Finite Group DatabasebetaFinite Group DatabasebetaOrder 1536.
23800 conjugacy classes of subgroups; only the 198 normal subgroups are shown.
This group has 23800 conjugacy classes of subgroups — too many to draw. Only its 198 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 | C22×#5934 | C2 | 1 | 14 | yes | maximal, direct product factor |
| 2 | C23×#997 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C23×B(4,𝔽2) | — | 3 | 1 | no | maximal, Sylow |
| 4 | C24×S4 | — | 4 | 3 | no | maximal |
| 5 | C22×#997 | C22 | 1 | 7 | yes | complemented |
| 6 | C2×#5934 | C22 | 1 | 28 | yes | direct product factor |
| 7 | C23×H(5,𝔽2) | D3 | 1 | 1 | yes | characteristic, complemented |
| 8 | C23×#120 | — | 3 | 3 | no | — |
| 9 | C22×B(4,𝔽2) | — | 3 | 42 | no | — |
| 10 | C22×B(4,𝔽2) | — | 3 | 14 | no | — |
| 11 | C23×#99 | — | 3 | 3 | no | — |
| 12 | C24×A4 | — | 4 | 3 | no | — |
| 13 | C23×S4 | — | 4 | 42 | no | — |
| 14 | C23×S4 | — | 4 | 48 | no | — |
| 15 | C2×#997 | C23 | 1 | 7 | yes | complemented |
| 16 | C23×SL(2,𝔽3) | — | 8 | 1 | no | — |
| 17 | #5934 | C23 | 1 | 8 | yes | direct product factor |
| 18 | C27 | — | 3 | 1 | no | — |
| 19 | C22×H(5,𝔽2) | — | 3 | 21 | no | — |
| 20 | C22×H(5,𝔽2) | — | 6 | 7 | no | — |
| 21 | C22×H(5,𝔽2) | C2×D3 | 1 | 7 | yes | complemented |
| 22 | C23×#41 | — | 6 | 1 | no | — |
| 23 | C24×D4 | — | 3 | 3 | no | — |
| 24 | C24×D4 | — | 6 | 3 | no | — |
| 25 | C22×#120 | — | 3 | 42 | no | — |
| 26 | C22×#120 | — | 3 | 42 | no | — |
| 27 | C22×#120 | — | 6 | 21 | no | — |
| 28 | C22×#120 | — | 6 | 21 | no | — |
| 29 | C23×C22⋊C4 | — | 3 | 3 | no | — |
| 30 | C23×C22⋊C4 | — | 6 | 3 | no | — |
| 31 | C2×B(4,𝔽2) | — | 3 | 252 | no | — |
| 32 | C2×B(4,𝔽2) | — | 3 | 168 | no | — |
| 33 | C2×B(4,𝔽2) | — | 3 | 28 | no | — |
| 34 | C22×#99 | — | 3 | 42 | no | — |
| 35 | C22×#99 | — | 3 | 42 | no | — |
| 36 | C24×D3 | — | 16 | 1 | no | — |
| 37 | C23×A4 | — | 4 | 24 | no | — |
| 38 | C23×A4 | — | 4 | 21 | no | — |
| 39 | C22×S4 | — | 4 | 84 | no | — |
| 40 | C22×S4 | — | 4 | 336 | no | — |
| 41 | #997 | C24 | 1 | 1 | yes | characteristic, complemented |
| 42 | C22×SL(2,𝔽3) | — | 8 | 7 | no | — |
| 43 | C26 | — | 3 | 1 | no | — |
| 44 | C26 | — | 3 | 14 | no | — |
| 45 | C26 | — | 6 | 3 | no | — |
| 46 | C26 | — | 6 | 21 | no | — |
