Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
368 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 368 conjugacy classes of subgroups — too many to draw. Only its 51 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 | #5605 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3×#447 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #5223 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #5217 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #4537 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #4470 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #4468 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #1595 | — | 3 | 1 | no | maximal, Sylow |
| 9 | C22×C3×D4 | C22 | 1 | 1 | yes | characteristic, complemented |
| 10 | C22×#67 | — | 2 | 1 | no | — |
| 11 | C2×D3×D4 | — | 2 | 1 | no | — |
| 12 | C2×C3×D8 | — | 2 | 1 | no | — |
| 13 | C3×#121 | C22 | 1 | 1 | yes | characteristic |
| 14 | C2×#217 | — | 2 | 1 | no | — |
| 15 | #939 | — | 2 | 1 | no | — |
| 16 | #937 | — | 2 | 1 | no | — |
| 17 | C2×#213 | — | 2 | 1 | no | — |
| 18 | #930 | C22 | 1 | 1 | yes | characteristic |
| 19 | C2×(C2×D3)⋊C4 | C22 | 1 | 1 | yes | characteristic |
| 20 | #894 | — | 2 | 1 | no | — |
| 21 | #883 | — | 2 | 1 | no | — |
| 22 | #882 | — | 2 | 1 | no | — |
| 23 | C3×#102 | — | 2 | 1 | no | — |
| 24 | C3×C22⋊C8 | C22 | 1 | 1 | yes | characteristic, complemented |
| 25 | #832 | — | 2 | 1 | no | — |
| 26 | C3⋊#95 | C22 | 1 | 1 | yes | characteristic, complemented |
| 27 | #830 | C22 | 1 | 1 | yes | characteristic, complemented |
| 28 | #534 | — | 3 | 1 | no | — |
| 29 | #447 | — | 3 | 1 | no | — |
| 30 | #447 | D3 | 1 | 1 | yes | characteristic, complemented |
| 31 | #392 | — | 3 | 1 | no | — |
| 32 | #327 | — | 3 | 1 | no | — |
| 33 | #327 | — | 3 | 1 | no | — |
| 34 | #323 | — | 3 | 1 | no | — |
| 35 | C24×C3 | — | 2 | 1 | no | — |
| 36 | C23×D3 | — | 2 | 1 | no | — |
| 37 | C23×D3 | D4 | 1 | 1 | yes | characteristic |
| 38 | C2×C3×D4 | — | 2 | 1 | no | — |
| 39 | C2×C3×D4 | — | 2 | 1 | no | — |
| 40 | C2×C3×D4 | — | 2 | 1 | no | — |
| 41 | C2×C3×D4 | — | 4 | 2 | no | — |
| 42 | C2×C3×D4 | D4 | 1 | 1 | yes | characteristic, complemented |
| 43 | C22×C3×C4 | C23 | 1 | 1 | yes | characteristic |
| 44 | C2×#67 | — | 2 | 1 | no | — |
| 45 | C2×#67 | — | 2 | 1 | no | — |
| 46 | C2×#67 | — | 2 | 1 | no | — |
| 47 | C2×#67 | — | 4 | 2 | no | — |
| 48 | C2×#67 | — | 4 | 2 | no | — |
| 49 | C2×#67 | — | 4 | 2 | no | — |
| 50 | C22×Dic3 | — | 2 | 1 | no | — |
| 51 | C22×Dic3 | D4 | 1 | 1 | yes | characteristic |
| 52 | D3×D4 | — | 4 | 2 | no | — |
| 53 | D3×D4 | — | 4 | 2 | no | — |
| 54 | C2×D12 | — | 2 | 1 | no | — |
