Finite Group DatabasebetaFinite Group DatabasebetaOrder 768.
10698 conjugacy classes of subgroups; only the 3815 normal subgroups are shown.
This group has 10698 conjugacy classes of subgroups — too many to draw. Only its 3815 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 | #87572 | C2 | 1 | 2 | yes | maximal, complemented |
| 2 | C3×#3573 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #87168 | C2 | 1 | 30 | yes | maximal, complemented |
| 4 | C2×#5753 | C2 | 1 | 15 | yes | maximal, complemented |
| 5 | #87163 | C2 | 1 | 15 | yes | maximal, complemented |
| 6 | #63093 | — | 3 | 1 | no | maximal, Sylow |
| 7 | C3×#585 | C22 | 1 | 1 | yes | characteristic |
| 8 | D3×2−1+4 | — | 2 | 12 | no | — |
| 9 | #5966 | — | 2 | 20 | no | — |
| 10 | #5964 | C22 | 1 | 30 | yes | complemented |
| 11 | C2×#1010 | C22 | 1 | 30 | yes | complemented |
| 12 | C3×#568 | C22 | 1 | 15 | yes | complemented |
| 13 | C2×C3×#131 | C22 | 1 | 15 | yes | complemented |
| 14 | #5753 | C22 | 1 | 60 | yes | complemented |
| 15 | #5753 | C22 | 1 | 120 | yes | complemented |
| 16 | #5752 | C22 | 1 | 20 | yes | complemented |
| 17 | #5751 | C22 | 1 | 120 | yes | complemented |
| 18 | #5748 | C22 | 1 | 90 | yes | complemented |
| 19 | C2×#909 | C22 | 1 | 90 | yes | complemented |
| 20 | C22×Dic12 | C22 | 1 | 15 | yes | complemented |
| 21 | C2×#904 | C22 | 1 | 45 | yes | complemented |
| 22 | 2−1+6 | — | 3 | 2 | no | — |
| 23 | #3586 | — | 3 | 30 | no | — |
| 24 | #3584 | — | 3 | 15 | no | — |
| 25 | C2×#578 | — | 3 | 15 | no | — |
| 26 | #3573 | D3 | 1 | 1 | yes | characteristic, complemented |
| 27 | C3×2−1+4 | D4 | 1 | 6 | yes | — |
| 28 | C3×H(5,𝔽2) | D4 | 1 | 10 | yes | — |
| 29 | C2×C3×#41 | C23 | 1 | 15 | yes | — |
| 30 | #1010 | — | 2 | 120 | no | — |
| 31 | #1010 | C23 | 1 | 120 | yes | — |
| 32 | #1009 | C23 | 1 | 40 | yes | — |
| 33 | D3×#41 | — | 2 | 120 | no | — |
| 34 | #1007 | — | 2 | 30 | no | — |
| 35 | C2×D3×Q8 | — | 2 | 30 | no | — |
| 36 | #1004 | — | 2 | 90 | no | — |
| 37 | C2×#237 | — | 2 | 90 | no | — |
| 38 | C2×#235 | C23 | 1 | 90 | yes | — |
| 39 | C22×Dic6 | C23 | 1 | 30 | yes | — |
| 40 | C3×#131 | C23 | 1 | 20 | yes | — |
| 41 | C3×#131 | C23 | 1 | 60 | yes | — |
| 42 | C2×C3×M16 | C23 | 1 | 45 | yes | — |
| 43 | C22×C3×C8 | C23 | 1 | 15 | yes | — |
| 44 | #909 | C23 | 1 | 360 | yes | — |
| 45 | #908 | C23 | 1 | 120 | yes | — |
