Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
736 conjugacy classes of subgroups; only the 145 normal subgroups are shown.
This group has 736 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 | #5964 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #5830 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | C2×#908 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | C3×#413 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | #5139 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | #4753 | C2 | 1 | 4 | yes | maximal, complemented |
| 7 | #4747 | C2 | 1 | 4 | yes | maximal, complemented |
| 8 | #3021 | — | 3 | 1 | no | maximal, Sylow |
| 9 | C2×C3×#41 | C22 | 1 | 1 | yes | characteristic |
| 10 | #1010 | — | 2 | 2 | no | — |
| 11 | #1009 | — | 2 | 2 | no | — |
| 12 | D3×#41 | — | 2 | 4 | no | — |
| 13 | #1007 | C22 | 1 | 2 | yes | complemented |
| 14 | #1004 | — | 2 | 2 | no | — |
| 15 | #1004 | C22 | 1 | 2 | yes | complemented |
| 16 | C2×#235 | — | 2 | 2 | no | — |
| 17 | C2×#235 | C22 | 1 | 2 | yes | — |
| 18 | C22×D12 | C22 | 1 | 1 | yes | characteristic, complemented |
| 19 | C2×C3×M16 | C22 | 1 | 2 | yes | complemented |
| 20 | #947 | C22 | 1 | 2 | yes | complemented |
| 21 | #940 | C22 | 1 | 2 | yes | complemented |
| 22 | #937 | — | 2 | 2 | no | — |
| 23 | #930 | — | 2 | 2 | no | — |
| 24 | #926 | C22 | 1 | 1 | yes | characteristic, complemented |
| 25 | #908 | — | 2 | 4 | no | — |
| 26 | #908 | C22 | 1 | 8 | yes | complemented |
| 27 | C2×D24 | — | 2 | 2 | no | — |
| 28 | C2×#204 | — | 2 | 2 | no | — |
| 29 | C3×#101 | C22 | 1 | 2 | yes | complemented |
| 30 | C3×#100 | C22 | 1 | 2 | yes | complemented |
| 31 | #825 | C22 | 1 | 8 | yes | complemented |
| 32 | #585 | — | 3 | 1 | no | — |
| 33 | C2×AGL(1,ℤ/8) | — | 3 | 2 | no | — |
| 34 | #535 | — | 3 | 1 | no | — |
| 35 | #454 | — | 3 | 4 | no | — |
| 36 | #453 | — | 3 | 4 | no | — |
| 37 | #421 | — | 3 | 2 | no | — |
| 38 | #413 | D3 | 1 | 1 | yes | characteristic, complemented |
| 39 | C23×D3 | — | 2 | 1 | no | — |
| 40 | C3×#41 | — | 2 | 2 | no | — |
| 41 | C3×#41 | — | 2 | 2 | no | — |
| 42 | C2×C3×Q8 | C23 | 1 | 1 | yes | characteristic |
| 43 | C2×C3×D4 | D4 | 1 | 2 | yes | — |
| 44 | C2×C3×D4 | C23 | 1 | 1 | yes | characteristic |
| 45 | C22×C3×C4 | D4 | 1 | 2 | yes | — |
| 46 | C22×C3×C4 | C23 | 1 | 1 | yes | characteristic |
| 47 | C2×#67 | — | 2 | 4 | no | — |
| 48 | C2×#67 | — | 2 | 2 | no | — |
| 49 | C2×#67 | — | 2 | 4 | no | — |
