Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
419 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 419 conjugacy classes of subgroups — too many to draw. Only its 93 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 | #85629 | C2 | 1 | 1 | yes | maximal, characteristic |
| 2 | #73119 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | C2×#4628 | C2 | 1 | 3 | yes | maximal, complemented |
| 4 | #33340 | — | 3 | 1 | no | maximal, Sylow |
| 5 | #5945 | C4 | 1 | 1 | yes | characteristic |
| 6 | #568⋊C3 | C4 | 1 | 3 | yes | — |
| 7 | C2×#131⋊C3 | C22 | 1 | 3 | yes | — |
| 8 | #5140 | — | 4 | 1 | no | maximal |
| 9 | #4628 | C22 | 1 | 1 | yes | characteristic, complemented |
| 10 | #4628 | C22 | 1 | 3 | yes | complemented |
| 11 | #3573 | D3 | 1 | 1 | yes | characteristic |
| 12 | #3407 | — | 3 | 1 | no | — |
| 13 | C2×#504 | — | 3 | 1 | no | — |
| 14 | #2117 | — | 3 | 3 | no | — |
| 15 | #2117 | — | 3 | 3 | no | — |
| 16 | C2×#350 | — | 3 | 3 | no | — |
| 17 | C2×#350 | — | 3 | 3 | no | — |
| 18 | 2−1+4⋊C3 | C8 | 1 | 3 | yes | — |
| 19 | #994 | C8 | 1 | 1 | yes | characteristic |
| 20 | C2×#41⋊C3 | C2×C4 | 1 | 3 | yes | — |
| 21 | C3×#131 | — | 4 | 1 | no | — |
| 22 | #131⋊C3 | C2×C4 | 1 | 3 | yes | — |
| 23 | #131⋊C3 | C23 | 1 | 1 | yes | characteristic |
| 24 | #812 | — | 4 | 3 | no | — |
| 25 | C2×C3⋊C16 | — | 4 | 3 | no | — |
| 26 | #585 | Dic3 | 1 | 1 | yes | characteristic |
| 27 | #568 | — | 3 | 1 | no | — |
| 28 | #568 | — | 3 | 3 | no | — |
| 29 | #568 | Dic3 | 1 | 3 | yes | — |
| 30 | C2×#131 | — | 3 | 1 | no | — |
| 31 | C2×#131 | — | 3 | 3 | no | — |
| 32 | C2×#131 | C2×D3 | 1 | 3 | yes | — |
| 33 | #504 | — | 6 | 1 | no | — |
| 34 | #504 | — | 6 | 3 | no | — |
| 35 | #504 | — | 6 | 3 | no | — |
| 36 | #504 | — | 6 | 1 | no | — |
| 37 | C2×M32 | — | 3 | 3 | no | — |
| 38 | C2×M32 | — | 3 | 3 | no | — |
| 39 | C2×M32 | — | 3 | 3 | no | — |
| 40 | C22×C16 | — | 3 | 3 | no | — |
| 41 | #350 | — | 3 | 3 | no | — |
| 42 | #350 | — | 3 | 6 | no | — |
| 43 | #350 | — | 3 | 3 | no | — |
| 44 | #350 | — | 3 | 1 | no | — |
| 45 | #350 | — | 3 | 3 | no | — |
| 46 | C3×#41 | — | 4 | 1 | no | — |
| 47 | #41⋊C3 | C2×C8 | 1 | 3 | yes | — |
| 48 | #41⋊C3 | C22×C4 | 1 | 1 | yes | characteristic |
| 49 | C2×SL(2,𝔽3) | C2×C8 | 1 | 3 | yes | — |
| 50 | C3×M16 | — | 4 | 3 | no | — |
| 51 | C2×C3×C8 | — | 4 | 3 | no | — |
| 52 | C3⋊C16 | — | 4 | 1 | no | — |
| 53 | C3⋊C16 | — | 4 | 3 | no | — |
| 54 | 2−1+4 | — | 3 | 1 | no | — |
| 55 | 2−1+4 | C3⋊C8 | 1 | 3 | yes | — |
