Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
5741 conjugacy classes of subgroups; only the 559 normal subgroups are shown.
This group has 5741 conjugacy classes of subgroups — too many to draw. Only its 559 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 | C24×SL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C22×#983 | C2 | 1 | 28 | yes | maximal, direct product factor |
| 3 | C23×GL(2,𝔽3) | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 4 | C23×#226 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C23×#137 | — | 3 | 1 | no | maximal, Sylow |
| 6 | C23×#67 | — | 4 | 1 | no | maximal |
| 7 | C23×SL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic |
| 8 | C23×SL(2,𝔽3) | C22 | 1 | 14 | yes | complemented |
| 9 | C2×#983 | C22 | 1 | 112 | yes | direct product factor |
| 10 | C22×GL(2,𝔽3) | C22 | 1 | 14 | yes | complemented |
| 11 | C22×#226 | C22 | 1 | 14 | yes | complemented |
| 12 | C23×#41 | — | 3 | 1 | no | — |
| 13 | C24×Q8 | D3 | 1 | 1 | yes | characteristic, complemented |
| 14 | C22×#137 | — | 3 | 28 | no | — |
| 15 | C22×#137 | — | 3 | 28 | no | — |
| 16 | C23×Q16 | — | 3 | 1 | no | — |
| 17 | C23×Q16 | — | 3 | 1 | no | — |
| 18 | C23×SD16 | — | 3 | 1 | no | — |
| 19 | C23×SD16 | — | 3 | 1 | no | — |
| 20 | C23×M16 | — | 3 | 1 | no | — |
| 21 | C25×C3 | — | 4 | 1 | no | — |
| 22 | C24×D3 | — | 4 | 1 | no | — |
| 23 | C22×#67 | — | 4 | 28 | no | — |
| 24 | C23×Dic3 | — | 4 | 1 | no | — |
| 25 | C22×SL(2,𝔽3) | C23 | 1 | 7 | yes | — |
| 26 | C22×SL(2,𝔽3) | C23 | 1 | 28 | yes | complemented |
| 27 | #983 | C23 | 1 | 64 | yes | direct product factor |
| 28 | C2×GL(2,𝔽3) | C23 | 1 | 28 | yes | complemented |
| 29 | C2×#226 | C23 | 1 | 28 | yes | complemented |
| 30 | C22×#41 | — | 3 | 28 | no | — |
| 31 | C22×#41 | — | 6 | 14 | no | — |
| 32 | C23×Q8 | — | 3 | 1 | no | — |
| 33 | C23×Q8 | — | 3 | 14 | no | — |
| 34 | C23×Q8 | — | 3 | 1 | no | — |
| 35 | C23×Q8 | C2×D3 | 1 | 1 | yes | characteristic |
| 36 | C23×Q8 | C2×D3 | 1 | 14 | yes | complemented |
| 37 | C23×D4 | — | 3 | 1 | no | — |
| 38 | C23×D4 | — | 6 | 1 | no | — |
| 39 | C24×C4 | — | 3 | 1 | no | — |
| 40 | C24×C4 | — | 6 | 1 | no | — |
| 41 | C2×#137 | — | 3 | 336 | no | — |
| 42 | C2×#137 | — | 3 | 112 | no | — |
| 43 | C22×Q16 | — | 3 | 14 | no | — |
| 44 | C22×Q16 | — | 3 | 28 | no | — |
| 45 | C22×Q16 | — | 3 | 14 | no | — |
| 46 | C22×SD16 | — | 3 | 14 | no | — |
| 47 | C22×SD16 | — | 3 | 28 | no | — |
| 48 | C22×SD16 | — | 3 | 14 | no | — |
| 49 | C22×M16 | — | 3 | 28 | no | — |
| 50 | C23×C8 | — | 3 | 1 | no | — |
| 51 | C23×C8 | — | 3 | 1 | no | — |
| 52 | C24×C3 | — | 4 | 14 | no | — |
| 53 | C24×C3 | — | 4 | 1 | no | — |
| 54 | C24×C3 | — | 8 | 8 | no | — |
| 55 | C23×D3 | — | 4 | 14 | no | — |
| 56 | C23×D3 | — | 8 | 8 | no | — |
| 57 | C2×#67 | — | 4 | 112 | no | — |
| 58 | C22×Dic3 | — | 4 | 14 | no | — |
| 59 | C2×SL(2,𝔽3) | C24 | 1 | 7 | yes | — |
| 60 | C2×SL(2,𝔽3) | C24 | 1 | 8 | yes | complemented |
| 61 | GL(2,𝔽3) | C24 | 1 | 8 | yes | complemented |
| 62 | #226 | C24 | 1 | 8 | yes | complemented |
