Finite Group DatabasebetaFinite Group DatabasebetaOrder 1536.
346 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 346 conjugacy classes of subgroups — too many to draw. Only its 44 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×#73035 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #11711538 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | A4×M64 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #152637 | — | 3 | 1 | no | maximal, Sylow |
| 5 | C2×#5402 | — | 2 | 1 | no | — |
| 6 | C2×A4×C16 | C22 | 1 | 1 | yes | characteristic |
| 7 | #73105 | — | 2 | 1 | no | — |
| 8 | #73035 | — | 2 | 1 | no | — |
| 9 | #73035 | C4 | 1 | 2 | yes | — |
| 10 | A4×C32 | — | 2 | 1 | no | — |
| 11 | #67604 | — | 4 | 1 | no | maximal |
| 12 | C22×M64 | D3 | 1 | 1 | yes | characteristic, complemented |
| 13 | C2×#2215 | — | 3 | 1 | no | — |
| 14 | C2×#1419 | — | 3 | 1 | no | — |
| 15 | C2×#1419 | — | 3 | 1 | no | — |
| 16 | #12837 | — | 3 | 1 | no | — |
| 17 | #7478 | — | 3 | 1 | no | — |
| 18 | #7478 | — | 3 | 1 | no | — |
| 19 | C2×C4⋊S4 | — | 4 | 1 | no | — |
| 20 | C2×C8×A4 | D4 | 1 | 1 | yes | characteristic |
| 21 | #5409 | — | 4 | 1 | no | — |
| 22 | #5402 | — | 2 | 2 | no | — |
| 23 | #5402 | — | 4 | 1 | no | — |
| 24 | C2×D48 | — | 4 | 1 | no | — |
| 25 | A4×C16 | C2×C4 | 1 | 1 | yes | characteristic |
| 26 | A4×C16 | D4 | 1 | 1 | yes | characteristic |
| 27 | C3×M64 | — | 4 | 1 | no | — |
| 28 | #4506 | — | 4 | 1 | no | — |
| 29 | C22×D16 | — | 3 | 1 | no | — |
| 30 | C23×C16 | C2×D3 | 1 | 1 | yes | characteristic |
| 31 | C2×#466 | — | 6 | 1 | no | — |
| 32 | C2×M64 | — | 3 | 1 | no | — |
| 33 | C2×M64 | — | 3 | 1 | no | — |
| 34 | C2×M64 | — | 6 | 1 | no | — |
| 35 | C22×C32 | — | 2 | 1 | no | — |
| 36 | #2215 | — | 3 | 2 | no | — |
| 37 | #2215 | — | 3 | 2 | no | — |
| 38 | #2215 | — | 6 | 1 | no | — |
| 39 | #2215 | — | 6 | 1 | no | — |
| 40 | C2×#368 | — | 3 | 1 | no | — |
| 41 | C2×#368 | — | 3 | 1 | no | — |
| 42 | C2×#366 | — | 3 | 1 | no | — |
| 43 | C2×#357 | — | 6 | 1 | no | — |
| 44 | #1810 | — | 6 | 1 | no | — |
| 45 | #1419 | — | 6 | 2 | no | — |
| 46 | #1419 | — | 6 | 1 | no | — |
| 47 | #1419 | — | 6 | 1 | no | — |
| 48 | #1381 | — | 6 | 1 | no | — |
| 49 | C22×S4 | — | 8 | 1 | no | — |
| 50 | C2×C4×A4 | D8 | 1 | 1 | yes | characteristic |
| 51 | C4⋊S4 | — | 4 | 2 | no | — |
| 52 | C4⋊S4 | — | 8 | 1 | no | — |