| 55 | C2×D12 | D4 | 1 | 1 | yes | characteristic, complemented |
| 56 | C2×C4×D3 | — | 2 | 1 | no | — |
| 57 | C3×D8 | — | 4 | 2 | no | — |
| 58 | C2×C3×C8 | — | 2 | 1 | no | — |
| 59 | C3×C4⋊C4 | D4 | 1 | 1 | yes | characteristic, complemented |
| 60 | C3×C22⋊C4 | — | 2 | 1 | no | — |
| 61 | #217 | — | 2 | 1 | no | — |
| 62 | #217 | — | 4 | 2 | no | — |
| 63 | #213 | — | 4 | 2 | no | — |
| 64 | (C2×D3)⋊C4 | — | 2 | 1 | no | — |
| 65 | (C2×D3)⋊C4 | — | 2 | 1 | no | — |
| 66 | (C2×D3)⋊C4 | — | 2 | 1 | no | — |
| 67 | (C2×D3)⋊C4 | — | 2 | 1 | no | — |
| 68 | C4⋊Dic3 | D4 | 1 | 1 | yes | characteristic, complemented |
| 69 | Dic3⋊C4 | — | 2 | 1 | no | — |
| 70 | C2×C3⋊C8 | — | 2 | 1 | no | — |
| 71 | C22×D4 | — | 6 | 1 | no | — |
| 72 | C22×D4 | — | 6 | 1 | no | — |
| 73 | C22×D4 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 74 | C2×D8 | — | 2 | 1 | no | — |
| 75 | C2×D8 | — | 6 | 1 | no | — |
| 76 | #123 | — | 6 | 1 | no | — |
| 77 | #123 | — | 6 | 1 | no | — |
| 78 | #121 | — | 3 | 1 | no | — |
| 79 | #121 | — | 6 | 1 | no | — |
| 80 | #121 | C2×D3 | 1 | 1 | yes | characteristic |
| 81 | #120 | — | 6 | 1 | no | — |
| 82 | #120 | — | 6 | 1 | no | — |
| 83 | C2×C22⋊C4 | — | 3 | 1 | no | — |
| 84 | C2×C22⋊C4 | — | 6 | 1 | no | — |
| 85 | #102 | — | 2 | 1 | no | — |
| 86 | #102 | — | 6 | 1 | no | — |
| 87 | C22⋊C8 | — | 3 | 1 | no | — |
| 88 | C22⋊C8 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 89 | #95 | — | 3 | 1 | no | — |
| 90 | C23×C3 | — | 2 | 2 | no | — |
| 91 | C23×C3 | — | 2 | 1 | no | — |
| 92 | C23×C3 | — | 4 | 1 | no | — |
| 93 | C23×C3 | — | 4 | 1 | no | — |
| 94 | C23×C3 | — | 4 | 1 | no | — |
| 95 | C23×C3 | — | 4 | 1 | no | — |
| 96 | C23×C3 | — | 4 | 1 | no | — |
| 97 | C23×C3 | — | 4 | 1 | no | — |
| 98 | C23×C3 | C2×D4 | 1 | 1 | yes | characteristic |
| 99 | C22×D3 | — | 2 | 1 | no | — |
| 100 | C22×D3 | — | 2 | 1 | no | — |
| 101 | C22×D3 | — | 2 | 1 | no | — |
| 102 | C22×D3 | — | 4 | 2 | no | — |
| 103 | C22×D3 | — | 4 | 1 | no | — |
| 104 | C22×D3 | — | 4 | 1 | no | — |
| 105 | C22×D3 | — | 8 | 1 | no | — |
| 106 | C22×D3 | — | 8 | 1 | no | — |
| 107 | C22×D3 | D8 | 1 | 2 | yes | complemented |
| 108 | C3×D4 | — | 2 | 2 | no | — |
| 109 | C3×D4 | — | 2 | 2 | no | — |
| 110 | C3×D4 | — | 4 | 1 | no | — |
| 111 | C3×D4 | — | 4 | 1 | no | — |
| 112 | C3×D4 | — | 4 | 1 | no | — |
| 113 | C3×D4 | — | 4 | 1 | no | — |
| 114 | C3×D4 | — | 4 | 1 | no | — |
| 115 | C2×C3×C4 | — | 2 | 1 | no | — |
| 116 | C2×C3×C4 | — | 4 | 1 | no | — |
| 117 | C2×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 118 | C2×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 119 | #67 | — | 4 | 2 | no | — |
