Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
5313 conjugacy classes of subgroups; only the 519 normal subgroups are shown.
This group has 5313 conjugacy classes of subgroups — too many to draw. Only its 519 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 | C22×#994 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | C2×#5924 | C2 | 1 | 24 | yes | maximal, direct product factor |
| 3 | C22×#985 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | C22×#984 | C2 | 1 | 3 | yes | maximal, complemented |
| 5 | C22×#577 | — | 3 | 1 | no | maximal, Sylow |
| 6 | C22×D3×Q8 | — | 4 | 1 | no | maximal |
| 7 | C2×#994 | C22 | 1 | 12 | yes | complemented |
| 8 | C22×#41⋊C3 | C22 | 1 | 3 | yes | — |
| 9 | #5924 | C22 | 1 | 64 | yes | direct product factor |
| 10 | C2×#985 | C22 | 1 | 36 | yes | complemented |
| 11 | C2×#984 | C22 | 1 | 36 | yes | complemented |
| 12 | C22×GL(2,𝔽3) | C22 | 1 | 1 | yes | characteristic, complemented |
| 13 | C22×#226 | C22 | 1 | 3 | yes | complemented |
| 14 | C22×2−1+4 | — | 3 | 1 | no | — |
| 15 | C22×H(5,𝔽2) | D3 | 1 | 1 | yes | characteristic, complemented |
| 16 | C2×#577 | — | 3 | 24 | no | — |
| 17 | C2×#577 | — | 3 | 24 | no | — |
| 18 | C22×#137 | — | 3 | 3 | no | — |
| 19 | C22×AGL(1,ℤ/8) | — | 3 | 3 | no | — |
| 20 | C22×#135 | — | 3 | 3 | no | — |
| 21 | C23×SD16 | — | 3 | 3 | no | — |
| 22 | C22×#131 | — | 3 | 1 | no | — |
| 23 | C22×C3×Q8 | — | 4 | 1 | no | — |
| 24 | C2×D3×Q8 | — | 4 | 24 | no | — |
| 25 | C22×C4×D3 | — | 4 | 3 | no | — |
| 26 | C22×Dic6 | — | 4 | 3 | no | — |
| 27 | #994 | C23 | 1 | 16 | yes | complemented |
| 28 | C2×#41⋊C3 | C23 | 1 | 18 | yes | — |
| 29 | C22×SL(2,𝔽3) | C23 | 1 | 1 | yes | characteristic |
| 30 | #985 | C23 | 1 | 48 | yes | complemented |
| 31 | #984 | C23 | 1 | 48 | yes | complemented |
| 32 | C2×GL(2,𝔽3) | C23 | 1 | 6 | yes | complemented |
| 33 | C2×#226 | C23 | 1 | 18 | yes | complemented |
| 34 | C2×2−1+4 | — | 3 | 24 | no | — |
| 35 | C2×2−1+4 | — | 6 | 12 | no | — |
| 36 | C2×H(5,𝔽2) | — | 3 | 12 | no | — |
| 37 | C2×H(5,𝔽2) | C2×D3 | 1 | 12 | yes | complemented |
| 38 | C22×#41 | — | 3 | 1 | no | — |
| 39 | C22×#41 | — | 3 | 3 | no | — |
| 40 | C22×#41 | — | 6 | 3 | no | — |
| 41 | C22×#41 | C2×D3 | 1 | 3 | yes | — |
| 42 | C23×Q8 | — | 3 | 3 | no | — |
| 43 | C23×Q8 | — | 6 | 1 | no | — |
| 44 | C23×D4 | — | 3 | 3 | no | — |
| 45 | #577 | — | 3 | 192 | no | — |
| 46 | #577 | — | 3 | 64 | no | — |
| 47 | C2×#137 | — | 3 | 36 | no | — |
| 48 | C2×#137 | — | 3 | 36 | no | — |
| 49 | C2×AGL(1,ℤ/8) | — | 3 | 72 | no | — |
| 50 | C2×#135 | — | 3 | 36 | no | — |
| 51 | C2×#135 | — | 3 | 36 | no | — |
| 52 | C22×Q16 | — | 3 | 3 | no | — |
