Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
762 conjugacy classes of subgroups; only the 77 normal subgroups are shown.
This group has 762 conjugacy classes of subgroups — too many to draw. Only its 77 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 | C3×#2840 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #69498 | C2 | 1 | 3 | yes | maximal |
| 3 | C3⋊#1304 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #10390 | — | 3 | 1 | no | maximal, Sylow |
| 5 | C23×#67 | C22 | 1 | 1 | yes | characteristic |
| 6 | C3×#393 | — | 2 | 6 | no | — |
| 7 | C3×#379 | C22 | 1 | 3 | yes | — |
| 8 | #5247 | C22 | 1 | 3 | yes | — |
| 9 | C2×#806 | — | 2 | 6 | no | — |
| 10 | #2840 | D3 | 1 | 1 | yes | characteristic, complemented |
| 11 | #1781 | — | 3 | 3 | no | — |
| 12 | #1304 | — | 3 | 3 | no | — |
| 13 | C25×C3 | C23 | 1 | 1 | yes | characteristic |
| 14 | C24×D3 | Q8 | 1 | 1 | yes | characteristic, complemented |
| 15 | C22×C3×Q8 | — | 2 | 2 | no | — |
| 16 | C22×#67 | — | 2 | 3 | no | — |
| 17 | C22×#67 | — | 2 | 3 | no | — |
| 18 | C22×#67 | — | 4 | 4 | no | — |
| 19 | C23×Dic3 | Q8 | 1 | 1 | yes | characteristic, complemented |
| 20 | C2×C3×C22⋊C4 | — | 2 | 6 | no | — |
| 21 | C2×C3×C22⋊C4 | D4 | 1 | 6 | yes | — |
| 22 | C2×#217 | — | 2 | 3 | no | — |
| 23 | C2×#217 | — | 2 | 6 | no | — |
| 24 | C3×#95 | — | 2 | 6 | no | — |
| 25 | #806 | — | 4 | 12 | no | — |
| 26 | C23×D4 | — | 3 | 1 | no | — |
| 27 | C2×#99 | — | 6 | 6 | no | — |
| 28 | #393 | — | 2 | 6 | no | — |
| 29 | #379 | — | 3 | 3 | no | — |
| 30 | #379 | C2×D3 | 1 | 3 | yes | — |
| 31 | C24×C3 | — | 4 | 2 | no | — |
| 32 | C24×C3 | — | 8 | 2 | no | — |
| 33 | C24×C3 | C2×D4 | 1 | 3 | yes | — |
| 34 | C24×C3 | C2×Q8 | 1 | 1 | yes | characteristic |
| 35 | C24×C3 | #41 | 1 | 3 | yes | — |
| 36 | C23×D3 | — | 2 | 3 | no | — |
| 37 | C23×D3 | — | 4 | 2 | no | — |
| 38 | C23×D3 | — | 8 | 2 | no | — |
| 39 | C2×C3×Q8 | — | 8 | 4 | no | — |
| 40 | C2×C3×Q8 | — | 8 | 2 | no | — |
| 41 | C22×C3×C4 | — | 2 | 6 | no | — |
| 42 | C2×#67 | — | 2 | 12 | no | — |
| 43 | C2×#67 | — | 4 | 6 | no | — |
| 44 | C2×#67 | — | 4 | 6 | no | — |
| 45 | C2×#67 | — | 4 | 6 | no | — |
| 46 | C2×#67 | — | 4 | 1 | no | — |
| 47 | C2×#67 | — | 4 | 3 | no | — |
| 48 | C22×Dic3 | — | 2 | 3 | no | — |
| 49 | C22×Dic3 | — | 4 | 2 | no | — |
| 50 | C22×Dic3 | — | 4 | 3 | no | — |
| 51 | C3×C22⋊C4 | — | 4 | 6 | no | — |
| 52 | C3×C22⋊C4 | — | 8 | 6 | no | — |
| 53 | #217 | — | 4 | 6 | no | — |
| 54 | #217 | — | 8 | 6 | no | — |
| 55 | C25 | — | 3 | 1 | no | — |
| 56 | C25 | C22×D3 | 1 | 1 | yes | characteristic |
| 57 | C22×Q8 | — | 2 | 2 | no | — |