| 120 | #67 | — | 4 | 8 | no | — |
| 121 | #67 | — | 4 | 1 | no | — |
| 122 | #67 | — | 4 | 1 | no | — |
| 123 | #67 | — | 8 | 1 | no | — |
| 124 | C2×Dic3 | — | 2 | 2 | no | — |
| 125 | C2×Dic3 | — | 2 | 1 | no | — |
| 126 | C2×Dic3 | — | 2 | 1 | no | — |
| 127 | C2×Dic3 | — | 4 | 1 | no | — |
| 128 | C2×Dic3 | — | 4 | 1 | no | — |
| 129 | C2×Dic3 | — | 4 | 1 | no | — |
| 130 | D12 | — | 4 | 1 | no | — |
| 131 | D12 | — | 4 | 2 | no | — |
| 132 | C4×D3 | — | 4 | 2 | no | — |
| 133 | C3×C8 | — | 4 | 1 | no | — |
| 134 | C3⋊C8 | — | 4 | 1 | no | — |
| 135 | C24 | — | 2 | 1 | no | — |
| 136 | C24 | — | 3 | 1 | no | — |
| 137 | C24 | — | 6 | 1 | no | — |
| 138 | C2×D4 | — | 2 | 1 | no | — |
| 139 | C2×D4 | — | 2 | 1 | no | — |
| 140 | C2×D4 | — | 2 | 1 | no | — |
| 141 | C2×D4 | — | 3 | 1 | no | — |
| 142 | C2×D4 | — | 4 | 2 | no | — |
| 143 | C2×D4 | — | 6 | 1 | no | — |
| 144 | C2×D4 | — | 6 | 1 | no | — |
| 145 | C2×D4 | — | 6 | 1 | no | — |
| 146 | C2×D4 | — | 6 | 1 | no | — |
| 147 | C2×D4 | — | 12 | 2 | no | — |
| 148 | C2×D4 | — | 12 | 2 | no | — |
| 149 | C2×D4 | — | 12 | 2 | no | — |
| 150 | C2×D4 | — | 12 | 2 | no | — |
| 151 | C2×D4 | — | 12 | 2 | no | — |
| 152 | C2×D4 | #67 | 1 | 1 | yes | characteristic, complemented |
| 153 | C22×C4 | — | 3 | 1 | no | — |
| 154 | C22×C4 | — | 6 | 1 | no | — |
| 155 | C22×C4 | — | 6 | 1 | no | — |
| 156 | C22×C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 157 | D8 | — | 4 | 2 | no | — |
| 158 | D8 | — | 12 | 2 | no | — |
| 159 | C2×C8 | — | 2 | 1 | no | — |
| 160 | C2×C8 | — | 6 | 1 | no | — |
| 161 | C4⋊C4 | — | 3 | 1 | no | — |
| 162 | C4⋊C4 | — | 6 | 1 | no | — |
| 163 | C4⋊C4 | #67 | 1 | 1 | yes | characteristic, complemented |
| 164 | C22⋊C4 | — | 2 | 1 | no | — |
| 165 | C22⋊C4 | — | 6 | 1 | no | — |
| 166 | C22⋊C4 | — | 6 | 1 | no | — |
| 167 | C22⋊C4 | — | 6 | 1 | no | — |
| 168 | C22⋊C4 | — | 6 | 1 | no | — |
| 169 | C22⋊C4 | — | 6 | 1 | no | — |
| 170 | C22⋊C4 | — | 12 | 2 | no | — |
| 171 | C22×C3 | — | 2 | 1 | no | — |
| 172 | C22×C3 | — | 2 | 1 | no | — |
| 173 | C22×C3 | — | 4 | 2 | no | — |
| 174 | C22×C3 | — | 4 | 2 | no | — |
| 175 | C22×C3 | — | 4 | 1 | no | — |
| 176 | C22×C3 | — | 4 | 1 | no | — |
| 177 | C22×C3 | — | 4 | 2 | no | — |
| 178 | C22×C3 | — | 4 | 1 | no | — |
| 179 | C22×C3 | — | 8 | 1 | no | — |
| 180 | C22×C3 | — | 8 | 1 | no | — |
| 181 | C22×C3 | — | 8 | 1 | no | — |
| 182 | C22×C3 | — | 8 | 1 | no | — |
| 183 | C22×C3 | #120 | 1 | 1 | yes | characteristic |
| 184 | C22×C3 | C2×D8 | 1 | 1 | yes | characteristic |
| 185 | C22×C3 | AGL(1,ℤ/8) | 1 | 1 | yes | characteristic |
| 186 | C2×D3 | — | 2 | 2 | no | — |