| 56 | H(5,𝔽2) | — | 3 | 3 | no | — |
| 57 | H(5,𝔽2) | C3⋊C8 | 1 | 1 | yes | characteristic |
| 58 | C2×#41 | — | 3 | 1 | no | — |
| 59 | C2×#41 | — | 3 | 3 | no | — |
| 60 | C2×#41 | C2×Dic3 | 1 | 3 | yes | — |
| 61 | #131 | — | 3 | 1 | no | — |
| 62 | #131 | — | 3 | 3 | no | — |
| 63 | #131 | — | 3 | 3 | no | — |
| 64 | #131 | — | 3 | 3 | no | — |
| 65 | #131 | — | 3 | 3 | no | — |
| 66 | #131 | — | 3 | 6 | no | — |
| 67 | #131 | — | 6 | 3 | no | — |
| 68 | #131 | C2×Dic3 | 1 | 3 | yes | — |
| 69 | #131 | S4 | 1 | 1 | yes | characteristic |
| 70 | #131 | C22×D3 | 1 | 1 | yes | characteristic |
| 71 | C2×M16 | — | 3 | 3 | no | — |
| 72 | C2×M16 | — | 3 | 3 | no | — |
| 73 | C2×M16 | — | 3 | 3 | no | — |
| 74 | C2×M16 | — | 6 | 3 | no | — |
| 75 | C22×C8 | — | 3 | 3 | no | — |
| 76 | C22×C8 | — | 6 | 1 | no | — |
| 77 | M32 | — | 3 | 3 | no | — |
| 78 | M32 | — | 3 | 1 | no | — |
| 79 | M32 | — | 6 | 3 | no | — |
| 80 | M32 | — | 6 | 3 | no | — |
| 81 | M32 | — | 6 | 3 | no | — |
| 82 | M32 | — | 6 | 3 | no | — |
| 83 | C2×C16 | — | 3 | 1 | no | — |
| 84 | C2×C16 | — | 3 | 3 | no | — |
| 85 | C2×C16 | — | 6 | 3 | no | — |
| 86 | C2×C16 | — | 6 | 3 | no | — |
| 87 | C2×C16 | — | 6 | 3 | no | — |
| 88 | C2×C16 | — | 6 | 3 | no | — |
| 89 | C3×Q8 | — | 4 | 1 | no | — |
| 90 | C3×D4 | — | 4 | 3 | no | — |
| 91 | C2×C3×C4 | — | 4 | 3 | no | — |
| 92 | SL(2,𝔽3) | C22×C8 | 1 | 1 | yes | characteristic |
| 93 | C3×C8 | — | 4 | 3 | no | — |
| 94 | C3×C8 | — | 4 | 1 | no | — |
| 95 | #41 | — | 3 | 1 | no | — |
| 96 | #41 | — | 3 | 3 | no | — |
| 97 | #41 | — | 3 | 3 | no | — |
| 98 | #41 | — | 3 | 3 | no | — |
| 99 | #41 | — | 3 | 3 | no | — |
| 100 | #41 | — | 3 | 6 | no | — |
| 101 | #41 | — | 6 | 3 | no | — |
| 102 | #41 | C2×C3⋊C8 | 1 | 3 | yes | — |
| 103 | #41 | SL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 104 | #41 | C22×Dic3 | 1 | 1 | yes | characteristic |
| 105 | C2×Q8 | — | 3 | 1 | no | — |
| 106 | C2×Q8 | — | 3 | 3 | no | — |
| 107 | C2×Q8 | C2×C3⋊C8 | 1 | 3 | yes | — |
| 108 | C2×D4 | — | 3 | 3 | no | — |
| 109 | C2×D4 | — | 3 | 3 | no | — |
| 110 | C2×D4 | — | 3 | 3 | no | — |
| 111 | C2×D4 | — | 6 | 3 | no | — |
| 112 | C22×C4 | — | 3 | 3 | no | — |
| 113 | C22×C4 | — | 6 | 1 | no | — |
| 114 | M16 | — | 3 | 3 | no | — |
| 115 | M16 | — | 3 | 6 | no | — |
| 116 | M16 | — | 3 | 3 | no | — |
| 117 | M16 | — | 3 | 1 | no | — |
| 118 | M16 | — | 6 | 3 | no | — |
| 119 | M16 | SL(2,ℤ/4) | 1 | 3 | yes | — |
| 120 | C2×C8 | — | 3 | 3 | no | — |
| 121 | C2×C8 | — | 3 | 3 | no | — |