| 50 | #239 | — | 2 | 4 | no | — |
| 51 | D3×Q8 | — | 2 | 4 | no | — |
| 52 | #237 | — | 2 | 8 | no | — |
| 53 | #237 | — | 2 | 4 | no | — |
| 54 | D3×D4 | — | 2 | 8 | no | — |
| 55 | D3×D4 | — | 2 | 4 | no | — |
| 56 | #235 | — | 2 | 4 | no | — |
| 57 | #235 | — | 2 | 4 | no | — |
| 58 | #235 | — | 2 | 2 | no | — |
| 59 | #235 | — | 2 | 4 | no | — |
| 60 | #235 | — | 2 | 2 | no | — |
| 61 | #235 | D4 | 1 | 4 | yes | complemented |
| 62 | #235 | C23 | 1 | 4 | yes | — |
| 63 | C2×D12 | — | 2 | 2 | no | — |
| 64 | C2×D12 | — | 2 | 2 | no | — |
| 65 | C2×D12 | — | 2 | 2 | no | — |
| 66 | C2×D12 | — | 2 | 1 | no | — |
| 67 | C2×D12 | D4 | 1 | 2 | yes | complemented |
| 68 | C2×D12 | C23 | 1 | 2 | yes | — |
| 69 | C2×C4×D3 | — | 2 | 4 | no | — |
| 70 | C2×C4×D3 | — | 2 | 2 | no | — |
| 71 | C2×Dic6 | — | 2 | 2 | no | — |
| 72 | C2×Dic6 | D4 | 1 | 2 | yes | complemented |
| 73 | C3×M16 | — | 2 | 2 | no | — |
| 74 | C3×M16 | C23 | 1 | 4 | yes | — |
| 75 | C2×C3×C8 | — | 2 | 2 | no | — |
| 76 | #217 | — | 2 | 1 | no | — |
| 77 | (C2×D3)⋊C4 | — | 2 | 4 | no | — |
| 78 | C4⋊Dic3 | — | 2 | 1 | no | — |
| 79 | C4×Dic3 | C23 | 1 | 2 | yes | — |
| 80 | D24 | — | 2 | 8 | no | — |
| 81 | #204 | — | 2 | 8 | no | — |
| 82 | 2−1+4 | — | 3 | 2 | no | — |
| 83 | 2−1+4 | — | 6 | 2 | no | — |
| 84 | H(5,𝔽2) | — | 3 | 2 | no | — |
| 85 | H(5,𝔽2) | — | 6 | 2 | no | — |
| 86 | H(5,𝔽2) | — | 6 | 2 | no | — |
| 87 | C2×#41 | — | 3 | 2 | no | — |
| 88 | C2×#41 | — | 6 | 4 | no | — |
| 89 | C2×#41 | — | 6 | 2 | no | — |
| 90 | C2×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 91 | C22×D4 | — | 3 | 1 | no | — |
| 92 | AGL(1,ℤ/8) | — | 3 | 8 | no | — |
| 93 | AGL(1,ℤ/8) | — | 6 | 4 | no | — |
| 94 | C2×SD16 | — | 6 | 2 | no | — |
| 95 | C2×D8 | — | 6 | 2 | no | — |
| 96 | C2×M16 | C2×D3 | 1 | 2 | yes | complemented |
| 97 | #127 | — | 3 | 2 | no | — |
| 98 | #124 | — | 3 | 2 | no | — |
| 99 | #121 | — | 6 | 2 | no | — |
| 100 | #120 | — | 6 | 2 | no | — |
| 101 | #117 | — | 3 | 1 | no | — |
| 102 | #104 | — | 3 | 8 | no | — |
| 103 | #101 | C2×D3 | 1 | 2 | yes | complemented |
| 104 | #100 | C2×D3 | 1 | 2 | yes | complemented |
| 105 | C23×C3 | C2×D4 | 1 | 2 | yes | — |
| 106 | C23×C3 | C2×D4 | 1 | 1 | yes | characteristic |
| 107 | C22×D3 | — | 2 | 4 | no | — |
| 108 | C22×D3 | — | 2 | 2 | no | — |
| 109 | C22×D3 | — | 2 | 2 | no | — |
| 110 | C22×D3 | — | 4 | 2 | no | — |
| 111 | C22×D3 | — | 8 | 2 | no | — |