| 47 | C26 | — | 12 | 16 | no | — |
| 48 | C26 | S4 | 1 | 3 | yes | complemented |
| 49 | C2×H(5,𝔽2) | — | 3 | 63 | no | — |
| 50 | C2×H(5,𝔽2) | — | 3 | 42 | no | — |
| 51 | C2×H(5,𝔽2) | — | 6 | 21 | no | — |
| 52 | C2×H(5,𝔽2) | — | 6 | 21 | no | — |
| 53 | C2×H(5,𝔽2) | — | 6 | 126 | no | — |
| 54 | C2×H(5,𝔽2) | — | 6 | 63 | no | — |
| 55 | C2×H(5,𝔽2) | — | 6 | 14 | no | — |
| 56 | C2×H(5,𝔽2) | C22×D3 | 1 | 7 | yes | complemented |
| 57 | C22×#41 | — | 6 | 7 | no | — |
| 58 | C22×#41 | — | 6 | 21 | no | — |
| 59 | C22×#41 | — | 6 | 21 | no | — |
| 60 | C22×#41 | — | 6 | 7 | no | — |
| 61 | C23×Q8 | — | 2 | 1 | no | — |
| 62 | C23×D4 | — | 3 | 21 | no | — |
| 63 | C23×D4 | — | 3 | 21 | no | — |
| 64 | C23×D4 | — | 3 | 21 | no | — |
| 65 | C23×D4 | — | 3 | 21 | no | — |
| 66 | C23×D4 | — | 6 | 3 | no | — |
| 67 | C23×D4 | — | 6 | 42 | no | — |
| 68 | C23×D4 | — | 6 | 42 | no | — |
| 69 | C23×D4 | — | 6 | 42 | no | — |
| 70 | C23×D4 | — | 12 | 48 | no | — |
| 71 | C24×C4 | — | 3 | 3 | no | — |
| 72 | C24×C4 | — | 6 | 3 | no | — |
| 73 | C2×#120 | — | 3 | 252 | no | — |
| 74 | C2×#120 | — | 3 | 84 | no | — |
| 75 | C2×#120 | — | 6 | 252 | no | — |
| 76 | C2×#120 | — | 6 | 126 | no | — |
| 77 | C2×#120 | — | 6 | 126 | no | — |
| 78 | C22×C22⋊C4 | — | 3 | 42 | no | — |
| 79 | C22×C22⋊C4 | — | 6 | 42 | no | — |
| 80 | C22×C22⋊C4 | — | 6 | 42 | no | — |
| 81 | C22×C22⋊C4 | — | 6 | 21 | no | — |
| 82 | B(4,𝔽2) | — | 3 | 168 | no | — |
| 83 | B(4,𝔽2) | — | 3 | 336 | no | — |
| 84 | B(4,𝔽2) | — | 3 | 8 | no | — |
| 85 | C2×#99 | — | 3 | 252 | no | — |
| 86 | C2×#99 | — | 3 | 84 | no | — |
| 87 | C24×C3 | — | 16 | 1 | no | — |
| 88 | C23×D3 | — | 16 | 16 | no | — |
| 89 | C23×D3 | — | 16 | 14 | no | — |
| 90 | C22×A4 | — | 4 | 84 | no | — |
| 91 | C22×A4 | — | 4 | 21 | no | — |
| 92 | C2×S4 | — | 4 | 24 | no | — |
| 93 | C2×S4 | — | 4 | 336 | no | — |
| 94 | C2×SL(2,𝔽3) | — | 8 | 7 | no | — |
| 95 | C25 | — | 3 | 7 | no | — |
| 96 | C25 | — | 3 | 21 | no | — |
| 97 | C25 | — | 3 | 3 | no | — |
| 98 | C25 | — | 3 | 21 | no | — |
| 99 | C25 | — | 3 | 28 | no | — |
| 100 | C25 | — | 4 | 24 | no | — |
| 101 | C25 | — | 6 | 42 | no | — |
| 102 | C25 | — | 6 | 126 | no | — |
| 103 | C25 | — | 6 | 48 | no | — |
| 104 | C25 | — | 12 | 42 | no | — |
| 105 | C25 | — | 12 | 21 | no | — |
