Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
762 conjugacy classes of subgroups; only the 145 normal subgroups are shown.
This group has 762 conjugacy classes of subgroups — too many to draw. Only its 145 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 | C2×#1004 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #5805 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #5748 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C2×C3×#100 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #5139 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #4747 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #4746 | C2 | 1 | 4 | yes | maximal, complemented |
| 8 | #3019 | — | 3 | 1 | no | maximal, Sylow |
| 9 | C22×C3×D4 | C22 | 1 | 1 | yes | characteristic |
| 10 | C22×#67 | — | 2 | 2 | no | — |
| 11 | #1004 | — | 2 | 2 | no | — |
| 12 | #1004 | — | 2 | 4 | no | — |
| 13 | #1004 | C22 | 1 | 4 | yes | — |
| 14 | C2×#237 | — | 2 | 2 | no | — |
| 15 | C2×D3×D4 | — | 2 | 2 | no | — |
| 16 | C2×#235 | C22 | 1 | 2 | yes | — |
| 17 | C2×#235 | C22 | 1 | 1 | yes | characteristic, complemented |
| 18 | C2×C3×M16 | C22 | 1 | 2 | yes | complemented |
| 19 | #953 | — | 2 | 2 | no | — |
| 20 | #940 | C22 | 1 | 2 | yes | complemented |
| 21 | #938 | — | 2 | 2 | no | — |
| 22 | #936 | C22 | 1 | 2 | yes | complemented |
| 23 | #926 | C22 | 1 | 1 | yes | characteristic, complemented |
| 24 | #909 | — | 2 | 2 | no | — |
| 25 | #909 | C22 | 1 | 4 | yes | complemented |
| 26 | #908 | — | 2 | 2 | no | — |
| 27 | #908 | C22 | 1 | 4 | yes | complemented |
| 28 | #904 | — | 2 | 4 | no | — |
| 29 | C3×#100 | C22 | 1 | 4 | yes | complemented |
| 30 | #825 | C22 | 1 | 8 | yes | complemented |
| 31 | C2×H(5,𝔽2) | — | 3 | 1 | no | — |
| 32 | #575 | — | 3 | 2 | no | — |
| 33 | #535 | — | 3 | 1 | no | — |
| 34 | #455 | — | 3 | 4 | no | — |
| 35 | #453 | — | 3 | 4 | no | — |
| 36 | #421 | — | 3 | 2 | no | — |
| 37 | C2×#100 | D3 | 1 | 1 | yes | characteristic, complemented |
| 38 | C24×C3 | D4 | 1 | 2 | yes | — |
| 39 | C23×D3 | — | 2 | 2 | no | — |
| 40 | C2×C3×D4 | — | 2 | 2 | no | — |
| 41 | C2×C3×D4 | — | 2 | 2 | no | — |
| 42 | C2×C3×D4 | D4 | 1 | 2 | yes | — |
| 43 | C2×C3×D4 | C23 | 1 | 2 | yes | — |
| 44 | C22×C3×C4 | C23 | 1 | 1 | yes | characteristic |
| 45 | C2×#67 | — | 2 | 4 | no | — |
| 46 | C2×#67 | — | 2 | 8 | no | — |
