Finite Group DatabasebetaFinite Group DatabasebetaOrder 1440.
624 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 624 conjugacy classes of subgroups — too many to draw. Only its 140 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 | #10624222 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #10624197 | C2 | 1 | 2 | yes | maximal, complemented |
| 3 | C5×#3798 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #10624152 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C3×D3×Dic10 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #10624051 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #10624048 | C2 | 1 | 2 | yes | maximal, complemented |
| 8 | #10624040 | C2 | 1 | 2 | yes | maximal, complemented |
| 9 | #92004 | — | 3 | 2 | no | maximal |
| 10 | C4⋊(C3⋊D15) | C22 | 1 | 1 | yes | characteristic |
| 11 | C4×C5×C3⋊D3 | C22 | 1 | 1 | yes | characteristic |
| 12 | C3×Dic30 | C22 | 1 | 2 | yes | — |
| 13 | C3×C4×C5×D3 | C22 | 1 | 2 | yes | — |
| 14 | C32×Dic10 | C22 | 1 | 1 | yes | characteristic |
| 15 | #67158 | C22 | 1 | 2 | yes | — |
| 16 | #67155 | C22 | 1 | 2 | yes | — |
| 17 | #67152 | C22 | 1 | 4 | yes | — |
| 18 | #67149 | C22 | 1 | 4 | yes | — |
| 19 | C5×#627 | C22 | 1 | 1 | yes | characteristic |
| 20 | C5×#626 | C22 | 1 | 2 | yes | — |
| 21 | C5×#625 | C22 | 1 | 1 | yes | characteristic |
| 22 | #67140 | C22 | 1 | 2 | yes | — |
| 23 | #67137 | C22 | 1 | 2 | yes | — |
| 24 | C3×#1204 | C22 | 1 | 4 | yes | — |
| 25 | C3×D3×Dic5 | C22 | 1 | 4 | yes | — |
| 26 | #64316 | — | 5 | 1 | no | maximal, Hall |
| 27 | #6817 | — | 3 | 2 | no | — |
| 28 | C5×#235 | — | 3 | 2 | no | — |
| 29 | C2×C3×Dic10 | — | 3 | 2 | no | — |
| 30 | #6782 | — | 3 | 4 | no | — |
| 31 | #6782 | — | 3 | 4 | no | — |
| 32 | D15⋊Q8 | — | 3 | 4 | no | — |
| 33 | #6769 | — | 3 | 2 | no | — |
| 34 | #6769 | — | 3 | 2 | no | — |
| 35 | #6769 | — | 6 | 1 | no | — |
| 36 | #6768 | — | 3 | 4 | no | — |
| 37 | D3×Dic10 | — | 3 | 2 | no | — |
| 38 | D3×Dic10 | D3 | 1 | 2 | yes | complemented |
| 39 | C2×C3⋊D15 | C23 | 1 | 2 | yes | — |
| 40 | C2×C5×C3⋊D3 | C23 | 1 | 1 | yes | characteristic |
| 41 | C2×C3×C5×D3 | C23 | 1 | 2 | yes | — |
| 42 | C32×C4×C5 | C23 | 1 | 1 | yes | characteristic |
| 43 | C5×C3⋊Dic3 | C23 | 1 | 1 | yes | characteristic |
| 44 | C3×Dic15 | C23 | 1 | 4 | yes | — |
| 45 | C3×C5×Dic3 | C23 | 1 | 2 | yes | — |
| 46 | C32×Dic5 | C23 | 1 | 2 | yes | — |
| 47 | #4150 | — | 9 | 1 | no | Hall |
| 48 | #3832 | — | 5 | 1 | no | — |
| 49 | C3×D3×Q8 | — | 5 | 2 | no | — |
| 50 | #3799 | — | 5 | 2 | no | — |