| 112 | C3×Q8 | — | 2 | 2 | no | — |
| 113 | C3×D4 | — | 2 | 4 | no | — |
| 114 | C3×D4 | — | 2 | 2 | no | — |
| 115 | C2×C3×C4 | — | 2 | 2 | no | — |
| 116 | C2×C3×C4 | — | 2 | 2 | no | — |
| 117 | C2×C3×C4 | C2×D4 | 1 | 2 | yes | — |
| 118 | C2×C3×C4 | C2×D4 | 1 | 1 | yes | characteristic |
| 119 | C2×C3×C4 | C2×D4 | 1 | 2 | yes | — |
| 120 | C2×C3×C4 | C2×D4 | 1 | 2 | yes | — |
| 121 | C2×C3×C4 | C24 | 1 | 1 | yes | characteristic |
| 122 | #67 | — | 2 | 8 | no | — |
| 123 | #67 | — | 2 | 4 | no | — |
| 124 | #67 | — | 2 | 4 | no | — |
| 125 | #67 | — | 8 | 4 | no | — |
| 126 | C2×Dic3 | — | 2 | 4 | no | — |
| 127 | C2×Dic3 | — | 2 | 2 | no | — |
| 128 | C2×Dic3 | — | 2 | 2 | no | — |
| 129 | D12 | — | 2 | 4 | no | — |
| 130 | D12 | — | 2 | 2 | no | — |
| 131 | D12 | — | 2 | 4 | no | — |
| 132 | D12 | — | 4 | 1 | no | — |
| 133 | D12 | — | 4 | 1 | no | — |
| 134 | D12 | C2×D4 | 1 | 4 | yes | — |
| 135 | C4×D3 | — | 2 | 8 | no | — |
| 136 | C4×D3 | — | 2 | 4 | no | — |
| 137 | C4×D3 | — | 2 | 4 | no | — |
| 138 | Dic6 | — | 2 | 4 | no | — |
| 139 | Dic6 | — | 2 | 2 | no | — |
| 140 | Dic6 | C2×D4 | 1 | 4 | yes | — |
| 141 | C3×C8 | — | 2 | 4 | no | — |
| 142 | C24 | — | 6 | 1 | no | — |
| 143 | #41 | — | 2 | 2 | no | — |
| 144 | #41 | — | 2 | 2 | no | — |
| 145 | #41 | — | 3 | 4 | no | — |
| 146 | #41 | — | 3 | 4 | no | — |
| 147 | #41 | — | 6 | 4 | no | — |
| 148 | #41 | — | 6 | 4 | no | — |
| 149 | #41 | — | 6 | 4 | no | — |
| 150 | #41 | — | 6 | 8 | no | — |
| 151 | #41 | — | 6 | 4 | no | — |
| 152 | #41 | — | 6 | 2 | no | — |
| 153 | #41 | — | 6 | 4 | no | — |
| 154 | #41 | — | 6 | 2 | no | — |
| 155 | C2×Q8 | — | 3 | 2 | no | — |
| 156 | C2×Q8 | — | 6 | 2 | no | — |
| 157 | C2×Q8 | — | 6 | 4 | no | — |
| 158 | C2×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 159 | C2×D4 | — | 3 | 2 | no | — |
| 160 | C2×D4 | — | 3 | 2 | no | — |
| 161 | C2×D4 | — | 6 | 2 | no | — |
| 162 | C2×D4 | — | 6 | 4 | no | — |
| 163 | C2×D4 | — | 6 | 8 | no | — |
| 164 | C2×D4 | — | 6 | 2 | no | — |
| 165 | C2×D4 | — | 6 | 4 | no | — |
| 166 | C2×D4 | — | 6 | 4 | no | — |
| 167 | C2×D4 | — | 6 | 1 | no | — |
| 168 | C2×D4 | — | 6 | 2 | no | — |
| 169 | C2×D4 | — | 6 | 2 | no | — |
| 170 | C2×D4 | D12 | 1 | 2 | yes | — |
| 171 | C2×D4 | C22×D3 | 1 | 1 | yes | characteristic |
| 172 | C22×C4 | — | 6 | 4 | no | — |
| 173 | C22×C4 | — | 6 | 2 | no | — |
| 174 | C22×C4 | D12 | 1 | 2 | yes | — |
| 175 | C22×C4 | C22×D3 | 1 | 1 | yes | characteristic |
