Finite Group DatabasebetaFinite Group DatabasebetaOrder 15552.
419 conjugacy classes of subgroups; only the 21 normal subgroups are shown.
This group has 419 conjugacy classes of subgroups — too many to draw. Only its 21 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 | #72220479 | — | 4 | 2 | no | maximal |
| 2 | PSU(3,𝔽2)2 | C3 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #157×PSU(3,𝔽2) | — | 3 | 2 | no | — |
| 4 | #72213277 | — | 3 | 3 | no | — |
| 5 | #72210295 | — | 4 | 3 | no | maximal |
| 6 | #14519481 | — | 4 | 3 | no | — |
| 7 | PSU(3,𝔽2)⋊H(3,𝔽3) | — | 4 | 2 | no | — |
| 8 | Q8⋊(PSU(3,𝔽2)⋊C3) | — | 9 | 2 | no | maximal |
| 9 | #13049818 | — | 3 | 1 | no | — |
| 10 | #13049818 | A4 | 1 | 3 | yes | — |
| 11 | C3⋊D3×PSU(3,𝔽2) | A4 | 1 | 2 | yes | — |
| 12 | #13049783 | — | 3 | 6 | no | — |
| 13 | #1572 | — | 3 | 3 | no | — |
| 14 | D3×PSU(3,𝔽2)⋊C3 | — | 12 | 2 | no | — |
| 15 | #11757702 | — | 16 | 1 | no | — |
| 16 | #10620211 | — | 4 | 3 | no | — |
| 17 | #10620211 | — | 12 | 1 | no | — |
| 18 | #10620209 | — | 3 | 2 | no | — |
| 19 | C32×PSU(3,𝔽2) | SL(2,𝔽3) | 1 | 2 | yes | complemented |
| 20 | #10620198 | — | 3 | 3 | no | — |
| 21 | C3⋊D3×#157 | — | 3 | 2 | no | — |
| 22 | C3×PSU(3,𝔽2)⋊C3 | — | 4 | 2 | no | — |
| 23 | C3×PSU(3,𝔽2)⋊C3 | — | 8 | 2 | no | — |
| 24 | Q8×PSU(3,𝔽2) | — | 9 | 2 | no | — |
| 25 | #92301 | — | 16 | 1 | no | — |
| 26 | (C3⋊D3)⋊H(3,𝔽3) | — | 16 | 2 | no | — |
| 27 | D3×PSU(3,𝔽2) | — | 12 | 2 | no | — |
| 28 | C2×PSU(3,𝔽2)⋊C3 | — | 36 | 2 | no | — |
| 29 | C2×PSU(3,𝔽2)⋊C3 | — | 36 | 6 | no | — |
| 30 | #88623 | — | 36 | 2 | no | — |
| 31 | C3⋊D32 | C22⋊A4 | 1 | 1 | yes | characteristic |
| 32 | #66472 | — | 6 | 3 | no | — |
| 33 | C3⋊(C3⋊#157) | — | 3 | 2 | no | — |
| 34 | C32×#157 | — | 3 | 2 | no | — |
| 35 | D3×(C3⋊D3)⋊C3 | — | 48 | 2 | no | — |
| 36 | Q8×#157 | — | 27 | 2 | no | — |
| 37 | C4⋊PSU(3,𝔽2) | — | 27 | 6 | no | — |
| 38 | C4×PSU(3,𝔽2) | — | 27 | 2 | no | — |
| 39 | C32⋊H(3,𝔽3) | — | 16 | 1 | no | Sylow |
| 40 | C3⋊PSU(3,𝔽2) | — | 36 | 2 | no | — |
| 41 | C3×PSU(3,𝔽2) | — | 4 | 2 | no | — |
| 42 | D3×#157 | — | 36 | 2 | no | — |
| 43 | PSU(3,𝔽2)⋊C3 | — | 12 | 2 | no | — |
