Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
11151 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 11151 conjugacy classes of subgroups — too many to draw. Only its 134 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 | #87543 | C2 | 1 | 4 | yes | maximal, direct product factor |
| 2 | C2×#5948 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C23×H(5,𝔽2) | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 4 | #5948 | C22 | 1 | 6 | yes | complemented |
| 5 | C2×C22⋊SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 6 | C2×#994 | — | 4 | 4 | no | maximal |
| 7 | C2×Q8×A4 | — | 4 | 1 | no | maximal |
| 8 | C22×H(5,𝔽2) | — | 3 | 16 | no | — |
| 9 | C22×H(5,𝔽2) | — | 3 | 12 | no | — |
| 10 | C22×H(5,𝔽2) | — | 3 | 4 | no | — |
| 11 | C22×H(5,𝔽2) | — | 3 | 4 | no | — |
| 12 | C22×H(5,𝔽2) | C2×C3 | 1 | 4 | yes | complemented |
| 13 | C23×#41 | — | 3 | 1 | no | — |
| 14 | C23×#41 | C2×C3 | 1 | 3 | yes | — |
| 15 | C24×D4 | — | 3 | 3 | no | — |
| 16 | C22⋊SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic, complemented |
| 17 | #994 | — | 4 | 16 | no | — |
| 18 | C2×#41⋊C3 | — | 4 | 12 | no | — |
| 19 | Q8×A4 | — | 4 | 4 | no | — |
| 20 | C2×C4×A4 | — | 4 | 3 | no | — |
| 21 | C26 | — | 3 | 2 | no | — |
| 22 | C2×H(5,𝔽2) | — | 3 | 48 | no | — |
| 23 | C2×H(5,𝔽2) | — | 3 | 48 | no | — |
| 24 | C2×H(5,𝔽2) | — | 3 | 48 | no | — |
| 25 | C2×H(5,𝔽2) | — | 3 | 24 | no | — |
| 26 | C2×H(5,𝔽2) | — | 3 | 144 | no | — |
| 27 | C2×H(5,𝔽2) | — | 3 | 144 | no | — |
| 28 | C2×H(5,𝔽2) | — | 3 | 48 | no | — |
| 29 | C2×H(5,𝔽2) | — | 3 | 8 | no | — |
| 30 | C2×H(5,𝔽2) | — | 3 | 4 | no | — |
| 31 | C2×H(5,𝔽2) | — | 3 | 36 | no | — |
| 32 | C2×H(5,𝔽2) | — | 3 | 16 | no | — |
| 33 | C2×H(5,𝔽2) | — | 3 | 16 | no | — |
| 34 | C2×H(5,𝔽2) | — | 3 | 12 | no | — |
| 35 | C2×H(5,𝔽2) | A4 | 1 | 4 | yes | complemented |
| 36 | C22×#41 | — | 3 | 24 | no | — |
| 37 | C22×#41 | — | 3 | 12 | no | — |
| 38 | C22×#41 | — | 3 | 24 | no | — |
| 39 | C22×#41 | — | 3 | 18 | no | — |
| 40 | C22×#41 | — | 3 | 6 | no | — |
| 41 | C22×#41 | — | 3 | 12 | no | — |
| 42 | C22×#41 | — | 3 | 4 | no | — |
| 43 | C22×#41 | — | 3 | 4 | no | — |
| 44 | C22×#41 | — | 3 | 6 | no | — |
| 45 | C22×#41 | C22×C3 | 1 | 6 | yes | — |
| 46 | C23×Q8 | A4 | 1 | 1 | yes | characteristic |
| 47 | C23×Q8 | C22×C3 | 1 | 1 | yes | characteristic |
| 48 | C23×D4 | — | 3 | 3 | no | — |
| 49 | C23×D4 | — | 3 | 36 | no | — |
| 50 | C23×D4 | — | 3 | 72 | no | — |
| 51 | C23×D4 | — | 3 | 36 | no | — |
| 52 | C23×D4 | — | 3 | 12 | no | — |
| 53 | C23×D4 | — | 3 | 12 | no | — |