| 47 | C2×#67 | — | 2 | 2 | no | — |
| 48 | C2×#67 | — | 2 | 8 | no | — |
| 49 | C2×#67 | — | 2 | 1 | no | — |
| 50 | C2×#67 | — | 2 | 4 | no | — |
| 51 | C22×Dic3 | — | 2 | 2 | no | — |
| 52 | #237 | — | 2 | 8 | no | — |
| 53 | #237 | — | 2 | 8 | no | — |
| 54 | D3×D4 | — | 2 | 8 | no | — |
| 55 | D3×D4 | — | 2 | 8 | no | — |
| 56 | #235 | — | 2 | 2 | no | — |
| 57 | #235 | — | 2 | 2 | no | — |
| 58 | #235 | — | 2 | 2 | no | — |
| 59 | #235 | — | 2 | 2 | no | — |
| 60 | #235 | D4 | 1 | 4 | yes | complemented |
| 61 | #235 | C23 | 1 | 4 | yes | — |
| 62 | C2×D12 | D4 | 1 | 2 | yes | complemented |
| 63 | C2×D12 | C23 | 1 | 1 | yes | characteristic, complemented |
| 64 | C2×C4×D3 | — | 2 | 2 | no | — |
| 65 | C2×C4×D3 | — | 2 | 1 | no | — |
| 66 | C2×Dic6 | D4 | 1 | 2 | yes | complemented |
| 67 | C2×Dic6 | C23 | 1 | 1 | yes | characteristic, complemented |
| 68 | C3×M16 | — | 2 | 2 | no | — |
| 69 | C3×M16 | C23 | 1 | 4 | yes | complemented |
| 70 | C2×C3×C8 | — | 2 | 2 | no | — |
| 71 | #217 | — | 2 | 1 | no | — |
| 72 | #217 | — | 2 | 4 | no | — |
| 73 | C4⋊Dic3 | — | 2 | 1 | no | — |
| 74 | C4×Dic3 | C23 | 1 | 2 | yes | complemented |
| 75 | Dic12 | — | 2 | 4 | no | — |
| 76 | D24 | — | 2 | 4 | no | — |
| 77 | #204 | — | 2 | 4 | no | — |
| 78 | #204 | — | 2 | 4 | no | — |
| 79 | H(5,𝔽2) | — | 3 | 4 | no | — |
| 80 | H(5,𝔽2) | — | 6 | 4 | no | — |
| 81 | H(5,𝔽2) | — | 6 | 2 | no | — |
| 82 | C2×#41 | — | 3 | 2 | no | — |
| 83 | C2×#41 | — | 3 | 1 | no | — |
| 84 | C2×#41 | — | 6 | 2 | no | — |
| 85 | C22×D4 | — | 6 | 2 | no | — |
| 86 | C22×D4 | — | 6 | 2 | no | — |
| 87 | C22×D4 | C2×D3 | 1 | 1 | yes | characteristic |
| 88 | #137 | — | 3 | 4 | no | — |
| 89 | #137 | — | 6 | 2 | no | — |
| 90 | AGL(1,ℤ/8) | — | 3 | 4 | no | — |
| 91 | AGL(1,ℤ/8) | — | 6 | 2 | no | — |
| 92 | #135 | — | 6 | 4 | no | — |
| 93 | C2×M16 | C2×D3 | 1 | 2 | yes | complemented |
| 94 | #127 | — | 3 | 2 | no | — |
| 95 | #124 | — | 3 | 2 | no | — |
| 96 | #121 | — | 6 | 2 | no | — |
| 97 | #120 | — | 6 | 2 | no | — |
| 98 | #117 | — | 3 | 1 | no | — |
| 99 | #104 | — | 3 | 8 | no | — |
| 100 | #100 | C2×D3 | 1 | 4 | yes | complemented |
| 101 | C23×C3 | — | 2 | 2 | no | — |
| 102 | C23×C3 | — | 2 | 2 | no | — |
| 103 | C23×C3 | — | 8 | 2 | no | — |
| 104 | C23×C3 | C2×D4 | 1 | 4 | yes | — |
| 105 | C23×C3 | C2×D4 | 1 | 1 | yes | characteristic |
| 106 | C22×D3 | — | 2 | 2 | no | — |
