Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
270 conjugacy classes of subgroups; only the 41 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 41 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 | #5594 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C3×B(4,𝔽2) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #5244 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | #5242 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #4536 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | #4473 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 7 | #4471 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 8 | #2199 | — | 3 | 1 | no | maximal, Sylow |
| 9 | C3×H(5,𝔽2) | C22 | 1 | 1 | yes | characteristic |
| 10 | C3×#120 | — | 2 | 1 | no | — |
| 11 | C3×#120 | C22 | 1 | 1 | yes | characteristic |
| 12 | #953 | — | 2 | 1 | no | — |
| 13 | #951 | — | 2 | 1 | no | — |
| 14 | #937 | — | 2 | 1 | no | — |
| 15 | #937 | — | 2 | 1 | no | — |
| 16 | #937 | C22 | 1 | 1 | yes | characteristic |
| 17 | #936 | — | 2 | 1 | no | — |
| 18 | #936 | C22 | 1 | 1 | yes | characteristic |
| 19 | #885 | — | 2 | 1 | no | — |
| 20 | #885 | — | 2 | 1 | no | — |
| 21 | #885 | — | 2 | 1 | no | — |
| 22 | C3×#99 | — | 2 | 1 | no | — |
| 23 | C3×#99 | C22 | 1 | 1 | yes | characteristic |
| 24 | #837 | — | 2 | 1 | no | — |
| 25 | #834 | — | 2 | 1 | no | — |
| 26 | #834 | C22 | 1 | 1 | yes | characteristic |
| 27 | #833 | C22 | 1 | 1 | yes | characteristic |
| 28 | H(3,𝔽4) | — | 3 | 1 | no | — |
| 29 | B(4,𝔽2) | — | 3 | 1 | no | — |
| 30 | B(4,𝔽2) | D3 | 1 | 1 | yes | characteristic, complemented |
| 31 | #455 | — | 3 | 1 | no | — |
| 32 | #354 | — | 3 | 1 | no | — |
| 33 | AΣL(1,𝔽16) | — | 3 | 1 | no | — |
| 34 | AΣL(1,𝔽16) | — | 3 | 1 | no | — |
| 35 | C24×C3 | D4 | 1 | 1 | yes | characteristic, complemented |
| 36 | C23×D3 | D4 | 1 | 1 | yes | characteristic, complemented |
| 37 | C3×#41 | — | 2 | 1 | no | — |
| 38 | C3×#41 | — | 4 | 1 | no | — |
| 39 | C2×C3×D4 | — | 2 | 1 | no | — |
| 40 | C2×C3×D4 | — | 2 | 1 | no | — |
| 41 | C2×C3×D4 | — | 2 | 1 | no | — |
| 42 | C2×C3×D4 | — | 4 | 1 | no | — |
| 43 | C2×C3×D4 | — | 4 | 1 | no | — |
| 44 | C2×C3×D4 | C23 | 1 | 1 | yes | characteristic |
| 45 | C2×#67 | — | 2 | 1 | no | — |
| 46 | C2×#67 | — | 2 | 1 | no | — |
| 47 | C2×#67 | — | 2 | 1 | no | — |
| 48 | C2×#67 | — | 4 | 1 | no | — |
| 49 | C2×Dic6 | — | 2 | 1 | no | — |
| 50 | C2×Dic6 | D4 | 1 | 1 | yes | characteristic, complemented |
| 51 | C3×C22⋊C4 | — | 2 | 1 | no | — |
| 52 | C3×C22⋊C4 | — | 2 | 1 | no | — |
| 53 | C3×C22⋊C4 | — | 4 | 1 | no | — |
| 54 | C3×C22⋊C4 | D4 | 1 | 1 | yes | characteristic, complemented |
| 55 | #217 | — | 2 | 1 | no | — |
| 56 | #217 | — | 2 | 1 | no | — |
| 57 | #217 | — | 2 | 1 | no | — |
| 58 | #217 | — | 2 | 1 | no | — |