| 46 | C2×Dic12 | C23 | 1 | 45 | yes | — |
| 47 | C2×Dic12 | C23 | 1 | 90 | yes | — |
| 48 | #904 | C23 | 1 | 60 | yes | — |
| 49 | #904 | C23 | 1 | 180 | yes | — |
| 50 | C2×D24 | C23 | 1 | 15 | yes | — |
| 51 | C2×#204 | C23 | 1 | 90 | yes | — |
| 52 | #585 | — | 3 | 30 | no | — |
| 53 | #585 | — | 6 | 20 | no | — |
| 54 | #585 | C2×D3 | 1 | 1 | yes | characteristic |
| 55 | C2×2−1+4 | — | 3 | 30 | no | — |
| 56 | C2×2−1+4 | — | 6 | 12 | no | — |
| 57 | #578 | — | 3 | 60 | no | — |
| 58 | #578 | — | 3 | 120 | no | — |
| 59 | #577 | — | 3 | 120 | no | — |
| 60 | #576 | — | 3 | 20 | no | — |
| 61 | #575 | — | 3 | 90 | no | — |
| 62 | C2×#137 | — | 3 | 90 | no | — |
| 63 | C2×#135 | — | 3 | 45 | no | — |
| 64 | C22×Q16 | — | 3 | 15 | no | — |
| 65 | #568 | C2×D3 | 1 | 15 | yes | complemented |
| 66 | C2×#131 | C2×D3 | 1 | 15 | yes | complemented |
| 67 | C3×#41 | C2×D4 | 1 | 60 | yes | — |
| 68 | C3×#41 | C24 | 1 | 20 | yes | — |
| 69 | C2×C3×Q8 | C2×D4 | 1 | 15 | yes | — |
| 70 | C2×C3×D4 | C2×D4 | 1 | 45 | yes | — |
| 71 | C22×C3×C4 | C24 | 1 | 15 | yes | — |
| 72 | C2×#67 | — | 2 | 90 | no | — |
| 73 | C22×Dic3 | — | 2 | 30 | no | — |
| 74 | #239 | — | 2 | 40 | no | — |
| 75 | D3×Q8 | — | 2 | 120 | no | — |
| 76 | #237 | — | 2 | 360 | no | — |
| 77 | D3×D4 | — | 2 | 120 | no | — |
| 78 | #235 | — | 2 | 180 | no | — |
| 79 | #235 | C24 | 1 | 120 | yes | — |
| 80 | C2×D12 | C24 | 1 | 30 | yes | — |
| 81 | C2×C4×D3 | — | 2 | 90 | no | — |
| 82 | C2×Dic6 | — | 2 | 90 | no | — |
| 83 | C2×Dic6 | C24 | 1 | 90 | yes | — |
| 84 | C3×M16 | C24 | 1 | 60 | yes | — |
| 85 | C2×C3×C8 | C24 | 1 | 45 | yes | — |
| 86 | C2×C3×C8 | C24 | 1 | 15 | yes | — |
| 87 | Dic12 | C24 | 1 | 10 | yes | — |
| 88 | Dic12 | C24 | 1 | 90 | yes | — |
| 89 | D24 | C24 | 1 | 6 | yes | — |
| 90 | D24 | C24 | 1 | 30 | yes | — |
| 91 | #204 | C24 | 1 | 120 | yes | — |
| 92 | 2−1+4 | — | 3 | 120 | no | — |
| 93 | 2−1+4 | — | 6 | 120 | no | — |
| 94 | 2−1+4 | — | 6 | 30 | no | — |
| 95 | 2−1+4 | D12 | 1 | 6 | yes | — |
| 96 | H(5,𝔽2) | — | 3 | 40 | no | — |
| 97 | H(5,𝔽2) | — | 6 | 90 | no | — |
| 98 | H(5,𝔽2) | D12 | 1 | 10 | yes | — |
| 99 | C2×#41 | — | 3 | 90 | no | — |
| 100 | C2×#41 | — | 6 | 120 | no | — |
| 101 | C2×#41 | — | 6 | 90 | no | — |
| 102 | C2×#41 | C22×D3 | 1 | 15 | yes | — |
| 103 | C22×Q8 | — | 3 | 30 | no | — |
| 104 | C22×Q8 | — | 6 | 30 | no | — |