| 53 | C2×D24 | — | 8 | 1 | no | — |
| 54 | C8×A4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 55 | C8×A4 | SD16 | 1 | 1 | yes | characteristic |
| 56 | A4⋊C8 | — | 8 | 1 | no | — |
| 57 | C2×C3×C16 | — | 4 | 1 | no | — |
| 58 | #819 | — | 8 | 1 | no | — |
| 59 | D48 | — | 4 | 2 | no | — |
| 60 | D48 | — | 8 | 1 | no | — |
| 61 | C3×C32 | — | 8 | 1 | no | — |
| 62 | C22×D8 | — | 6 | 1 | no | — |
| 63 | C23×C8 | D12 | 1 | 1 | yes | characteristic |
| 64 | C2×#121 | — | 12 | 1 | no | — |
| 65 | C2×D16 | — | 3 | 2 | no | — |
| 66 | C2×D16 | — | 6 | 1 | no | — |
| 67 | C2×D16 | — | 6 | 1 | no | — |
| 68 | C2×D16 | — | 6 | 1 | no | — |
| 69 | C2×D16 | — | 6 | 1 | no | — |
| 70 | C2×D16 | — | 6 | 1 | no | — |
| 71 | C22×C16 | — | 3 | 1 | no | — |
| 72 | C22×C16 | — | 3 | 1 | no | — |
| 73 | C22×C16 | — | 3 | 1 | no | — |
| 74 | C22×C16 | — | 3 | 1 | no | — |
| 75 | C22×C16 | C4×D3 | 1 | 1 | yes | characteristic |
| 76 | C22×C16 | #67 | 1 | 1 | yes | characteristic |
| 77 | #466 | — | 6 | 2 | no | — |
| 78 | #466 | — | 6 | 2 | no | — |
| 79 | #466 | — | 12 | 1 | no | — |
| 80 | #466 | — | 12 | 1 | no | — |
| 81 | C2×#108 | — | 6 | 1 | no | — |
| 82 | C2×#108 | — | 6 | 1 | no | — |
| 83 | C2×#107 | — | 6 | 1 | no | — |
| 84 | C2×#102 | — | 12 | 1 | no | — |
| 85 | #407 | — | 12 | 1 | no | — |
| 86 | M64 | — | 3 | 2 | no | — |
| 87 | M64 | — | 3 | 1 | no | — |
| 88 | M64 | — | 6 | 1 | no | — |
| 89 | M64 | S4 | 1 | 1 | yes | characteristic, complemented |
| 90 | C2×C32 | — | 6 | 1 | no | — |
| 91 | C2×C32 | — | 6 | 1 | no | — |
| 92 | #368 | — | 6 | 1 | no | — |
| 93 | #368 | — | 6 | 1 | no | — |
| 94 | #368 | — | 6 | 1 | no | — |
| 95 | #368 | — | 6 | 1 | no | — |
| 96 | #366 | — | 3 | 2 | no | — |
| 97 | #366 | — | 6 | 1 | no | — |
| 98 | #357 | — | 6 | 2 | no | — |
| 99 | #357 | — | 6 | 2 | no | — |
| 100 | #341 | — | 12 | 1 | no | — |
| 101 | C22×A4 | SD32 | 1 | 1 | yes | characteristic |
| 102 | C2×S4 | — | 8 | 2 | no | — |
| 103 | C2×S4 | — | 16 | 2 | no | — |
| 104 | C2×D12 | — | 16 | 1 | no | — |
| 105 | C4×A4 | #102 | 1 | 1 | yes | characteristic |
| 106 | C4×A4 | D16 | 1 | 1 | yes | characteristic |
| 107 | C2×C3×C8 | — | 4 | 1 | no | — |
| 108 | #208 | — | 16 | 1 | no | — |
| 109 | D24 | — | 8 | 2 | no | — |
| 110 | D24 | — | 16 | 1 | no | — |
| 111 | C3×C16 | — | 4 | 1 | no | — |
| 112 | C3×C16 | — | 4 | 1 | no | — |
