Finite Group DatabasebetaFinite Group DatabasebetaOrder 1920.
574 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 574 conjugacy classes of subgroups — too many to draw. Only its 81 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×#91955 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C4×C5×#41⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #11752143 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C4×C5⋊GL(2,𝔽3) | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C4×C5⋊#226 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C4×#4096 | — | 3 | 1 | no | maximal, Hall |
| 7 | C2×C5×#41⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 8 | #91955 | C4 | 1 | 4 | yes | direct product factor |
| 9 | C2×C5⋊GL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 10 | C2×C5⋊#226 | C22 | 1 | 1 | yes | characteristic |
| 11 | C42×D15 | — | 4 | 1 | no | maximal |
| 12 | C4×C5×SL(2,𝔽3) | C22 | 1 | 2 | yes | — |
| 13 | #91184 | C22 | 1 | 2 | yes | — |
| 14 | C4×#985 | — | 5 | 1 | no | maximal, Hall |
| 15 | C4×C5×#41 | D3 | 1 | 1 | yes | characteristic, complemented |
| 16 | C2×#4096 | — | 3 | 1 | no | — |
| 17 | C4×#709 | — | 3 | 1 | no | — |
| 18 | #65522 | — | 3 | 1 | no | — |
| 19 | C4×#689 | — | 3 | 2 | no | — |
| 20 | C4×#688 | — | 3 | 2 | no | — |
| 21 | C4×#687 | — | 3 | 2 | no | — |
| 22 | C4×#686 | — | 3 | 2 | no | — |
| 23 | #65291 | — | 3 | 1 | no | — |
| 24 | #65287 | — | 3 | 1 | no | — |
| 25 | C2×C4×#168 | — | 3 | 1 | no | — |
| 26 | C2×C4×D15 | — | 4 | 1 | no | — |
| 27 | C2×C4×D15 | — | 4 | 2 | no | — |
| 28 | C5×#41⋊C3 | C2×C4 | 1 | 2 | yes | complemented |
| 29 | C2×C5×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 30 | C5⋊GL(2,𝔽3) | C2×C4 | 1 | 2 | yes | complemented |
| 31 | C5⋊#226 | C2×C4 | 1 | 2 | yes | complemented |
| 32 | C3×C42×C5 | — | 4 | 1 | no | — |
| 33 | C4×Dic15 | — | 4 | 2 | no | — |
| 34 | C4×Dic15 | — | 4 | 1 | no | — |
| 35 | C2×#985 | — | 5 | 1 | no | — |
| 36 | C4×#41⋊C3 | D5 | 1 | 1 | yes | characteristic, complemented |
| 37 | #5424 | — | 5 | 1 | no | — |
| 38 | C4×GL(2,𝔽3) | — | 5 | 2 | no | — |
| 39 | C4×#226 | — | 5 | 2 | no | — |
| 40 | C2×C5×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 41 | C2×#709 | — | 3 | 1 | no | — |
| 42 | C4×C5×Q8 | C2×D3 | 1 | 2 | yes | — |
| 43 | C4×C5×D4 | — | 3 | 2 | no | — |
