Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
264 conjugacy classes of subgroups; only the 34 normal subgroups are shown.
This group has 264 conjugacy classes of subgroups — too many to draw. Only its 34 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 | #5914 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | A4×SD16 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #5416 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #5414 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #5400 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #5398 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #3266 | — | 3 | 1 | no | maximal, Sylow |
| 9 | Q8×A4 | C22 | 1 | 1 | yes | characteristic |
| 10 | D4×A4 | C22 | 1 | 1 | yes | characteristic |
| 11 | GL(2,ℤ/4) | — | 2 | 1 | no | — |
| 12 | C2×SL(2,ℤ/4) | — | 2 | 1 | no | — |
| 13 | C4×S4 | — | 2 | 1 | no | — |
| 14 | C4×S4 | C22 | 1 | 1 | yes | characteristic |
| 15 | A4⋊Q8 | — | 2 | 1 | no | — |
| 16 | A4⋊Q8 | C22 | 1 | 1 | yes | characteristic |
| 17 | A4⋊Q8 | C22 | 1 | 1 | yes | characteristic |
| 18 | #915 | — | 4 | 1 | no | maximal |
| 19 | C8×A4 | C22 | 1 | 1 | yes | characteristic |
| 20 | A4⋊C8 | C22 | 1 | 1 | yes | characteristic |
| 21 | C2×#137 | — | 3 | 1 | no | — |
| 22 | C22×SD16 | D3 | 1 | 1 | yes | characteristic, complemented |
| 23 | D4×Q8 | — | 3 | 1 | no | — |
| 24 | #546 | — | 3 | 1 | no | — |
| 25 | #497 | — | 3 | 1 | no | — |
| 26 | #468 | — | 3 | 1 | no | — |
| 27 | #467 | — | 3 | 1 | no | — |
| 28 | #464 | — | 3 | 1 | no | — |
| 29 | #461 | — | 3 | 1 | no | — |
| 30 | #452 | — | 3 | 1 | no | — |
| 31 | #451 | — | 3 | 1 | no | — |
| 32 | #450 | — | 3 | 1 | no | — |
| 33 | #448 | — | 3 | 1 | no | — |
| 34 | #440 | — | 3 | 1 | no | — |
| 35 | #435 | — | 3 | 1 | no | — |
| 36 | C22×A4 | — | 2 | 1 | no | — |
| 37 | C2×S4 | D4 | 1 | 1 | yes | characteristic |
| 38 | D3×Q8 | — | 4 | 1 | no | — |
| 39 | #237 | — | 4 | 1 | no | — |
| 40 | C4×A4 | — | 2 | 1 | no | — |
| 41 | C4×A4 | C23 | 1 | 1 | yes | characteristic |
| 42 | SL(2,ℤ/4) | — | 2 | 1 | no | — |
| 43 | SL(2,ℤ/4) | — | 2 | 1 | no | — |
| 44 | SL(2,ℤ/4) | D4 | 1 | 1 | yes | characteristic |
| 45 | C3×SD16 | — | 4 | 1 | no | — |
| 46 | #216 | — | 4 | 1 | no | — |
| 47 | #214 | — | 4 | 1 | no | — |
| 48 | Dic12 | — | 4 | 1 | no | — |
| 49 | ΓL(1,𝔽25) | — | 4 | 1 | no | — |
| 50 | C2×#41 | — | 3 | 1 | no | — |
| 51 | C22×Q8 | — | 3 | 1 | no | — |
| 52 | C22×Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 53 | C22×D4 | C2×D3 | 1 | 1 | yes | characteristic |
| 54 | #137 | — | 6 | 1 | no | — |