| 63 | C25 | — | 6 | 1 | no | — |
| 64 | C25 | S4 | 1 | 1 | yes | characteristic |
| 65 | C2×#41 | — | 3 | 112 | no | — |
| 66 | C2×#41 | — | 6 | 168 | no | — |
| 67 | C22×Q8 | — | 3 | 7 | no | — |
| 68 | C22×Q8 | — | 3 | 14 | no | — |
| 69 | C22×Q8 | — | 3 | 84 | no | — |
| 70 | C22×Q8 | — | 3 | 14 | no | — |
| 71 | C22×Q8 | — | 6 | 7 | no | — |
| 72 | C22×Q8 | — | 6 | 28 | no | — |
| 73 | C22×Q8 | — | 6 | 7 | no | — |
| 74 | C22×Q8 | C22×D3 | 1 | 7 | yes | — |
| 75 | C22×Q8 | C22×D3 | 1 | 28 | yes | complemented |
| 76 | C22×D4 | — | 3 | 14 | no | — |
| 77 | C22×D4 | — | 6 | 7 | no | — |
| 78 | C22×D4 | — | 6 | 28 | no | — |
| 79 | C23×C4 | — | 3 | 14 | no | — |
| 80 | C23×C4 | — | 3 | 14 | no | — |
| 81 | C23×C4 | — | 3 | 1 | no | — |
| 82 | C23×C4 | — | 3 | 1 | no | — |
| 83 | C23×C4 | — | 6 | 28 | no | — |
| 84 | C23×C4 | — | 6 | 1 | no | — |
| 85 | #137 | — | 3 | 448 | no | — |
| 86 | #137 | — | 3 | 64 | no | — |
| 87 | C2×Q16 | — | 3 | 84 | no | — |
| 88 | C2×Q16 | — | 3 | 112 | no | — |
| 89 | C2×Q16 | — | 3 | 28 | no | — |
| 90 | C2×SD16 | — | 3 | 84 | no | — |
| 91 | C2×SD16 | — | 3 | 112 | no | — |
| 92 | C2×SD16 | — | 3 | 28 | no | — |
| 93 | C2×M16 | — | 3 | 112 | no | — |
| 94 | C22×C8 | — | 3 | 14 | no | — |
| 95 | C22×C8 | — | 3 | 14 | no | — |
| 96 | C23×C3 | — | 4 | 8 | no | — |
| 97 | C23×C3 | — | 4 | 7 | no | — |
| 98 | C23×C3 | — | 4 | 28 | no | — |
| 99 | C23×C3 | — | 8 | 56 | no | — |
| 100 | C22×D3 | — | 4 | 28 | no | — |
| 101 | C22×D3 | — | 8 | 56 | no | — |
| 102 | #67 | — | 4 | 64 | no | — |
| 103 | C2×Dic3 | — | 4 | 28 | no | — |
| 104 | SL(2,𝔽3) | C25 | 1 | 1 | yes | characteristic |
| 105 | C24 | — | 2 | 8 | no | — |
| 106 | C24 | — | 6 | 14 | no | — |
| 107 | C24 | — | 12 | 8 | no | — |
| 108 | C24 | C2×S4 | 1 | 14 | yes | — |
| 109 | C24 | C2×S4 | 1 | 1 | yes | characteristic |
| 110 | #41 | — | 3 | 64 | no | — |
| 111 | #41 | — | 6 | 224 | no | — |
| 112 | C2×Q8 | — | 3 | 21 | no | — |
| 113 | C2×Q8 | — | 3 | 28 | no | — |
| 114 | C2×Q8 | — | 3 | 56 | no | — |
| 115 | C2×Q8 | — | 3 | 28 | no | — |
| 116 | C2×Q8 | — | 6 | 42 | no | — |
| 117 | C2×Q8 | — | 6 | 7 | no | — |
| 118 | C2×Q8 | — | 6 | 56 | no | — |
| 119 | C2×Q8 | — | 6 | 42 | no | — |
| 120 | C2×Q8 | C23×D3 | 1 | 7 | yes | — |
| 121 | C2×Q8 | C23×D3 | 1 | 8 | yes | complemented |
| 122 | C2×D4 | — | 3 | 28 | no | — |
| 123 | C2×D4 | — | 6 | 42 | no | — |
| 124 | C2×D4 | — | 6 | 112 | no | — |
| 125 | C22×C4 | — | 3 | 7 | no | — |
| 126 | C22×C4 | — | 3 | 28 | no | — |
| 127 | C22×C4 | — | 3 | 14 | no | — |
| 128 | C22×C4 | — | 3 | 7 | no | — |
| 129 | C22×C4 | — | 3 | 84 | no | — |
| 130 | C22×C4 | — | 6 | 112 | no | — |
| 131 | C22×C4 | — | 6 | 14 | no | — |
| 132 | Q16 | — | 3 | 56 | no | — |
| 133 | Q16 | — | 3 | 64 | no | — |
| 134 | Q16 | — | 3 | 8 | no | — |
| 135 | SD16 | — | 3 | 56 | no | — |
| 136 | SD16 | — | 3 | 64 | no | — |
| 137 | SD16 | — | 3 | 8 | no | — |
| 138 | M16 | — | 3 | 64 | no | — |
| 139 | C2×C8 | — | 3 | 28 | no | — |
| 140 | C2×C8 | — | 3 | 28 | no | — |
| 141 | C22×C3 | — | 4 | 28 | no | — |
