Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
381 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 381 conjugacy classes of subgroups — too many to draw. Only its 67 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 | C4×#984 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | C8×#41⋊C3 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #73071 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #226⋊C8 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C8×#226 | C2 | 1 | 2 | yes | maximal, complemented |
| 6 | C8×#137 | — | 3 | 1 | no | maximal, Sylow |
| 7 | C2×#984 | C4 | 1 | 1 | yes | characteristic |
| 8 | C4×#41⋊C3 | C22 | 1 | 1 | yes | characteristic |
| 9 | #5426 | C4 | 1 | 1 | yes | characteristic |
| 10 | #5388 | C4 | 1 | 2 | yes | — |
| 11 | C4×#226 | C22 | 1 | 2 | yes | — |
| 12 | C8×Dic6 | — | 4 | 1 | no | maximal |
| 13 | C8×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 14 | C8×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 15 | SL(2,𝔽3)⋊C8 | C22 | 1 | 2 | yes | — |
| 16 | C8×#41 | D3 | 1 | 1 | yes | characteristic |
| 17 | C2×C8×Q8 | — | 3 | 1 | no | — |
| 18 | C4×#137 | — | 3 | 1 | no | — |
| 19 | #1579 | — | 3 | 2 | no | — |
| 20 | SD16⋊C8 | — | 3 | 2 | no | — |
| 21 | C8×Q16 | — | 3 | 2 | no | — |
| 22 | C8×SD16 | — | 3 | 2 | no | — |
| 23 | #1565 | — | 3 | 1 | no | — |
| 24 | #1479 | — | 3 | 1 | no | — |
| 25 | #1477 | — | 3 | 1 | no | — |
| 26 | C8×M16 | — | 3 | 1 | no | — |
| 27 | C2×#41⋊C3 | C2×C4 | 1 | 1 | yes | characteristic |
| 28 | #984 | C8 | 1 | 4 | yes | direct product factor |
| 29 | C2×#226 | C2×C4 | 1 | 2 | yes | — |
| 30 | C4×Dic6 | — | 4 | 1 | no | — |
| 31 | C4×SL(2,𝔽3) | C2×C4 | 1 | 1 | yes | characteristic |
| 32 | C4×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 33 | #859 | C2×C4 | 1 | 2 | yes | — |
| 34 | C3×C4×C8 | — | 4 | 1 | no | — |
| 35 | Dic3⋊C8 | — | 4 | 2 | no | — |
| 36 | C8×Dic3 | — | 4 | 2 | no | — |
| 37 | C4⋊(C3⋊C8) | — | 4 | 1 | no | — |
| 38 | C2×#137 | — | 3 | 1 | no | — |
| 39 | C4×#41 | C2×D3 | 1 | 1 | yes | characteristic |
| 40 | C2×C4×Q8 | — | 3 | 1 | no | — |
| 41 | C8×Q8 | — | 3 | 2 | no | — |
| 42 | C8×Q8 | — | 3 | 2 | no | — |
| 43 | C8×Q8 | — | 6 | 1 | no | — |
| 44 | C8×Q8 | — | 6 | 1 | no | — |
| 45 | C8×Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 46 | C8×Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 47 | #441 | — | 3 | 2 | no | — |
| 48 | #440 | — | 3 | 2 | no | — |
| 49 | C4×Q16 | — | 3 | 2 | no | — |
| 50 | C4×SD16 | — | 3 | 2 | no | — |
| 51 | C8×D4 | — | 3 | 1 | no | — |