| 176 | SD16 | — | 6 | 8 | no | — |
| 177 | D8 | — | 6 | 8 | no | — |
| 178 | M16 | — | 2 | 2 | no | — |
| 179 | M16 | C22×D3 | 1 | 4 | yes | — |
| 180 | C2×C8 | — | 2 | 2 | no | — |
| 181 | C4⋊C4 | — | 6 | 1 | no | — |
| 182 | C22⋊C4 | — | 6 | 1 | no | — |
| 183 | C22⋊C4 | — | 6 | 4 | no | — |
| 184 | C42 | — | 3 | 2 | no | — |
| 185 | C22×C3 | — | 2 | 2 | no | — |
| 186 | C22×C3 | — | 4 | 1 | no | — |
| 187 | C22×C3 | — | 4 | 2 | no | — |
| 188 | C22×C3 | #120 | 1 | 2 | yes | — |
| 189 | C22×C3 | C22×D4 | 1 | 1 | yes | characteristic |
| 190 | C2×D3 | — | 2 | 4 | no | — |
| 191 | C2×D3 | — | 4 | 4 | no | — |
| 192 | C2×D3 | — | 4 | 2 | no | — |
| 193 | C2×D3 | — | 8 | 2 | no | — |
| 194 | C2×D3 | — | 8 | 4 | no | — |
| 195 | C2×D3 | — | 8 | 4 | no | — |
| 196 | C3×C4 | — | 2 | 2 | no | — |
| 197 | C3×C4 | #120 | 1 | 1 | yes | characteristic |
| 198 | C3×C4 | #120 | 1 | 1 | yes | characteristic |
| 199 | C3×C4 | C22×D4 | 1 | 2 | yes | — |
| 200 | Dic3 | — | 2 | 4 | no | — |
| 201 | Dic3 | — | 4 | 2 | no | — |
| 202 | C23 | — | 6 | 4 | no | — |
| 203 | C23 | — | 6 | 2 | no | — |
| 204 | C23 | — | 6 | 2 | no | — |
| 205 | C23 | — | 12 | 2 | no | — |
| 206 | C23 | — | 24 | 2 | no | — |
| 207 | C23 | C2×D12 | 1 | 2 | yes | — |
| 208 | C23 | C2×D12 | 1 | 1 | yes | characteristic |
| 209 | Q8 | — | 2 | 2 | no | — |
| 210 | Q8 | — | 3 | 4 | no | — |
| 211 | Q8 | — | 6 | 4 | no | — |
| 212 | Q8 | — | 6 | 2 | no | — |
| 213 | D4 | — | 2 | 4 | no | — |
| 214 | D4 | — | 2 | 2 | no | — |
| 215 | D4 | — | 3 | 4 | no | — |
| 216 | D4 | — | 6 | 4 | no | — |
| 217 | D4 | — | 6 | 8 | no | — |
| 218 | D4 | — | 6 | 4 | no | — |
| 219 | D4 | — | 6 | 4 | no | — |
| 220 | D4 | — | 6 | 2 | no | — |
| 221 | D4 | — | 6 | 4 | no | — |
| 222 | D4 | — | 12 | 1 | no | — |
| 223 | D4 | — | 12 | 1 | no | — |
| 224 | D4 | — | 24 | 4 | no | — |
| 225 | C2×C4 | — | 2 | 2 | no | — |
| 226 | C2×C4 | — | 2 | 2 | no | — |
| 227 | C2×C4 | — | 6 | 8 | no | — |
| 228 | C2×C4 | — | 6 | 4 | no | — |
| 229 | C2×C4 | — | 6 | 2 | no | — |
| 230 | C2×C4 | — | 6 | 4 | no | — |
| 231 | C2×C4 | — | 6 | 4 | no | — |
| 232 | C2×C4 | — | 6 | 2 | no | — |
| 233 | C2×C4 | C2×D12 | 1 | 2 | yes | — |
| 234 | C2×C4 | C2×D12 | 1 | 1 | yes | characteristic |
| 235 | C2×C4 | D3×D4 | 1 | 2 | yes | — |
| 236 | C2×C4 | D3×D4 | 1 | 2 | yes | — |
| 237 | C2×C4 | C23×D3 | 1 | 1 | yes | characteristic |
| 238 | C8 | — | 2 | 4 | no | — |
| 239 | C2×C3 | — | 2 | 1 | no | — |