| 58 | C22×D4 | — | 6 | 3 | no | — |
| 59 | C22×D4 | — | 6 | 3 | no | — |
| 60 | C22×D4 | — | 12 | 4 | no | — |
| 61 | C23×C4 | — | 3 | 1 | no | — |
| 62 | C2×C22⋊C4 | — | 2 | 6 | no | — |
| 63 | C2×C22⋊C4 | — | 6 | 3 | no | — |
| 64 | C2×C22⋊C4 | — | 6 | 6 | no | — |
| 65 | C2×C22⋊C4 | #67 | 1 | 6 | yes | — |
| 66 | #99 | — | 12 | 12 | no | — |
| 67 | #95 | — | 2 | 6 | no | — |
| 68 | C23×C3 | — | 2 | 6 | no | — |
| 69 | C23×C3 | — | 4 | 1 | no | — |
| 70 | C23×C3 | — | 4 | 2 | no | — |
| 71 | C23×C3 | — | 4 | 6 | no | — |
| 72 | C23×C3 | — | 4 | 6 | no | — |
| 73 | C23×C3 | — | 4 | 3 | no | — |
| 74 | C23×C3 | — | 8 | 8 | no | — |
| 75 | C23×C3 | #120 | 1 | 1 | yes | characteristic |
| 76 | C23×C3 | C22⋊Q8 | 1 | 3 | yes | — |
| 77 | C23×C3 | #124 | 1 | 3 | yes | — |
| 78 | C22×D3 | — | 2 | 6 | no | — |
| 79 | C22×D3 | — | 4 | 3 | no | — |
| 80 | C22×D3 | — | 4 | 1 | no | — |
| 81 | C22×D3 | — | 8 | 8 | no | — |
| 82 | C22×D3 | — | 8 | 6 | no | — |
| 83 | C3×Q8 | — | 8 | 8 | no | — |
| 84 | C2×C3×C4 | — | 4 | 12 | no | — |
| 85 | C2×C3×C4 | — | 4 | 6 | no | — |
| 86 | #67 | — | 4 | 12 | no | — |
| 87 | #67 | — | 4 | 4 | no | — |
| 88 | C2×Dic3 | — | 2 | 6 | no | — |
| 89 | C2×Dic3 | — | 4 | 3 | no | — |
| 90 | C2×Dic3 | — | 4 | 1 | no | — |
| 91 | C2×Dic3 | — | 8 | 3 | no | — |
| 92 | C2×Dic3 | — | 8 | 6 | no | — |
| 93 | C24 | — | 4 | 2 | no | — |
| 94 | C24 | — | 6 | 3 | no | — |
| 95 | C24 | — | 8 | 2 | no | — |
| 96 | C24 | — | 12 | 2 | no | — |
| 97 | C24 | — | 24 | 2 | no | — |
| 98 | C24 | D3×Q8 | 1 | 1 | yes | characteristic |
| 99 | C24 | #239 | 1 | 3 | yes | — |
| 100 | C24 | C2×#67 | 1 | 3 | yes | — |
| 101 | C2×Q8 | — | 8 | 4 | no | — |
| 102 | C2×Q8 | — | 8 | 2 | no | — |
| 103 | C2×D4 | — | 6 | 12 | no | — |
| 104 | C2×D4 | — | 12 | 6 | no | — |
| 105 | C2×D4 | — | 12 | 6 | no | — |
| 106 | C2×D4 | — | 12 | 6 | no | — |
| 107 | C2×D4 | — | 12 | 3 | no | — |
| 108 | C2×D4 | — | 12 | 1 | no | — |
| 109 | C22×C4 | — | 2 | 6 | no | — |
| 110 | C22×C4 | — | 6 | 3 | no | — |
| 111 | C22×C4 | — | 12 | 2 | no | — |
| 112 | C22×C4 | — | 12 | 3 | no | — |
| 113 | C22⋊C4 | — | 4 | 6 | no | — |
| 114 | C22⋊C4 | — | 8 | 6 | no | — |
| 115 | C22⋊C4 | — | 12 | 6 | no | — |
| 116 | C22⋊C4 | — | 24 | 6 | no | — |
| 117 | C22×C3 | — | 2 | 2 | no | — |
| 118 | C22×C3 | — | 2 | 3 | no | — |
| 119 | C22×C3 | — | 2 | 3 | no | — |
| 120 | C22×C3 | — | 2 | 6 | no | — |
| 121 | C22×C3 | — | 2 | 6 | no | — |
| 122 | C22×C3 | — | 4 | 4 | no | — |
| 123 | C22×C3 | — | 8 | 6 | no | — |
| 124 | C22×C3 | — | 8 | 6 | no | — |
| 125 | C22×C3 | #393 | 1 | 1 | yes | characteristic |
| 126 | C22×C3 | B(4,𝔽2) | 1 | 2 | yes | — |