| 44 | C5×#117 | — | 3 | 1 | no | — |
| 45 | C2×C42×C5 | — | 3 | 1 | no | — |
| 46 | #4096 | — | 3 | 4 | no | — |
| 47 | #4096 | — | 3 | 4 | no | — |
| 48 | C2×#689 | — | 3 | 1 | no | — |
| 49 | C2×#688 | — | 3 | 1 | no | — |
| 50 | C2×#687 | — | 3 | 1 | no | — |
| 51 | C2×#686 | — | 3 | 1 | no | — |
| 52 | C4×C22⋊D5 | — | 6 | 2 | no | — |
| 53 | #4072 | — | 3 | 1 | no | — |
| 54 | C22×#168 | — | 3 | 1 | no | — |
| 55 | C22×#168 | — | 3 | 1 | no | — |
| 56 | C4×D20 | — | 3 | 2 | no | — |
| 57 | #4018 | — | 6 | 1 | no | — |
| 58 | C42×D5 | — | 6 | 1 | no | — |
| 59 | C4×Dic10 | — | 3 | 2 | no | — |
| 60 | #3967 | — | 3 | 2 | no | — |
| 61 | #3964 | — | 3 | 2 | no | — |
| 62 | #3942 | — | 3 | 2 | no | — |
| 63 | #3941 | — | 3 | 2 | no | — |
| 64 | #3940 | — | 3 | 2 | no | — |
| 65 | #3939 | — | 3 | 2 | no | — |
| 66 | C4×#168 | — | 3 | 2 | no | — |
| 67 | C4×#168 | — | 3 | 2 | no | — |
| 68 | C4×#135 | — | 15 | 1 | no | Sylow |
| 69 | C22×D15 | — | 4 | 1 | no | — |
| 70 | C2×C3×C4×C5 | — | 4 | 1 | no | — |
| 71 | C2×C3×C4×C5 | — | 4 | 2 | no | — |
| 72 | C2×Dic15 | — | 4 | 1 | no | — |
| 73 | C2×Dic15 | — | 4 | 2 | no | — |
| 74 | C4×D15 | — | 4 | 4 | no | — |
| 75 | C4×D15 | — | 4 | 8 | no | — |
| 76 | C5×SL(2,𝔽3) | C22×C4 | 1 | 1 | yes | characteristic |
| 77 | C2×#41⋊C3 | C2×D5 | 1 | 1 | yes | characteristic |
| 78 | #985 | — | 5 | 4 | no | — |
| 79 | C2×GL(2,𝔽3) | — | 5 | 1 | no | — |
| 80 | C2×#226 | — | 5 | 1 | no | — |
| 81 | C42×D3 | — | 20 | 1 | no | — |
| 82 | C4×SL(2,𝔽3) | C2×D5 | 1 | 2 | yes | — |
| 83 | #859 | — | 5 | 2 | no | — |
| 84 | C5×#41 | — | 3 | 2 | no | — |
| 85 | C5×#41 | C4×D3 | 1 | 2 | yes | complemented |
| 86 | C2×C5×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 87 | C2×C5×D4 | — | 3 | 1 | no | — |
| 88 | C22×C4×C5 | — | 3 | 1 | no | — |
| 89 | C22×C4×C5 | — | 3 | 2 | no | — |
| 90 | C2×C22⋊D5 | — | 6 | 1 | no | — |
| 91 | #709 | — | 3 | 4 | no | — |
| 92 | #709 | — | 6 | 2 | no | — |
| 93 | C2×D20 | — | 3 | 1 | no | — |
| 94 | C2×C4×D5 | — | 6 | 1 | no | — |
| 95 | C2×C4×D5 | — | 6 | 2 | no | — |
| 96 | C2×Dic10 | — | 3 | 1 | no | — |
| 97 | C5×C4⋊C4 | — | 3 | 2 | no | — |
| 98 | C5×C22⋊C4 | — | 6 | 1 | no | — |
| 99 | C42×C5 | — | 3 | 1 | no | — |
| 100 | C42×C5 | — | 3 | 2 | no | — |
| 101 | C42×C5 | S4 | 1 | 1 | yes | characteristic |
