Finite Group DatabasebetaFinite Group DatabasebetaOrder 384.
950 conjugacy classes of subgroups; only the 449 normal subgroups are shown.
This group has 950 conjugacy classes of subgroups — too many to draw. Only its 449 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 | C3×#564 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #5700 | C2 | 1 | 3 | yes | maximal, complemented |
| 3 | #5697 | C2 | 1 | 6 | yes | maximal, complemented |
| 4 | #5695 | C2 | 1 | 6 | yes | maximal, complemented |
| 5 | #5693 | C2 | 1 | 6 | yes | maximal, complemented |
| 6 | #5692 | C2 | 1 | 6 | yes | maximal, complemented |
| 7 | #5691 | C2 | 1 | 3 | yes | maximal, complemented |
| 8 | #3567 | — | 3 | 1 | no | maximal, Sylow |
| 9 | C3×#125 | C22 | 1 | 3 | yes | — |
| 10 | C3×#125 | C22 | 1 | 6 | yes | — |
| 11 | C3×#125 | C22 | 1 | 6 | yes | — |
| 12 | #898 | C22 | 1 | 6 | yes | — |
| 13 | #898 | C22 | 1 | 6 | yes | — |
| 14 | #897 | C22 | 1 | 3 | yes | — |
| 15 | #896 | C22 | 1 | 6 | yes | — |
| 16 | #896 | C22 | 1 | 6 | yes | — |
| 17 | #895 | C22 | 1 | 3 | yes | — |
| 18 | #895 | C22 | 1 | 6 | yes | — |
| 19 | #895 | C22 | 1 | 6 | yes | — |
| 20 | #895 | C22 | 1 | 6 | yes | — |
| 21 | #895 | C22 | 1 | 6 | yes | — |
| 22 | #894 | C22 | 1 | 6 | yes | — |
| 23 | #894 | C22 | 1 | 6 | yes | — |
| 24 | #894 | C22 | 1 | 6 | yes | — |
| 25 | #894 | C22 | 1 | 6 | yes | — |
| 26 | #894 | C22 | 1 | 6 | yes | — |
| 27 | #893 | C22 | 1 | 6 | yes | — |
| 28 | #893 | C22 | 1 | 3 | yes | — |
| 29 | #893 | C22 | 1 | 6 | yes | — |
| 30 | #892 | C22 | 1 | 6 | yes | — |
| 31 | D3×C4⋊C4 | C22 | 1 | 6 | yes | — |
| 32 | #889 | C22 | 1 | 6 | yes | — |
| 33 | #887 | C22 | 1 | 3 | yes | — |
| 34 | #876 | C22 | 1 | 2 | yes | — |
| 35 | #875 | C22 | 1 | 3 | yes | — |
| 36 | #874 | C22 | 1 | 2 | yes | — |
| 37 | C4×D12 | C22 | 1 | 6 | yes | — |
| 38 | #872 | C22 | 1 | 6 | yes | — |
| 39 | C42×D3 | C22 | 1 | 1 | yes | characteristic |
| 40 | #564 | D3 | 1 | 1 | yes | characteristic, complemented |
| 41 | #562 | — | 3 | 3 | no | — |
| 42 | #553 | — | 3 | 6 | no | — |
| 43 | #552 | — | 3 | 6 | no | — |
| 44 | #540 | — | 3 | 6 | no | — |
| 45 | #539 | — | 3 | 6 | no | — |
| 46 | #529 | — | 3 | 3 | no | — |
| 47 | C2×D12 | C23 | 1 | 6 | yes | — |
| 48 | C2×D12 | C23 | 1 | 6 | yes | — |
| 49 | C2×D12 | C23 | 1 | 6 | yes | — |
| 50 | C2×D12 | C23 | 1 | 3 | yes | — |
| 51 | C2×C4×D3 | C23 | 1 | 3 | yes | — |
| 52 | C2×C4×D3 | C23 | 1 | 6 | yes | — |
| 53 | C2×C4×D3 | C23 | 1 | 6 | yes | — |
| 54 | C2×C4×D3 | C23 | 1 | 6 | yes | — |
| 55 | C2×Dic6 | C23 | 1 | 3 | yes | — |
| 56 | C3×C4⋊C4 | C23 | 1 | 3 | yes | — |
| 57 | C3×C4⋊C4 | C23 | 1 | 6 | yes | — |
| 58 | C3×C4⋊C4 | C23 | 1 | 6 | yes | — |
| 59 | C3×C4⋊C4 | C23 | 1 | 6 | yes | — |
| 60 | C3×C4⋊C4 | C23 | 1 | 3 | yes | — |
| 61 | C3×C4⋊C4 | C23 | 1 | 6 | yes | — |
| 62 | C3×C42 | C23 | 1 | 1 | yes | characteristic |
| 63 | C3×C42 | C23 | 1 | 2 | yes | — |