| 55 | #137 | — | 6 | 1 | no | — |
| 56 | #137 | — | 6 | 1 | no | — |
| 57 | #137 | — | 6 | 1 | no | — |
| 58 | C2×Q16 | — | 3 | 1 | no | — |
| 59 | C2×Q16 | — | 3 | 1 | no | — |
| 60 | C2×Q16 | — | 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 | — | 3 | 1 | no | — |
| 66 | C2×SD16 | — | 3 | 1 | no | — |
| 67 | C2×M16 | — | 3 | 1 | no | — |
| 68 | C22×C8 | C2×D3 | 1 | 1 | yes | characteristic |
| 69 | C4⋊Q8 | — | 3 | 1 | no | — |
| 70 | C4⋊Q8 | — | 6 | 1 | no | — |
| 71 | #124 | — | 3 | 1 | no | — |
| 72 | #123 | — | 6 | 1 | no | — |
| 73 | C22⋊Q8 | — | 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 | — | 6 | 1 | no | — |
| 80 | #120 | — | 6 | 1 | no | — |
| 81 | C4×Q8 | — | 3 | 1 | no | — |
| 82 | C4×D4 | — | 3 | 1 | no | — |
| 83 | C4×D4 | — | 3 | 1 | no | — |
| 84 | C4×D4 | — | 6 | 1 | no | — |
| 85 | C2×C22⋊C4 | — | 6 | 1 | no | — |
| 86 | #107 | — | 3 | 1 | no | — |
| 87 | C4⋊C8 | — | 3 | 1 | no | — |
| 88 | #103 | — | 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 | C22⋊C8 | — | 3 | 1 | no | — |
| 95 | C22⋊C8 | — | 3 | 1 | no | — |
| 96 | #97 | — | 3 | 1 | no | — |
| 97 | C2×A4 | — | 4 | 1 | no | — |
| 98 | C2×A4 | C2×D4 | 1 | 1 | yes | characteristic |
| 99 | S4 | — | 2 | 1 | no | — |
| 100 | C3×Q8 | — | 4 | 1 | no | — |
| 101 | C3×D4 | — | 4 | 1 | no | — |
| 102 | #67 | — | 8 | 1 | no | — |
| 103 | C2×Dic3 | — | 8 | 1 | no | — |
| 104 | C4×D3 | — | 4 | 1 | no | — |
| 105 | C4×D3 | — | 8 | 1 | no | — |
| 106 | Dic6 | — | 4 | 1 | no | — |
| 107 | Dic6 | — | 4 | 1 | no | — |
| 108 | Dic6 | — | 8 | 1 | no | — |
| 109 | C3×C8 | — | 4 | 1 | no | — |
| 110 | C3⋊C8 | — | 4 | 1 | no | — |
| 111 | C24 | — | 2 | 1 | no | — |
| 112 | #41 | — | 6 | 1 | no | — |
| 113 | #41 | — | 6 | 1 | no | — |
| 114 | #41 | — | 12 | 1 | no | — |
| 115 | C2×Q8 | — | 3 | 1 | no | — |
| 116 | C2×Q8 | — | 3 | 1 | no | — |
| 117 | C2×Q8 | — | 3 | 1 | no | — |
| 118 | C2×Q8 | — | 3 | 1 | no | — |
| 119 | C2×Q8 | — | 6 | 1 | no | — |
| 120 | C2×Q8 | — | 6 | 1 | no | — |
| 121 | C2×Q8 | — | 6 | 1 | no | — |
| 122 | C2×Q8 | — | 6 | 1 | no | — |
| 123 | C2×Q8 | — | 12 | 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 | C22×C4 | — | 2 | 1 | no | — |
| 131 | C22×C4 | — | 3 | 1 | no | — |
| 132 | C22×C4 | — | 6 | 1 | no | — |
| 133 | C22×C4 | — | 6 | 1 | no | — |
| 134 | C22×C4 | — | 6 | 1 | no | — |
| 135 | C22×C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 136 | Q16 | — | 6 | 1 | no | — |
| 137 | Q16 | — | 6 | 1 | no | — |
| 138 | Q16 | — | 6 | 1 | no | — |
| 139 | Q16 | — | 6 | 1 | no | — |
| 140 | Q16 | — | 12 | 1 | no | — |
| 141 | SD16 | — | 3 | 1 | no | — |