| 142 | C22×C3 | — | 4 | 7 | no | — |
| 143 | C22×C3 | — | 4 | 8 | no | — |
| 144 | C22×C3 | — | 8 | 56 | no | — |
| 145 | C2×D3 | — | 4 | 8 | no | — |
| 146 | C2×D3 | — | 8 | 56 | no | — |
| 147 | Dic3 | — | 4 | 8 | no | — |
| 148 | C23 | — | 2 | 56 | no | — |
| 149 | C23 | — | 6 | 28 | no | — |
| 150 | C23 | — | 12 | 56 | no | — |
| 151 | C23 | #983 | 1 | 8 | yes | direct product factor |
| 152 | C23 | C22×S4 | 1 | 28 | yes | — |
| 153 | C23 | C22×S4 | 1 | 7 | yes | — |
| 154 | Q8 | — | 3 | 7 | no | — |
| 155 | Q8 | — | 3 | 8 | no | — |
| 156 | Q8 | — | 3 | 8 | no | — |
| 157 | Q8 | — | 6 | 28 | no | — |
| 158 | Q8 | — | 6 | 7 | no | — |
| 159 | Q8 | — | 6 | 28 | no | — |
| 160 | Q8 | C24×D3 | 1 | 1 | yes | characteristic |
| 161 | D4 | — | 3 | 8 | no | — |
| 162 | D4 | — | 6 | 28 | no | — |
| 163 | D4 | — | 6 | 64 | no | — |
| 164 | C2×C4 | — | 3 | 7 | no | — |
| 165 | C2×C4 | — | 3 | 8 | no | — |
| 166 | C2×C4 | — | 3 | 28 | no | — |
| 167 | C2×C4 | — | 3 | 21 | no | — |
| 168 | C2×C4 | — | 3 | 56 | no | — |
| 169 | C2×C4 | — | 6 | 64 | no | — |
| 170 | C2×C4 | — | 6 | 28 | no | — |
| 171 | C8 | — | 3 | 8 | no | — |
| 172 | C8 | — | 3 | 8 | no | — |
| 173 | C2×C3 | — | 4 | 1 | no | — |
| 174 | C2×C3 | — | 4 | 14 | no | — |
| 175 | C2×C3 | — | 8 | 8 | no | — |
| 176 | D3 | — | 8 | 8 | no | — |
| 177 | C22 | — | 2 | 56 | no | — |
| 178 | C22 | — | 6 | 8 | no | — |
| 179 | C22 | — | 12 | 56 | no | — |
| 180 | C22 | C2×#983 | 1 | 28 | yes | direct product factor |
| 181 | C22 | C23×S4 | 1 | 8 | yes | — |
| 182 | C22 | C23×S4 | 1 | 7 | yes | — |
| 183 | C4 | — | 3 | 1 | no | — |
| 184 | C4 | — | 3 | 8 | no | — |
| 185 | C4 | — | 3 | 7 | no | — |
| 186 | C4 | — | 6 | 8 | no | — |
| 187 | C3 | — | 4 | 1 | no | Sylow |
| 188 | C2 | — | 2 | 8 | no | — |
| 189 | C2 | — | 12 | 8 | no | — |
| 190 | C2 | C22×#983 | 1 | 14 | yes | direct product factor |
| 191 | C2 | C24×S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C25 |
| Frattini | C2 | C24×S4 |
| Fitting | C24×Q8 | D3 |
| hypercenter | C25 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C24 ◁ C25 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×SL(2,𝔽3) | C2 | semidirect |
| 2. C22×#983 | C2 | direct (N × H) |
| 3. C23×GL(2,𝔽3) | C2 | semidirect |
| 4. C23×#226 | C2 | semidirect |
| 8. C23×SL(2,𝔽3) | C22 | semidirect |
| 9. C2×#983 | C22 | direct (N × H) |
| 10. C22×GL(2,𝔽3) | C22 | semidirect |
| 11. C22×#226 | C22 | semidirect |
| 13. C24×Q8 | D3 | semidirect |
| 26. C22×SL(2,𝔽3) | C23 | semidirect |
| 27. #983 | C23 | direct (N × H) |
| 28. C2×GL(2,𝔽3) | C23 | semidirect |
| 29. C2×#226 | C23 | semidirect |
| 36. C23×Q8 | C2×D3 | semidirect |
| 60. C2×SL(2,𝔽3) | C24 | semidirect |
| 61. GL(2,𝔽3) | C24 | semidirect |
| 62. #226 | C24 | semidirect |
| 75. C22×Q8 | C22×D3 | semidirect |
| 121. C2×Q8 | C23×D3 | semidirect |
| 151. C23 | #983 | direct (N × H) |
| 180. C22 | C2×#983 | direct (N × H) |
| 190. C2 | C22×#983 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×SL(2,𝔽3) | 2 |
| C22×#983 | 2 |
| C23×GL(2,𝔽3) | 2 |
| C23×#226 | 2 |
| C23×#137 | 3 |
| C23×#67 | 4 |