| 107 | C22×D3 | — | 2 | 8 | no | — |
| 108 | C22×D3 | — | 2 | 1 | no | — |
| 109 | C22×D3 | — | 8 | 4 | no | — |
| 110 | C3×D4 | — | 2 | 4 | no | — |
| 111 | C3×D4 | — | 2 | 4 | no | — |
| 112 | C2×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 113 | C2×C3×C4 | C2×D4 | 1 | 2 | yes | — |
| 114 | C2×C3×C4 | C2×D4 | 1 | 2 | yes | — |
| 115 | C2×C3×C4 | C24 | 1 | 1 | yes | characteristic |
| 116 | #67 | — | 2 | 16 | no | — |
| 117 | #67 | — | 2 | 4 | no | — |
| 118 | #67 | — | 2 | 4 | no | — |
| 119 | #67 | — | 4 | 1 | no | — |
| 120 | #67 | — | 4 | 1 | no | — |
| 121 | #67 | — | 8 | 4 | no | — |
| 122 | C2×Dic3 | — | 2 | 8 | no | — |
| 123 | C2×Dic3 | — | 2 | 2 | no | — |
| 124 | C2×Dic3 | — | 2 | 2 | no | — |
| 125 | C2×Dic3 | — | 2 | 1 | no | — |
| 126 | D12 | — | 2 | 2 | no | — |
| 127 | D12 | — | 2 | 2 | no | — |
| 128 | D12 | C2×D4 | 1 | 4 | yes | — |
| 129 | C4×D3 | — | 2 | 4 | no | — |
| 130 | C4×D3 | — | 2 | 4 | no | — |
| 131 | C4×D3 | — | 4 | 2 | no | — |
| 132 | Dic6 | — | 2 | 2 | no | — |
| 133 | Dic6 | — | 2 | 2 | no | — |
| 134 | Dic6 | C2×D4 | 1 | 4 | yes | — |
| 135 | C3×C8 | — | 2 | 4 | no | — |
| 136 | C24 | — | 6 | 2 | no | — |
| 137 | C24 | D12 | 1 | 2 | yes | — |
| 138 | #41 | — | 3 | 4 | no | — |
| 139 | #41 | — | 3 | 4 | no | — |
| 140 | #41 | — | 6 | 8 | no | — |
| 141 | #41 | — | 6 | 8 | no | — |
| 142 | #41 | — | 6 | 2 | no | — |
| 143 | #41 | — | 6 | 2 | no | — |
| 144 | #41 | — | 6 | 2 | no | — |
| 145 | #41 | — | 6 | 2 | no | — |
| 146 | C2×Q8 | — | 3 | 2 | no | — |
| 147 | C2×Q8 | — | 3 | 1 | no | — |
| 148 | C2×D4 | — | 2 | 2 | no | — |
| 149 | C2×D4 | — | 2 | 2 | no | — |
| 150 | C2×D4 | — | 3 | 2 | no | — |
| 151 | C2×D4 | — | 3 | 1 | no | — |
| 152 | C2×D4 | — | 6 | 8 | no | — |
| 153 | C2×D4 | — | 6 | 8 | no | — |
| 154 | C2×D4 | — | 6 | 4 | no | — |
| 155 | C2×D4 | — | 6 | 8 | no | — |
| 156 | C2×D4 | — | 6 | 2 | no | — |
| 157 | C2×D4 | — | 6 | 8 | no | — |
| 158 | C2×D4 | — | 6 | 1 | no | — |
| 159 | C2×D4 | — | 6 | 4 | no | — |
| 160 | C2×D4 | D12 | 1 | 2 | yes | — |
| 161 | C2×D4 | C22×D3 | 1 | 2 | yes | — |
| 162 | C22×C4 | — | 6 | 2 | no | — |
| 163 | C22×C4 | — | 6 | 2 | no | — |
| 164 | C22×C4 | — | 6 | 1 | no | — |
| 165 | C22×C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 166 | Q16 | — | 6 | 4 | no | — |
| 167 | SD16 | — | 6 | 4 | no | — |
| 168 | SD16 | — | 6 | 4 | no | — |