| 64 | C3×C42 | C23 | 1 | 2 | yes | — |
| 65 | (C2×D3)⋊C4 | C23 | 1 | 6 | yes | — |
| 66 | (C2×D3)⋊C4 | C23 | 1 | 6 | yes | — |
| 67 | (C2×D3)⋊C4 | C23 | 1 | 6 | yes | — |
| 68 | (C2×D3)⋊C4 | C23 | 1 | 6 | yes | — |
| 69 | (C2×D3)⋊C4 | C23 | 1 | 6 | yes | — |
| 70 | (C2×D3)⋊C4 | C23 | 1 | 6 | yes | — |
| 71 | (C2×D3)⋊C4 | C23 | 1 | 6 | yes | — |
| 72 | C4⋊Dic3 | C23 | 1 | 6 | yes | — |
| 73 | Dic3⋊C4 | C23 | 1 | 6 | yes | — |
| 74 | Dic3⋊C4 | C23 | 1 | 6 | yes | — |
| 75 | Dic3⋊C4 | C23 | 1 | 6 | yes | — |
| 76 | C4×Dic3 | C23 | 1 | 3 | yes | — |
| 77 | C4×Dic3 | C23 | 1 | 6 | yes | — |
| 78 | #127 | — | 3 | 2 | no | — |
| 79 | #126 | — | 3 | 6 | no | — |
| 80 | #126 | — | 3 | 6 | no | — |
| 81 | #126 | — | 3 | 2 | no | — |
| 82 | #125 | — | 3 | 6 | no | — |
| 83 | #125 | C2×D3 | 1 | 3 | yes | — |
| 84 | #125 | C2×D3 | 1 | 6 | yes | — |
| 85 | #125 | C2×D3 | 1 | 6 | yes | — |
| 86 | #124 | — | 3 | 3 | no | — |
| 87 | #123 | — | 3 | 6 | no | — |
| 88 | #123 | — | 3 | 6 | no | — |
| 89 | #123 | — | 3 | 6 | no | — |
| 90 | #123 | — | 3 | 6 | no | — |
| 91 | #123 | — | 3 | 6 | no | — |
| 92 | C22⋊Q8 | — | 3 | 3 | no | — |
| 93 | C22⋊Q8 | — | 3 | 6 | no | — |
| 94 | C22⋊Q8 | — | 3 | 6 | no | — |
| 95 | #121 | — | 3 | 3 | no | — |
| 96 | #121 | — | 3 | 6 | no | — |
| 97 | #121 | — | 3 | 6 | no | — |
| 98 | #121 | — | 3 | 6 | no | — |
| 99 | #121 | — | 3 | 6 | no | — |
| 100 | C4×Q8 | — | 3 | 3 | no | — |
| 101 | C4×D4 | — | 3 | 6 | no | — |
| 102 | C4×D4 | — | 3 | 3 | no | — |
| 103 | C4×D4 | — | 3 | 6 | no | — |
| 104 | C4×D4 | — | 3 | 6 | no | — |
| 105 | #117 | — | 3 | 6 | no | — |
| 106 | #117 | — | 3 | 6 | no | — |
| 107 | C2×C4⋊C4 | — | 3 | 6 | no | — |
| 108 | C2×C42 | — | 3 | 1 | no | — |
| 109 | C22×D3 | C24 | 1 | 6 | yes | — |
| 110 | C22×D3 | C24 | 1 | 1 | yes | characteristic |
| 111 | C2×C3×C4 | C24 | 1 | 6 | yes | — |
| 112 | C2×C3×C4 | C24 | 1 | 3 | yes | — |
| 113 | C2×C3×C4 | C24 | 1 | 6 | yes | — |
| 114 | C2×Dic3 | C24 | 1 | 3 | yes | — |
| 115 | C2×Dic3 | C24 | 1 | 6 | yes | — |
| 116 | D12 | — | 4 | 3 | no | — |
| 117 | D12 | — | 4 | 6 | no | — |
| 118 | D12 | — | 4 | 6 | no | — |
| 119 | D12 | — | 4 | 6 | no | — |
| 120 | C4×D3 | — | 4 | 6 | no | — |
| 121 | C4×D3 | — | 4 | 6 | no | — |
| 122 | C4×D3 | — | 4 | 3 | no | — |
| 123 | C4×D3 | — | 4 | 6 | no | — |
| 124 | Dic6 | — | 4 | 3 | no | — |
| 125 | C2×Q8 | — | 3 | 3 | no | — |
| 126 | C2×D4 | — | 3 | 6 | no | — |
| 127 | C2×D4 | — | 3 | 3 | no | — |
| 128 | C2×D4 | — | 3 | 6 | no | — |
| 129 | C2×D4 | — | 3 | 6 | no | — |
| 130 | C22×C4 | — | 3 | 3 | no | — |
| 131 | C22×C4 | — | 3 | 6 | no | — |
| 132 | C22×C4 | — | 3 | 6 | no | — |
| 133 | C22×C4 | — | 3 | 6 | no | — |
| 134 | C4⋊C4 | — | 3 | 6 | no | — |
| 135 | C4⋊C4 | — | 3 | 6 | no | — |
| 136 | C4⋊C4 | — | 3 | 6 | no | — |
| 137 | C4⋊C4 | — | 3 | 6 | no | — |
| 138 | C4⋊C4 | C22×D3 | 1 | 6 | yes | — |
| 139 | C4⋊C4 | C22×D3 | 1 | 6 | yes | — |