| 106 | C25 | — | 12 | 8 | no | — |
| 107 | C25 | — | 12 | 112 | no | — |
| 108 | C25 | — | 12 | 24 | no | — |
| 109 | C25 | — | 12 | 336 | no | — |
| 110 | C25 | — | 12 | 1 | no | — |
| 111 | C25 | C2×S4 | 1 | 21 | yes | complemented |
| 112 | H(5,𝔽2) | — | 3 | 21 | no | — |
| 113 | H(5,𝔽2) | — | 3 | 42 | no | — |
| 114 | H(5,𝔽2) | — | 6 | 7 | no | — |
| 115 | H(5,𝔽2) | — | 6 | 21 | no | — |
| 116 | H(5,𝔽2) | — | 6 | 126 | no | — |
| 117 | H(5,𝔽2) | — | 6 | 63 | no | — |
| 118 | H(5,𝔽2) | — | 6 | 252 | no | — |
| 119 | H(5,𝔽2) | — | 6 | 14 | no | — |
| 120 | H(5,𝔽2) | — | 6 | 168 | no | — |
| 121 | H(5,𝔽2) | C23×D3 | 1 | 1 | yes | characteristic, complemented |
| 122 | C2×#41 | — | 6 | 7 | no | — |
| 123 | C2×#41 | — | 6 | 63 | no | — |
| 124 | C2×#41 | — | 6 | 126 | no | — |
| 125 | C2×#41 | — | 6 | 63 | no | — |
| 126 | C2×#41 | — | 6 | 21 | no | — |
| 127 | C2×#41 | — | 6 | 42 | no | — |
| 128 | C2×#41 | — | 6 | 126 | no | — |
| 129 | C22×Q8 | — | 2 | 7 | no | — |
| 130 | C22×Q8 | — | 6 | 7 | no | — |
| 131 | C22×D4 | — | 3 | 63 | no | — |
| 132 | C22×D4 | — | 3 | 126 | no | — |
| 133 | C22×D4 | — | 3 | 63 | no | — |
| 134 | C22×D4 | — | 3 | 21 | no | — |
| 135 | C22×D4 | — | 3 | 63 | no | — |
| 136 | C22×D4 | — | 6 | 21 | no | — |
| 137 | C22×D4 | — | 6 | 21 | no | — |
| 138 | C22×D4 | — | 6 | 42 | no | — |
| 139 | C22×D4 | — | 6 | 126 | no | — |
| 140 | C22×D4 | — | 6 | 126 | no | — |
| 141 | C22×D4 | — | 6 | 126 | no | — |
| 142 | C22×D4 | — | 6 | 126 | no | — |
| 143 | C22×D4 | — | 6 | 252 | no | — |
| 144 | C22×D4 | — | 6 | 84 | no | — |
| 145 | C22×D4 | — | 12 | 336 | no | — |
| 146 | C22×D4 | — | 12 | 336 | no | — |
| 147 | C23×C4 | — | 3 | 21 | no | — |
| 148 | C23×C4 | — | 3 | 21 | no | — |
| 149 | C23×C4 | — | 6 | 21 | no | — |
| 150 | C23×C4 | — | 6 | 1 | no | — |
| 151 | C23×C4 | — | 6 | 42 | no | — |
| 152 | C23×C4 | — | 12 | 24 | no | — |
| 153 | #120 | — | 3 | 168 | no | — |
| 154 | #120 | — | 3 | 24 | no | — |
| 155 | #120 | — | 6 | 504 | no | — |
| 156 | #120 | — | 6 | 84 | no | — |
| 157 | #120 | — | 6 | 84 | no | — |
| 158 | C2×C22⋊C4 | — | 3 | 84 | no | — |
| 159 | C2×C22⋊C4 | — | 6 | 84 | no | — |
| 160 | C2×C22⋊C4 | — | 6 | 252 | no | — |
| 161 | C2×C22⋊C4 | — | 6 | 126 | no | — |
| 162 | #99 | — | 3 | 168 | no | — |
| 163 | #99 | — | 3 | 24 | no | — |