| 54 | C23×D4 | — | 3 | 3 | no | — |
| 55 | C24×C4 | — | 3 | 3 | no | — |
| 56 | C22×A4 | — | 4 | 1 | no | — |
| 57 | C2×C3×Q8 | — | 16 | 1 | no | — |
| 58 | #41⋊C3 | — | 4 | 24 | no | — |
| 59 | C2×SL(2,𝔽3) | — | 4 | 4 | no | — |
| 60 | C4×A4 | — | 4 | 6 | no | — |
| 61 | C25 | — | 3 | 24 | no | — |
| 62 | C25 | — | 3 | 24 | no | — |
| 63 | C25 | — | 3 | 8 | no | — |
| 64 | C25 | — | 3 | 3 | no | — |
| 65 | C25 | — | 12 | 12 | no | — |
| 66 | C25 | — | 12 | 4 | no | — |
| 67 | H(5,𝔽2) | — | 3 | 16 | no | — |
| 68 | H(5,𝔽2) | — | 3 | 96 | no | — |
| 69 | H(5,𝔽2) | — | 3 | 288 | no | — |
| 70 | H(5,𝔽2) | — | 3 | 96 | no | — |
| 71 | H(5,𝔽2) | — | 3 | 288 | no | — |
| 72 | H(5,𝔽2) | — | 3 | 144 | no | — |
| 73 | H(5,𝔽2) | — | 3 | 32 | no | — |
| 74 | H(5,𝔽2) | — | 3 | 48 | no | — |
| 75 | H(5,𝔽2) | — | 3 | 16 | no | — |
| 76 | H(5,𝔽2) | — | 3 | 144 | no | — |
| 77 | H(5,𝔽2) | — | 3 | 144 | no | — |
| 78 | H(5,𝔽2) | — | 3 | 48 | no | — |
| 79 | H(5,𝔽2) | C2×A4 | 1 | 16 | yes | complemented |
| 80 | C2×#41 | — | 3 | 24 | no | — |
| 81 | C2×#41 | — | 3 | 36 | no | — |
| 82 | C2×#41 | — | 3 | 72 | no | — |
| 83 | C2×#41 | — | 3 | 72 | no | — |
| 84 | C2×#41 | — | 3 | 72 | no | — |
| 85 | C2×#41 | — | 3 | 72 | no | — |
| 86 | C2×#41 | — | 3 | 144 | no | — |
| 87 | C2×#41 | — | 3 | 24 | no | — |
| 88 | C2×#41 | — | 3 | 72 | no | — |
| 89 | C2×#41 | — | 3 | 72 | no | — |
| 90 | C2×#41 | — | 3 | 24 | no | — |
| 91 | C2×#41 | — | 3 | 24 | no | — |
| 92 | C2×#41 | — | 3 | 24 | no | — |
| 93 | C2×#41 | — | 3 | 48 | no | — |
| 94 | C2×#41 | — | 3 | 36 | no | — |
| 95 | C2×#41 | — | 3 | 12 | no | — |
| 96 | C2×#41 | — | 3 | 24 | no | — |
| 97 | C2×#41 | — | 3 | 24 | no | — |
| 98 | C2×#41 | — | 3 | 12 | no | — |
| 99 | C2×#41 | — | 3 | 4 | no | — |
| 100 | C2×#41 | C2×A4 | 1 | 12 | yes | — |
| 101 | C22×Q8 | — | 3 | 3 | no | — |
| 102 | C22×Q8 | — | 3 | 4 | no | — |
| 103 | C22×Q8 | — | 3 | 4 | no | — |
| 104 | C22×Q8 | — | 3 | 4 | no | — |
| 105 | C22×Q8 | — | 3 | 1 | no | — |
| 106 | C22×Q8 | — | 3 | 1 | no | — |
| 107 | C22×Q8 | C2×A4 | 1 | 4 | yes | — |
| 108 | C22×Q8 | C23×C3 | 1 | 1 | yes | characteristic |
| 109 | C22×D4 | — | 3 | 6 | no | — |
| 110 | C22×D4 | — | 3 | 36 | no | — |
| 111 | C22×D4 | — | 3 | 36 | no | — |
| 112 | C22×D4 | — | 3 | 6 | no | — |
| 113 | C22×D4 | — | 3 | 18 | no | — |
| 114 | C22×D4 | — | 3 | 18 | no | — |
| 115 | C22×D4 | — | 3 | 36 | no | — |
| 116 | C22×D4 | — | 3 | 72 | no | — |
| 117 | C22×D4 | — | 3 | 288 | no | — |
| 118 | C22×D4 | — | 3 | 144 | no | — |
| 119 | C22×D4 | — | 3 | 144 | no | — |