| 51 | #3799 | — | 5 | 4 | no | — |
| 52 | #3798 | — | 5 | 2 | no | — |
| 53 | #3798 | D5 | 1 | 1 | yes | characteristic, complemented |
| 54 | #3797 | — | 5 | 1 | no | — |
| 55 | #3797 | — | 5 | 2 | no | — |
| 56 | C2×C3×C4×C5 | — | 3 | 2 | no | — |
| 57 | #1220 | — | 3 | 4 | no | — |
| 58 | D60 | — | 3 | 2 | no | — |
| 59 | D60 | — | 6 | 1 | no | — |
| 60 | C4×D15 | — | 3 | 4 | no | — |
| 61 | Dic30 | C2×D3 | 1 | 2 | yes | — |
| 62 | C5×#67 | — | 3 | 2 | no | — |
| 63 | C5×#67 | — | 3 | 2 | no | — |
| 64 | C5×D12 | — | 3 | 2 | no | — |
| 65 | C4×C5×D3 | — | 3 | 2 | no | — |
| 66 | C4×C5×D3 | — | 6 | 1 | no | — |
| 67 | C4×C5×D3 | C2×D3 | 1 | 2 | yes | — |
| 68 | C5×Dic6 | — | 3 | 2 | no | — |
| 69 | C2×C3×Dic5 | — | 3 | 4 | no | — |
| 70 | C3×Dic10 | — | 2 | 1 | no | — |
| 71 | C3×Dic10 | — | 3 | 2 | no | — |
| 72 | C3×Dic10 | — | 3 | 4 | no | — |
| 73 | C3×Dic10 | C2×D3 | 1 | 2 | yes | — |
| 74 | #1204 | — | 3 | 4 | no | — |
| 75 | #1204 | C2×D3 | 1 | 4 | yes | — |
| 76 | #1203 | — | 3 | 4 | no | — |
| 77 | #1203 | — | 3 | 4 | no | — |
| 78 | #1203 | — | 6 | 2 | no | — |
| 79 | #1200 | — | 3 | 4 | no | — |
| 80 | #1200 | — | 3 | 4 | no | — |
| 81 | #1200 | — | 6 | 2 | no | — |
| 82 | D3×Dic5 | — | 3 | 4 | no | — |
| 83 | D3×Dic5 | C2×D3 | 1 | 4 | yes | — |
| 84 | #1007 | — | 15 | 2 | no | — |
| 85 | C2×C32×C5 | C24 | 1 | 1 | yes | characteristic |
| 86 | C3⋊D15 | — | 2 | 2 | no | — |
| 87 | C5×C3⋊D3 | — | 2 | 1 | no | — |
| 88 | C3×C5×D3 | — | 2 | 2 | no | — |
| 89 | C5×#41 | — | 9 | 1 | no | — |
| 90 | Q8×D5 | — | 9 | 2 | no | — |
| 91 | #711 | — | 9 | 4 | no | — |
| 92 | #711 | — | 9 | 2 | no | — |
| 93 | #709 | — | 9 | 2 | no | — |
| 94 | #709 | — | 9 | 1 | no | — |
| 95 | C2×Dic10 | — | 3 | 2 | no | — |
| 96 | C2×Dic10 | — | 9 | 1 | no | — |
| 97 | C32×Q8 | — | 5 | 1 | no | — |
| 98 | C4⋊(C3⋊D3) | — | 5 | 2 | no | — |
| 99 | C4⋊(C3⋊D3) | — | 5 | 1 | no | — |
| 100 | C4×C3⋊D3 | — | 5 | 2 | no | — |
| 101 | C4×C3⋊D3 | C2×D5 | 1 | 1 | yes | characteristic |
| 102 | C3×C4×D3 | — | 5 | 4 | no | — |
| 103 | C3×C4×D3 | C2×D5 | 1 | 2 | yes | — |
| 104 | C3×Dic6 | — | 5 | 2 | no | — |
| 105 | C3×Dic6 | — | 5 | 4 | no | — |
| 106 | #627 | — | 5 | 2 | no | — |
| 107 | #627 | C2×D5 | 1 | 1 | yes | characteristic |
| 108 | #626 | — | 5 | 4 | no | — |
| 109 | #626 | C2×D5 | 1 | 2 | yes | — |
| 110 | #625 | C2×D5 | 1 | 1 | yes | characteristic |
| 111 | #624 | — | 5 | 2 | no | — |
| 112 | #624 | — | 5 | 4 | no | — |
| 113 | C22×C3×C5 | — | 3 | 2 | no | — |
| 114 | C2×D15 | — | 3 | 4 | no | — |
| 115 | C2×D15 | — | 6 | 2 | no | — |
