Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
298 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 298 conjugacy classes of subgroups — too many to draw. Only its 36 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 | Q8×S4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | D4×S4 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | A4×SD16 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #5417 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #5414 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #5401 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | C8×S4 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | D4×SD16 | — | 3 | 1 | no | maximal, Sylow |
| 9 | C22×S4 | — | 2 | 1 | no | — |
| 10 | Q8×A4 | C22 | 1 | 1 | yes | characteristic, complemented |
| 11 | D4×A4 | C22 | 1 | 1 | yes | characteristic |
| 12 | GL(2,ℤ/4) | — | 2 | 1 | no | — |
| 13 | C4⋊S4 | C22 | 1 | 1 | yes | characteristic |
| 14 | C4×S4 | — | 2 | 1 | no | — |
| 15 | C4×S4 | C22 | 1 | 1 | yes | characteristic |
| 16 | A4⋊Q8 | — | 2 | 1 | no | — |
| 17 | A4⋊Q8 | C22 | 1 | 1 | yes | characteristic, complemented |
| 18 | D3×SD16 | — | 4 | 1 | no | maximal |
| 19 | C8×A4 | C22 | 1 | 1 | yes | characteristic, complemented |
| 20 | A4⋊C8 | C22 | 1 | 1 | yes | characteristic, complemented |
| 21 | C22×SD16 | — | 3 | 1 | no | — |
| 22 | C22×SD16 | D3 | 1 | 1 | yes | characteristic, complemented |
| 23 | D4×Q8 | — | 3 | 1 | no | — |
| 24 | D42 | — | 3 | 1 | no | — |
| 25 | #492 | — | 3 | 1 | no | — |
| 26 | #465 | — | 3 | 1 | no | — |
| 27 | #465 | — | 3 | 1 | no | — |
| 28 | #461 | — | 3 | 1 | no | — |
| 29 | #460 | — | 3 | 1 | no | — |
| 30 | #450 | — | 3 | 1 | no | — |
| 31 | #450 | — | 3 | 1 | no | — |
| 32 | #448 | — | 3 | 1 | no | — |
| 33 | #448 | — | 3 | 1 | no | — |
| 34 | C4×SD16 | — | 3 | 1 | no | — |
| 35 | C8×D4 | — | 3 | 1 | no | — |
| 36 | C22×A4 | — | 2 | 1 | no | — |
| 37 | C2×S4 | — | 2 | 1 | no | — |
| 38 | C2×S4 | — | 4 | 2 | no | — |
| 39 | C2×S4 | D4 | 1 | 1 | yes | characteristic |
| 40 | D3×Q8 | — | 4 | 1 | no | — |
| 41 | D3×D4 | — | 4 | 1 | no | — |
| 42 | C4×A4 | — | 2 | 1 | no | — |
| 43 | C4×A4 | C23 | 1 | 1 | yes | characteristic |
| 44 | SL(2,ℤ/4) | — | 2 | 1 | no | — |
| 45 | SL(2,ℤ/4) | D4 | 1 | 1 | yes | characteristic |
| 46 | C3×SD16 | — | 4 | 1 | no | — |
| 47 | #215 | — | 4 | 1 | no | — |
| 48 | #214 | — | 4 | 1 | no | — |