| 105 | #137 | — | 3 | 360 | no | — |
| 106 | AGL(1,ℤ/8) | — | 3 | 120 | no | — |
| 107 | #135 | — | 3 | 60 | no | — |
| 108 | #135 | — | 3 | 180 | no | — |
| 109 | C2×Q16 | — | 3 | 45 | no | — |
| 110 | C2×Q16 | — | 3 | 90 | no | — |
| 111 | C2×SD16 | — | 3 | 90 | no | — |
| 112 | C2×D8 | — | 3 | 15 | no | — |
| 113 | #131 | C22×D3 | 1 | 20 | yes | — |
| 114 | #131 | C22×D3 | 1 | 60 | yes | — |
| 115 | C2×M16 | C22×D3 | 1 | 45 | yes | — |
| 116 | C22×C8 | C22×D3 | 1 | 15 | yes | — |
| 117 | C23×C3 | C22×D4 | 1 | 15 | yes | — |
| 118 | C22×D3 | — | 2 | 30 | no | — |
| 119 | C3×Q8 | C22×D4 | 1 | 20 | yes | — |
| 120 | C3×D4 | C22×D4 | 1 | 60 | yes | — |
| 121 | C2×C3×C4 | C22×D4 | 1 | 45 | yes | — |
| 122 | C2×C3×C4 | C25 | 1 | 15 | yes | — |
| 123 | #67 | — | 2 | 120 | no | — |
| 124 | C2×Dic3 | — | 2 | 90 | no | — |
| 125 | D12 | — | 2 | 30 | no | — |
| 126 | D12 | C25 | 1 | 12 | yes | — |
| 127 | C4×D3 | — | 2 | 120 | no | — |
| 128 | Dic6 | — | 2 | 90 | no | — |
| 129 | Dic6 | C25 | 1 | 20 | yes | — |
| 130 | C3×C8 | C25 | 1 | 15 | yes | — |
| 131 | C3×C8 | C25 | 1 | 1 | yes | characteristic |
| 132 | #41 | — | 3 | 120 | no | — |
| 133 | #41 | — | 6 | 40 | no | — |
| 134 | #41 | — | 6 | 360 | no | — |
| 135 | #41 | — | 6 | 180 | no | — |
| 136 | #41 | C2×D12 | 1 | 60 | yes | — |
| 137 | #41 | C23×D3 | 1 | 20 | yes | — |
| 138 | C2×Q8 | — | 3 | 90 | no | — |
| 139 | C2×Q8 | — | 6 | 120 | no | — |
| 140 | C2×Q8 | — | 6 | 90 | no | — |
| 141 | C2×Q8 | C2×D12 | 1 | 15 | yes | — |
| 142 | C2×D4 | — | 3 | 30 | no | — |
| 143 | C2×D4 | — | 6 | 90 | no | — |
| 144 | C2×D4 | — | 6 | 120 | no | — |
| 145 | C2×D4 | C2×D12 | 1 | 45 | yes | — |
| 146 | C22×C4 | — | 6 | 30 | no | — |
| 147 | C22×C4 | — | 6 | 90 | no | — |
| 148 | C22×C4 | C23×D3 | 1 | 15 | yes | — |
| 149 | Q16 | — | 3 | 10 | no | — |
| 150 | Q16 | — | 3 | 90 | no | — |
| 151 | SD16 | — | 3 | 120 | no | — |
| 152 | D8 | — | 3 | 6 | no | — |
| 153 | D8 | — | 3 | 30 | no | — |
| 154 | M16 | C23×D3 | 1 | 60 | yes | — |
| 155 | C2×C8 | C23×D3 | 1 | 15 | yes | — |
| 156 | C2×C8 | C23×D3 | 1 | 45 | yes | — |
| 157 | C22×C3 | — | 4 | 15 | no | — |
| 158 | C22×C3 | C23×D4 | 1 | 15 | yes | — |
| 159 | C2×D3 | — | 2 | 12 | no | — |
| 160 | C2×D3 | — | 8 | 30 | no | — |
| 161 | C3×C4 | C23×D4 | 1 | 15 | yes | — |
| 162 | C3×C4 | C26 | 1 | 1 | yes | characteristic |
| 163 | Dic3 | — | 2 | 20 | no | — |