| 113 | C22×D4 | — | 12 | 1 | no | — |
| 114 | C22×D4 | — | 24 | 1 | no | — |
| 115 | C23×C4 | D24 | 1 | 1 | yes | characteristic |
| 116 | C2×D8 | — | 6 | 2 | no | — |
| 117 | C2×D8 | — | 12 | 1 | no | — |
| 118 | C2×D8 | — | 12 | 1 | no | — |
| 119 | C2×D8 | — | 12 | 1 | no | — |
| 120 | C2×D8 | — | 12 | 1 | no | — |
| 121 | C2×D8 | — | 12 | 1 | no | — |
| 122 | C2×M16 | — | 12 | 1 | no | — |
| 123 | C2×M16 | — | 12 | 1 | no | — |
| 124 | C22×C8 | — | 3 | 1 | no | — |
| 125 | C22×C8 | — | 3 | 1 | no | — |
| 126 | C22×C8 | — | 3 | 1 | no | — |
| 127 | C22×C8 | — | 3 | 1 | no | — |
| 128 | C22×C8 | #204 | 1 | 1 | yes | characteristic |
| 129 | C22×C8 | (C2×D3)⋊C4 | 1 | 1 | yes | characteristic |
| 130 | #121 | — | 12 | 2 | no | — |
| 131 | #121 | — | 12 | 2 | no | — |
| 132 | #121 | — | 24 | 1 | no | — |
| 133 | #121 | — | 24 | 1 | no | — |
| 134 | C2×C4⋊C4 | — | 12 | 1 | no | — |
| 135 | C2×C22⋊C4 | — | 24 | 1 | no | — |
| 136 | D16 | — | 6 | 2 | no | — |
| 137 | D16 | — | 6 | 2 | no | — |
| 138 | D16 | — | 12 | 1 | no | — |
| 139 | D16 | — | 12 | 1 | no | — |
| 140 | C2×C16 | — | 3 | 1 | no | — |
| 141 | C2×C16 | — | 3 | 1 | no | — |
| 142 | C2×C16 | — | 3 | 1 | no | — |
| 143 | C2×C16 | — | 3 | 1 | no | — |
| 144 | C2×C16 | — | 3 | 1 | no | — |
| 145 | C2×C16 | — | 3 | 1 | no | — |
| 146 | C2×C16 | — | 3 | 1 | no | — |
| 147 | C2×C16 | — | 6 | 1 | no | — |
| 148 | C2×C16 | C2×S4 | 1 | 1 | yes | characteristic |
| 149 | #108 | — | 12 | 1 | no | — |
| 150 | #108 | — | 12 | 1 | no | — |
| 151 | #108 | — | 12 | 1 | no | — |
| 152 | #108 | — | 12 | 1 | no | — |
| 153 | #107 | — | 6 | 2 | no | — |
| 154 | #107 | — | 12 | 1 | no | — |
| 155 | #102 | — | 12 | 2 | no | — |
| 156 | #102 | — | 12 | 2 | no | — |
| 157 | C22⋊C8 | — | 24 | 1 | no | — |
| 158 | C22⋊C8 | — | 24 | 1 | no | — |
| 159 | C32 | — | 2 | 1 | no | — |
| 160 | C32 | — | 6 | 1 | no | — |
| 161 | C22×D3 | — | 32 | 1 | no | — |
| 162 | C2×A4 | — | 2 | 1 | no | — |
| 163 | C2×A4 | #357 | 1 | 1 | yes | characteristic |
| 164 | S4 | — | 16 | 2 | no | — |
| 165 | C2×C3×C4 | — | 4 | 1 | no | — |
| 166 | D12 | — | 16 | 2 | no | — |
| 167 | D12 | — | 32 | 1 | no | — |
| 168 | C3×C8 | — | 4 | 1 | no | — |
| 169 | C3×C8 | — | 4 | 1 | no | — |
| 170 | C3⋊C8 | — | 32 | 1 | no | — |
| 171 | C24 | — | 24 | 1 | no | — |
| 172 | C24 | #800 | 1 | 1 | yes | characteristic |