| 102 | C22⋊Dic5 | — | 6 | 1 | no | — |
| 103 | #689 | — | 3 | 2 | no | — |
| 104 | #689 | — | 3 | 2 | no | — |
| 105 | #688 | — | 3 | 2 | no | — |
| 106 | #688 | — | 3 | 2 | no | — |
| 107 | #687 | — | 3 | 4 | no | — |
| 108 | #686 | — | 3 | 4 | no | — |
| 109 | #685 | — | 6 | 2 | no | — |
| 110 | C4⋊Dic5 | — | 3 | 2 | no | — |
| 111 | #683 | — | 6 | 2 | no | — |
| 112 | C4×Dic5 | — | 6 | 2 | no | — |
| 113 | C4×Dic5 | — | 6 | 1 | no | — |
| 114 | C2×#168 | — | 3 | 4 | no | — |
| 115 | C2×#168 | — | 3 | 1 | no | — |
| 116 | C2×#168 | — | 3 | 1 | no | — |
| 117 | C2×#168 | — | 3 | 2 | no | — |
| 118 | C2×#168 | — | 6 | 2 | no | — |
| 119 | C2×#135 | — | 15 | 1 | no | — |
| 120 | C4×#41 | — | 15 | 1 | no | — |
| 121 | C4×#41 | D15 | 1 | 1 | yes | characteristic, complemented |
| 122 | C4×Q16 | — | 15 | 2 | no | — |
| 123 | C4×SD16 | — | 15 | 2 | no | — |
| 124 | C4×SD16 | — | 15 | 2 | no | — |
| 125 | C4×D8 | — | 15 | 2 | no | — |
| 126 | #427 | — | 15 | 1 | no | — |
| 127 | #416 | — | 15 | 1 | no | — |
| 128 | #416 | — | 15 | 1 | no | — |
| 129 | C2×C4×C8 | — | 15 | 1 | no | — |
| 130 | C22×C3×C5 | — | 4 | 1 | no | — |
| 131 | C2×D15 | — | 4 | 2 | no | — |
| 132 | C2×D15 | — | 4 | 4 | no | — |
| 133 | C3×C4×C5 | — | 4 | 2 | no | — |
| 134 | C3×C4×C5 | — | 4 | 4 | no | — |
| 135 | Dic15 | — | 4 | 2 | no | — |
| 136 | Dic15 | — | 4 | 4 | no | — |
| 137 | C2×C4×D3 | — | 20 | 2 | no | — |
| 138 | C2×C4×D3 | — | 20 | 1 | no | — |
| 139 | #41⋊C3 | C4×D5 | 1 | 2 | yes | complemented |
| 140 | C2×SL(2,𝔽3) | C22×D5 | 1 | 1 | yes | characteristic |
| 141 | GL(2,𝔽3) | — | 5 | 2 | no | — |
| 142 | #226 | — | 5 | 2 | no | — |
| 143 | C3×C42 | — | 4 | 1 | no | — |
| 144 | C4×Dic3 | — | 20 | 2 | no | — |
| 145 | C4×Dic3 | — | 20 | 1 | no | — |
| 146 | C23×C5 | — | 3 | 1 | no | — |
| 147 | C22×D5 | — | 6 | 1 | no | — |
| 148 | C5×Q8 | — | 3 | 1 | no | — |
| 149 | C5×Q8 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 150 | C5×D4 | — | 3 | 2 | no | — |
| 151 | C5×D4 | — | 6 | 1 | no | — |
| 152 | C2×C4×C5 | — | 3 | 1 | no | — |
| 153 | C2×C4×C5 | — | 3 | 2 | no | — |
| 154 | C2×C4×C5 | — | 3 | 2 | no | — |
| 155 | C2×C4×C5 | — | 3 | 2 | no | — |
| 156 | C2×C4×C5 | — | 6 | 4 | no | — |
| 157 | C2×C4×C5 | C2×S4 | 1 | 1 | yes | characteristic |
| 158 | C2×C4×C5 | C2×S4 | 1 | 2 | yes | — |