| 164 | C23×C3 | — | 16 | 8 | no | — |
| 165 | C23×C3 | — | 16 | 7 | no | — |
| 166 | C22×D3 | — | 16 | 28 | no | — |
| 167 | C22×D3 | — | 16 | 112 | no | — |
| 168 | C2×A4 | — | 4 | 42 | no | — |
| 169 | C2×A4 | — | 4 | 3 | no | — |
| 170 | S4 | — | 4 | 48 | no | — |
| 171 | SL(2,𝔽3) | — | 8 | 1 | no | — |
| 172 | C24 | — | 3 | 7 | no | — |
| 173 | C24 | — | 3 | 63 | no | — |
| 174 | C24 | — | 3 | 42 | no | — |
| 175 | C24 | — | 3 | 21 | no | — |
| 176 | C24 | — | 3 | 21 | no | — |
| 177 | C24 | — | 3 | 63 | no | — |
| 178 | C24 | — | 3 | 8 | no | — |
| 179 | C24 | — | 4 | 84 | no | — |
| 180 | C24 | — | 6 | 21 | no | — |
| 181 | C24 | — | 6 | 84 | no | — |
| 182 | C24 | — | 6 | 84 | no | — |
| 183 | C24 | — | 6 | 24 | no | — |
| 184 | C24 | — | 6 | 336 | no | — |
| 185 | C24 | — | 12 | 84 | no | — |
| 186 | C24 | — | 12 | 126 | no | — |
| 187 | C24 | — | 12 | 28 | no | — |
| 188 | C24 | — | 12 | 84 | no | — |
| 189 | C24 | — | 12 | 112 | no | — |
| 190 | C24 | — | 12 | 84 | no | — |
| 191 | C24 | — | 12 | 168 | no | — |
| 192 | C24 | — | 12 | 336 | no | — |
| 193 | C24 | — | 12 | 1008 | no | — |
| 194 | C24 | — | 12 | 672 | no | — |
| 195 | C24 | — | 12 | 16 | no | — |
| 196 | C24 | — | 12 | 14 | no | — |
| 197 | C24 | C22×S4 | 1 | 21 | yes | complemented |
| 198 | C24 | #1020 | 1 | 1 | yes | characteristic |
| 199 | #41 | — | 6 | 1 | no | — |
| 200 | #41 | — | 6 | 21 | no | — |
| 201 | #41 | — | 6 | 21 | no | — |
| 202 | #41 | — | 6 | 126 | no | — |
| 203 | #41 | — | 6 | 7 | no | — |
| 204 | #41 | — | 6 | 126 | no | — |
| 205 | #41 | — | 6 | 42 | no | — |
| 206 | #41 | — | 6 | 168 | no | — |
| 207 | C2×Q8 | — | 2 | 7 | no | — |
| 208 | C2×Q8 | — | 6 | 14 | no | — |
| 209 | C2×Q8 | — | 6 | 21 | no | — |
| 210 | C2×D4 | — | 3 | 21 | no | — |
| 211 | C2×D4 | — | 3 | 126 | no | — |
| 212 | C2×D4 | — | 3 | 21 | no | — |
| 213 | C2×D4 | — | 3 | 3 | no | — |
| 214 | C2×D4 | — | 3 | 21 | no | — |
| 215 | C2×D4 | — | 6 | 21 | no | — |
| 216 | C2×D4 | — | 6 | 63 | no | — |
| 217 | C2×D4 | — | 6 | 126 | no | — |
| 218 | C2×D4 | — | 6 | 126 | no | — |
| 219 | C2×D4 | — | 6 | 42 | no | — |
| 220 | C2×D4 | — | 6 | 126 | no | — |
| 221 | C2×D4 | — | 6 | 126 | no | — |
| 222 | C2×D4 | — | 6 | 126 | no | — |
| 223 | C2×D4 | — | 6 | 252 | no | — |
| 224 | C2×D4 | — | 6 | 168 | no | — |