| 142 | SD16 | — | 3 | 1 | no | — |
| 143 | SD16 | — | 3 | 1 | no | — |
| 144 | SD16 | — | 6 | 1 | no | — |
| 145 | SD16 | — | 6 | 1 | no | — |
| 146 | SD16 | — | 6 | 1 | no | — |
| 147 | SD16 | — | 12 | 1 | no | — |
| 148 | SD16 | S4 | 1 | 1 | yes | characteristic, complemented |
| 149 | M16 | — | 6 | 1 | no | — |
| 150 | M16 | — | 6 | 1 | no | — |
| 151 | C2×C8 | — | 3 | 1 | no | — |
| 152 | C2×C8 | — | 3 | 1 | no | — |
| 153 | C2×C8 | — | 3 | 1 | no | — |
| 154 | C2×C8 | — | 3 | 1 | no | — |
| 155 | C4⋊C4 | — | 3 | 1 | no | — |
| 156 | C4⋊C4 | — | 3 | 1 | no | — |
| 157 | C4⋊C4 | — | 3 | 1 | no | — |
| 158 | C4⋊C4 | — | 6 | 1 | no | — |
| 159 | C4⋊C4 | — | 6 | 1 | no | — |
| 160 | C4⋊C4 | — | 6 | 1 | no | — |
| 161 | C4⋊C4 | — | 6 | 1 | no | — |
| 162 | C4⋊C4 | — | 6 | 1 | no | — |
| 163 | C4⋊C4 | — | 6 | 1 | no | — |
| 164 | C22⋊C4 | — | 3 | 1 | no | — |
| 165 | C22⋊C4 | — | 3 | 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 | — | 6 | 1 | no | — |
| 171 | C22⋊C4 | — | 6 | 1 | no | — |
| 172 | C22⋊C4 | — | 6 | 1 | no | — |
| 173 | C42 | — | 3 | 1 | no | — |
| 174 | C42 | — | 6 | 1 | no | — |
| 175 | C22×C3 | — | 8 | 1 | no | — |
| 176 | C2×D3 | — | 4 | 1 | no | — |
| 177 | A4 | #137 | 1 | 1 | yes | characteristic, complemented |
| 178 | C3×C4 | — | 4 | 1 | no | — |
| 179 | C3×C4 | — | 8 | 1 | no | — |
| 180 | Dic3 | — | 4 | 1 | no | — |
| 181 | Dic3 | — | 8 | 1 | no | — |
| 182 | Dic3 | — | 8 | 1 | no | — |
| 183 | C23 | — | 3 | 1 | no | — |
| 184 | C23 | — | 4 | 1 | no | — |
| 185 | C23 | — | 6 | 1 | no | — |
| 186 | C23 | — | 6 | 1 | no | — |
| 187 | C23 | — | 12 | 1 | no | — |
| 188 | C23 | D3×D4 | 1 | 1 | yes | characteristic |
| 189 | Q8 | — | 3 | 1 | no | — |
| 190 | Q8 | — | 6 | 1 | no | — |
| 191 | Q8 | — | 6 | 1 | no | — |
| 192 | Q8 | — | 6 | 1 | no | — |
| 193 | Q8 | — | 6 | 1 | no | — |
| 194 | Q8 | — | 6 | 1 | no | — |
| 195 | Q8 | — | 6 | 1 | no | — |
| 196 | Q8 | — | 12 | 1 | no | — |
| 197 | Q8 | — | 12 | 1 | no | — |
| 198 | Q8 | C2×S4 | 1 | 1 | yes | characteristic |
| 199 | D4 | — | 3 | 1 | no | — |
| 200 | D4 | — | 6 | 1 | no | — |
| 201 | D4 | — | 6 | 1 | no | — |
| 202 | D4 | — | 6 | 1 | no | — |
| 203 | D4 | — | 6 | 1 | no | — |
| 204 | D4 | — | 12 | 1 | no | — |
| 205 | D4 | — | 12 | 1 | no | — |
| 206 | D4 | — | 24 | 1 | no | — |
| 207 | D4 | C2×S4 | 1 | 1 | yes | characteristic |
| 208 | C2×C4 | — | 3 | 1 | no | — |
| 209 | C2×C4 | — | 3 | 1 | no | — |
| 210 | C2×C4 | — | 3 | 1 | no | — |
| 211 | C2×C4 | — | 3 | 1 | no | — |
| 212 | C2×C4 | — | 3 | 1 | no | — |
| 213 | C2×C4 | — | 6 | 1 | no | — |