| 59 | #217 | — | 2 | 1 | no | — |
| 60 | #217 | D4 | 1 | 1 | yes | characteristic, complemented |
| 61 | #216 | — | 4 | 1 | no | — |
| 62 | #214 | — | 4 | 1 | no | — |
| 63 | (C2×D3)⋊C4 | — | 2 | 1 | no | — |
| 64 | (C2×D3)⋊C4 | — | 2 | 1 | no | — |
| 65 | (C2×D3)⋊C4 | — | 2 | 1 | no | — |
| 66 | (C2×D3)⋊C4 | — | 4 | 1 | no | — |
| 67 | C4×Dic3 | — | 2 | 1 | no | — |
| 68 | C4×Dic3 | D4 | 1 | 1 | yes | characteristic, complemented |
| 69 | #208 | — | 2 | 1 | no | — |
| 70 | H(5,𝔽2) | C2×D3 | 1 | 1 | yes | characteristic |
| 71 | #137 | — | 6 | 1 | no | — |
| 72 | #124 | — | 3 | 1 | no | — |
| 73 | #124 | — | 6 | 1 | no | — |
| 74 | #124 | — | 6 | 1 | no | — |
| 75 | #124 | — | 6 | 1 | no | — |
| 76 | #124 | — | 6 | 1 | no | — |
| 77 | #120 | — | 2 | 1 | no | — |
| 78 | #120 | — | 3 | 1 | no | — |
| 79 | #120 | — | 6 | 1 | no | — |
| 80 | #120 | — | 6 | 1 | no | — |
| 81 | #120 | — | 6 | 1 | no | — |
| 82 | #120 | C2×D3 | 1 | 1 | yes | characteristic |
| 83 | #104 | — | 6 | 1 | no | — |
| 84 | #100 | — | 3 | 1 | no | — |
| 85 | #99 | — | 2 | 1 | no | — |
| 86 | #99 | — | 3 | 1 | no | — |
| 87 | #99 | — | 6 | 1 | no | — |
| 88 | #99 | C2×D3 | 1 | 1 | yes | characteristic |
| 89 | C23×C3 | — | 2 | 1 | no | — |
| 90 | C23×C3 | — | 2 | 1 | no | — |
| 91 | C23×C3 | — | 2 | 1 | no | — |
| 92 | C23×C3 | — | 4 | 1 | no | — |
| 93 | C23×C3 | — | 8 | 1 | no | — |
| 94 | C23×C3 | C2×D4 | 1 | 1 | yes | characteristic |
| 95 | C23×C3 | C2×D4 | 1 | 1 | yes | characteristic |
| 96 | C22×D3 | — | 2 | 1 | no | — |
| 97 | C22×D3 | — | 4 | 1 | no | — |
| 98 | C22×D3 | — | 8 | 1 | no | — |
| 99 | C3×Q8 | — | 2 | 1 | no | — |
| 100 | C3×D4 | — | 2 | 1 | no | — |
| 101 | C3×D4 | — | 4 | 1 | no | — |
| 102 | C3×D4 | — | 4 | 1 | no | — |
| 103 | C3×D4 | — | 4 | 1 | no | — |
| 104 | C3×D4 | — | 4 | 1 | no | — |
| 105 | C3×D4 | — | 8 | 1 | no | — |
| 106 | C3×D4 | — | 8 | 2 | no | — |
| 107 | C2×C3×C4 | — | 2 | 1 | no | — |
| 108 | C2×C3×C4 | — | 2 | 1 | no | — |
| 109 | C2×C3×C4 | — | 2 | 1 | no | — |
| 110 | C2×C3×C4 | — | 4 | 1 | no | — |
| 111 | C2×C3×C4 | — | 4 | 1 | no | — |
| 112 | C2×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 113 | #67 | — | 8 | 1 | no | — |
| 114 | #67 | — | 8 | 1 | no | — |
| 115 | #67 | — | 8 | 1 | no | — |
| 116 | #67 | — | 8 | 2 | no | — |
| 117 | C2×Dic3 | — | 2 | 1 | no | — |
| 118 | C2×Dic3 | — | 2 | 1 | no | — |
| 119 | C2×Dic3 | — | 2 | 1 | no | — |
| 120 | C2×Dic3 | — | 4 | 1 | no | — |
| 121 | Dic6 | — | 4 | 1 | no | — |
| 122 | Dic6 | — | 8 | 1 | no | — |
| 123 | C3⋊C8 | — | 4 | 1 | no | — |
| 124 | C24 | — | 3 | 1 | no | — |
| 125 | C24 | #67 | 1 | 1 | yes | characteristic, complemented |
| 126 | #41 | — | 2 | 1 | no | — |
| 127 | #41 | — | 4 | 1 | no | — |
| 128 | C2×Q8 | — | 3 | 1 | no | — |
| 129 | C2×Q8 | — | 6 | 1 | no | — |
| 130 | C2×D4 | — | 2 | 1 | no | — |