| 49 | #204 | — | 4 | 1 | no | — |
| 50 | D3×C8 | — | 4 | 1 | no | — |
| 51 | C22×Q8 | — | 3 | 1 | no | — |
| 52 | C22×Q8 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 53 | C22×D4 | — | 3 | 1 | no | — |
| 54 | C22×D4 | — | 6 | 1 | no | — |
| 55 | C22×D4 | C2×D3 | 1 | 1 | yes | characteristic |
| 56 | C2×SD16 | — | 3 | 1 | no | — |
| 57 | C2×SD16 | — | 3 | 1 | no | — |
| 58 | C2×SD16 | — | 3 | 1 | no | — |
| 59 | C2×SD16 | — | 3 | 1 | no | — |
| 60 | C2×SD16 | — | 3 | 1 | no | — |
| 61 | C2×SD16 | — | 3 | 1 | no | — |
| 62 | C2×SD16 | — | 3 | 1 | no | — |
| 63 | C2×SD16 | — | 3 | 1 | no | — |
| 64 | C2×SD16 | — | 3 | 1 | no | — |
| 65 | C2×SD16 | — | 6 | 1 | no | — |
| 66 | C2×SD16 | — | 6 | 1 | no | — |
| 67 | C2×SD16 | — | 6 | 1 | no | — |
| 68 | C2×SD16 | — | 6 | 1 | no | — |
| 69 | C22×C8 | — | 3 | 1 | no | — |
| 70 | C22×C8 | C2×D3 | 1 | 1 | yes | characteristic, complemented |
| 71 | C4⋊Q8 | — | 3 | 1 | no | — |
| 72 | C4⋊Q8 | — | 6 | 1 | no | — |
| 73 | #127 | — | 3 | 1 | no | — |
| 74 | C22⋊Q8 | — | 3 | 1 | no | — |
| 75 | C22⋊Q8 | — | 3 | 1 | no | — |
| 76 | C22⋊Q8 | — | 6 | 1 | no | — |
| 77 | C22⋊Q8 | — | 6 | 1 | no | — |
| 78 | #121 | — | 3 | 1 | no | — |
| 79 | #121 | — | 3 | 1 | no | — |
| 80 | #121 | — | 6 | 1 | no | — |
| 81 | #120 | — | 6 | 1 | no | — |
| 82 | #120 | — | 6 | 1 | no | — |
| 83 | C4×Q8 | — | 3 | 1 | no | — |
| 84 | C4×D4 | — | 3 | 1 | no | — |
| 85 | C4×D4 | — | 3 | 1 | no | — |
| 86 | C4×D4 | — | 6 | 1 | no | — |
| 87 | #106 | — | 3 | 1 | no | — |
| 88 | C4⋊C8 | — | 3 | 1 | no | — |
| 89 | #103 | — | 3 | 1 | no | — |
| 90 | #103 | — | 3 | 1 | no | — |
| 91 | #103 | — | 3 | 1 | no | — |
| 92 | #102 | — | 3 | 1 | no | — |
| 93 | #102 | — | 3 | 1 | no | — |
| 94 | #102 | — | 3 | 1 | no | — |
| 95 | C22⋊C8 | — | 3 | 1 | no | — |
| 96 | C22⋊C8 | — | 3 | 1 | no | — |
| 97 | C4×C8 | — | 3 | 1 | no | — |
| 98 | C22×D3 | — | 8 | 1 | no | — |
| 99 | C2×A4 | — | 4 | 1 | no | — |
| 100 | C2×A4 | C2×D4 | 1 | 1 | yes | characteristic |
| 101 | S4 | — | 4 | 1 | no | — |
| 102 | S4 | SD16 | 1 | 2 | yes | direct product factor |
| 103 | C3×Q8 | — | 4 | 1 | no | — |
| 104 | C3×D4 | — | 4 | 1 | no | — |
| 105 | #67 | — | 8 | 1 | no | — |
| 106 | D12 | — | 4 | 1 | no | — |
| 107 | C4×D3 | — | 4 | 1 | no | — |
| 108 | C4×D3 | — | 8 | 1 | no | — |
| 109 | Dic6 | — | 4 | 1 | no | — |
| 110 | Dic6 | — | 8 | 1 | no | — |