| 140 | C4⋊C4 | C22×D3 | 1 | 6 | yes | — |
| 141 | C4⋊C4 | C22×D3 | 1 | 6 | yes | — |
| 142 | C4⋊C4 | C22×D3 | 1 | 3 | yes | — |
| 143 | C4⋊C4 | C22×D3 | 1 | 3 | yes | — |
| 144 | C22⋊C4 | — | 3 | 6 | no | — |
| 145 | C22⋊C4 | — | 3 | 6 | no | — |
| 146 | C22⋊C4 | — | 3 | 6 | no | — |
| 147 | C22⋊C4 | — | 3 | 6 | no | — |
| 148 | C22⋊C4 | — | 3 | 6 | no | — |
| 149 | C22⋊C4 | — | 3 | 6 | no | — |
| 150 | C22⋊C4 | — | 3 | 6 | no | — |
| 151 | C42 | — | 3 | 3 | no | — |
| 152 | C42 | — | 3 | 6 | no | — |
| 153 | C42 | C22×D3 | 1 | 1 | yes | characteristic |
| 154 | C42 | C22×D3 | 1 | 2 | yes | — |
| 155 | C42 | C22×D3 | 1 | 2 | yes | — |
| 156 | C22×C3 | C25 | 1 | 1 | yes | characteristic |
| 157 | C2×D3 | — | 2 | 6 | no | — |
| 158 | C2×D3 | — | 2 | 6 | no | — |
| 159 | C2×D3 | — | 2 | 3 | no | — |
| 160 | C2×D3 | — | 2 | 6 | no | — |
| 161 | C3×C4 | — | 2 | 3 | no | — |
| 162 | C3×C4 | — | 2 | 6 | no | — |
| 163 | C3×C4 | — | 2 | 6 | no | — |
| 164 | Dic3 | — | 2 | 6 | no | — |
| 165 | Dic3 | — | 2 | 3 | no | — |
| 166 | C23 | — | 3 | 6 | no | — |
| 167 | C23 | — | 3 | 1 | no | — |
| 168 | Q8 | — | 12 | 3 | no | — |
| 169 | D4 | — | 12 | 6 | no | — |
| 170 | D4 | — | 12 | 6 | no | — |
| 171 | D4 | — | 12 | 6 | no | — |
| 172 | D4 | — | 12 | 3 | no | — |
| 173 | C2×C4 | — | 3 | 3 | no | — |
| 174 | C2×C4 | — | 3 | 6 | no | — |
| 175 | C2×C4 | — | 12 | 6 | no | — |
| 176 | C2×C4 | — | 12 | 6 | no | — |
| 177 | C2×C4 | — | 12 | 6 | no | — |
| 178 | C2×C4 | — | 12 | 3 | no | — |
| 179 | C2×C4 | C23×D3 | 1 | 3 | yes | — |
| 180 | C2×C4 | C23×D3 | 1 | 6 | yes | — |
| 181 | C2×C4 | C23×D3 | 1 | 6 | yes | — |
| 182 | C2×C3 | #585 | 1 | 3 | yes | — |
| 183 | D3 | — | 4 | 6 | no | — |
| 184 | D3 | — | 4 | 1 | no | — |
| 185 | C22 | — | 6 | 6 | no | — |
| 186 | C22 | — | 6 | 6 | no | — |
| 187 | C22 | — | 6 | 3 | no | — |
| 188 | C22 | — | 6 | 6 | no | — |
| 189 | C22 | C24×D3 | 1 | 1 | yes | characteristic |
| 190 | C4 | — | 2 | 6 | no | — |
| 191 | C4 | — | 2 | 6 | no | — |
| 192 | C4 | — | 2 | 3 | no | — |
| 193 | C4 | — | 6 | 3 | no | — |
| 194 | C4 | — | 6 | 6 | no | — |
| 195 | C3 | #3567 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 196 | C2 | — | 12 | 6 | no | — |
| 197 | C2 | — | 12 | 1 | no | — |
| 198 | C2 | #5968 | 1 | 3 | yes | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×#564 | C2 |
| hypercenter | #564 | D3 |
| hypocenter | C3 | #3567 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ #564 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C3×#564 | C2 | semidirect |
| 2. #5700 | C2 | semidirect |
| 3. #5697 | C2 | semidirect |
| 4. #5695 | C2 | semidirect |
| 5. #5693 | C2 | semidirect |
| 6. #5692 | C2 | semidirect |
| 7. #5691 | C2 | semidirect |
| 40. #564 | D3 | semidirect |
| 195. C3 | #3567 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×#564 | 2 |
| #5700 | 2 |
| #5697 | 2 |
| #5695 | 2 |
| #5693 | 2 |
| #5692 | 2 |
| #5691 | 2 |
| #3567 | 3 |