Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
11481 conjugacy classes of subgroups; only the 179 normal subgroups are shown.
This group has 11481 conjugacy classes of subgroups — too many to draw. Only its 179 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×#997 | C2 | 1 | 7 | yes | maximal, direct product factor |
| 2 | C23×H(5,𝔽2) | C3 | 1 | 1 | yes | maximal, characteristic, complemented, Sylow |
| 3 | C24×A4 | — | 4 | 3 | no | maximal |
| 4 | C2×#997 | C22 | 1 | 7 | yes | direct product factor |
| 5 | C23×SL(2,𝔽3) | — | 4 | 2 | no | maximal |
| 6 | C22×H(5,𝔽2) | — | 3 | 14 | no | — |
| 7 | C22×H(5,𝔽2) | — | 3 | 21 | no | — |
| 8 | C22×H(5,𝔽2) | C2×C3 | 1 | 7 | yes | complemented |
| 9 | C23×#41 | — | 3 | 2 | no | — |
| 10 | C24×D4 | — | 3 | 3 | no | — |
| 11 | C23×A4 | — | 4 | 24 | no | — |
| 12 | C23×A4 | — | 4 | 21 | no | — |
| 13 | #997 | C23 | 1 | 1 | yes | characteristic, direct product factor |
| 14 | C22×SL(2,𝔽3) | — | 4 | 14 | no | — |
| 15 | C26 | — | 3 | 1 | no | — |
| 16 | C26 | A4 | 1 | 3 | yes | complemented |
| 17 | C2×H(5,𝔽2) | — | 3 | 126 | no | — |
| 18 | C2×H(5,𝔽2) | — | 3 | 252 | no | — |
| 19 | C2×H(5,𝔽2) | — | 3 | 42 | no | — |
| 20 | C2×H(5,𝔽2) | — | 3 | 28 | no | — |
| 21 | C2×H(5,𝔽2) | — | 3 | 42 | no | — |
| 22 | C2×H(5,𝔽2) | — | 3 | 42 | no | — |
| 23 | C2×H(5,𝔽2) | — | 3 | 63 | no | — |
| 24 | C2×H(5,𝔽2) | C22×C3 | 1 | 7 | yes | complemented |
| 25 | C22×#41 | — | 3 | 42 | no | — |
| 26 | C22×#41 | — | 3 | 42 | no | — |
| 27 | C22×#41 | — | 3 | 14 | no | — |
| 28 | C22×#41 | — | 3 | 14 | no | — |
| 29 | C23×Q8 | A4 | 1 | 2 | yes | complemented |
| 30 | C23×D4 | — | 3 | 6 | no | — |
| 31 | C23×D4 | — | 3 | 21 | no | — |
| 32 | C23×D4 | — | 3 | 84 | no | — |
| 33 | C23×D4 | — | 3 | 21 | no | — |
| 34 | C23×D4 | — | 3 | 21 | no | — |
| 35 | C23×D4 | — | 3 | 21 | no | — |
| 36 | C24×C4 | — | 3 | 3 | no | — |
| 37 | C24×C3 | — | 16 | 1 | no | — |
| 38 | C22×A4 | — | 4 | 84 | no | — |
| 39 | C22×A4 | — | 4 | 21 | no | — |
| 40 | C2×SL(2,𝔽3) | — | 4 | 14 | no | — |
| 41 | C25 | — | 3 | 21 | no | — |
| 42 | C25 | — | 3 | 7 | no | — |
| 43 | C25 | — | 3 | 3 | no | — |
| 44 | C25 | — | 3 | 21 | no | — |
| 45 | C25 | — | 4 | 24 | no | — |
| 46 | C25 | — | 12 | 8 | no | — |
| 47 | C25 | C2×A4 | 1 | 21 | yes | complemented |
| 48 | H(5,𝔽2) | — | 3 | 14 | no | — |
| 49 | H(5,𝔽2) | — | 3 | 336 | no | — |
| 50 | H(5,𝔽2) | — | 3 | 504 | no | — |
| 51 | H(5,𝔽2) | — | 3 | 126 | no | — |
| 52 | H(5,𝔽2) | — | 3 | 252 | no | — |
| 53 | H(5,𝔽2) | — | 3 | 42 | no | — |
| 54 | H(5,𝔽2) | — | 3 | 28 | no | — |
| 55 | H(5,𝔽2) | — | 3 | 42 | no | — |
| 56 | H(5,𝔽2) | — | 3 | 21 | no | — |
| 57 | H(5,𝔽2) | C23×C3 | 1 | 1 | yes | characteristic, complemented |
| 58 | C2×#41 | — | 3 | 252 | no | — |
| 59 | C2×#41 | — | 3 | 84 | no | — |
| 60 | C2×#41 | — | 3 | 252 | no | — |
| 61 | C2×#41 | — | 3 | 126 | no | — |
| 62 | C2×#41 | — | 3 | 126 | no | — |