| 159 | C22⋊D5 | — | 6 | 4 | no | — |
| 160 | C2×Dic5 | — | 6 | 1 | no | — |
| 161 | C2×Dic5 | — | 6 | 2 | no | — |
| 162 | D20 | — | 3 | 2 | no | — |
| 163 | D20 | — | 6 | 1 | no | — |
| 164 | C4×D5 | — | 6 | 4 | no | — |
| 165 | C4×D5 | — | 12 | 4 | no | — |
| 166 | Dic10 | — | 3 | 2 | no | — |
| 167 | Dic10 | — | 6 | 1 | no | — |
| 168 | #168 | — | 3 | 2 | no | — |
| 169 | #168 | — | 3 | 2 | no | — |
| 170 | #168 | — | 6 | 2 | no | — |
| 171 | C2×#41 | — | 15 | 1 | no | — |
| 172 | C2×#41 | C2×D15 | 1 | 1 | yes | characteristic |
| 173 | #135 | — | 15 | 4 | no | — |
| 174 | #135 | — | 15 | 4 | no | — |
| 175 | C2×Q16 | — | 15 | 1 | no | — |
| 176 | C2×SD16 | — | 15 | 1 | no | — |
| 177 | C2×SD16 | — | 15 | 1 | no | — |
| 178 | C2×D8 | — | 15 | 1 | no | — |
| 179 | C22×C8 | — | 15 | 1 | no | — |
| 180 | C22×C8 | — | 15 | 1 | no | — |
| 181 | C4×Q8 | — | 15 | 2 | no | — |
| 182 | C4×Q8 | C2×D15 | 1 | 2 | yes | — |
| 183 | C4×D4 | — | 3 | 2 | no | — |
| 184 | C4×D4 | — | 15 | 2 | no | — |
| 185 | C4×D4 | — | 30 | 2 | no | — |
| 186 | #117 | — | 3 | 1 | no | — |
| 187 | #117 | — | 15 | 1 | no | — |
| 188 | #117 | — | 30 | 1 | no | — |
| 189 | C2×C42 | — | 3 | 1 | no | — |
| 190 | C2×C42 | — | 30 | 1 | no | — |
| 191 | #107 | — | 15 | 2 | no | — |
| 192 | #106 | — | 15 | 2 | no | — |
| 193 | #103 | — | 15 | 2 | no | — |
| 194 | #103 | — | 15 | 2 | no | — |
| 195 | #102 | — | 15 | 2 | no | — |
| 196 | #102 | — | 15 | 2 | no | — |
| 197 | C4×C8 | — | 15 | 2 | no | — |
| 198 | C4×C8 | — | 15 | 2 | no | — |
| 199 | C2×C3×C5 | — | 4 | 1 | no | — |
| 200 | C2×C3×C5 | — | 4 | 2 | no | — |
| 201 | D15 | — | 4 | 4 | no | — |
| 202 | C22×D3 | — | 20 | 1 | no | — |
| 203 | C2×C3×C4 | — | 4 | 2 | no | — |
| 204 | C2×C3×C4 | — | 4 | 1 | no | — |
| 205 | C2×Dic3 | — | 20 | 2 | no | — |
| 206 | C2×Dic3 | — | 20 | 1 | no | — |
| 207 | C4×D3 | — | 20 | 4 | no | — |
| 208 | C4×D3 | — | 20 | 8 | no | — |
| 209 | SL(2,𝔽3) | C2×C4×D5 | 1 | 1 | yes | characteristic |
| 210 | C22×C5 | — | 3 | 2 | no | — |
| 211 | C22×C5 | — | 6 | 2 | no | — |
| 212 | C22×C5 | C22×S4 | 1 | 1 | yes | characteristic |
| 213 | C2×D5 | — | 6 | 2 | no | — |
| 214 | C2×D5 | — | 12 | 2 | no | — |
| 215 | C4×C5 | — | 3 | 1 | no | — |
| 216 | C4×C5 | — | 3 | 1 | no | — |
| 217 | C4×C5 | — | 6 | 2 | no | — |