| 120 | C22×D4 | — | 3 | 144 | no | — |
| 121 | C22×D4 | — | 3 | 72 | no | — |
| 122 | C22×D4 | — | 3 | 144 | no | — |
| 123 | C22×D4 | — | 3 | 144 | no | — |
| 124 | C22×D4 | — | 3 | 144 | no | — |
| 125 | C22×D4 | — | 3 | 36 | no | — |
| 126 | C22×D4 | — | 3 | 12 | no | — |
| 127 | C22×D4 | — | 3 | 12 | no | — |
| 128 | C23×C4 | — | 3 | 36 | no | — |
| 129 | C23×C4 | — | 3 | 18 | no | — |
| 130 | C23×C4 | — | 3 | 18 | no | — |
| 131 | C23×C4 | — | 3 | 1 | no | — |
| 132 | C23×C4 | — | 3 | 6 | no | — |
| 133 | C23×C4 | — | 3 | 6 | no | — |
| 134 | C23×C4 | C2×A4 | 1 | 3 | yes | — |
| 135 | C2×A4 | — | 4 | 2 | no | — |
| 136 | C2×A4 | — | 4 | 1 | no | — |
| 137 | C3×Q8 | — | 16 | 4 | no | — |
| 138 | C2×C3×C4 | — | 16 | 3 | no | — |
| 139 | SL(2,𝔽3) | — | 4 | 4 | no | — |
| 140 | C24 | — | 3 | 18 | no | — |
| 141 | C24 | — | 3 | 18 | no | — |
| 142 | C24 | — | 3 | 96 | no | — |
| 143 | C24 | — | 3 | 96 | no | — |
| 144 | C24 | — | 3 | 24 | no | — |
| 145 | C24 | — | 3 | 8 | no | — |
| 146 | C24 | — | 3 | 6 | no | — |
| 147 | C24 | — | 6 | 18 | no | — |
| 148 | C24 | — | 6 | 6 | no | — |
| 149 | C24 | — | 12 | 48 | no | — |
| 150 | C24 | — | 12 | 24 | no | — |
| 151 | C24 | — | 12 | 48 | no | — |
| 152 | C24 | — | 12 | 48 | no | — |
| 153 | C24 | — | 12 | 48 | no | — |
| 154 | C24 | — | 12 | 8 | no | — |
| 155 | C24 | C22×A4 | 1 | 1 | yes | characteristic |
| 156 | #41 | — | 3 | 72 | no | — |
| 157 | #41 | — | 3 | 72 | no | — |
| 158 | #41 | — | 3 | 72 | no | — |
| 159 | #41 | — | 3 | 96 | no | — |
| 160 | #41 | — | 3 | 144 | no | — |
| 161 | #41 | — | 3 | 96 | no | — |
| 162 | #41 | — | 3 | 144 | no | — |
| 163 | #41 | — | 3 | 144 | no | — |
| 164 | #41 | — | 3 | 72 | no | — |
| 165 | #41 | — | 3 | 24 | no | — |
| 166 | #41 | — | 3 | 24 | no | — |
| 167 | #41 | — | 3 | 16 | no | — |
| 168 | #41 | — | 3 | 16 | no | — |
| 169 | #41 | — | 3 | 24 | no | — |
| 170 | #41 | C22×A4 | 1 | 24 | yes | — |
| 171 | C2×Q8 | — | 3 | 12 | no | — |
| 172 | C2×Q8 | — | 3 | 12 | no | — |
| 173 | C2×Q8 | — | 3 | 12 | no | — |
| 174 | C2×Q8 | — | 3 | 12 | no | — |
| 175 | C2×Q8 | — | 3 | 4 | no | — |
| 176 | C2×Q8 | — | 3 | 2 | no | — |
| 177 | C2×Q8 | — | 3 | 4 | no | — |
| 178 | C2×Q8 | — | 3 | 4 | no | — |
| 179 | C2×Q8 | — | 3 | 4 | no | — |
| 180 | C2×Q8 | — | 3 | 4 | no | — |
| 181 | C2×Q8 | — | 3 | 3 | no | — |
| 182 | C2×Q8 | C22×A4 | 1 | 4 | yes | — |
| 183 | C2×Q8 | C22⋊A4 | 1 | 1 | yes | characteristic |
| 184 | C2×D4 | — | 3 | 12 | no | — |
| 185 | C2×D4 | — | 3 | 72 | no | — |
| 186 | C2×D4 | — | 3 | 72 | no | — |