| 63 | C2×#41 | — | 3 | 42 | no | — |
| 64 | C2×#41 | — | 3 | 14 | no | — |
| 65 | C22×Q8 | — | 3 | 14 | no | — |
| 66 | C22×Q8 | C2×A4 | 1 | 14 | yes | complemented |
| 67 | C22×D4 | — | 3 | 84 | no | — |
| 68 | C22×D4 | — | 3 | 42 | no | — |
| 69 | C22×D4 | — | 3 | 42 | no | — |
| 70 | C22×D4 | — | 3 | 252 | no | — |
| 71 | C22×D4 | — | 3 | 252 | no | — |
| 72 | C22×D4 | — | 3 | 252 | no | — |
| 73 | C22×D4 | — | 3 | 126 | no | — |
| 74 | C22×D4 | — | 3 | 63 | no | — |
| 75 | C22×D4 | — | 3 | 252 | no | — |
| 76 | C22×D4 | — | 3 | 63 | no | — |
| 77 | C22×D4 | — | 3 | 63 | no | — |
| 78 | C22×D4 | — | 3 | 21 | no | — |
| 79 | C23×C4 | — | 3 | 21 | no | — |
| 80 | C23×C4 | — | 3 | 42 | no | — |
| 81 | C23×C4 | — | 3 | 21 | no | — |
| 82 | C23×C4 | — | 3 | 2 | no | — |
| 83 | C23×C3 | — | 16 | 8 | no | — |
| 84 | C23×C3 | — | 16 | 7 | no | — |
| 85 | C2×A4 | — | 4 | 42 | no | — |
| 86 | C2×A4 | — | 4 | 3 | no | — |
| 87 | SL(2,𝔽3) | — | 4 | 2 | no | — |
| 88 | C24 | — | 3 | 42 | no | — |
| 89 | C24 | — | 3 | 42 | no | — |
| 90 | C24 | — | 3 | 63 | no | — |
| 91 | C24 | — | 3 | 7 | no | — |
| 92 | C24 | — | 3 | 21 | no | — |
| 93 | C24 | — | 3 | 21 | no | — |
| 94 | C24 | — | 3 | 63 | no | — |
| 95 | C24 | — | 4 | 84 | no | — |
| 96 | C24 | — | 6 | 24 | no | — |
| 97 | C24 | — | 12 | 28 | no | — |
| 98 | C24 | — | 12 | 84 | no | — |
| 99 | C24 | — | 12 | 84 | no | — |
| 100 | C24 | C22×A4 | 1 | 21 | yes | complemented |
| 101 | C24 | C22⋊A4 | 1 | 1 | yes | characteristic |
| 102 | #41 | — | 3 | 252 | no | — |
| 103 | #41 | — | 3 | 42 | no | — |
| 104 | #41 | — | 3 | 42 | no | — |
| 105 | #41 | — | 3 | 336 | no | — |
| 106 | #41 | — | 3 | 252 | no | — |
| 107 | #41 | — | 3 | 84 | no | — |
| 108 | #41 | — | 3 | 14 | no | — |
| 109 | #41 | — | 3 | 2 | no | — |
| 110 | C2×Q8 | — | 3 | 28 | no | — |
| 111 | C2×Q8 | — | 3 | 42 | no | — |
| 112 | C2×Q8 | C22×A4 | 1 | 14 | yes | complemented |
| 113 | C2×D4 | — | 3 | 252 | no | — |
| 114 | C2×D4 | — | 3 | 252 | no | — |
| 115 | C2×D4 | — | 3 | 126 | no | — |
| 116 | C2×D4 | — | 3 | 42 | no | — |
| 117 | C2×D4 | — | 3 | 504 | no | — |
| 118 | C2×D4 | — | 3 | 252 | no | — |
| 119 | C2×D4 | — | 3 | 252 | no | — |
| 120 | C2×D4 | — | 3 | 252 | no | — |
| 121 | C2×D4 | — | 3 | 126 | no | — |
| 122 | C2×D4 | — | 3 | 21 | no | — |
| 123 | C2×D4 | — | 3 | 84 | no | — |
| 124 | C2×D4 | — | 3 | 21 | no | — |
| 125 | C2×D4 | — | 3 | 21 | no | — |
| 126 | C2×D4 | — | 3 | 3 | no | — |
| 127 | C22×C4 | — | 3 | 126 | no | — |
| 128 | C22×C4 | — | 3 | 63 | no | — |
| 129 | C22×C4 | — | 3 | 126 | no | — |
| 130 | C22×C4 | — | 3 | 21 | no | — |
| 131 | C22×C4 | — | 3 | 14 | no | — |
| 132 | C22×C4 | — | 3 | 14 | no | — |
| 133 | C22×C3 | — | 16 | 28 | no | — |
| 134 | C22×C3 | — | 16 | 7 | no | — |
| 135 | A4 | — | 4 | 3 | no | — |
| 136 | C23 | — | 3 | 42 | no | — |
| 137 | C23 | — | 3 | 42 | no | — |
| 138 | C23 | — | 3 | 21 | no | — |