| 116 | C2×C5×D3 | — | 3 | 2 | no | — |
| 117 | C2×C5×D3 | — | 6 | 1 | no | — |
| 118 | C2×C5×D3 | C22×D3 | 1 | 2 | yes | — |
| 119 | C3×C4×C5 | — | 2 | 1 | no | — |
| 120 | C3×C4×C5 | — | 3 | 2 | no | — |
| 121 | C3×C4×C5 | C22×D3 | 1 | 2 | yes | — |
| 122 | Dic15 | C22×D3 | 1 | 4 | yes | — |
| 123 | C3×Dic5 | — | 2 | 2 | no | — |
| 124 | C3×Dic5 | — | 3 | 4 | no | — |
| 125 | C3×Dic5 | C22×D3 | 1 | 4 | yes | — |
| 126 | C5×Dic3 | — | 3 | 2 | no | — |
| 127 | C5×Dic3 | — | 6 | 1 | no | — |
| 128 | C5×Dic3 | C22×D3 | 1 | 2 | yes | — |
| 129 | C2×C3×Q8 | — | 15 | 2 | no | — |
| 130 | #239 | — | 15 | 2 | no | — |
| 131 | #239 | — | 15 | 4 | no | — |
| 132 | #239 | — | 15 | 2 | no | — |
| 133 | #239 | — | 30 | 1 | no | — |
| 134 | D3×Q8 | — | 5 | 2 | no | — |
| 135 | D3×Q8 | — | 15 | 4 | no | — |
| 136 | D3×Q8 | — | 15 | 2 | no | — |
| 137 | #235 | — | 3 | 2 | no | — |
| 138 | #235 | — | 15 | 4 | no | — |
| 139 | #235 | — | 15 | 2 | no | — |
| 140 | #235 | — | 15 | 4 | no | — |
| 141 | C32×C5 | 2−1+4 | 1 | 1 | yes | characteristic, complemented, Hall |
| 142 | C5×Q8 | — | 9 | 1 | no | — |
| 143 | C5×D4 | — | 9 | 2 | no | — |
| 144 | C5×D4 | — | 9 | 1 | no | — |
| 145 | C2×C4×C5 | — | 3 | 2 | no | — |
| 146 | C2×C4×C5 | — | 9 | 1 | no | — |
| 147 | C22⋊D5 | — | 9 | 4 | no | — |
| 148 | C22⋊D5 | — | 9 | 2 | no | — |
| 149 | C2×Dic5 | — | 3 | 4 | no | — |
| 150 | C2×Dic5 | — | 9 | 2 | no | — |
| 151 | D20 | — | 9 | 1 | no | — |
| 152 | C4×D5 | — | 9 | 4 | no | — |
| 153 | C4×D5 | — | 9 | 2 | no | — |
| 154 | Dic10 | — | 3 | 2 | no | — |
| 155 | Dic10 | — | 3 | 4 | no | — |
| 156 | Dic10 | — | 9 | 2 | no | — |
| 157 | Dic10 | D32 | 1 | 1 | yes | characteristic |
| 158 | C2×C3⋊D3 | — | 5 | 2 | no | — |
| 159 | C2×C3⋊D3 | C22×D5 | 1 | 1 | yes | characteristic |
| 160 | C2×C3×D3 | C22×D5 | 1 | 2 | yes | — |
| 161 | C32×C4 | — | 5 | 2 | no | — |
| 162 | C32×C4 | C22×D5 | 1 | 1 | yes | characteristic |
| 163 | C3⋊Dic3 | C22×D5 | 1 | 1 | yes | characteristic |
| 164 | C3×Dic3 | — | 5 | 4 | no | — |
| 165 | C3×Dic3 | C22×D5 | 1 | 2 | yes | — |
| 166 | 2−1+4 | — | 45 | 1 | no | Sylow |
| 167 | C2×C3×C5 | — | 2 | 1 | no | — |
| 168 | C2×C3×C5 | — | 6 | 2 | no | — |
| 169 | C2×C3×C5 | C23×D3 | 1 | 2 | yes | — |
| 170 | D15 | — | 6 | 4 | no | — |
| 171 | D15 | — | 12 | 2 | no | — |
| 172 | C5×D3 | — | 2 | 2 | no | — |
| 173 | C5×D3 | — | 6 | 2 | no | — |
| 174 | C5×D3 | — | 12 | 1 | no | — |
| 175 | C3×Q8 | — | 5 | 2 | no | — |
| 176 | C3×Q8 | — | 10 | 1 | no | — |
| 177 | C3×Q8 | — | 15 | 4 | no | — |