| 173 | C2×D4 | — | 12 | 2 | no | — |
| 174 | C2×D4 | — | 24 | 1 | no | — |
| 175 | C2×D4 | — | 24 | 1 | no | — |
| 176 | C2×D4 | — | 24 | 1 | no | — |
| 177 | C2×D4 | — | 24 | 1 | no | — |
| 178 | C2×D4 | — | 24 | 2 | no | — |
| 179 | C2×D4 | — | 24 | 2 | no | — |
| 180 | C2×D4 | — | 24 | 1 | no | — |
| 181 | C2×D4 | — | 48 | 2 | no | — |
| 182 | C2×D4 | — | 48 | 2 | no | — |
| 183 | C22×C4 | — | 3 | 1 | no | — |
| 184 | C22×C4 | — | 3 | 1 | no | — |
| 185 | C22×C4 | — | 3 | 1 | no | — |
| 186 | C22×C4 | — | 3 | 1 | no | — |
| 187 | C22×C4 | — | 24 | 1 | no | — |
| 188 | C22×C4 | D48 | 1 | 1 | yes | characteristic |
| 189 | C22×C4 | #821 | 1 | 1 | yes | characteristic |
| 190 | D8 | — | 12 | 2 | no | — |
| 191 | D8 | — | 12 | 2 | no | — |
| 192 | D8 | — | 24 | 1 | no | — |
| 193 | D8 | — | 24 | 1 | no | — |
| 194 | M16 | — | 24 | 1 | no | — |
| 195 | M16 | — | 24 | 1 | no | — |
| 196 | M16 | — | 24 | 1 | no | — |
| 197 | M16 | — | 24 | 1 | no | — |
| 198 | C2×C8 | — | 3 | 1 | no | — |
| 199 | C2×C8 | — | 3 | 1 | no | — |
| 200 | C2×C8 | — | 3 | 1 | no | — |
| 201 | C2×C8 | — | 3 | 1 | no | — |
| 202 | C2×C8 | — | 3 | 1 | no | — |
| 203 | C2×C8 | — | 3 | 1 | no | — |
| 204 | C2×C8 | — | 3 | 1 | no | — |
| 205 | C2×C8 | — | 6 | 1 | no | — |
| 206 | C2×C8 | — | 24 | 1 | no | — |
| 207 | C2×C8 | — | 24 | 1 | no | — |
| 208 | C2×C8 | C4⋊S4 | 1 | 1 | yes | characteristic |
| 209 | C4⋊C4 | — | 12 | 2 | no | — |
| 210 | C4⋊C4 | — | 24 | 1 | no | — |
| 211 | C22⋊C4 | — | 24 | 2 | no | — |
| 212 | C22⋊C4 | — | 24 | 2 | no | — |
| 213 | C16 | — | 3 | 1 | no | — |
| 214 | C16 | — | 3 | 1 | no | — |
| 215 | C16 | C4×S4 | 1 | 1 | yes | characteristic |
| 216 | C16 | GL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 217 | C22×C3 | — | 4 | 1 | no | — |
| 218 | C2×D3 | — | 32 | 2 | no | — |
| 219 | C2×D3 | — | 64 | 2 | no | — |
| 220 | A4 | #1419 | 1 | 1 | yes | characteristic, complemented |
| 221 | C3×C4 | — | 4 | 1 | no | — |
| 222 | C3×C4 | — | 4 | 1 | no | — |
| 223 | C23 | — | 2 | 1 | no | — |
| 224 | C23 | — | 3 | 1 | no | — |
| 225 | C23 | — | 3 | 1 | no | — |
| 226 | C23 | — | 6 | 1 | no | — |
| 227 | C23 | — | 24 | 2 | no | — |
| 228 | C23 | — | 48 | 1 | no | — |
| 229 | C23 | — | 48 | 2 | no | — |
| 230 | C23 | — | 48 | 2 | no | — |
| 231 | C23 | — | 48 | 1 | no | — |
| 232 | C23 | #4509 | 1 | 1 | yes | characteristic |