| 139 | C23 | — | 3 | 1 | no | — |
| 140 | C23 | — | 3 | 63 | no | — |
| 141 | C23 | — | 3 | 21 | no | — |
| 142 | C23 | — | 3 | 21 | no | — |
| 143 | C23 | — | 4 | 42 | no | — |
| 144 | C23 | — | 6 | 84 | no | — |
| 145 | C23 | — | 6 | 84 | no | — |
| 146 | C23 | — | 12 | 14 | no | — |
| 147 | C23 | — | 12 | 126 | no | — |
| 148 | C23 | — | 12 | 84 | no | — |
| 149 | C23 | — | 12 | 84 | no | — |
| 150 | C23 | — | 12 | 126 | no | — |
| 151 | C23 | #997 | 1 | 8 | yes | direct product factor |
| 152 | C23 | C23×A4 | 1 | 3 | yes | complemented |
| 153 | C23 | C2×C22⋊A4 | 1 | 7 | yes | — |
| 154 | Q8 | — | 3 | 14 | no | — |
| 155 | Q8 | — | 3 | 28 | no | — |
| 156 | Q8 | C23×A4 | 1 | 2 | yes | complemented |
| 157 | D4 | — | 3 | 252 | no | — |
| 158 | D4 | — | 3 | 84 | no | — |
| 159 | D4 | — | 3 | 42 | no | — |
| 160 | D4 | — | 3 | 6 | no | — |
| 161 | C2×C4 | — | 3 | 126 | no | — |
| 162 | C2×C4 | — | 3 | 21 | no | — |
| 163 | C2×C4 | — | 3 | 42 | no | — |
| 164 | C2×C4 | — | 3 | 3 | no | — |
| 165 | C2×C4 | — | 3 | 42 | no | — |
| 166 | C2×C4 | — | 3 | 14 | no | — |
| 167 | C2×C3 | — | 16 | 1 | no | — |
| 168 | C2×C3 | — | 16 | 14 | no | — |
| 169 | C22 | — | 3 | 21 | no | — |
| 170 | C22 | — | 3 | 3 | no | — |
| 171 | C22 | — | 4 | 3 | no | — |
| 172 | C22 | — | 6 | 42 | no | — |
| 173 | C22 | — | 6 | 126 | no | — |
| 174 | C22 | — | 12 | 21 | no | — |
| 175 | C22 | — | 12 | 1 | no | — |
| 176 | C22 | — | 12 | 21 | no | — |
| 177 | C22 | — | 12 | 42 | no | — |
| 178 | C22 | — | 12 | 42 | no | — |
| 179 | C22 | C2×#997 | 1 | 28 | yes | direct product factor |
| 180 | C22 | C22×C22⋊A4 | 1 | 7 | yes | — |
| 181 | C4 | — | 3 | 14 | no | — |
| 182 | C4 | — | 3 | 2 | no | — |
| 183 | C3 | — | 16 | 1 | no | Sylow |
| 184 | C2 | — | 6 | 3 | no | — |
| 185 | C2 | — | 6 | 21 | no | — |
| 186 | C2 | C22×#997 | 1 | 14 | yes | direct product factor |
| 187 | C2 | C23×C22⋊A4 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C24 | C22⋊A4 |
| commutator, cosocle, hypocenter | H(5,𝔽2) | C23×C3 |
| Frattini | C2 | C23×C22⋊A4 |
| Fitting | C23×H(5,𝔽2) | C3 |
| LCS | G ▷ H(5,𝔽2) | |
| UCS | C1 ◁ C24 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#997 | C2 | direct (N × H) |
| 2. C23×H(5,𝔽2) | C3 | semidirect |
| 4. C2×#997 | C22 | direct (N × H) |
| 8. C22×H(5,𝔽2) | C2×C3 | semidirect |
| 13. #997 | C23 | direct (N × H) |
| 16. C26 | A4 | semidirect |
| 24. C2×H(5,𝔽2) | C22×C3 | semidirect |
| 29. C23×Q8 | A4 | semidirect |
| 47. C25 | C2×A4 | semidirect |
| 57. H(5,𝔽2) | C23×C3 | semidirect |
| 66. C22×Q8 | C2×A4 | semidirect |
| 100. C24 | C22×A4 | semidirect |
| 112. C2×Q8 | C22×A4 | semidirect |
| 151. C23 | #997 | direct (N × H) |
| 152. C23 | C23×A4 | semidirect |
| 156. Q8 | C23×A4 | semidirect |
| 179. C22 | C2×#997 | direct (N × H) |
| 186. C2 | C22×#997 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#997 | 2 |
| C23×H(5,𝔽2) | 3 |
| C24×A4 | 4 |
| C23×SL(2,𝔽3) | 4 |