| 187 | C2×D3 | — | 4 | 1 | no | — |
| 188 | C2×D3 | — | 4 | 1 | no | — |
| 189 | C2×D3 | — | 4 | 2 | no | — |
| 190 | C2×D3 | — | 4 | 1 | no | — |
| 191 | C2×D3 | — | 4 | 1 | no | — |
| 192 | C2×D3 | — | 4 | 1 | no | — |
| 193 | C2×D3 | — | 8 | 1 | no | — |
| 194 | C2×D3 | — | 8 | 4 | no | — |
| 195 | C3×C4 | — | 2 | 1 | no | — |
| 196 | C3×C4 | — | 2 | 1 | no | — |
| 197 | C3×C4 | — | 4 | 1 | no | — |
| 198 | Dic3 | — | 4 | 2 | no | — |
| 199 | Dic3 | — | 4 | 1 | no | — |
| 200 | Dic3 | — | 4 | 1 | no | — |
| 201 | C23 | — | 2 | 2 | no | — |
| 202 | C23 | — | 2 | 1 | no | — |
| 203 | C23 | — | 3 | 2 | no | — |
| 204 | C23 | — | 4 | 1 | no | — |
| 205 | C23 | — | 4 | 1 | no | — |
| 206 | C23 | — | 4 | 1 | no | — |
| 207 | C23 | — | 4 | 1 | no | — |
| 208 | C23 | — | 4 | 1 | no | — |
| 209 | C23 | — | 4 | 1 | no | — |
| 210 | C23 | — | 6 | 1 | no | — |
| 211 | C23 | — | 6 | 1 | no | — |
| 212 | C23 | — | 6 | 1 | no | — |
| 213 | C23 | — | 12 | 1 | no | — |
| 214 | C23 | — | 12 | 1 | no | — |
| 215 | C23 | — | 12 | 2 | no | — |
| 216 | C23 | — | 24 | 1 | no | — |
| 217 | C23 | — | 24 | 1 | no | — |
| 218 | C23 | D3×D4 | 1 | 1 | yes | characteristic |
| 219 | D4 | — | 2 | 2 | no | — |
| 220 | D4 | — | 2 | 2 | no | — |
| 221 | D4 | — | 4 | 1 | no | — |
| 222 | D4 | — | 4 | 1 | no | — |
| 223 | D4 | — | 4 | 1 | no | — |
| 224 | D4 | — | 4 | 1 | no | — |
| 225 | D4 | — | 4 | 1 | no | — |
| 226 | D4 | — | 12 | 1 | no | — |
| 227 | D4 | — | 12 | 2 | no | — |
| 228 | D4 | — | 12 | 2 | no | — |
| 229 | D4 | — | 12 | 8 | no | — |
| 230 | D4 | — | 12 | 1 | no | — |
| 231 | D4 | — | 12 | 1 | no | — |
| 232 | D4 | — | 24 | 1 | no | — |
| 233 | C2×C4 | — | 2 | 1 | no | — |
| 234 | C2×C4 | — | 4 | 1 | no | — |
| 235 | C2×C4 | — | 6 | 2 | no | — |
| 236 | C2×C4 | — | 6 | 1 | no | — |
| 237 | C2×C4 | — | 6 | 1 | no | — |
| 238 | C2×C4 | — | 12 | 2 | no | — |
| 239 | C2×C4 | — | 12 | 1 | no | — |
| 240 | C2×C4 | — | 12 | 1 | no | — |
| 241 | C2×C4 | — | 12 | 1 | no | — |
| 242 | C2×C4 | D3×D4 | 1 | 1 | yes | characteristic |
| 243 | C2×C4 | C2×#67 | 1 | 1 | yes | characteristic |
| 244 | C8 | — | 4 | 1 | no | — |
| 245 | C8 | — | 12 | 1 | no | — |
| 246 | C2×C3 | — | 2 | 1 | no | — |
| 247 | C2×C3 | — | 2 | 1 | no | — |
| 248 | C2×C3 | — | 8 | 1 | no | — |
| 249 | C2×C3 | — | 8 | 2 | no | — |
| 250 | C2×C3 | #447 | 1 | 1 | yes | characteristic |
| 251 | C2×C3 | #453 | 1 | 1 | yes | characteristic |
| 252 | C2×C3 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 253 | D3 | — | 4 | 2 | no | — |
| 254 | D3 | — | 8 | 1 | no | — |
| 255 | C22 | — | 2 | 1 | no | — |
| 256 | C22 | — | 2 | 1 | no | — |
| 257 | C22 | — | 4 | 2 | no | — |