| 178 | C3×Q8 | — | 15 | 2 | no | — |
| 179 | C2×C3×C4 | — | 3 | 2 | no | — |
| 180 | C2×C3×C4 | — | 15 | 4 | no | — |
| 181 | #67 | — | 3 | 2 | no | — |
| 182 | #67 | — | 3 | 2 | no | — |
| 183 | #67 | — | 15 | 4 | no | — |
| 184 | D12 | — | 3 | 2 | no | — |
| 185 | D12 | — | 15 | 4 | no | — |
| 186 | D12 | — | 15 | 4 | no | — |
| 187 | D12 | — | 15 | 2 | no | — |
| 188 | D12 | — | 30 | 1 | no | — |
| 189 | D12 | — | 30 | 2 | no | — |
| 190 | C4×D3 | — | 3 | 2 | no | — |
| 191 | C4×D3 | — | 5 | 4 | no | — |
| 192 | C4×D3 | — | 6 | 1 | no | — |
| 193 | C4×D3 | — | 15 | 4 | no | — |
| 194 | C4×D3 | — | 15 | 4 | no | — |
| 195 | C4×D3 | — | 15 | 4 | no | — |
| 196 | C4×D3 | — | 15 | 4 | no | — |
| 197 | C4×D3 | — | 30 | 2 | no | — |
| 198 | C4×D3 | D3×D5 | 1 | 2 | yes | — |
| 199 | Dic6 | — | 3 | 2 | no | — |
| 200 | Dic6 | — | 5 | 2 | no | — |
| 201 | Dic6 | — | 5 | 4 | no | — |
| 202 | Dic6 | — | 15 | 4 | no | — |
| 203 | C22×C5 | — | 3 | 2 | no | — |
| 204 | C22×C5 | — | 9 | 1 | no | — |
| 205 | C2×D5 | — | 9 | 2 | no | — |
| 206 | C4×C5 | — | 3 | 2 | no | — |
| 207 | C4×C5 | — | 9 | 1 | no | — |
| 208 | C4×C5 | C2×D32 | 1 | 1 | yes | characteristic |
| 209 | Dic5 | — | 3 | 4 | no | — |
| 210 | Dic5 | C2×D32 | 1 | 2 | yes | — |
| 211 | C2×C32 | C23×D5 | 1 | 1 | yes | characteristic |
| 212 | C3⋊D3 | — | 2 | 1 | no | — |
| 213 | C3⋊D3 | — | 10 | 2 | no | — |
| 214 | C3×D3 | — | 2 | 2 | no | — |
| 215 | #41 | — | 9 | 1 | no | — |
| 216 | #41 | — | 45 | 2 | no | — |
| 217 | #41 | — | 45 | 1 | no | — |
| 218 | #41 | — | 45 | 2 | no | — |
| 219 | #41 | — | 45 | 4 | no | — |
| 220 | C2×Q8 | — | 15 | 2 | no | — |
| 221 | C2×Q8 | — | 45 | 1 | no | — |
| 222 | C2×Q8 | — | 45 | 2 | no | — |
| 223 | C3×C5 | — | 2 | 1 | no | — |
| 224 | C3×C5 | #1007 | 1 | 2 | yes | complemented |
| 225 | C22×C3 | — | 3 | 2 | no | — |
| 226 | C2×D3 | — | 3 | 2 | no | — |
| 227 | C2×D3 | — | 6 | 1 | no | — |
| 228 | C2×D3 | — | 15 | 4 | no | — |
| 229 | C2×D3 | — | 30 | 2 | no | — |
| 230 | C2×D3 | C2×D3×D5 | 1 | 2 | yes | — |
| 231 | C3×C4 | — | 2 | 1 | no | — |
| 232 | C3×C4 | — | 3 | 2 | no | — |
| 233 | C3×C4 | — | 5 | 4 | no | — |
| 234 | C3×C4 | — | 10 | 2 | no | — |
| 235 | C3×C4 | — | 15 | 4 | no | — |
| 236 | C3×C4 | C2×D3×D5 | 1 | 2 | yes | — |
| 237 | Dic3 | — | 3 | 2 | no | — |
| 238 | Dic3 | — | 5 | 4 | no | — |
| 239 | Dic3 | — | 6 | 1 | no | — |
| 240 | Dic3 | C2×D3×D5 | 1 | 2 | yes | — |
| 241 | C2×C5 | — | 6 | 2 | no | — |
| 242 | C2×C5 | — | 18 | 1 | no | — |
| 243 | C2×C5 | C22×D32 | 1 | 1 | yes | characteristic |
| 244 | D5 | — | 18 | 2 | no | — |