| 187 | C2×D4 | — | 3 | 72 | no | — |
| 188 | C2×D4 | — | 3 | 72 | no | — |
| 189 | C2×D4 | — | 3 | 72 | no | — |
| 190 | C2×D4 | — | 3 | 36 | no | — |
| 191 | C2×D4 | — | 3 | 36 | no | — |
| 192 | C2×D4 | — | 3 | 72 | no | — |
| 193 | C2×D4 | — | 3 | 72 | no | — |
| 194 | C2×D4 | — | 3 | 72 | no | — |
| 195 | C2×D4 | — | 3 | 288 | no | — |
| 196 | C2×D4 | — | 3 | 288 | no | — |
| 197 | C2×D4 | — | 3 | 288 | no | — |
| 198 | C2×D4 | — | 3 | 288 | no | — |
| 199 | C2×D4 | — | 3 | 144 | no | — |
| 200 | C2×D4 | — | 3 | 48 | no | — |
| 201 | C2×D4 | — | 3 | 144 | no | — |
| 202 | C2×D4 | — | 3 | 48 | no | — |
| 203 | C2×D4 | — | 3 | 12 | no | — |
| 204 | C22×C4 | — | 3 | 6 | no | — |
| 205 | C22×C4 | — | 3 | 6 | no | — |
| 206 | C22×C4 | — | 3 | 72 | no | — |
| 207 | C22×C4 | — | 3 | 36 | no | — |
| 208 | C22×C4 | — | 3 | 36 | no | — |
| 209 | C22×C4 | — | 3 | 36 | no | — |
| 210 | C22×C4 | — | 3 | 36 | no | — |
| 211 | C22×C4 | — | 3 | 72 | no | — |
| 212 | C22×C4 | — | 3 | 36 | no | — |
| 213 | C22×C4 | — | 3 | 1 | no | — |
| 214 | C22×C4 | — | 3 | 1 | no | — |
| 215 | C22×C4 | — | 3 | 3 | no | — |
| 216 | C22×C4 | — | 3 | 12 | no | — |
| 217 | C22×C4 | — | 3 | 3 | no | — |
| 218 | C22×C4 | — | 3 | 6 | no | — |
| 219 | C22×C4 | C22×A4 | 1 | 6 | yes | — |
| 220 | C22×C3 | — | 16 | 1 | no | — |
| 221 | A4 | — | 4 | 1 | no | — |
| 222 | C3×C4 | — | 16 | 6 | no | — |
| 223 | C23 | — | 3 | 1 | no | — |
| 224 | C23 | — | 3 | 2 | no | — |
| 225 | C23 | — | 3 | 1 | no | — |
| 226 | C23 | — | 3 | 36 | no | — |
| 227 | C23 | — | 3 | 36 | no | — |
| 228 | C23 | — | 3 | 96 | no | — |
| 229 | C23 | — | 3 | 32 | no | — |
| 230 | C23 | — | 3 | 12 | no | — |
| 231 | C23 | — | 6 | 36 | no | — |
| 232 | C23 | — | 6 | 18 | no | — |
| 233 | C23 | — | 6 | 36 | no | — |
| 234 | C23 | — | 6 | 36 | no | — |
| 235 | C23 | — | 6 | 36 | no | — |
| 236 | C23 | — | 6 | 6 | no | — |
| 237 | C23 | — | 12 | 192 | no | — |
| 238 | C23 | — | 12 | 96 | no | — |
| 239 | C23 | — | 12 | 96 | no | — |
| 240 | C23 | — | 12 | 48 | no | — |
| 241 | C23 | — | 12 | 16 | no | — |
| 242 | C23 | #994 | 1 | 2 | yes | complemented |
| 243 | C23 | C23×A4 | 1 | 1 | yes | characteristic |
| 244 | Q8 | — | 3 | 4 | no | — |
| 245 | Q8 | — | 3 | 12 | no | — |
| 246 | Q8 | — | 3 | 4 | no | — |
| 247 | Q8 | — | 3 | 8 | no | — |
| 248 | Q8 | — | 3 | 12 | no | — |
| 249 | Q8 | C23×A4 | 1 | 4 | yes | — |
| 250 | Q8 | C2×C22⋊A4 | 1 | 4 | yes | — |
| 251 | D4 | — | 3 | 24 | no | — |
| 252 | D4 | — | 3 | 24 | no | — |
| 253 | D4 | — | 3 | 72 | no | — |
| 254 | D4 | — | 3 | 72 | no | — |