| 233 | D4 | — | 24 | 2 | no | — |
| 234 | D4 | — | 24 | 2 | no | — |
| 235 | D4 | — | 48 | 1 | no | — |
| 236 | D4 | — | 48 | 2 | no | — |
| 237 | D4 | — | 48 | 4 | no | — |
| 238 | D4 | — | 48 | 2 | no | — |
| 239 | D4 | — | 48 | 1 | no | — |
| 240 | C2×C4 | — | 3 | 1 | no | — |
| 241 | C2×C4 | — | 3 | 1 | no | — |
| 242 | C2×C4 | — | 3 | 1 | no | — |
| 243 | C2×C4 | — | 3 | 1 | no | — |
| 244 | C2×C4 | — | 3 | 1 | no | — |
| 245 | C2×C4 | — | 3 | 1 | no | — |
| 246 | C2×C4 | — | 3 | 1 | no | — |
| 247 | C2×C4 | — | 6 | 1 | no | — |
| 248 | C2×C4 | — | 24 | 2 | no | — |
| 249 | C2×C4 | — | 48 | 2 | no | — |
| 250 | C2×C4 | #5402 | 1 | 1 | yes | characteristic |
| 251 | C8 | — | 3 | 1 | no | — |
| 252 | C8 | — | 3 | 1 | no | — |
| 253 | C8 | — | 48 | 1 | no | — |
| 254 | C8 | — | 48 | 1 | no | — |
| 255 | C8 | #5401 | 1 | 1 | yes | characteristic |
| 256 | C8 | (C2×S4)⋊C4 | 1 | 1 | yes | characteristic |
| 257 | C2×C3 | — | 4 | 1 | no | — |
| 258 | C2×C3 | — | 8 | 1 | no | — |
| 259 | D3 | — | 64 | 2 | no | — |
| 260 | C22 | — | 3 | 1 | no | — |
| 261 | C22 | — | 3 | 1 | no | — |
| 262 | C22 | — | 3 | 1 | no | — |
| 263 | C22 | — | 6 | 1 | no | — |
| 264 | C22 | — | 6 | 1 | no | — |
| 265 | C22 | — | 6 | 1 | no | — |
| 266 | C22 | — | 6 | 1 | no | — |
| 267 | C22 | — | 48 | 2 | no | — |
| 268 | C22 | — | 48 | 2 | no | — |
| 269 | C22 | — | 48 | 2 | no | — |
| 270 | C22 | — | 96 | 2 | no | — |
| 271 | C22 | — | 96 | 2 | no | — |
| 272 | C22 | #67604 | 1 | 1 | yes | characteristic, complemented |
| 273 | C22 | #73036 | 1 | 1 | yes | characteristic |
| 274 | C4 | — | 3 | 1 | no | — |
| 275 | C4 | — | 3 | 1 | no | — |
| 276 | C4 | — | 48 | 2 | no | — |
| 277 | C4 | #73035 | 1 | 1 | yes | characteristic |
| 278 | C4 | #73113 | 1 | 1 | yes | characteristic |
| 279 | C3 | — | 4 | 1 | no | Sylow |
| 280 | C2 | — | 2 | 1 | no | — |
| 281 | C2 | — | 3 | 1 | no | — |
| 282 | C2 | — | 3 | 1 | no | — |
| 283 | C2 | — | 6 | 1 | no | — |
| 284 | C2 | — | 96 | 2 | no | — |
| 285 | C2 | #11711598 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #11711538 | C2 | semidirect |
| 3. A4×M64 | C2 | semidirect |
| 12. C22×M64 | D3 | semidirect |
| 89. M64 | S4 | semidirect |
| 220. A4 | #1419 | semidirect |
| 272. C22 | #67604 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#73035 | 2 |
| #11711538 | 2 |
| A4×M64 | 2 |
| #152637 | 3 |
| #67604 | 4 |