| 52 | C8×D4 | — | 3 | 1 | no | — |
| 53 | #431 | — | 3 | 1 | no | — |
| 54 | #431 | — | 3 | 1 | no | — |
| 55 | #428 | — | 3 | 1 | no | — |
| 56 | C2×C4⋊C8 | — | 3 | 1 | no | — |
| 57 | C2×C4⋊C8 | — | 6 | 1 | no | — |
| 58 | #419 | — | 3 | 1 | no | — |
| 59 | #417 | — | 3 | 1 | no | — |
| 60 | C4×M16 | — | 3 | 1 | no | — |
| 61 | C2×C4×C8 | — | 3 | 1 | no | — |
| 62 | C2×C4×C8 | — | 6 | 1 | no | — |
| 63 | #335 | — | 3 | 2 | no | — |
| 64 | #334 | — | 3 | 2 | no | — |
| 65 | Q8⋊C8 | — | 3 | 2 | no | — |
| 66 | Q8⋊C8 | — | 3 | 2 | no | — |
| 67 | Q8⋊C8 | — | 3 | 2 | no | — |
| 68 | D4⋊C8 | — | 3 | 2 | no | — |
| 69 | #322 | — | 3 | 2 | no | — |
| 70 | C82 | — | 3 | 2 | no | — |
| 71 | C2×Dic6 | — | 4 | 1 | no | — |
| 72 | #41⋊C3 | C2×C8 | 1 | 2 | yes | — |
| 73 | C2×SL(2,𝔽3) | C22×C4 | 1 | 1 | yes | characteristic |
| 74 | #226 | C2×C8 | 1 | 4 | yes | complemented |
| 75 | C2×C3×C8 | — | 4 | 1 | no | — |
| 76 | C2×C3×C8 | — | 4 | 1 | no | — |
| 77 | C3×C42 | — | 4 | 1 | no | — |
| 78 | C4⋊Dic3 | — | 4 | 1 | no | — |
| 79 | Dic3⋊C4 | — | 4 | 2 | no | — |
| 80 | C4×Dic3 | — | 4 | 2 | no | — |
| 81 | C2×C3⋊C8 | — | 4 | 2 | no | — |
| 82 | C2×#41 | C4×D3 | 1 | 1 | yes | characteristic |
| 83 | C22×Q8 | — | 3 | 1 | no | — |
| 84 | #137 | — | 3 | 4 | no | — |
| 85 | #137 | — | 3 | 4 | no | — |
| 86 | C2×Q16 | — | 3 | 2 | no | — |
| 87 | C2×SD16 | — | 3 | 2 | no | — |
| 88 | C2×M16 | — | 3 | 1 | no | — |
| 89 | C2×M16 | — | 3 | 1 | no | — |
| 90 | C22×C8 | — | 3 | 1 | no | — |
| 91 | C22×C8 | — | 3 | 1 | no | — |
| 92 | C22×C8 | — | 6 | 1 | no | — |
| 93 | C4×Q8 | — | 3 | 2 | no | — |
| 94 | C4×Q8 | — | 3 | 2 | no | — |
| 95 | C4×Q8 | — | 6 | 1 | no | — |
| 96 | C4×Q8 | — | 6 | 1 | no | — |
| 97 | C4×Q8 | C4×D3 | 1 | 1 | yes | characteristic |
| 98 | C4×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 99 | C4×D4 | — | 3 | 1 | no | — |
| 100 | C4×D4 | — | 3 | 1 | no | — |
| 101 | #117 | — | 3 | 1 | no | — |
| 102 | C2×C4⋊C4 | — | 3 | 1 | no | — |
| 103 | C2×C4⋊C4 | — | 6 | 1 | no | — |
| 104 | C2×C42 | — | 3 | 1 | no | — |
| 105 | C2×C42 | — | 6 | 1 | no | — |
| 106 | #107 | — | 3 | 2 | no | — |
| 107 | #106 | — | 3 | 2 | no | — |
| 108 | C4⋊C8 | — | 3 | 1 | no | — |
| 109 | C4⋊C8 | — | 3 | 1 | no | — |
| 110 | C4⋊C8 | — | 3 | 2 | no | — |
| 111 | C4⋊C8 | — | 6 | 1 | no | — |
| 112 | C4⋊C8 | — | 6 | 2 | no | — |
| 113 | C4⋊C8 | — | 6 | 2 | no | — |
| 114 | C4⋊C8 | — | 6 | 1 | no | — |
| 115 | #103 | — | 3 | 2 | no | — |
| 116 | #103 | — | 3 | 2 | no | — |
| 117 | #103 | — | 3 | 2 | no | — |
| 118 | #102 | — | 3 | 2 | no | — |