| 255 | D4 | — | 3 | 144 | no | — |
| 256 | D4 | — | 3 | 48 | no | — |
| 257 | C2×C4 | — | 3 | 6 | no | — |
| 258 | C2×C4 | — | 3 | 12 | no | — |
| 259 | C2×C4 | — | 3 | 6 | no | — |
| 260 | C2×C4 | — | 3 | 6 | no | — |
| 261 | C2×C4 | — | 3 | 72 | no | — |
| 262 | C2×C4 | — | 3 | 72 | no | — |
| 263 | C2×C4 | — | 3 | 4 | no | — |
| 264 | C2×C4 | — | 3 | 3 | no | — |
| 265 | C2×C4 | — | 3 | 6 | no | — |
| 266 | C2×C4 | — | 3 | 6 | no | — |
| 267 | C2×C4 | — | 3 | 24 | no | — |
| 268 | C2×C4 | — | 3 | 24 | no | — |
| 269 | C2×C4 | — | 3 | 6 | no | — |
| 270 | C2×C4 | C2×C22⋊A4 | 1 | 3 | yes | — |
| 271 | C2×C3 | — | 16 | 1 | no | — |
| 272 | C2×C3 | — | 16 | 2 | no | — |
| 273 | C22 | — | 3 | 2 | no | — |
| 274 | C22 | — | 3 | 1 | no | — |
| 275 | C22 | — | 3 | 2 | no | — |
| 276 | C22 | — | 3 | 1 | no | — |
| 277 | C22 | — | 3 | 1 | no | — |
| 278 | C22 | — | 3 | 2 | no | — |
| 279 | C22 | — | 3 | 2 | no | — |
| 280 | C22 | — | 3 | 24 | no | — |
| 281 | C22 | — | 6 | 72 | no | — |
| 282 | C22 | — | 6 | 36 | no | — |
| 283 | C22 | — | 6 | 36 | no | — |
| 284 | C22 | — | 6 | 24 | no | — |
| 285 | C22 | — | 12 | 96 | no | — |
| 286 | C22 | — | 12 | 32 | no | — |
| 287 | C22 | C2×#994 | 1 | 1 | yes | characteristic, complemented |
| 288 | C22 | C22×C22⋊A4 | 1 | 1 | yes | characteristic |
| 289 | C4 | — | 3 | 4 | no | — |
| 290 | C4 | — | 3 | 4 | no | — |
| 291 | C4 | — | 3 | 6 | no | — |
| 292 | C4 | C22×C22⋊A4 | 1 | 6 | yes | — |
| 293 | C3 | — | 16 | 1 | no | Sylow |
| 294 | C2 | — | 3 | 2 | no | — |
| 295 | C2 | — | 3 | 1 | no | — |
| 296 | C2 | — | 3 | 1 | no | — |
| 297 | C2 | — | 6 | 24 | no | — |
| 298 | C2 | #87543 | 1 | 2 | yes | direct product factor |
| 299 | C2 | C23×C22⋊A4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×C22⋊A4 |
| commutator, cosocle, hypocenter | C22×Q8 | C23×C3 |
| Frattini | C2 | C23×C22⋊A4 |
| Fitting | C23×H(5,𝔽2) | C3 |
| socle | C24 | C22×A4 |
| hypercenter | C2×Q8 | C22⋊A4 |
| LCS | G ▷ C22×Q8 | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #87543 | C2 | direct (N × H) |
| 2. C2×#5948 | C2 | semidirect |
| 3. C23×H(5,𝔽2) | C3 | semidirect |
| 4. #5948 | C22 | semidirect |
| 5. C2×C22⋊SL(2,𝔽3) | C22 | semidirect |
| 12. C22×H(5,𝔽2) | C2×C3 | semidirect |
| 16. C22⋊SL(2,𝔽3) | C23 | semidirect |
| 35. C2×H(5,𝔽2) | A4 | semidirect |
| 79. H(5,𝔽2) | C2×A4 | semidirect |
| 242. C23 | #994 | semidirect |
| 287. C22 | C2×#994 | semidirect |
| 298. C2 | #87543 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #87543 | 2 |
| C2×#5948 | 2 |
| C23×H(5,𝔽2) | 3 |
| C2×#994 | 4 |
| C2×Q8×A4 | 4 |