| 240 | C2×C3 | — | 2 | 2 | no | — |
| 241 | C2×C3 | — | 4 | 2 | no | — |
| 242 | C2×C3 | C2×#120 | 1 | 1 | yes | characteristic |
| 243 | D3 | — | 4 | 4 | no | — |
| 244 | D3 | — | 8 | 2 | no | — |
| 245 | C22 | — | 2 | 2 | no | — |
| 246 | C22 | — | 4 | 1 | no | — |
| 247 | C22 | — | 4 | 2 | no | — |
| 248 | C22 | — | 6 | 4 | no | — |
| 249 | C22 | — | 12 | 4 | no | — |
| 250 | C22 | — | 12 | 2 | no | — |
| 251 | C22 | — | 24 | 2 | no | — |
| 252 | C22 | — | 24 | 4 | no | — |
| 253 | C22 | — | 24 | 4 | no | — |
| 254 | C22 | #882 | 1 | 2 | yes | — |
| 255 | C22 | C22×D12 | 1 | 1 | yes | characteristic |
| 256 | C4 | — | 2 | 2 | no | — |
| 257 | C4 | — | 6 | 4 | no | — |
| 258 | C4 | — | 12 | 2 | no | — |
| 259 | C4 | #882 | 1 | 1 | yes | characteristic |
| 260 | C4 | #882 | 1 | 1 | yes | characteristic |
| 261 | C4 | C2×D3×D4 | 1 | 2 | yes | — |
| 262 | C3 | #3021 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 263 | C2 | — | 2 | 1 | no | — |
| 264 | C2 | — | 2 | 2 | no | — |
| 265 | C2 | — | 4 | 2 | no | — |
| 266 | C2 | — | 12 | 4 | no | — |
| 267 | C2 | — | 24 | 2 | no | — |
| 268 | C2 | C2×#882 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#882 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#413 | C2 |
| socle | C2×C3 | C2×#120 |
| hypercenter | #413 | D3 |
| hypocenter | C3 | #3021 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #413 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #5964 | C2 | semidirect |
| 2. #5830 | C2 | semidirect |
| 3. C2×#908 | C2 | semidirect |
| 4. C3×#413 | C2 | semidirect |
| 5. #5139 | C2 | semidirect |
| 6. #4753 | C2 | semidirect |
| 7. #4747 | C2 | semidirect |
| 13. #1007 | C22 | semidirect |
| 15. #1004 | C22 | semidirect |
| 18. C22×D12 | C22 | semidirect |
| 19. C2×C3×M16 | C22 | semidirect |
| 20. #947 | C22 | semidirect |
| 21. #940 | C22 | semidirect |
| 24. #926 | C22 | semidirect |
| 26. #908 | C22 | semidirect |
| 29. C3×#101 | C22 | semidirect |
| 30. C3×#100 | C22 | semidirect |
| 31. #825 | C22 | semidirect |
| 38. #413 | D3 | semidirect |
| 61. #235 | D4 | semidirect |
| 67. C2×D12 | D4 | semidirect |
| 72. C2×Dic6 | D4 | semidirect |
| 96. C2×M16 | C2×D3 | semidirect |
| 103. #101 | C2×D3 | semidirect |
| 104. #100 | C2×D3 | semidirect |
| 262. C3 | #3021 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #5964 | 2 |
| #5830 | 2 |
| C2×#908 | 2 |
| C3×#413 | 2 |
| #5139 | 2 |
| #4753 | 2 |
| #4747 | 2 |
| #3021 | 3 |