| 44 | PSU(3,𝔽2)⋊C3 | — | 24 | 2 | no | — |
| 45 | PSU(3,𝔽2)⋊C3 | — | 36 | 6 | no | — |
| 46 | PSU(3,𝔽2)⋊C3 | — | 72 | 3 | no | — |
| 47 | Q8⋊H(3,𝔽3) | — | 36 | 2 | no | — |
| 48 | Q8⋊SL(2,𝔽3) | — | 81 | 1 | no | — |
| 49 | C3⋊(C3⋊(C3⋊D3)) | #997 | 1 | 1 | yes | characteristic |
| 50 | C32×C3⋊D3 | C22⋊SL(2,𝔽3) | 1 | 2 | yes | — |
| 51 | C3⋊((C3⋊D3)⋊C3) | — | 48 | 2 | no | — |
| 52 | C3⋊((C3⋊D3)⋊C3) | — | 96 | 1 | no | — |
| 53 | D3×H(3,𝔽3) | — | 48 | 2 | no | — |
| 54 | C3×(C3⋊D3)⋊C3 | — | 16 | 2 | no | — |
| 55 | C3×(C3⋊D3)⋊C3 | — | 32 | 2 | no | — |
| 56 | C2×PSU(3,𝔽2) | — | 9 | 2 | no | — |
| 57 | C2×PSU(3,𝔽2) | — | 9 | 6 | no | — |
| 58 | C2×PSU(3,𝔽2) | — | 27 | 2 | no | — |
| 59 | Q8×C3⋊D3 | — | 9 | 2 | no | — |
| 60 | C4⋊#157 | — | 27 | 6 | no | — |
| 61 | C4×#157 | — | 27 | 6 | no | — |
| 62 | D3×SL(2,𝔽3) | — | 108 | 2 | no | — |
| 63 | #3777 | — | 27 | 6 | no | — |
| 64 | D3×C3⋊D3 | — | 12 | 2 | no | — |
| 65 | C3×D32 | — | 144 | 1 | no | — |
| 66 | C3⋊#157 | — | 36 | 2 | no | — |
| 67 | C3×#157 | — | 12 | 2 | no | — |
| 68 | C2×(C3⋊D3)⋊C3 | — | 144 | 2 | no | — |
| 69 | C34 | Q8⋊SL(2,𝔽3) | 1 | 1 | yes | characteristic, complemented |
| 70 | C3×H(3,𝔽3) | — | 16 | 2 | no | — |
| 71 | C3×H(3,𝔽3) | — | 32 | 1 | no | — |
| 72 | C3×H(3,𝔽3) | — | 32 | 2 | no | — |
| 73 | C3×H(3,𝔽3) | — | 64 | 1 | no | — |
| 74 | C2×#157 | — | 27 | 2 | no | — |
| 75 | C2×#157 | — | 27 | 6 | no | — |
| 76 | PSU(3,𝔽2) | — | 27 | 2 | no | — |
| 77 | PSU(3,𝔽2) | — | 36 | 6 | no | — |
| 78 | PSU(3,𝔽2) | — | 72 | 3 | no | — |
| 79 | PSU(3,𝔽2) | — | 108 | 2 | no | — |
| 80 | PSU(3,𝔽2) | — | 216 | 1 | no | — |
| 81 | PSU(3,𝔽2) | PSU(3,𝔽2)⋊C3 | 1 | 2 | yes | complemented |
| 82 | C32×Q8 | — | 9 | 2 | no | — |
| 83 | C4×C3⋊D3 | — | 27 | 2 | no | — |
| 84 | #634 | — | 27 | 2 | no | — |
| 85 | C3×SL(2,𝔽3) | — | 36 | 2 | no | — |
| 86 | C3×SL(2,𝔽3) | — | 72 | 2 | no | — |
| 87 | Q82 | — | 81 | 1 | no | Sylow |
| 88 | C3⋊(C3⋊D3) | — | 12 | 2 | no | — |
| 89 | C3⋊(C3⋊D3) | — | 96 | 1 | no | — |
| 90 | C3×C3⋊D3 | — | 4 | 2 | no | — |
| 91 | C3×C3⋊D3 | — | 144 | 1 | no | — |
| 92 | C32×D3 | — | 12 | 2 | no | — |
| 93 | C32×D3 | — | 48 | 2 | no | — |