| 164 | C23 | — | 6 | 30 | no | — |
| 165 | C23 | C22×D12 | 1 | 15 | yes | — |
| 166 | Q8 | — | 3 | 20 | no | — |
| 167 | Q8 | — | 6 | 90 | no | — |
| 168 | Q8 | C22×D12 | 1 | 20 | yes | — |
| 169 | D4 | — | 3 | 12 | no | — |
| 170 | D4 | — | 6 | 120 | no | — |
| 171 | D4 | — | 6 | 30 | no | — |
| 172 | D4 | C22×D12 | 1 | 60 | yes | — |
| 173 | C2×C4 | — | 6 | 90 | no | — |
| 174 | C2×C4 | — | 6 | 120 | no | — |
| 175 | C2×C4 | C22×D12 | 1 | 45 | yes | — |
| 176 | C2×C4 | C24×D3 | 1 | 15 | yes | — |
| 177 | C8 | C24×D3 | 1 | 1 | yes | characteristic |
| 178 | C8 | C24×D3 | 1 | 15 | yes | — |
| 179 | C2×C3 | — | 2 | 15 | no | — |
| 180 | C2×C3 | C24×D4 | 1 | 1 | yes | characteristic |
| 181 | D3 | — | 4 | 12 | no | — |
| 182 | C22 | — | 4 | 15 | no | — |
| 183 | C22 | — | 6 | 12 | no | — |
| 184 | C22 | — | 24 | 30 | no | — |
| 185 | C22 | C23×D12 | 1 | 15 | yes | — |
| 186 | C4 | — | 6 | 20 | no | — |
| 187 | C4 | C23×D12 | 1 | 15 | yes | — |
| 188 | C4 | C25×D3 | 1 | 1 | yes | characteristic |
| 189 | C3 | #63093 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 190 | C2 | — | 2 | 15 | no | — |
| 191 | C2 | — | 12 | 12 | no | — |
| 192 | C2 | C24×D12 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C24×D12 |
| commutator, cosocle | C3×C4 | C26 |
| Frattini | C4 | C25×D3 |
| Fitting | C3×#3573 | C2 |
| socle | C2×C3 | C24×D4 |
| hypercenter | #3573 | D3 |
| hypocenter | C3 | #63093 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ #585 ◁ #3573 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #87572 | C2 | semidirect |
| 2. C3×#3573 | C2 | semidirect |
| 3. #87168 | C2 | semidirect |
| 4. C2×#5753 | C2 | semidirect |
| 5. #87163 | C2 | semidirect |
| 10. #5964 | C22 | semidirect |
| 11. C2×#1010 | C22 | semidirect |
| 12. C3×#568 | C22 | semidirect |
| 13. C2×C3×#131 | C22 | semidirect |
| 14. #5753 | C22 | semidirect |
| 15. #5753 | C22 | semidirect |
| 16. #5752 | C22 | semidirect |
| 17. #5751 | C22 | semidirect |
| 18. #5748 | C22 | semidirect |
| 19. C2×#909 | C22 | semidirect |
| 20. C22×Dic12 | C22 | semidirect |
| 21. C2×#904 | C22 | semidirect |
| 26. #3573 | D3 | semidirect |
| 65. #568 | C2×D3 | semidirect |
| 66. C2×#131 | C2×D3 | semidirect |
| 189. C3 | #63093 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #87572 | 2 |
| C3×#3573 | 2 |
| #87168 | 2 |
| C2×#5753 | 2 |
| #87163 | 2 |
| #63093 | 3 |