| 111 | C3×C8 | — | 4 | 1 | no | — |
| 112 | C3⋊C8 | — | 4 | 1 | no | — |
| 113 | C24 | — | 2 | 1 | no | — |
| 114 | C24 | — | 6 | 1 | no | — |
| 115 | C2×Q8 | — | 3 | 1 | no | — |
| 116 | C2×Q8 | — | 3 | 1 | no | — |
| 117 | C2×Q8 | — | 3 | 1 | no | — |
| 118 | C2×Q8 | — | 6 | 1 | no | — |
| 119 | C2×Q8 | — | 6 | 1 | no | — |
| 120 | C2×Q8 | — | 6 | 1 | no | — |
| 121 | C2×Q8 | — | 6 | 1 | no | — |
| 122 | C2×Q8 | — | 12 | 1 | no | — |
| 123 | C2×D4 | — | 3 | 1 | no | — |
| 124 | C2×D4 | — | 3 | 1 | no | — |
| 125 | C2×D4 | — | 3 | 1 | no | — |
| 126 | C2×D4 | — | 3 | 1 | no | — |
| 127 | C2×D4 | — | 6 | 1 | no | — |
| 128 | C2×D4 | — | 6 | 1 | no | — |
| 129 | C2×D4 | — | 6 | 1 | no | — |
| 130 | C2×D4 | — | 6 | 1 | no | — |
| 131 | C2×D4 | — | 6 | 1 | no | — |
| 132 | C2×D4 | — | 6 | 1 | no | — |
| 133 | C2×D4 | — | 6 | 1 | no | — |
| 134 | C2×D4 | — | 6 | 1 | no | — |
| 135 | C2×D4 | — | 12 | 1 | no | — |
| 136 | C2×D4 | — | 12 | 2 | no | — |
| 137 | C2×D4 | — | 12 | 2 | no | — |
| 138 | C22×C4 | — | 2 | 1 | no | — |
| 139 | C22×C4 | — | 3 | 1 | no | — |
| 140 | C22×C4 | — | 6 | 1 | no | — |
| 141 | C22×C4 | — | 6 | 1 | no | — |
| 142 | C22×C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 143 | SD16 | — | 3 | 1 | no | — |
| 144 | SD16 | — | 3 | 1 | no | — |
| 145 | SD16 | — | 3 | 1 | no | — |
| 146 | SD16 | — | 6 | 1 | no | — |
| 147 | SD16 | — | 6 | 1 | no | — |
| 148 | SD16 | — | 6 | 1 | no | — |
| 149 | SD16 | — | 6 | 1 | no | — |
| 150 | SD16 | — | 6 | 1 | no | — |
| 151 | SD16 | — | 6 | 1 | no | — |
| 152 | SD16 | — | 6 | 1 | no | — |
| 153 | SD16 | — | 12 | 1 | no | — |
| 154 | SD16 | — | 12 | 1 | no | — |
| 155 | SD16 | S4 | 1 | 1 | yes | characteristic, direct product factor |
| 156 | C2×C8 | — | 3 | 1 | no | — |
| 157 | C2×C8 | — | 3 | 1 | no | — |
| 158 | C2×C8 | — | 3 | 1 | no | — |
| 159 | C2×C8 | — | 3 | 1 | no | — |
| 160 | C2×C8 | — | 6 | 1 | no | — |
| 161 | C2×C8 | — | 6 | 1 | no | — |
| 162 | C4⋊C4 | — | 3 | 1 | no | — |
| 163 | C4⋊C4 | — | 3 | 1 | no | — |
| 164 | C4⋊C4 | — | 3 | 1 | no | — |
| 165 | C4⋊C4 | — | 6 | 1 | no | — |
| 166 | C4⋊C4 | — | 6 | 1 | no | — |
| 167 | C4⋊C4 | — | 6 | 1 | no | — |
| 168 | C4⋊C4 | — | 6 | 1 | no | — |
| 169 | C4⋊C4 | — | 6 | 1 | no | — |
| 170 | C22⋊C4 | — | 3 | 1 | no | — |
| 171 | C22⋊C4 | — | 3 | 1 | no | — |
| 172 | C22⋊C4 | — | 6 | 1 | no | — |
| 173 | C22⋊C4 | — | 6 | 1 | no | — |