| 127 | C2×D3 | — | 4 | 2 | no | — |
| 128 | C2×D3 | — | 8 | 6 | no | — |
| 129 | C2×D3 | — | 8 | 2 | no | — |
| 130 | C2×D3 | — | 8 | 6 | no | — |
| 131 | C3×C4 | — | 8 | 6 | no | — |
| 132 | Dic3 | — | 4 | 2 | no | — |
| 133 | Dic3 | — | 16 | 3 | no | — |
| 134 | C23 | — | 2 | 6 | no | — |
| 135 | C23 | — | 4 | 1 | no | — |
| 136 | C23 | — | 4 | 2 | no | — |
| 137 | C23 | — | 4 | 6 | no | — |
| 138 | C23 | — | 4 | 6 | no | — |
| 139 | C23 | — | 4 | 3 | no | — |
| 140 | C23 | — | 6 | 6 | no | — |
| 141 | C23 | — | 8 | 8 | no | — |
| 142 | C23 | — | 12 | 3 | no | — |
| 143 | C23 | — | 12 | 1 | no | — |
| 144 | C23 | — | 24 | 6 | no | — |
| 145 | C23 | — | 24 | 8 | no | — |
| 146 | C23 | #946 | 1 | 3 | yes | — |
| 147 | C23 | #947 | 1 | 3 | yes | — |
| 148 | C23 | #953 | 1 | 1 | yes | characteristic |
| 149 | Q8 | — | 8 | 8 | no | — |
| 150 | D4 | — | 12 | 12 | no | — |
| 151 | D4 | — | 12 | 4 | no | — |
| 152 | C2×C4 | — | 4 | 12 | no | — |
| 153 | C2×C4 | — | 4 | 6 | no | — |
| 154 | C2×C4 | — | 6 | 6 | no | — |
| 155 | C2×C4 | — | 12 | 3 | no | — |
| 156 | C2×C4 | — | 12 | 1 | no | — |
| 157 | C2×C4 | — | 24 | 3 | no | — |
| 158 | C2×C4 | — | 24 | 6 | no | — |
| 159 | C2×C3 | — | 2 | 2 | no | — |
| 160 | C2×C3 | — | 4 | 3 | no | — |
| 161 | C2×C3 | — | 4 | 3 | no | — |
| 162 | C2×C3 | #2029 | 1 | 1 | yes | characteristic |
| 163 | C2×C3 | #2032 | 1 | 2 | yes | — |
| 164 | D3 | — | 8 | 2 | no | — |
| 165 | C22 | — | 2 | 2 | no | — |
| 166 | C22 | — | 2 | 6 | no | — |
| 167 | C22 | — | 2 | 6 | no | — |
| 168 | C22 | — | 2 | 3 | no | — |
| 169 | C22 | — | 2 | 3 | no | — |
| 170 | C22 | — | 4 | 4 | no | — |
| 171 | C22 | — | 8 | 6 | no | — |
| 172 | C22 | — | 8 | 6 | no | — |
| 173 | C22 | — | 12 | 2 | no | — |
| 174 | C22 | — | 24 | 6 | no | — |
| 175 | C22 | — | 24 | 6 | no | — |
| 176 | C22 | — | 24 | 2 | no | — |
| 177 | C22 | #5231 | 1 | 1 | yes | characteristic |
| 178 | C22 | #5244 | 1 | 2 | yes | — |
| 179 | C4 | — | 8 | 6 | no | — |
| 180 | C4 | — | 12 | 2 | no | — |
| 181 | C4 | — | 48 | 3 | no | — |
| 182 | C3 | #10390 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 183 | C2 | — | 2 | 2 | no | — |
| 184 | C2 | — | 4 | 3 | no | — |
| 185 | C2 | — | 4 | 3 | no | — |
| 186 | C2 | — | 24 | 2 | no | — |
| 187 | C2 | #71896 | 1 | 2 | yes | — |
| 188 | C2 | #71897 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#2840 | C2 | semidirect |
| 3. C3⋊#1304 | C2 | semidirect |
| 10. #2840 | D3 | semidirect |
| 14. C24×D3 | Q8 | semidirect |
| 19. C23×Dic3 | Q8 | semidirect |
| 182. C3 | #10390 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#2840 | 2 |
| #69498 | 2 |
| C3⋊#1304 | 2 |
| #10390 | 3 |