| 131 | C2×D4 | — | 2 | 1 | no | — |
| 132 | C2×D4 | — | 2 | 1 | no | — |
| 133 | C2×D4 | — | 4 | 1 | no | — |
| 134 | C2×D4 | — | 4 | 1 | no | — |
| 135 | C2×D4 | — | 6 | 1 | no | — |
| 136 | C2×D4 | — | 6 | 1 | no | — |
| 137 | C2×D4 | — | 6 | 1 | no | — |
| 138 | C2×D4 | — | 12 | 1 | no | — |
| 139 | C2×D4 | C22×D3 | 1 | 1 | yes | characteristic |
| 140 | Q16 | — | 12 | 1 | no | — |
| 141 | SD16 | — | 12 | 1 | no | — |
| 142 | M16 | — | 6 | 1 | no | — |
| 143 | C22⋊C4 | — | 2 | 1 | no | — |
| 144 | C22⋊C4 | — | 2 | 1 | no | — |
| 145 | C22⋊C4 | — | 3 | 1 | no | — |
| 146 | C22⋊C4 | — | 4 | 1 | no | — |
| 147 | C22⋊C4 | — | 6 | 1 | no | — |
| 148 | C22⋊C4 | — | 6 | 1 | no | — |
| 149 | C22⋊C4 | — | 6 | 1 | no | — |
| 150 | C22⋊C4 | — | 6 | 1 | no | — |
| 151 | C22⋊C4 | — | 6 | 1 | no | — |
| 152 | C22⋊C4 | — | 6 | 1 | no | — |
| 153 | C22⋊C4 | — | 6 | 1 | no | — |
| 154 | C22⋊C4 | — | 6 | 1 | no | — |
| 155 | C22⋊C4 | — | 12 | 1 | no | — |
| 156 | C22⋊C4 | #67 | 1 | 1 | yes | characteristic, complemented |
| 157 | C42 | — | 3 | 1 | no | — |
| 158 | C42 | — | 6 | 1 | no | — |
| 159 | C22×C3 | — | 2 | 1 | no | — |
| 160 | C22×C3 | — | 2 | 1 | no | — |
| 161 | C22×C3 | — | 4 | 1 | no | — |
| 162 | C22×C3 | — | 4 | 1 | no | — |
| 163 | C22×C3 | — | 4 | 1 | no | — |
| 164 | C22×C3 | — | 4 | 1 | no | — |
| 165 | C22×C3 | — | 4 | 2 | no | — |
| 166 | C22×C3 | — | 4 | 1 | no | — |
| 167 | C22×C3 | — | 4 | 1 | no | — |
| 168 | C22×C3 | — | 8 | 1 | no | — |
| 169 | C22×C3 | — | 8 | 1 | no | — |
| 170 | C22×C3 | — | 8 | 1 | no | — |
| 171 | C22×C3 | #120 | 1 | 1 | yes | characteristic |
| 172 | C2×D3 | — | 4 | 1 | no | — |
| 173 | C2×D3 | — | 4 | 1 | no | — |
| 174 | C2×D3 | — | 4 | 2 | no | — |
| 175 | C2×D3 | — | 4 | 1 | no | — |
| 176 | C2×D3 | — | 8 | 1 | no | — |
| 177 | C3×C4 | — | 2 | 1 | no | — |
| 178 | C3×C4 | — | 4 | 1 | no | — |
| 179 | C3×C4 | — | 4 | 1 | no | — |
| 180 | C3×C4 | — | 8 | 1 | no | — |
| 181 | Dic3 | — | 4 | 1 | no | — |
| 182 | Dic3 | — | 4 | 1 | no | — |
| 183 | Dic3 | — | 4 | 1 | no | — |
| 184 | Dic3 | — | 8 | 1 | no | — |
| 185 | C23 | — | 2 | 1 | no | — |
| 186 | C23 | — | 2 | 1 | no | — |
| 187 | C23 | — | 2 | 1 | no | — |
| 188 | C23 | — | 4 | 1 | no | — |
| 189 | C23 | — | 6 | 1 | no | — |
| 190 | C23 | — | 8 | 1 | no | — |
| 191 | C23 | — | 12 | 1 | no | — |
| 192 | C23 | — | 24 | 1 | no | — |
| 193 | C23 | D3×D4 | 1 | 1 | yes | characteristic |
| 194 | C23 | C2×#67 | 1 | 1 | yes | characteristic |
| 195 | Q8 | — | 2 | 1 | no | — |
| 196 | Q8 | — | 12 | 1 | no | — |
| 197 | Q8 | — | 24 | 1 | no | — |
| 198 | D4 | — | 2 | 1 | no | — |
| 199 | D4 | — | 4 | 1 | no | — |
| 200 | D4 | — | 4 | 1 | no | — |
| 201 | D4 | — | 4 | 1 | no | — |
| 202 | D4 | — | 4 | 1 | no | — |
| 203 | D4 | — | 8 | 1 | no | — |
| 204 | D4 | — | 8 | 2 | no | — |