| 174 | C22⋊C4 | — | 6 | 1 | no | — |
| 175 | C22⋊C4 | — | 6 | 1 | no | — |
| 176 | C22⋊C4 | — | 6 | 1 | no | — |
| 177 | C42 | — | 3 | 1 | no | — |
| 178 | C42 | — | 6 | 1 | no | — |
| 179 | C22×C3 | — | 8 | 1 | no | — |
| 180 | C2×D3 | — | 4 | 1 | no | — |
| 181 | C2×D3 | — | 8 | 1 | no | — |
| 182 | C2×D3 | — | 16 | 2 | no | — |
| 183 | A4 | C2×SD16 | 1 | 1 | yes | characteristic, complemented |
| 184 | C3×C4 | — | 4 | 1 | no | — |
| 185 | C3×C4 | — | 8 | 1 | no | — |
| 186 | Dic3 | — | 4 | 1 | no | — |
| 187 | Dic3 | — | 8 | 1 | no | — |
| 188 | C23 | — | 3 | 1 | no | — |
| 189 | C23 | — | 4 | 1 | no | — |
| 190 | C23 | — | 6 | 1 | no | — |
| 191 | C23 | — | 6 | 1 | no | — |
| 192 | C23 | — | 6 | 1 | no | — |
| 193 | C23 | — | 12 | 1 | no | — |
| 194 | C23 | — | 12 | 1 | no | — |
| 195 | C23 | — | 12 | 2 | no | — |
| 196 | C23 | — | 12 | 1 | no | — |
| 197 | C23 | — | 24 | 1 | no | — |
| 198 | C23 | D3×D4 | 1 | 1 | yes | characteristic |
| 199 | Q8 | — | 3 | 1 | no | — |
| 200 | Q8 | — | 6 | 1 | no | — |
| 201 | Q8 | — | 6 | 1 | no | — |
| 202 | Q8 | — | 6 | 1 | no | — |
| 203 | Q8 | — | 6 | 1 | no | — |
| 204 | Q8 | — | 12 | 1 | no | — |
| 205 | Q8 | — | 12 | 1 | no | — |
| 206 | Q8 | C2×S4 | 1 | 1 | yes | characteristic, complemented |
| 207 | D4 | — | 3 | 2 | no | — |
| 208 | D4 | — | 3 | 2 | no | — |
| 209 | D4 | — | 3 | 1 | no | — |
| 210 | D4 | — | 6 | 1 | no | — |
| 211 | D4 | — | 6 | 1 | no | — |
| 212 | D4 | — | 6 | 1 | no | — |
| 213 | D4 | — | 6 | 1 | no | — |
| 214 | D4 | — | 12 | 2 | no | — |
| 215 | D4 | — | 12 | 1 | no | — |
| 216 | D4 | — | 12 | 1 | no | — |
| 217 | D4 | — | 12 | 1 | no | — |
| 218 | D4 | — | 12 | 2 | no | — |
| 219 | D4 | — | 12 | 1 | no | — |
| 220 | D4 | — | 24 | 1 | no | — |
| 221 | D4 | C2×S4 | 1 | 1 | yes | characteristic |
| 222 | C2×C4 | — | 3 | 1 | no | — |
| 223 | C2×C4 | — | 3 | 1 | no | — |
| 224 | C2×C4 | — | 3 | 1 | no | — |
| 225 | C2×C4 | — | 3 | 1 | no | — |
| 226 | C2×C4 | — | 3 | 1 | no | — |
| 227 | C2×C4 | — | 6 | 1 | no | — |
| 228 | C2×C4 | — | 6 | 1 | no | — |
| 229 | C2×C4 | — | 6 | 1 | no | — |
| 230 | C2×C4 | — | 6 | 1 | no | — |
| 231 | C2×C4 | — | 6 | 1 | no | — |
| 232 | C2×C4 | — | 6 | 1 | no | — |
| 233 | C2×C4 | — | 6 | 1 | no | — |
| 234 | C2×C4 | — | 12 | 1 | no | — |
| 235 | C2×C4 | — | 12 | 2 | no | — |
| 236 | C2×C4 | — | 12 | 1 | no | — |
| 237 | C8 | — | 3 | 1 | no | — |