| 53 | C22×SD16 | — | 3 | 3 | no | — |
| 54 | C22×SD16 | — | 3 | 6 | no | — |
| 55 | C22×SD16 | — | 3 | 72 | no | — |
| 56 | C22×SD16 | — | 3 | 1 | no | — |
| 57 | C22×D8 | — | 3 | 3 | no | — |
| 58 | C2×#131 | — | 3 | 24 | no | — |
| 59 | C22×M16 | — | 3 | 3 | no | — |
| 60 | C23×C8 | — | 3 | 3 | no | — |
| 61 | C23×D3 | — | 4 | 1 | no | — |
| 62 | C2×C3×Q8 | — | 4 | 12 | no | — |
| 63 | C22×C3×C4 | — | 4 | 3 | no | — |
| 64 | C22×Dic3 | — | 4 | 3 | no | — |
| 65 | D3×Q8 | — | 4 | 64 | no | — |
| 66 | C2×C4×D3 | — | 4 | 36 | no | — |
| 67 | C2×Dic6 | — | 4 | 36 | no | — |
| 68 | #41⋊C3 | C24 | 1 | 12 | yes | — |
| 69 | C2×SL(2,𝔽3) | C24 | 1 | 3 | yes | — |
| 70 | GL(2,𝔽3) | C24 | 1 | 4 | yes | complemented |
| 71 | #226 | C24 | 1 | 12 | yes | complemented |
| 72 | C25 | — | 6 | 1 | no | — |
| 73 | 2−1+4 | — | 3 | 64 | no | — |
| 74 | 2−1+4 | — | 6 | 96 | no | — |
| 75 | H(5,𝔽2) | — | 3 | 48 | no | — |
| 76 | H(5,𝔽2) | — | 6 | 16 | no | — |
| 77 | H(5,𝔽2) | C22×D3 | 1 | 16 | yes | complemented |
| 78 | C2×#41 | — | 3 | 12 | no | — |
| 79 | C2×#41 | — | 3 | 12 | no | — |
| 80 | C2×#41 | — | 3 | 18 | no | — |
| 81 | C2×#41 | — | 3 | 36 | no | — |
| 82 | C2×#41 | — | 6 | 18 | no | — |
| 83 | C2×#41 | — | 6 | 72 | no | — |
| 84 | C2×#41 | C22×D3 | 1 | 18 | yes | — |
| 85 | C22×Q8 | — | 3 | 3 | no | — |
| 86 | C22×Q8 | — | 3 | 36 | no | — |
| 87 | C22×Q8 | — | 6 | 18 | no | — |
| 88 | C22×Q8 | — | 6 | 24 | no | — |
| 89 | C22×Q8 | — | 6 | 3 | no | — |
| 90 | C22×Q8 | S4 | 1 | 1 | yes | characteristic |
| 91 | C22×Q8 | C22×D3 | 1 | 1 | yes | characteristic |
| 92 | C22×D4 | — | 3 | 36 | no | — |
| 93 | C22×D4 | — | 3 | 3 | no | — |
| 94 | C22×D4 | — | 3 | 1 | no | — |
| 95 | C22×D4 | — | 3 | 3 | no | — |
| 96 | C22×D4 | — | 6 | 18 | no | — |
| 97 | C22×D4 | — | 6 | 3 | no | — |
| 98 | C23×C4 | — | 3 | 3 | no | — |
| 99 | C23×C4 | — | 6 | 3 | no | — |
| 100 | C23×C4 | — | 6 | 3 | no | — |
| 101 | #137 | — | 3 | 144 | no | — |
| 102 | #137 | — | 3 | 48 | no | — |
| 103 | AGL(1,ℤ/8) | — | 3 | 192 | no | — |
| 104 | #135 | — | 3 | 144 | no | — |
| 105 | #135 | — | 3 | 48 | no | — |
| 106 | C2×Q16 | — | 3 | 18 | no | — |
| 107 | C2×Q16 | — | 3 | 18 | no | — |
| 108 | C2×SD16 | — | 3 | 36 | no | — |
| 109 | C2×SD16 | — | 3 | 72 | no | — |
| 110 | C2×SD16 | — | 3 | 192 | no | — |
| 111 | C2×SD16 | — | 3 | 6 | no | — |
| 112 | C2×SD16 | — | 3 | 6 | no | — |
| 113 | C2×D8 | — | 3 | 36 | no | — |
| 114 | #131 | — | 3 | 64 | no | — |
| 115 | C2×M16 | — | 3 | 36 | no | — |
| 116 | C22×C8 | — | 3 | 1 | no | — |
| 117 | C22×C8 | — | 3 | 3 | no | — |
| 118 | C22×C8 | — | 3 | 36 | no | — |
| 119 | C23×C3 | — | 4 | 1 | no | — |
| 120 | C22×D3 | — | 4 | 6 | no | — |