| 245 | C32 | #4150 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 246 | Q8 | — | 5 | 1 | no | — |
| 247 | Q8 | — | 9 | 1 | no | — |
| 248 | Q8 | — | 15 | 4 | no | — |
| 249 | Q8 | — | 15 | 2 | no | — |
| 250 | Q8 | — | 45 | 2 | no | — |
| 251 | D4 | — | 9 | 2 | no | — |
| 252 | D4 | — | 9 | 1 | no | — |
| 253 | D4 | — | 45 | 2 | no | — |
| 254 | D4 | — | 45 | 4 | no | — |
| 255 | D4 | — | 45 | 1 | no | — |
| 256 | C2×C4 | — | 3 | 2 | no | — |
| 257 | C2×C4 | — | 9 | 1 | no | — |
| 258 | C2×C4 | — | 15 | 4 | no | — |
| 259 | C2×C4 | — | 45 | 4 | no | — |
| 260 | C2×C4 | — | 45 | 2 | no | — |
| 261 | C2×C4 | — | 45 | 2 | no | — |
| 262 | C2×C3 | — | 2 | 1 | no | — |
| 263 | C2×C3 | — | 6 | 2 | no | — |
| 264 | C2×C3 | C22×D3×D5 | 1 | 2 | yes | — |
| 265 | D3 | — | 2 | 2 | no | — |
| 266 | D3 | — | 6 | 2 | no | — |
| 267 | D3 | — | 12 | 1 | no | — |
| 268 | D3 | — | 30 | 4 | no | — |
| 269 | D3 | — | 60 | 2 | no | — |
| 270 | C5 | #64316 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 271 | C22 | — | 3 | 2 | no | — |
| 272 | C22 | — | 9 | 1 | no | — |
| 273 | C22 | — | 45 | 2 | no | — |
| 274 | C4 | — | 3 | 2 | no | — |
| 275 | C4 | — | 5 | 2 | no | — |
| 276 | C4 | — | 9 | 1 | no | — |
| 277 | C4 | — | 15 | 4 | no | — |
| 278 | C4 | D32×D5 | 1 | 1 | yes | characteristic |
| 279 | C3 | — | 2 | 1 | no | — |
| 280 | C3 | #92004 | 1 | 2 | yes | complemented |
| 281 | C2 | — | 6 | 2 | no | — |
| 282 | C2 | — | 18 | 1 | no | — |
| 283 | C2 | — | 90 | 2 | no | — |
| 284 | C2 | C2×D32×D5 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D32×D5 |
| commutator, socle, cosocle | C2×C32×C5 | C24 |
| Fitting | C32×C4×C5 | C23 |
| hypercenter | C4 | D32×D5 |
| hypocenter | C32×C5 | 2−1+4 |
| LCS | G ▷ C2×C32×C5 ▷ C32×C5 | |
| UCS | C1 ◁ C2 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10624222 | C2 | semidirect |
| 2. #10624197 | C2 | semidirect |
| 3. C5×#3798 | C2 | semidirect |
| 4. #10624152 | C2 | semidirect |
| 5. C3×D3×Dic10 | C2 | semidirect |
| 6. #10624051 | C2 | semidirect |
| 7. #10624048 | C2 | semidirect |
| 8. #10624040 | C2 | semidirect |
| 38. D3×Dic10 | D3 | semidirect |
| 53. #3798 | D5 | semidirect |
| 141. C32×C5 | 2−1+4 | semidirect |
| 224. C3×C5 | #1007 | semidirect |
| 245. C32 | #4150 | semidirect |
| 270. C5 | #64316 | semidirect |
| 280. C3 | #92004 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10624222 | 2 |
| #10624197 | 2 |
| C5×#3798 | 2 |
| #10624152 | 2 |
| C3×D3×Dic10 | 2 |
| #10624051 | 2 |
| #10624048 | 2 |
| #10624040 | 2 |
| #92004 | 3 |
| #64316 | 5 |