| 169 | D8 | — | 6 | 4 | no | — |
| 170 | M16 | — | 2 | 2 | no | — |
| 171 | M16 | C22×D3 | 1 | 4 | yes | complemented |
| 172 | C2×C8 | — | 2 | 2 | no | — |
| 173 | C4⋊C4 | — | 6 | 1 | no | — |
| 174 | C22⋊C4 | — | 6 | 1 | no | — |
| 175 | C22⋊C4 | — | 6 | 4 | no | — |
| 176 | C42 | — | 3 | 2 | no | — |
| 177 | C22×C3 | — | 2 | 4 | no | — |
| 178 | C22×C3 | — | 4 | 1 | no | — |
| 179 | C22×C3 | — | 4 | 4 | no | — |
| 180 | C22×C3 | — | 4 | 4 | no | — |
| 181 | C22×C3 | — | 8 | 2 | no | — |
| 182 | C22×C3 | #120 | 1 | 1 | yes | characteristic |
| 183 | C22×C3 | #120 | 1 | 1 | yes | characteristic |
| 184 | C22×C3 | C22×D4 | 1 | 1 | yes | characteristic |
| 185 | C2×D3 | — | 2 | 4 | no | — |
| 186 | C2×D3 | — | 4 | 4 | no | — |
| 187 | C2×D3 | — | 4 | 2 | no | — |
| 188 | C2×D3 | — | 4 | 1 | no | — |
| 189 | C2×D3 | — | 8 | 8 | no | — |
| 190 | C3×C4 | #120 | 1 | 2 | yes | — |
| 191 | C3×C4 | C22×D4 | 1 | 2 | yes | — |
| 192 | Dic3 | — | 2 | 4 | no | — |
| 193 | Dic3 | — | 4 | 2 | no | — |
| 194 | Dic3 | — | 4 | 1 | no | — |
| 195 | C23 | — | 2 | 2 | no | — |
| 196 | C23 | — | 2 | 2 | no | — |
| 197 | C23 | — | 6 | 8 | no | — |
| 198 | C23 | — | 6 | 2 | no | — |
| 199 | C23 | — | 6 | 1 | no | — |
| 200 | C23 | — | 8 | 2 | no | — |
| 201 | C23 | — | 24 | 4 | no | — |
| 202 | C23 | C2×D12 | 1 | 4 | yes | — |
| 203 | C23 | C2×D12 | 1 | 1 | yes | characteristic |
| 204 | Q8 | — | 3 | 4 | no | — |
| 205 | Q8 | — | 6 | 2 | no | — |
| 206 | Q8 | — | 6 | 2 | no | — |
| 207 | D4 | — | 2 | 4 | no | — |
| 208 | D4 | — | 2 | 4 | no | — |
| 209 | D4 | — | 3 | 4 | no | — |
| 210 | D4 | — | 6 | 16 | no | — |
| 211 | D4 | — | 6 | 4 | no | — |
| 212 | D4 | — | 6 | 4 | no | — |
| 213 | D4 | — | 6 | 2 | no | — |
| 214 | D4 | — | 6 | 2 | no | — |
| 215 | D4 | — | 12 | 1 | no | — |
| 216 | D4 | — | 12 | 1 | no | — |
| 217 | D4 | — | 24 | 4 | no | — |
| 218 | C2×C4 | — | 6 | 8 | no | — |
| 219 | C2×C4 | — | 6 | 2 | no | — |
| 220 | C2×C4 | — | 6 | 4 | no | — |
| 221 | C2×C4 | — | 6 | 4 | no | — |
| 222 | C2×C4 | — | 6 | 2 | no | — |
| 223 | C2×C4 | — | 6 | 1 | no | — |
| 224 | C2×C4 | — | 12 | 2 | no | — |
| 225 | C2×C4 | C2×D12 | 1 | 1 | yes | characteristic |
| 226 | C2×C4 | D3×D4 | 1 | 2 | yes | — |
| 227 | C2×C4 | D3×D4 | 1 | 2 | yes | — |
| 228 | C2×C4 | C23×D3 | 1 | 1 | yes | characteristic |
| 229 | C8 | — | 2 | 4 | no | — |
| 230 | C2×C3 | — | 2 | 1 | no | — |