| 121 | C22×D3 | — | 4 | 8 | no | — |
| 122 | C3×Q8 | — | 4 | 16 | no | — |
| 123 | C2×C3×C4 | — | 4 | 18 | no | — |
| 124 | C2×Dic3 | — | 4 | 18 | no | — |
| 125 | C4×D3 | — | 4 | 48 | no | — |
| 126 | Dic6 | — | 4 | 48 | no | — |
| 127 | SL(2,𝔽3) | C25 | 1 | 1 | yes | characteristic |
| 128 | C24 | — | 3 | 3 | no | — |
| 129 | C24 | — | 6 | 12 | no | — |
| 130 | C24 | — | 6 | 1 | no | — |
| 131 | C24 | — | 24 | 4 | no | — |
| 132 | #41 | — | 3 | 16 | no | — |
| 133 | #41 | — | 3 | 48 | no | — |
| 134 | #41 | — | 3 | 36 | no | — |
| 135 | #41 | — | 3 | 48 | no | — |
| 136 | #41 | — | 6 | 12 | no | — |
| 137 | #41 | — | 6 | 72 | no | — |
| 138 | #41 | — | 6 | 192 | no | — |
| 139 | #41 | C23×D3 | 1 | 12 | yes | — |
| 140 | C2×Q8 | — | 3 | 3 | no | — |
| 141 | C2×Q8 | — | 3 | 18 | no | — |
| 142 | C2×Q8 | — | 3 | 48 | no | — |
| 143 | C2×Q8 | — | 6 | 9 | no | — |
| 144 | C2×Q8 | — | 6 | 72 | no | — |
| 145 | C2×Q8 | — | 6 | 64 | no | — |
| 146 | C2×Q8 | — | 6 | 36 | no | — |
| 147 | C2×Q8 | C2×S4 | 1 | 12 | yes | — |
| 148 | C2×Q8 | C23×D3 | 1 | 3 | yes | — |
| 149 | C2×D4 | — | 3 | 48 | no | — |
| 150 | C2×D4 | — | 3 | 18 | no | — |
| 151 | C2×D4 | — | 3 | 6 | no | — |
| 152 | C2×D4 | — | 3 | 36 | no | — |
| 153 | C2×D4 | — | 6 | 72 | no | — |
| 154 | C2×D4 | — | 6 | 9 | no | — |
| 155 | C2×D4 | — | 6 | 3 | no | — |
| 156 | C2×D4 | — | 6 | 36 | no | — |
| 157 | C22×C4 | — | 3 | 1 | no | — |
| 158 | C22×C4 | — | 3 | 18 | no | — |
| 159 | C22×C4 | — | 3 | 18 | no | — |
| 160 | C22×C4 | — | 6 | 36 | no | — |
| 161 | C22×C4 | — | 6 | 3 | no | — |
| 162 | C22×C4 | — | 6 | 36 | no | — |
| 163 | C22×C4 | C2×S4 | 1 | 3 | yes | — |
| 164 | Q16 | — | 3 | 36 | no | — |
| 165 | Q16 | — | 3 | 12 | no | — |
| 166 | SD16 | — | 3 | 48 | no | — |
| 167 | SD16 | — | 3 | 96 | no | — |
| 168 | SD16 | — | 3 | 12 | no | — |
| 169 | SD16 | — | 3 | 4 | no | — |
| 170 | D8 | — | 3 | 48 | no | — |
| 171 | M16 | — | 3 | 48 | no | — |
| 172 | C2×C8 | — | 3 | 6 | no | — |
| 173 | C2×C8 | — | 3 | 18 | no | — |
| 174 | C2×C8 | — | 3 | 48 | no | — |
| 175 | C22×C3 | — | 4 | 4 | no | — |
| 176 | C22×C3 | — | 4 | 3 | no | — |
| 177 | C2×D3 | — | 4 | 4 | no | — |
| 178 | C2×D3 | — | 4 | 24 | no | — |
| 179 | C3×C4 | — | 4 | 12 | no | — |
| 180 | Dic3 | — | 4 | 12 | no | — |
| 181 | C23 | — | 3 | 18 | no | — |
| 182 | C23 | — | 6 | 16 | no | — |
| 183 | C23 | — | 6 | 12 | no | — |
| 184 | C23 | — | 6 | 6 | no | — |
| 185 | C23 | — | 12 | 4 | no | — |
| 186 | C23 | — | 24 | 24 | no | — |
| 187 | C23 | C22×S4 | 1 | 1 | yes | characteristic |
| 188 | Q8 | — | 3 | 3 | no | — |
| 189 | Q8 | — | 3 | 12 | no | — |
| 190 | Q8 | — | 6 | 1 | no | — |