| 214 | C2×C4 | — | 6 | 1 | no | — |
| 215 | C2×C4 | — | 6 | 1 | no | — |
| 216 | C2×C4 | — | 6 | 1 | no | — |
| 217 | C2×C4 | — | 6 | 1 | no | — |
| 218 | C2×C4 | — | 6 | 1 | no | — |
| 219 | C2×C4 | — | 6 | 1 | no | — |
| 220 | C2×C4 | — | 6 | 1 | no | — |
| 221 | C2×C4 | — | 12 | 1 | no | — |
| 222 | C2×C4 | — | 12 | 1 | no | — |
| 223 | C2×C4 | — | 12 | 1 | no | — |
| 224 | C2×C4 | — | 12 | 1 | no | — |
| 225 | C2×C4 | — | 24 | 1 | no | — |
| 226 | C8 | — | 3 | 1 | no | — |
| 227 | C8 | — | 6 | 1 | no | — |
| 228 | C8 | — | 6 | 1 | no | — |
| 229 | C8 | C2×S4 | 1 | 1 | yes | characteristic |
| 230 | C2×C3 | — | 4 | 1 | no | — |
| 231 | C2×C3 | — | 16 | 1 | no | — |
| 232 | D3 | — | 8 | 1 | no | — |
| 233 | C22 | — | 2 | 1 | no | — |
| 234 | C22 | — | 3 | 1 | no | — |
| 235 | C22 | — | 3 | 1 | no | — |
| 236 | C22 | — | 6 | 1 | no | — |
| 237 | C22 | — | 6 | 1 | no | — |
| 238 | C22 | — | 6 | 1 | no | — |
| 239 | C22 | — | 6 | 1 | no | — |
| 240 | C22 | — | 12 | 1 | no | — |
| 241 | C22 | — | 12 | 1 | no | — |
| 242 | C22 | — | 12 | 1 | no | — |
| 243 | C22 | — | 12 | 1 | no | — |
| 244 | C22 | #915 | 1 | 1 | yes | characteristic, complemented |
| 245 | C4 | — | 2 | 1 | no | — |
| 246 | C4 | — | 3 | 1 | no | — |
| 247 | C4 | — | 6 | 1 | no | — |
| 248 | C4 | — | 6 | 1 | no | — |
| 249 | C4 | — | 6 | 1 | no | — |
| 250 | C4 | — | 6 | 1 | no | — |
| 251 | C4 | — | 12 | 1 | no | — |
| 252 | C4 | — | 12 | 1 | no | — |
| 253 | C4 | — | 12 | 1 | no | — |
| 254 | C4 | — | 12 | 1 | no | — |
| 255 | C4 | C22×S4 | 1 | 1 | yes | characteristic |
| 256 | C3 | — | 4 | 1 | no | Sylow |
| 257 | C2 | — | 3 | 1 | no | — |
| 258 | C2 | — | 3 | 1 | no | — |
| 259 | C2 | — | 4 | 1 | no | — |
| 260 | C2 | — | 12 | 1 | no | — |
| 261 | C2 | — | 12 | 1 | no | — |
| 262 | C2 | D4×S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D4×S4 |
| commutator, cosocle | C4×A4 | C23 |
| Frattini | C4 | C22×S4 |
| Fitting | C22×SD16 | D3 |
| socle | C23 | D3×D4 |
| prime radical | #5914 | C2 |
| hypercenter | SD16 | S4 |
| hypocenter | A4 | #137 |
| LCS | G ▷ C4×A4 ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ SD16 | |
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. #5416 | C2 | semidirect |
| 5. #5414 | C2 | semidirect |
| 6. #5400 | C2 | semidirect |
| 7. #5398 | C2 | semidirect |
| 22. C22×SD16 | D3 | semidirect |
| 148. SD16 | S4 | semidirect |
| 177. A4 | #137 | semidirect |
| 244. C22 | #915 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×S4 | 2 |
| #5914 | 2 |
| A4×SD16 | 2 |
| #5416 | 2 |
| #5414 | 2 |
| #5400 | 2 |
| #5398 | 2 |
| #3266 | 3 |
| #915 | 4 |