| 238 | C8 | — | 6 | 1 | no | — |
| 239 | C8 | — | 6 | 1 | no | — |
| 240 | C8 | C2×S4 | 1 | 1 | yes | characteristic, complemented |
| 241 | C2×C3 | — | 4 | 1 | no | — |
| 242 | C2×C3 | — | 16 | 1 | no | — |
| 243 | D3 | — | 4 | 2 | no | — |
| 244 | D3 | — | 16 | 1 | no | — |
| 245 | C22 | — | 2 | 1 | no | — |
| 246 | C22 | — | 3 | 1 | no | — |
| 247 | C22 | — | 3 | 1 | no | — |
| 248 | C22 | — | 3 | 2 | no | — |
| 249 | C22 | — | 6 | 1 | no | — |
| 250 | C22 | — | 6 | 1 | no | — |
| 251 | C22 | — | 6 | 1 | no | — |
| 252 | C22 | — | 12 | 1 | no | — |
| 253 | C22 | — | 12 | 1 | no | — |
| 254 | C22 | — | 12 | 1 | no | — |
| 255 | C22 | — | 12 | 1 | no | — |
| 256 | C22 | — | 12 | 1 | no | — |
| 257 | C22 | — | 12 | 1 | no | — |
| 258 | C22 | — | 12 | 1 | no | — |
| 259 | C22 | — | 24 | 2 | no | — |
| 260 | C22 | — | 24 | 2 | no | — |
| 261 | C22 | D3×SD16 | 1 | 1 | yes | characteristic, complemented |
| 262 | C4 | — | 2 | 1 | no | — |
| 263 | C4 | — | 3 | 2 | no | — |
| 264 | C4 | — | 3 | 1 | no | — |
| 265 | C4 | — | 6 | 1 | no | — |
| 266 | C4 | — | 6 | 1 | no | — |
| 267 | C4 | — | 6 | 1 | no | — |
| 268 | C4 | — | 12 | 1 | no | — |
| 269 | C4 | — | 12 | 1 | no | — |
| 270 | C4 | — | 12 | 1 | no | — |
| 271 | C4 | C22×S4 | 1 | 1 | yes | characteristic |
| 272 | C3 | — | 4 | 1 | no | Sylow |
| 273 | C2 | — | 3 | 1 | no | — |
| 274 | C2 | — | 3 | 1 | no | — |
| 275 | C2 | — | 4 | 1 | no | — |
| 276 | C2 | — | 6 | 2 | no | — |
| 277 | C2 | — | 12 | 1 | no | — |
| 278 | C2 | — | 24 | 1 | no | — |
| 279 | C2 | D4×S4 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q8×S4 | C2 | semidirect |
| 3. A4×SD16 | C2 | semidirect |
| 4. #5417 | C2 | semidirect |
| 5. #5414 | C2 | semidirect |
| 6. #5401 | C2 | semidirect |
| 7. C8×S4 | C2 | semidirect |
| 10. Q8×A4 | C22 | semidirect |
| 17. A4⋊Q8 | C22 | semidirect |
| 19. C8×A4 | C22 | semidirect |
| 20. A4⋊C8 | C22 | semidirect |
| 22. C22×SD16 | D3 | semidirect |
| 52. C22×Q8 | C2×D3 | semidirect |
| 70. C22×C8 | C2×D3 | semidirect |
| 102. S4 | SD16 | direct (N × H) |
| 155. SD16 | S4 | direct (N × H) |
| 183. A4 | C2×SD16 | semidirect |
| 206. Q8 | C2×S4 | semidirect |
| 240. C8 | C2×S4 | semidirect |
| 261. C22 | D3×SD16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×S4 | 2 |
| D4×S4 | 2 |
| A4×SD16 | 2 |
| #5417 | 2 |
| #5414 | 2 |
| #5401 | 2 |
| C8×S4 | 2 |
| D4×SD16 | 3 |
| D3×SD16 | 4 |