| 218 | C4×C5 | C4×S4 | 1 | 2 | yes | — |
| 219 | C4×C5 | #985 | 1 | 4 | yes | complemented |
| 220 | Dic5 | — | 6 | 2 | no | — |
| 221 | Dic5 | — | 12 | 2 | no | — |
| 222 | #41 | — | 3 | 2 | no | — |
| 223 | #41 | — | 15 | 4 | no | — |
| 224 | #41 | — | 30 | 2 | no | — |
| 225 | #41 | C4×D15 | 1 | 2 | yes | complemented |
| 226 | C2×Q8 | — | 15 | 1 | no | — |
| 227 | C2×Q8 | C22×D15 | 1 | 1 | yes | characteristic |
| 228 | C2×D4 | — | 3 | 1 | no | — |
| 229 | C2×D4 | — | 15 | 1 | no | — |
| 230 | C2×D4 | — | 30 | 1 | no | — |
| 231 | C22×C4 | — | 3 | 1 | no | — |
| 232 | C22×C4 | — | 3 | 2 | no | — |
| 233 | C22×C4 | — | 30 | 1 | no | — |
| 234 | C22×C4 | — | 30 | 2 | no | — |
| 235 | Q16 | — | 15 | 2 | no | — |
| 236 | Q16 | — | 15 | 2 | no | — |
| 237 | SD16 | — | 15 | 4 | no | — |
| 238 | SD16 | — | 15 | 2 | no | — |
| 239 | SD16 | — | 15 | 2 | no | — |
| 240 | D8 | — | 15 | 4 | no | — |
| 241 | C2×C8 | — | 15 | 1 | no | — |
| 242 | C2×C8 | — | 15 | 1 | no | — |
| 243 | C2×C8 | — | 15 | 4 | no | — |
| 244 | C2×C8 | — | 15 | 2 | no | — |
| 245 | C2×C8 | — | 30 | 2 | no | — |
| 246 | C4⋊C4 | — | 3 | 2 | no | — |
| 247 | C4⋊C4 | — | 15 | 2 | no | — |
| 248 | C4⋊C4 | — | 30 | 2 | no | — |
| 249 | C22⋊C4 | — | 6 | 1 | no | — |
| 250 | C22⋊C4 | — | 30 | 2 | no | — |
| 251 | C22⋊C4 | — | 30 | 1 | no | — |
| 252 | C42 | — | 3 | 2 | no | — |
| 253 | C42 | — | 3 | 1 | no | — |
| 254 | C42 | — | 30 | 1 | no | — |
| 255 | C42 | — | 30 | 2 | no | — |
| 256 | C42 | C5⋊S4 | 1 | 1 | yes | characteristic |
| 257 | C3×C5 | — | 4 | 1 | no | Hall |
| 258 | C22×C3 | — | 4 | 1 | no | — |
| 259 | C2×D3 | — | 20 | 2 | no | — |
| 260 | C2×D3 | — | 20 | 4 | no | — |
| 261 | C3×C4 | — | 4 | 2 | no | — |
| 262 | C3×C4 | — | 4 | 4 | no | — |
| 263 | Dic3 | — | 20 | 2 | no | — |
| 264 | Dic3 | — | 20 | 4 | no | — |
| 265 | C2×C5 | — | 6 | 2 | no | — |
| 266 | C2×C5 | C2×C4×S4 | 1 | 1 | yes | characteristic |
| 267 | C2×C5 | C2×#985 | 1 | 2 | yes | — |
| 268 | D5 | — | 12 | 2 | no | — |
| 269 | C23 | — | 3 | 1 | no | — |
| 270 | C23 | — | 30 | 1 | no | — |
| 271 | Q8 | — | 3 | 1 | no | — |
| 272 | Q8 | — | 15 | 2 | no | — |
| 273 | Q8 | — | 30 | 1 | no | — |
| 274 | Q8 | C2×C4×D15 | 1 | 1 | yes | characteristic |
| 275 | D4 | — | 3 | 2 | no | — |
| 276 | D4 | — | 6 | 1 | no | — |
| 277 | D4 | — | 15 | 2 | no | — |