| 94 | C32×D3 | — | 96 | 2 | no | — |
| 95 | C2×H(3,𝔽3) | — | 144 | 2 | no | — |
| 96 | (C3⋊D3)⋊C3 | — | 288 | 3 | no | — |
| 97 | (C3⋊D3)⋊C3 | — | 288 | 2 | no | — |
| 98 | (C3⋊D3)⋊C3 | — | 48 | 2 | no | — |
| 99 | (C3⋊D3)⋊C3 | — | 96 | 2 | no | — |
| 100 | (C3⋊D3)⋊C3 | — | 144 | 2 | no | — |
| 101 | D3×Q8 | — | 108 | 2 | no | — |
| 102 | C2×SL(2,𝔽3) | — | 324 | 3 | no | — |
| 103 | C2×SL(2,𝔽3) | — | 324 | 2 | no | — |
| 104 | C2×C3⋊D3 | — | 9 | 2 | no | — |
| 105 | C2×C3×D3 | — | 432 | 2 | no | — |
| 106 | D32 | — | 144 | 1 | no | — |
| 107 | #157 | — | 3 | 2 | no | — |
| 108 | #157 | — | 27 | 2 | no | — |
| 109 | #157 | — | 54 | 6 | no | — |
| 110 | #157 | — | 216 | 3 | no | — |
| 111 | #157 | — | 216 | 3 | no | — |
| 112 | C32×C4 | — | 27 | 2 | no | — |
| 113 | C3⋊Dic3 | — | 27 | 2 | no | — |
| 114 | C4⋊Q8 | — | 243 | 3 | no | — |
| 115 | C4×Q8 | — | 243 | 2 | no | — |
| 116 | C33 | — | 4 | 2 | no | — |
| 117 | C33 | — | 16 | 1 | no | — |
| 118 | C33 | — | 32 | 1 | no | — |
| 119 | C33 | — | 32 | 2 | no | — |
| 120 | C33 | — | 64 | 1 | no | — |
| 121 | H(3,𝔽3) | — | 48 | 2 | no | — |
| 122 | H(3,𝔽3) | — | 96 | 3 | no | — |
| 123 | H(3,𝔽3) | — | 96 | 2 | no | — |
| 124 | H(3,𝔽3) | — | 96 | 2 | no | — |
| 125 | H(3,𝔽3) | — | 192 | 3 | no | — |
| 126 | H(3,𝔽3) | — | 192 | 2 | no | — |
| 127 | C3×Q8 | — | 36 | 2 | no | — |
| 128 | C4×D3 | — | 324 | 2 | no | — |
| 129 | Dic6 | — | 324 | 2 | no | — |
| 130 | SL(2,𝔽3) | — | 324 | 3 | no | — |
| 131 | SL(2,𝔽3) | — | 108 | 2 | no | — |
| 132 | SL(2,𝔽3) | — | 216 | 2 | no | — |
| 133 | C2×C32 | — | 9 | 2 | no | — |
| 134 | C2×C32 | — | 288 | 2 | no | — |
| 135 | C2×C32 | — | 144 | 2 | no | — |
| 136 | C3⋊D3 | — | 9 | 2 | no | — |
| 137 | C3⋊D3 | — | 288 | 2 | no | — |
| 138 | C3⋊D3 | — | 72 | 3 | no | — |
| 139 | C3⋊D3 | — | 144 | 1 | no | — |
| 140 | C3⋊D3 | — | 216 | 1 | no | — |
| 141 | C3⋊D3 | C22⋊(PSU(3,𝔽2)⋊C3) | 1 | 2 | yes | — |
| 142 | C3×D3 | — | 288 | 2 | no | — |
| 143 | C3×D3 | — | 48 | 2 | no | — |
| 144 | C3×D3 | — | 864 | 1 | no | — |
| 145 | C3×D3 | — | 144 | 2 | no | — |
| 146 | C3×D3 | — | 432 | 2 | no | — |
| 147 | C2×Q8 | — | 81 | 2 | no | — |
| 148 | C2×Q8 | — | 81 | 3 | no | — |