| 122 | C2×C8 | — | 3 | 3 | no | — |
| 123 | C2×C8 | — | 3 | 1 | no | — |
| 124 | C2×C8 | — | 3 | 3 | no | — |
| 125 | C2×C8 | — | 6 | 3 | no | — |
| 126 | C2×C8 | C2×S4 | 1 | 3 | yes | — |
| 127 | C16 | — | 6 | 1 | no | — |
| 128 | C16 | — | 6 | 3 | no | — |
| 129 | C16 | — | 6 | 3 | no | — |
| 130 | C16 | — | 6 | 1 | no | — |
| 131 | C22×C3 | — | 4 | 3 | no | — |
| 132 | C3×C4 | — | 4 | 3 | no | — |
| 133 | C3×C4 | — | 4 | 1 | no | — |
| 134 | C23 | — | 3 | 3 | no | — |
| 135 | C23 | — | 6 | 1 | no | — |
| 136 | Q8 | — | 3 | 3 | no | — |
| 137 | Q8 | — | 3 | 3 | no | — |
| 138 | Q8 | A4⋊C8 | 1 | 1 | yes | characteristic |
| 139 | Q8 | C22×C3⋊C8 | 1 | 1 | yes | characteristic |
| 140 | D4 | — | 3 | 3 | no | — |
| 141 | D4 | — | 3 | 1 | no | — |
| 142 | D4 | — | 3 | 6 | no | — |
| 143 | D4 | — | 3 | 3 | no | — |
| 144 | D4 | — | 6 | 3 | no | — |
| 145 | D4 | A4⋊C8 | 1 | 3 | yes | — |
| 146 | C2×C4 | — | 3 | 3 | no | — |
| 147 | C2×C4 | — | 3 | 3 | no | — |
| 148 | C2×C4 | — | 3 | 3 | no | — |
| 149 | C2×C4 | — | 3 | 1 | no | — |
| 150 | C2×C4 | — | 3 | 3 | no | — |
| 151 | C2×C4 | — | 6 | 3 | no | — |
| 152 | C2×C4 | C2×SL(2,ℤ/4) | 1 | 3 | yes | — |
| 153 | C8 | — | 3 | 1 | no | — |
| 154 | C8 | — | 3 | 3 | no | — |
| 155 | C8 | C2×SL(2,ℤ/4) | 1 | 3 | yes | — |
| 156 | C8 | C22×S4 | 1 | 1 | yes | characteristic |
| 157 | C2×C3 | — | 4 | 1 | no | — |
| 158 | C2×C3 | — | 8 | 3 | no | — |
| 159 | C22 | — | 3 | 1 | no | — |
| 160 | C22 | — | 3 | 3 | no | — |
| 161 | C22 | — | 12 | 3 | no | — |
| 162 | C22 | — | 24 | 1 | no | — |
| 163 | C22 | C2×A4⋊C8 | 1 | 3 | yes | — |
| 164 | C4 | — | 3 | 1 | no | — |
| 165 | C4 | — | 3 | 3 | no | — |
| 166 | C4 | C2×A4⋊C8 | 1 | 3 | yes | — |
| 167 | C4 | C22×SL(2,ℤ/4) | 1 | 1 | yes | characteristic |
| 168 | C3 | — | 4 | 1 | no | Sylow |
| 169 | C2 | — | 2 | 3 | no | — |
| 170 | C2 | — | 6 | 1 | no | — |
| 171 | C2 | — | 6 | 3 | no | — |
| 172 | C2 | C22×A4⋊C8 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C8 | C22×S4 |
| commutator, hypocenter | SL(2,𝔽3) | C22×C8 |
| Fitting | #3573 | D3 |
| socle | C2 | C22×A4⋊C8 |
| prime radical | #5945 | C4 |
| cosocle | #131⋊C3 | C23 |
| hypercenter | #131 | S4 |
| LCS | G ▷ SL(2,𝔽3) | |
| UCS | C1 ◁ C8 ◁ #131 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #73119 | C2 | semidirect |
| 3. C2×#4628 | C2 | semidirect |
| 9. #4628 | C22 | semidirect |
| 10. #4628 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #85629 | 2 |
| #73119 | 2 |
| C2×#4628 | 2 |
| #33340 | 3 |
| #5140 | 4 |