| 258 | C22 | — | 4 | 2 | no | — |
| 259 | C22 | — | 4 | 1 | no | — |
| 260 | C22 | — | 4 | 1 | no | — |
| 261 | C22 | — | 4 | 2 | no | — |
| 262 | C22 | — | 4 | 1 | no | — |
| 263 | C22 | — | 6 | 2 | no | — |
| 264 | C22 | — | 8 | 1 | no | — |
| 265 | C22 | — | 8 | 1 | no | — |
| 266 | C22 | — | 8 | 1 | no | — |
| 267 | C22 | — | 8 | 1 | no | — |
| 268 | C22 | — | 12 | 2 | no | — |
| 269 | C22 | — | 12 | 1 | no | — |
| 270 | C22 | — | 12 | 1 | no | — |
| 271 | C22 | — | 12 | 1 | no | — |
| 272 | C22 | — | 12 | 1 | no | — |
| 273 | C22 | — | 12 | 1 | no | — |
| 274 | C22 | — | 24 | 1 | no | — |
| 275 | C22 | — | 24 | 4 | no | — |
| 276 | C22 | D3×D8 | 1 | 1 | yes | characteristic |
| 277 | C22 | #911 | 1 | 1 | yes | characteristic |
| 278 | C22 | #937 | 1 | 1 | yes | characteristic |
| 279 | C4 | — | 2 | 1 | no | — |
| 280 | C4 | — | 2 | 1 | no | — |
| 281 | C4 | — | 4 | 1 | no | — |
| 282 | C4 | — | 12 | 2 | no | — |
| 283 | C4 | — | 12 | 1 | no | — |
| 284 | C4 | — | 12 | 1 | no | — |
| 285 | C3 | #1595 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 286 | C2 | — | 2 | 1 | no | — |
| 287 | C2 | — | 2 | 1 | no | — |
| 288 | C2 | — | 8 | 2 | no | — |
| 289 | C2 | — | 8 | 1 | no | — |
| 290 | C2 | — | 12 | 2 | no | — |
| 291 | C2 | — | 24 | 1 | no | — |
| 292 | C2 | #5032 | 1 | 1 | yes | characteristic |
| 293 | C2 | #5156 | 1 | 1 | yes | characteristic |
| 294 | C2 | #5198 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #937 |
| commutator, cosocle | C22×C3×C4 | C23 |
| Frattini | C22×C4 | C22×D3 |
| Fitting | C3×#447 | C2 |
| socle | C22×C3 | #120 |
| hypercenter | #447 | D3 |
| hypocenter | C3 | #1595 |
| LCS | G ▷ C22×C3×C4 ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #447 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #5605 | C2 | semidirect |
| 2. C3×#447 | C2 | semidirect |
| 3. #5223 | C2 | semidirect |
| 4. #5217 | C2 | semidirect |
| 5. #4537 | C2 | semidirect |
| 6. #4470 | C2 | semidirect |
| 7. #4468 | C2 | semidirect |
| 9. C22×C3×D4 | C22 | semidirect |
| 24. C3×C22⋊C8 | C22 | semidirect |
| 26. C3⋊#95 | C22 | semidirect |
| 27. #830 | C22 | semidirect |
| 30. #447 | D3 | semidirect |
| 42. C2×C3×D4 | D4 | semidirect |
| 55. C2×D12 | D4 | semidirect |
| 59. C3×C4⋊C4 | D4 | semidirect |
| 68. C4⋊Dic3 | D4 | semidirect |
| 73. C22×D4 | C2×D3 | semidirect |
| 88. C22⋊C8 | C2×D3 | semidirect |
| 107. C22×D3 | D8 | semidirect |
| 152. C2×D4 | #67 | semidirect |
| 163. C4⋊C4 | #67 | semidirect |
| 285. C3 | #1595 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5605 | 2 |
| C3×#447 | 2 |
| #5223 | 2 |
| #5217 | 2 |
| #4537 | 2 |
| #4470 | 2 |
| #4468 | 2 |
| #1595 | 3 |