| 231 | C2×C3 | — | 2 | 1 | no | — |
| 232 | C2×C3 | — | 2 | 1 | no | — |
| 233 | C2×C3 | — | 4 | 4 | no | — |
| 234 | C2×C3 | C2×#120 | 1 | 1 | yes | characteristic |
| 235 | D3 | — | 4 | 4 | no | — |
| 236 | D3 | — | 8 | 1 | no | — |
| 237 | C22 | — | 2 | 4 | no | — |
| 238 | C22 | — | 4 | 1 | no | — |
| 239 | C22 | — | 4 | 4 | no | — |
| 240 | C22 | — | 4 | 4 | no | — |
| 241 | C22 | — | 6 | 4 | no | — |
| 242 | C22 | — | 8 | 2 | no | — |
| 243 | C22 | — | 12 | 4 | no | — |
| 244 | C22 | — | 12 | 2 | no | — |
| 245 | C22 | — | 12 | 1 | no | — |
| 246 | C22 | — | 24 | 8 | no | — |
| 247 | C22 | #882 | 1 | 1 | yes | characteristic |
| 248 | C22 | #882 | 1 | 1 | yes | characteristic |
| 249 | C22 | C22×D12 | 1 | 1 | yes | characteristic |
| 250 | C4 | — | 6 | 4 | no | — |
| 251 | C4 | — | 12 | 2 | no | — |
| 252 | C4 | — | 12 | 1 | no | — |
| 253 | C4 | #882 | 1 | 2 | yes | — |
| 254 | C4 | C2×D3×D4 | 1 | 2 | yes | — |
| 255 | C3 | #3019 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 256 | C2 | — | 2 | 1 | no | — |
| 257 | C2 | — | 2 | 1 | no | — |
| 258 | C2 | — | 2 | 1 | no | — |
| 259 | C2 | — | 4 | 4 | no | — |
| 260 | C2 | — | 12 | 4 | no | — |
| 261 | C2 | — | 24 | 1 | no | — |
| 262 | C2 | C2×#882 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#1004 | C2 | semidirect |
| 2. #5805 | C2 | semidirect |
| 3. #5748 | C2 | semidirect |
| 4. C2×C3×#100 | C2 | semidirect |
| 5. #5139 | C2 | semidirect |
| 6. #4747 | C2 | semidirect |
| 7. #4746 | C2 | semidirect |
| 17. C2×#235 | C22 | semidirect |
| 18. C2×C3×M16 | C22 | semidirect |
| 20. #940 | C22 | semidirect |
| 22. #936 | C22 | semidirect |
| 23. #926 | C22 | semidirect |
| 25. #909 | C22 | semidirect |
| 27. #908 | C22 | semidirect |
| 29. C3×#100 | C22 | semidirect |
| 30. #825 | C22 | semidirect |
| 37. C2×#100 | D3 | semidirect |
| 60. #235 | D4 | semidirect |
| 62. C2×D12 | D4 | semidirect |
| 63. C2×D12 | C23 | semidirect |
| 66. C2×Dic6 | D4 | semidirect |
| 67. C2×Dic6 | C23 | semidirect |
| 69. C3×M16 | C23 | semidirect |
| 74. C4×Dic3 | C23 | semidirect |
| 93. C2×M16 | C2×D3 | semidirect |
| 100. #100 | C2×D3 | semidirect |
| 171. M16 | C22×D3 | semidirect |
| 255. C3 | #3019 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#1004 | 2 |
| #5805 | 2 |
| #5748 | 2 |
| C2×C3×#100 | 2 |
| #5139 | 2 |
| #4747 | 2 |
| #4746 | 2 |
| #3019 | 3 |