| 225 | C2×D4 | — | 6 | 24 | no | — |
| 226 | C2×D4 | — | 12 | 336 | no | — |
| 227 | C2×D4 | — | 12 | 1008 | no | — |
| 228 | C22×C4 | — | 3 | 21 | no | — |
| 229 | C22×C4 | — | 3 | 63 | no | — |
| 230 | C22×C4 | — | 6 | 63 | no | — |
| 231 | C22×C4 | — | 6 | 63 | no | — |
| 232 | C22×C4 | — | 6 | 7 | no | — |
| 233 | C22×C4 | — | 6 | 7 | no | — |
| 234 | C22×C4 | — | 6 | 84 | no | — |
| 235 | C22×C4 | — | 12 | 168 | no | — |
| 236 | C22⋊C4 | — | 3 | 24 | no | — |
| 237 | C22⋊C4 | — | 6 | 24 | no | — |
| 238 | C22⋊C4 | — | 6 | 168 | no | — |
| 239 | C22⋊C4 | — | 6 | 84 | no | — |
| 240 | C22×C3 | — | 16 | 7 | no | — |
| 241 | C22×C3 | — | 16 | 28 | no | — |
| 242 | C2×D3 | — | 16 | 8 | no | — |
| 243 | C2×D3 | — | 16 | 112 | no | — |
| 244 | A4 | — | 4 | 3 | no | — |
| 245 | C23 | — | 3 | 1 | no | — |
| 246 | C23 | — | 3 | 21 | no | — |
| 247 | C23 | — | 3 | 42 | no | — |
| 248 | C23 | — | 3 | 21 | no | — |
| 249 | C23 | — | 3 | 63 | no | — |
| 250 | C23 | — | 3 | 21 | no | — |
| 251 | C23 | — | 4 | 42 | no | — |
| 252 | C23 | — | 6 | 21 | no | — |
| 253 | C23 | — | 6 | 24 | no | — |
| 254 | C23 | — | 6 | 84 | no | — |
| 255 | C23 | — | 6 | 84 | no | — |
| 256 | C23 | — | 6 | 336 | no | — |
| 257 | C23 | — | 12 | 84 | no | — |
| 258 | C23 | — | 12 | 14 | no | — |
| 259 | C23 | — | 12 | 126 | no | — |
| 260 | C23 | — | 12 | 84 | no | — |
| 261 | C23 | — | 12 | 16 | no | — |
| 262 | C23 | — | 12 | 126 | no | — |
| 263 | C23 | — | 12 | 168 | no | — |
| 264 | C23 | — | 12 | 1008 | no | — |
| 265 | C23 | — | 12 | 336 | no | — |
| 266 | C23 | — | 12 | 672 | no | — |
| 267 | C23 | — | 12 | 28 | no | — |
| 268 | C23 | — | 12 | 112 | no | — |
| 269 | C23 | — | 24 | 42 | no | — |
| 270 | C23 | #5934 | 1 | 8 | yes | direct product factor |
| 271 | C23 | C23×S4 | 1 | 3 | yes | complemented |
| 272 | C23 | C2×#1020 | 1 | 7 | yes | — |
| 273 | Q8 | — | 2 | 1 | no | — |
| 274 | Q8 | — | 6 | 7 | no | — |
| 275 | Q8 | — | 6 | 14 | no | — |
| 276 | D4 | — | 6 | 3 | no | — |
| 277 | D4 | — | 6 | 21 | no | — |
| 278 | D4 | — | 6 | 42 | no | — |
| 279 | D4 | — | 6 | 126 | no | — |
| 280 | D4 | — | 12 | 48 | no | — |
| 281 | D4 | — | 12 | 336 | no | — |
| 282 | C2×C4 | — | 3 | 3 | no | — |
| 283 | C2×C4 | — | 3 | 21 | no | — |
| 284 | C2×C4 | — | 6 | 21 | no | — |
| 285 | C2×C4 | — | 6 | 63 | no | — |