| 149 | C2×Q8 | — | 243 | 1 | no | — |
| 150 | C4⋊C4 | — | 243 | 6 | no | — |
| 151 | C42 | — | 243 | 3 | no | — |
| 152 | C22×C3 | — | 1296 | 1 | no | — |
| 153 | C2×D3 | — | 108 | 2 | no | — |
| 154 | C3×C4 | — | 108 | 2 | no | — |
| 155 | Dic3 | — | 324 | 2 | no | — |
| 156 | C32 | — | 8 | 3 | no | — |
| 157 | C32 | — | 16 | 1 | no | — |
| 158 | C32 | — | 24 | 1 | no | — |
| 159 | C32 | — | 32 | 2 | no | — |
| 160 | C32 | — | 48 | 2 | no | — |
| 161 | C32 | — | 96 | 2 | no | — |
| 162 | C32 | — | 96 | 2 | no | — |
| 163 | C32 | — | 96 | 1 | no | — |
| 164 | C32 | — | 192 | 2 | no | — |
| 165 | C32 | — | 192 | 2 | no | — |
| 166 | C32 | — | 192 | 1 | no | — |
| 167 | C32 | — | 192 | 1 | no | — |
| 168 | C32 | Q8⋊(PSU(3,𝔽2)⋊C3) | 1 | 2 | yes | complemented |
| 169 | Q8 | — | 9 | 2 | no | — |
| 170 | Q8 | — | 324 | 3 | no | — |
| 171 | Q8 | — | 972 | 1 | no | — |
| 172 | Q8 | — | 243 | 2 | no | — |
| 173 | C2×C4 | — | 243 | 2 | no | — |
| 174 | C2×C4 | — | 243 | 3 | no | — |
| 175 | C2×C3 | — | 1296 | 1 | no | — |
| 176 | C2×C3 | — | 36 | 2 | no | — |
| 177 | C2×C3 | — | 864 | 2 | no | — |
| 178 | C2×C3 | — | 432 | 2 | no | — |
| 179 | D3 | — | 12 | 2 | no | — |
| 180 | D3 | — | 864 | 1 | no | — |
| 181 | D3 | — | 108 | 2 | no | — |
| 182 | C22 | — | 81 | 1 | no | — |
| 183 | C4 | — | 27 | 2 | no | — |
| 184 | C4 | — | 486 | 3 | no | — |
| 185 | C4 | — | 243 | 2 | no | — |
| 186 | C3 | — | 4 | 2 | no | — |
| 187 | C3 | — | 32 | 1 | no | — |
| 188 | C3 | — | 288 | 2 | no | — |
| 189 | C3 | — | 576 | 1 | no | — |
| 190 | C3 | — | 144 | 1 | no | — |
| 191 | C2 | — | 9 | 2 | no | — |
| 192 | C2 | — | 81 | 1 | no | — |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | PSU(3,𝔽2)2 | C3 |
| Fitting, socle | C34 | Q8⋊SL(2,𝔽3) |
| LCS | G ▷ PSU(3,𝔽2)2 | |
| UCS | C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. PSU(3,𝔽2)2 | C3 | semidirect |
| 19. C32×PSU(3,𝔽2) | SL(2,𝔽3) | semidirect |
| 69. C34 | Q8⋊SL(2,𝔽3) | semidirect |
| 81. PSU(3,𝔽2) | PSU(3,𝔽2)⋊C3 | semidirect |
| 168. C32 | Q8⋊(PSU(3,𝔽2)⋊C3) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72220479 | 4 |
| PSU(3,𝔽2)2 | 3 |
| #72210295 | 4 |
| Q8⋊(PSU(3,𝔽2)⋊C3) | 9 |