| 278 | D4 | — | 30 | 1 | no | — |
| 279 | D4 | — | 30 | 4 | no | — |
| 280 | C2×C4 | — | 3 | 1 | no | — |
| 281 | C2×C4 | — | 3 | 2 | no | — |
| 282 | C2×C4 | — | 3 | 2 | no | — |
| 283 | C2×C4 | — | 3 | 2 | no | — |
| 284 | C2×C4 | — | 6 | 4 | no | — |
| 285 | C2×C4 | — | 30 | 1 | no | — |
| 286 | C2×C4 | — | 30 | 4 | no | — |
| 287 | C2×C4 | — | 30 | 2 | no | — |
| 288 | C2×C4 | — | 60 | 4 | no | — |
| 289 | C2×C4 | C2×C5⋊S4 | 1 | 1 | yes | characteristic |
| 290 | C2×C4 | C2×C5⋊S4 | 1 | 2 | yes | — |
| 291 | C8 | — | 15 | 2 | no | — |
| 292 | C8 | — | 15 | 2 | no | — |
| 293 | C8 | — | 30 | 2 | no | — |
| 294 | C2×C3 | — | 4 | 1 | no | — |
| 295 | C2×C3 | — | 4 | 2 | no | — |
| 296 | D3 | — | 20 | 4 | no | — |
| 297 | C5 | C4×#985 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 298 | C22 | — | 3 | 2 | no | — |
| 299 | C22 | — | 6 | 2 | no | — |
| 300 | C22 | — | 30 | 2 | no | — |
| 301 | C22 | — | 60 | 2 | no | — |
| 302 | C22 | C22×C5⋊S4 | 1 | 1 | yes | characteristic |
| 303 | C4 | — | 3 | 1 | no | — |
| 304 | C4 | — | 3 | 1 | no | — |
| 305 | C4 | — | 6 | 2 | no | — |
| 306 | C4 | — | 30 | 2 | no | — |
| 307 | C4 | — | 60 | 2 | no | — |
| 308 | C4 | C4×C5⋊S4 | 1 | 2 | yes | — |
| 309 | C4 | #91955 | 1 | 4 | yes | direct product factor |
| 310 | C3 | — | 4 | 1 | no | Sylow |
| 311 | C2 | — | 6 | 2 | no | — |
| 312 | C2 | — | 60 | 2 | no | — |
| 313 | C2 | C2×C4×C5⋊S4 | 1 | 1 | yes | characteristic |
| 314 | C2 | C2×#91955 | 1 | 2 | yes | — |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C4×C5×#41⋊C3 | C2 | semidirect |
| 3. #11752143 | C2 | semidirect |
| 4. C4×C5⋊GL(2,𝔽3) | C2 | semidirect |
| 5. C4×C5⋊#226 | C2 | semidirect |
| 8. #91955 | C4 | direct (N × H) |
| 15. C4×C5×#41 | D3 | semidirect |
| 28. C5×#41⋊C3 | C2×C4 | semidirect |
| 30. C5⋊GL(2,𝔽3) | C2×C4 | semidirect |
| 31. C5⋊#226 | C2×C4 | semidirect |
| 36. C4×#41⋊C3 | D5 | semidirect |
| 85. C5×#41 | C4×D3 | semidirect |
| 121. C4×#41 | D15 | semidirect |
| 139. #41⋊C3 | C4×D5 | semidirect |
| 219. C4×C5 | #985 | semidirect |
| 225. #41 | C4×D15 | semidirect |
| 297. C5 | C4×#985 | semidirect |
| 309. C4 | #91955 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#91955 | 2 |
| C4×C5×#41⋊C3 | 2 |
| #11752143 | 2 |
| C4×C5⋊GL(2,𝔽3) | 2 |
| C4×C5⋊#226 | 2 |
| C4×#4096 | 3 |
| C42×D15 | 4 |
| C4×#985 | 5 |