| 119 | C22⋊C8 | — | 6 | 1 | no | — |
| 120 | C22⋊C8 | — | 6 | 1 | no | — |
| 121 | #97 | — | 3 | 2 | no | — |
| 122 | C4×C8 | — | 3 | 2 | no | — |
| 123 | C4×C8 | — | 3 | 1 | no | — |
| 124 | C4×C8 | — | 3 | 1 | no | — |
| 125 | C4×C8 | — | 3 | 2 | no | — |
| 126 | C4×C8 | — | 3 | 1 | no | — |
| 127 | C4×C8 | — | 6 | 2 | no | — |
| 128 | C4×C8 | — | 6 | 2 | no | — |
| 129 | C4×C8 | S4 | 1 | 1 | yes | characteristic |
| 130 | C2×C3×C4 | — | 4 | 1 | no | — |
| 131 | C2×C3×C4 | — | 4 | 1 | no | — |
| 132 | C2×C3×C4 | — | 4 | 1 | no | — |
| 133 | C2×Dic3 | — | 4 | 2 | no | — |
| 134 | C2×Dic3 | — | 4 | 2 | no | — |
| 135 | Dic6 | — | 4 | 4 | no | — |
| 136 | SL(2,𝔽3) | C22×C8 | 1 | 1 | yes | characteristic |
| 137 | C3×C8 | — | 4 | 2 | no | — |
| 138 | C3×C8 | — | 8 | 1 | no | — |
| 139 | C3⋊C8 | — | 8 | 2 | no | — |
| 140 | #41 | — | 3 | 2 | no | — |
| 141 | #41 | D3×C8 | 1 | 2 | yes | — |
| 142 | C2×Q8 | — | 3 | 2 | no | — |
| 143 | C2×Q8 | — | 3 | 4 | no | — |
| 144 | C2×Q8 | — | 6 | 2 | no | — |
| 145 | C2×Q8 | — | 6 | 1 | no | — |
| 146 | C2×Q8 | C2×C4×D3 | 1 | 1 | yes | characteristic |
| 147 | C2×D4 | — | 3 | 1 | no | — |
| 148 | C22×C4 | — | 3 | 1 | no | — |
| 149 | C22×C4 | — | 3 | 1 | no | — |
| 150 | C22×C4 | — | 3 | 1 | no | — |
| 151 | C22×C4 | — | 6 | 1 | no | — |
| 152 | C22×C4 | — | 6 | 1 | no | — |
| 153 | Q16 | — | 3 | 4 | no | — |
| 154 | Q16 | — | 3 | 4 | no | — |
| 155 | SD16 | — | 3 | 8 | no | — |
| 156 | M16 | — | 3 | 4 | no | — |
| 157 | M16 | — | 6 | 2 | no | — |
| 158 | C2×C8 | — | 3 | 2 | no | — |
| 159 | C2×C8 | — | 3 | 2 | no | — |
| 160 | C2×C8 | — | 3 | 1 | no | — |
| 161 | C2×C8 | — | 3 | 1 | no | — |
| 162 | C2×C8 | — | 6 | 2 | no | — |
| 163 | C2×C8 | — | 6 | 2 | no | — |
| 164 | C2×C8 | — | 6 | 2 | no | — |
| 165 | C2×C8 | — | 12 | 1 | no | — |
| 166 | C2×C8 | — | 12 | 2 | no | — |
| 167 | C2×C8 | C2×S4 | 1 | 1 | yes | characteristic |
| 168 | C2×C8 | C2×S4 | 1 | 1 | yes | characteristic |
| 169 | C4⋊C4 | — | 3 | 1 | no | — |
| 170 | C4⋊C4 | — | 3 | 1 | no | — |
| 171 | C4⋊C4 | — | 3 | 2 | no | — |
| 172 | C4⋊C4 | — | 6 | 2 | no | — |
| 173 | C4⋊C4 | — | 6 | 1 | no | — |
| 174 | C4⋊C4 | — | 6 | 2 | no | — |
| 175 | C22⋊C4 | — | 6 | 1 | no | — |
| 176 | C42 | — | 3 | 1 | no | — |
| 177 | C42 | — | 3 | 1 | no | — |
| 178 | C42 | — | 3 | 1 | no | — |
| 179 | C42 | — | 6 | 2 | no | — |
| 180 | C42 | — | 6 | 2 | no | — |
| 181 | C42 | C2×S4 | 1 | 1 | yes | characteristic |
| 182 | C22×C3 | — | 4 | 1 | no | — |
| 183 | C3×C4 | — | 4 | 2 | no | — |
| 184 | C3×C4 | — | 4 | 1 | no | — |
