4 | 1 | no | — |
| 237 | C2×D3 | — | 20 | 2 | no | — |
| 238 | C2×D3 | — | 20 | 2 | no | — |
| 239 | C2×D3 | — | 20 | 2 | no | — |
| 240 | C3×C4 | — | 4 | 2 | no | — |
| 241 | C3×C4 | — | 20 | 4 | no | — |
| 242 | Dic3 | — | 4 | 4 | no | — |
| 243 | Dic3 | — | 20 | 2 | no | — |
| 244 | C2×C5 | C22×GL(2,𝔽3) | 1 | 1 | yes | characteristic |
| 245 | C2×C5 | Q8×S4 | 1 | 1 | yes | characteristic |
| 246 | C2×C5 | #5924 | 1 | 1 | yes | characteristic |
| 247 | D5 | — | 12 | 2 | no | — |
| 248 | C23 | — | 30 | 1 | no | — |
| 249 | Q8 | — | 3 | 1 | no | — |
| 250 | Q8 | — | 5 | 4 | no | — |
| 251 | Q8 | — | 12 | 2 | no | — |
| 252 | Q8 | — | 60 | 4 | no | — |
| 253 | Q8 | — | 120 | 1 | no | — |
| 254 | Q8 | D15⋊Q8 | 1 | 1 | yes | characteristic, complemented |
| 255 | D4 | — | 15 | 2 | no | — |
| 256 | D4 | — | 30 | 1 | no | — |
| 257 | C2×C4 | — | 3 | 1 | no | — |
| 258 | C2×C4 | — | 3 | 1 | no | — |
| 259 | C2×C4 | — | 5 | 2 | no | — |
| 260 | C2×C4 | — | 6 | 2 | no | — |
| 261 | C2×C4 | — | 15 | 2 | no | — |
| 262 | C2×C4 | — | 30 | 1 | no | — |
| 263 | C2×C4 | — | 60 | 4 | no | — |
| 264 | C2×C4 | — | 60 | 2 | no | — |
| 265 | C2×C4 | D5×S4 | 1 | 1 | yes | characteristic |
| 266 | C8 | — | 6 | 2 | no | — |
| 267 | C8 | — | 15 | 2 | no | — |
| 268 | C8 | — | 30 | 1 | no | — |
| 269 | C2×C3 | — | 4 | 1 | no | — |
| 270 | C2×C3 | — | 4 | 1 | no | — |
| 271 | C2×C3 | — | 4 | 1 | no | — |
| 272 | D3 | — | 20 | 4 | no | — |
| 273 | C5 | Q8×GL(2,𝔽3) | 1 | 1 | yes | characteristic, complemented, Sylow |
| 274 | C22 | — | 30 | 2 | no | — |
| 275 | C22 | — | 60 | 1 | no | — |
| 276 | C22 | — | 60 | 1 | no | — |
| 277 | C22 | C2×D5×S4 | 1 | 1 | yes | characteristic |
| 278 | C4 | — | 3 | 1 | no | — |
| 279 | C4 | — | 3 | 1 | no | — |
| 280 | C4 | — | 5 | 4 | no | — |
| 281 | C4 | — | 6 | 1 | no | — |
| 282 | C4 | — | 12 | 2 | no | — |
| 283 | C4 | — | 30 | 2 | no | — |
| 284 | C4 | — | 60 | 1 | no | — |
| 285 | C4 | D5×GL(2,𝔽3) | 1 | 2 | yes | — |
| 286 | C3 | — | 4 | 1 | no | Sylow |
| 287 | C2 | — | 60 | 2 | no | — |
| 288 | C2 | (C5⋊S4)⋊Q8 | 1 | 1 | yes | characteristic |
| 289 | C2 | C2×D5×GL(2,𝔽3) | 1 | 1 | yes | characteristic |
| 290 | C2 | #11756776 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. SL(2,𝔽3)×Dic10 | C2 | semidirect |
| 3. C5×#5386 | C2 | semidirect |
| 5. #11751608 | C2 | semidirect |
| 15. Q8×Dic10 | D3 | semidirect |
| 32. C5⋊GL(2,𝔽3) | Q8 | semidirect |
| 37. #5386 | D5 | semidirect |
| 73. C5×SL(2,𝔽3) | C2×Q8 | semidirect |
| 138. C5×Q8 | D3×Q8 | semidirect |
| 152. Dic10 | GL(2,𝔽3) | semidirect |
| 196. SL(2,𝔽3) | Q8×D5 | semidirect |
| 254. Q8 | D15⋊Q8 | semidirect |
| 273. C5 | Q8×GL(2,𝔽3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| SL(2,𝔽3)×Dic10 | 2 |
| C4×C5⋊GL(2,𝔽3) | 2 |
| C5×#5386 | 2 |
| #11751610 | 2 |
| #11751608 | 2 |
| #10612770 | 3 |
| C2×D15⋊Q8 | 4 |
| Q8×GL(2,𝔽3) | 5 |