| 286 | C2×C4 | — | 6 | 7 | no | — |
| 287 | C2×C4 | — | 6 | 21 | no | — |
| 288 | C2×C4 | — | 6 | 24 | no | — |
| 289 | C2×C4 | — | 12 | 168 | no | — |
| 290 | C2×C3 | — | 16 | 1 | no | — |
| 291 | C2×C3 | — | 16 | 14 | no | — |
| 292 | D3 | — | 16 | 16 | no | — |
| 293 | C22 | — | 3 | 3 | no | — |
| 294 | C22 | — | 3 | 21 | no | — |
| 295 | C22 | — | 4 | 3 | no | — |
| 296 | C22 | — | 6 | 42 | no | — |
| 297 | C22 | — | 6 | 126 | no | — |
| 298 | C22 | — | 6 | 48 | no | — |
| 299 | C22 | — | 12 | 21 | no | — |
| 300 | C22 | — | 12 | 1 | no | — |
| 301 | C22 | — | 12 | 21 | no | — |
| 302 | C22 | — | 12 | 42 | no | — |
| 303 | C22 | — | 12 | 24 | no | — |
| 304 | C22 | — | 12 | 336 | no | — |
| 305 | C22 | — | 12 | 8 | no | — |
| 306 | C22 | — | 12 | 112 | no | — |
| 307 | C22 | — | 24 | 21 | no | — |
| 308 | C22 | C2×#5934 | 1 | 28 | yes | direct product factor |
| 309 | C22 | C22×#1020 | 1 | 7 | yes | — |
| 310 | C4 | — | 6 | 1 | no | — |
| 311 | C4 | — | 6 | 7 | no | — |
| 312 | C4 | — | 12 | 24 | no | — |
| 313 | C3 | — | 16 | 1 | no | Sylow |
| 314 | C2 | — | 6 | 3 | no | — |
| 315 | C2 | — | 6 | 21 | no | — |
| 316 | C2 | — | 12 | 16 | no | — |
| 317 | C2 | C22×#5934 | 1 | 14 | yes | direct product factor |
| 318 | C2 | C23×#1020 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C24 | #1020 |
| commutator, cosocle, hypocenter | #997 | C24 |
| Frattini | C2 | C23×#1020 |
| Fitting | C23×H(5,𝔽2) | D3 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C24 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#5934 | C2 | direct (N × H) |
| 2. C23×#997 | C2 | semidirect |
| 5. C22×#997 | C22 | semidirect |
| 6. C2×#5934 | C22 | direct (N × H) |
| 7. C23×H(5,𝔽2) | D3 | semidirect |
| 15. C2×#997 | C23 | semidirect |
| 17. #5934 | C23 | direct (N × H) |
| 21. C22×H(5,𝔽2) | C2×D3 | semidirect |
| 41. #997 | C24 | semidirect |
| 48. C26 | S4 | semidirect |
| 56. C2×H(5,𝔽2) | C22×D3 | semidirect |
| 111. C25 | C2×S4 | semidirect |
| 121. H(5,𝔽2) | C23×D3 | semidirect |
| 197. C24 | C22×S4 | semidirect |
| 270. C23 | #5934 | direct (N × H) |
| 271. C23 | C23×S4 | semidirect |
| 308. C22 | C2×#5934 | direct (N × H) |
| 317. C2 | C22×#5934 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#5934 | 2 |
| C23×#997 | 2 |
| C23×B(4,𝔽2) | 3 |
| C24×S4 | 4 |