| 191 | Q8 | — | 6 | 18 | no | — |
| 192 | Q8 | — | 6 | 48 | no | — |
| 193 | Q8 | C22×S4 | 1 | 16 | yes | — |
| 194 | Q8 | C24×D3 | 1 | 1 | yes | characteristic |
| 195 | D4 | — | 3 | 12 | no | — |
| 196 | D4 | — | 3 | 4 | no | — |
| 197 | D4 | — | 3 | 48 | no | — |
| 198 | D4 | — | 6 | 18 | no | — |
| 199 | D4 | — | 6 | 6 | no | — |
| 200 | D4 | — | 6 | 48 | no | — |
| 201 | C2×C4 | — | 3 | 3 | no | — |
| 202 | C2×C4 | — | 3 | 3 | no | — |
| 203 | C2×C4 | — | 3 | 12 | no | — |
| 204 | C2×C4 | — | 3 | 36 | no | — |
| 205 | C2×C4 | — | 6 | 48 | no | — |
| 206 | C2×C4 | — | 6 | 18 | no | — |
| 207 | C2×C4 | — | 6 | 48 | no | — |
| 208 | C2×C4 | C22×S4 | 1 | 18 | yes | — |
| 209 | C8 | — | 3 | 4 | no | — |
| 210 | C8 | — | 3 | 12 | no | — |
| 211 | C2×C3 | — | 4 | 1 | no | — |
| 212 | C2×C3 | — | 4 | 6 | no | — |
| 213 | D3 | — | 4 | 8 | no | — |
| 214 | C22 | — | 3 | 12 | no | — |
| 215 | C22 | — | 6 | 36 | no | — |
| 216 | C22 | — | 6 | 4 | no | — |
| 217 | C22 | — | 12 | 12 | no | — |
| 218 | C22 | — | 24 | 16 | no | — |
| 219 | C22 | #5924 | 1 | 4 | yes | direct product factor |
| 220 | C22 | C23×S4 | 1 | 3 | yes | — |
| 221 | C4 | — | 3 | 1 | no | — |
| 222 | C4 | — | 3 | 3 | no | — |
| 223 | C4 | — | 6 | 12 | no | — |
| 224 | C4 | C23×S4 | 1 | 12 | yes | — |
| 225 | C3 | — | 4 | 1 | no | Sylow |
| 226 | C2 | — | 6 | 12 | no | — |
| 227 | C2 | — | 12 | 4 | no | — |
| 228 | C2 | C2×#5924 | 1 | 6 | yes | direct product factor |
| 229 | C2 | C24×S4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×S4 |
| commutator, cosocle, hypocenter | SL(2,𝔽3) | C25 |
| Frattini | C2 | C24×S4 |
| Fitting | C22×H(5,𝔽2) | D3 |
| hypercenter | C22×Q8 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C23 ◁ C22×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#994 | C2 | semidirect |
| 2. C2×#5924 | C2 | direct (N × H) |
| 3. C22×#985 | C2 | semidirect |
| 4. C22×#984 | C2 | semidirect |
| 7. C2×#994 | C22 | semidirect |
| 9. #5924 | C22 | direct (N × H) |
| 10. C2×#985 | C22 | semidirect |
| 11. C2×#984 | C22 | semidirect |
| 12. C22×GL(2,𝔽3) | C22 | semidirect |
| 13. C22×#226 | C22 | semidirect |
| 15. C22×H(5,𝔽2) | D3 | semidirect |
| 27. #994 | C23 | semidirect |
| 30. #985 | C23 | semidirect |
| 31. #984 | C23 | semidirect |
| 32. C2×GL(2,𝔽3) | C23 | semidirect |
| 33. C2×#226 | C23 | semidirect |
| 37. C2×H(5,𝔽2) | C2×D3 | semidirect |
| 70. GL(2,𝔽3) | C24 | semidirect |
| 71. #226 | C24 | semidirect |
| 77. H(5,𝔽2) | C22×D3 | semidirect |
| 219. C22 | #5924 | direct (N × H) |
| 228. C2 | C2×#5924 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#994 | 2 |
| C2×#5924 | 2 |
| C22×#985 | 2 |
| C22×#984 | 2 |
| C22×#577 | 3 |
| C22×D3×Q8 | 4 |