| 205 | D4 | — | 24 | 1 | no | — |
| 206 | D4 | — | 24 | 1 | no | — |
| 207 | D4 | — | 24 | 1 | no | — |
| 208 | D4 | — | 24 | 2 | no | — |
| 209 | C2×C4 | — | 2 | 1 | no | — |
| 210 | C2×C4 | — | 2 | 1 | no | — |
| 211 | C2×C4 | — | 2 | 1 | no | — |
| 212 | C2×C4 | — | 4 | 1 | no | — |
| 213 | C2×C4 | — | 4 | 1 | no | — |
| 214 | C2×C4 | — | 6 | 1 | no | — |
| 215 | C2×C4 | — | 6 | 1 | no | — |
| 216 | C2×C4 | — | 6 | 1 | no | — |
| 217 | C2×C4 | — | 12 | 1 | no | — |
| 218 | C2×C4 | D3×D4 | 1 | 1 | yes | characteristic |
| 219 | C8 | — | 12 | 1 | no | — |
| 220 | C2×C3 | — | 2 | 1 | no | — |
| 221 | C2×C3 | — | 4 | 1 | no | — |
| 222 | C2×C3 | — | 4 | 1 | no | — |
| 223 | C2×C3 | — | 8 | 1 | no | — |
| 224 | C2×C3 | — | 8 | 1 | no | — |
| 225 | C2×C3 | B(4,𝔽2) | 1 | 1 | yes | characteristic |
| 226 | D3 | — | 8 | 1 | no | — |
| 227 | C22 | — | 2 | 1 | no | — |
| 228 | C22 | — | 2 | 1 | no | — |
| 229 | C22 | — | 4 | 1 | no | — |
| 230 | C22 | — | 4 | 1 | no | — |
| 231 | C22 | — | 4 | 1 | no | — |
| 232 | C22 | — | 4 | 1 | no | — |
| 233 | C22 | — | 4 | 2 | no | — |
| 234 | C22 | — | 4 | 1 | no | — |
| 235 | C22 | — | 4 | 1 | no | — |
| 236 | C22 | — | 8 | 1 | no | — |
| 237 | C22 | — | 8 | 1 | no | — |
| 238 | C22 | — | 8 | 1 | no | — |
| 239 | C22 | — | 12 | 1 | no | — |
| 240 | C22 | — | 12 | 1 | no | — |
| 241 | C22 | — | 12 | 2 | no | — |
| 242 | C22 | — | 12 | 1 | no | — |
| 243 | C22 | — | 24 | 1 | no | — |
| 244 | C22 | #937 | 1 | 1 | yes | characteristic |
| 245 | C4 | — | 2 | 1 | no | — |
| 246 | C4 | — | 4 | 1 | no | — |
| 247 | C4 | — | 4 | 1 | no | — |
| 248 | C4 | — | 8 | 1 | no | — |
| 249 | C4 | — | 12 | 1 | no | — |
| 250 | C4 | — | 12 | 1 | no | — |
| 251 | C4 | — | 12 | 1 | no | — |
| 252 | C4 | — | 24 | 1 | no | — |
| 253 | C3 | #2199 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 254 | C2 | — | 2 | 1 | no | — |
| 255 | C2 | — | 4 | 1 | no | — |
| 256 | C2 | — | 4 | 1 | no | — |
| 257 | C2 | — | 8 | 1 | no | — |
| 258 | C2 | — | 8 | 1 | no | — |
| 259 | C2 | — | 24 | 1 | no | — |
| 260 | C2 | #5032 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #5594 | C2 | semidirect |
| 2. C3×B(4,𝔽2) | C2 | semidirect |
| 3. #5244 | C2 | semidirect |
| 4. #5242 | C2 | semidirect |
| 5. #4536 | C2 | semidirect |
| 6. #4473 | C2 | semidirect |
| 7. #4471 | C2 | semidirect |
| 30. B(4,𝔽2) | D3 | semidirect |
| 35. C24×C3 | D4 | semidirect |
| 36. C23×D3 | D4 | semidirect |
| 50. C2×Dic6 | D4 | semidirect |
| 54. C3×C22⋊C4 | D4 | semidirect |
| 60. #217 | D4 | semidirect |
| 68. C4×Dic3 | D4 | semidirect |
| 125. C24 | #67 | semidirect |
| 156. C22⋊C4 | #67 | semidirect |
| 253. C3 | #2199 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5594 | 2 |
| C3×B(4,𝔽2) | 2 |
| #5244 | 2 |
| #5242 | 2 |
| #4536 | 2 |
| #4473 | 2 |
| #4471 | 2 |
| #2199 | 3 |