| 185 | C3×C4 | — | 4 | 1 | no | — |
| 186 | C3×C4 | — | 8 | 1 | no | — |
| 187 | Dic3 | — | 4 | 4 | no | — |
| 188 | Dic3 | — | 8 | 2 | no | — |
| 189 | C23 | — | 3 | 1 | no | — |
| 190 | Q8 | — | 3 | 1 | no | — |
| 191 | Q8 | — | 3 | 4 | no | — |
| 192 | Q8 | — | 6 | 2 | no | — |
| 193 | Q8 | — | 6 | 4 | no | — |
| 194 | Q8 | C2×D3×C8 | 1 | 1 | yes | characteristic |
| 195 | D4 | — | 3 | 2 | no | — |
| 196 | D4 | — | 6 | 1 | no | — |
| 197 | C2×C4 | — | 3 | 1 | no | — |
| 198 | C2×C4 | — | 3 | 2 | no | — |
| 199 | C2×C4 | — | 3 | 2 | no | — |
| 200 | C2×C4 | — | 3 | 1 | no | — |
| 201 | C2×C4 | — | 3 | 1 | no | — |
| 202 | C2×C4 | — | 6 | 4 | no | — |
| 203 | C2×C4 | — | 6 | 2 | no | — |
| 204 | C2×C4 | — | 6 | 1 | no | — |
| 205 | C2×C4 | — | 6 | 1 | no | — |
| 206 | C2×C4 | — | 6 | 2 | no | — |
| 207 | C2×C4 | — | 12 | 1 | no | — |
| 208 | C2×C4 | — | 12 | 2 | no | — |
| 209 | C2×C4 | C4×S4 | 1 | 1 | yes | characteristic |
| 210 | C2×C4 | C4×S4 | 1 | 1 | yes | characteristic |
| 211 | C2×C4 | C22×S4 | 1 | 1 | yes | characteristic |
| 212 | C8 | — | 2 | 1 | no | — |
| 213 | C8 | — | 3 | 4 | no | — |
| 214 | C8 | — | 6 | 2 | no | — |
| 215 | C8 | — | 6 | 1 | no | — |
| 216 | C8 | — | 6 | 1 | no | — |
| 217 | C8 | — | 12 | 2 | no | — |
| 218 | C8 | — | 12 | 2 | no | — |
| 219 | C8 | #984 | 1 | 2 | yes | direct product factor |
| 220 | C2×C3 | — | 4 | 1 | no | — |
| 221 | C2×C3 | — | 4 | 1 | no | — |
| 222 | C2×C3 | — | 4 | 1 | no | — |
| 223 | C22 | — | 3 | 2 | no | — |
| 224 | C22 | — | 6 | 1 | no | — |
| 225 | C22 | — | 6 | 1 | no | — |
| 226 | C22 | C2×C4×S4 | 1 | 1 | yes | characteristic |
| 227 | C4 | — | 2 | 1 | no | — |
| 228 | C4 | — | 3 | 1 | no | — |
| 229 | C4 | — | 3 | 1 | no | — |
| 230 | C4 | — | 6 | 1 | no | — |
| 231 | C4 | — | 6 | 4 | no | — |
| 232 | C4 | — | 6 | 1 | no | — |
| 233 | C4 | — | 12 | 2 | no | — |
| 234 | C4 | C8×S4 | 1 | 2 | yes | — |
| 235 | C4 | C2×#984 | 1 | 1 | yes | characteristic |
| 236 | C4 | #5923 | 1 | 1 | yes | characteristic |
| 237 | C3 | — | 4 | 1 | no | Sylow |
| 238 | C2 | — | 6 | 2 | no | — |
| 239 | C2 | C4×#984 | 1 | 1 | yes | characteristic |
| 240 | C2 | C2×C8×S4 | 1 | 1 | yes | characteristic |
| 241 | C2 | #85448 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #226⋊C8 | C2 | semidirect |
| 5. C8×#226 | C2 | semidirect |
| 28. #984 | C8 | direct (N × H) |
| 74. #226 | C2×C8 | semidirect |
| 219. C8 | #984 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×#984 | 2 |
| C8×#41⋊C3 | 2 |
| #73071 | 2 |
| #226⋊C8 | 2 |
| C8×#226 | 2 |
| C8×#137 | 3 |
| C8×Dic6 | 4 |