Finite Group DatabasebetaFinite Group DatabasebetaOrder 5184.
392 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 392 conjugacy classes of subgroups — too many to draw. Only its 43 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 | #72220235 | C2 | 1 | 4 | yes | maximal |
| 2 | #72220201 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | #72213866 | C2 | 1 | 2 | yes | maximal |
| 4 | C2×#10620211 | — | 2 | 2 | no | — |
| 5 | #13049467 | — | 2 | 2 | no | — |
| 6 | #13049411 | C22 | 1 | 4 | yes | — |
| 7 | #13049409 | C22 | 1 | 1 | yes | characteristic |
| 8 | #13049399 | C22 | 1 | 2 | yes | — |
| 9 | C2×#66472 | — | 2 | 4 | no | — |
| 10 | C2×#66472 | D4 | 1 | 1 | yes | characteristic |
| 11 | C2×#66472 | D4 | 1 | 1 | yes | characteristic |
| 12 | #10620211 | — | 4 | 4 | no | — |
| 13 | #10620150 | C23 | 1 | 1 | yes | characteristic |
| 14 | C3⋊(C3⋊#622) | Q8 | 1 | 2 | yes | complemented |
| 15 | C32×#622 | Q8 | 1 | 2 | yes | complemented |
| 16 | #10593536 | — | 9 | 2 | no | maximal |
| 17 | #88924 | — | 12 | 2 | no | — |
| 18 | C2×C3⋊(C3⋊(C3⋊D3)) | C2×D4 | 1 | 1 | yes | characteristic |
| 19 | #66472 | — | 4 | 4 | no | — |
| 20 | #66472 | D8 | 1 | 2 | yes | — |
| 21 | #66472 | Q16 | 1 | 2 | yes | — |
| 22 | C32×C3⋊Dic3 | C2×Q8 | 1 | 2 | yes | — |
| 23 | C4⋊PSU(3,𝔽2) | — | 9 | 4 | no | — |
| 24 | #63769 | — | 9 | 4 | no | — |
| 25 | C8×#157 | — | 9 | 2 | no | — |
| 26 | #63746 | — | 9 | 4 | no | — |
| 27 | #6289 | — | 12 | 2 | no | — |
| 28 | C3⋊#622 | — | 12 | 2 | no | — |
| 29 | C3×#622 | — | 4 | 2 | no | — |
| 30 | C2×C34 | C4⋊Q8 | 1 | 1 | yes | characteristic |
| 31 | C3⋊(C3⋊(C3⋊D3)) | C2×D8 | 1 | 1 | yes | characteristic |
| 32 | C3⋊(C3⋊(C3⋊D3)) | C2×Q16 | 1 | 1 | yes | characteristic |
| 33 | C2×PSU(3,𝔽2) | — | 18 | 4 | no | — |
| 34 | C2×PSU(3,𝔽2) | — | 36 | 2 | no | — |
| 35 | C4⋊#157 | — | 9 | 8 | no | — |
| 36 | C4×#157 | — | 9 | 2 | no | — |
| 37 | #3787 | — | 9 | 2 | no | — |
| 38 | #3777 | — | 18 | 4 | no | — |
| 39 | #3777 | — | 36 | 2 | no | — |
| 40 | C8×C3⋊D3 | — | 9 | 2 | no | — |
| 41 | C2×C3⋊(C3⋊D3) | — | 12 | 2 | no | — |
| 42 | C2×C3⋊(C3⋊D3) | — | 48 | 2 | no | — |
| 43 | C3×C3⋊Dic3 | — | 4 | 2 | no | — |
| 44 | C32×Dic3 | — | 12 | 2 | no | — |
| 45 | C34 | #500 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 46 | C2×#157 | — | 9 | 2 | no | — |
| 47 | C2×#157 | — | 9 | 2 | no | — |
| 48 | C2×#157 | — | 18 | 8 | no | — |
| 49 | C2×#157 | — | 36 | 2 | no | — |
| 50 | C2×#157 | — | 36 | 2 | no | — |
| 51 | C2×#157 | — | 36 | 4 | no | — |
| 52 | C2×#157 | — | 36 | 4 | no | — |
| 53 | C2×#157 | — | 72 | 4 | no | — |
| 54 | PSU(3,𝔽2) | — | 36 | 8 | no | — |
| 55 | PSU(3,𝔽2) | — | 36 | 8 | no | — |
| 56 | C4×C3⋊D3 | — | 9 | 2 | no | — |
| 57 | #624 | — | 72 | 2 | no | — |
| 58 | #622 | — | 9 | 2 | no | — |
| 59 | #622 | PSU(3,𝔽2) | 1 | 2 | yes | complemented |
| 60 | C32×C8 | — | 9 | 2 | no | — |
| 61 | C3⋊(C3⋊C8) | — | 9 | 2 | no | — |
| 62 | #500 | — | 81 | 1 | no | Sylow |
| 63 | C2×C33 | — | 4 | 2 | no | — |
| 64 | C2×C33 | — | 16 | 2 | no | — |
| 65 | C3⋊(C3⋊D3) | — | 12 | 2 | no | — |
| 66 | C3⋊(C3⋊D3) | — | 12 | 2 | no | — |
| 67 | C3⋊(C3⋊D3) | — | 48 | 2 | no | — |
| 68 | C3⋊(C3⋊D3) | — | 48 | 2 | no | — |
| 69 | D3×C8 | — | 108 | 2 | no | — |
| 70 | C2×C3⋊D3 | — | 9 | 2 | no | — |
| 71 | C2×C3⋊D3 | — | 36 | 2 | no | — |
| 72 | C2×C3⋊D3 | — | 36 | 2 | no | — |
| 73 | C2×C3⋊D3 | — | 72 | 2 | no | — |
| 74 | C2×C3⋊D3 | — | 72 | 4 | no | — |
| 75 | C2×C3⋊D3 | — | 144 | 4 | no | — |
| 76 | #157 | — | 9 | 4 | no | — |
| 77 | #157 | — | 9 | 4 | no | — |
| 78 | #157 | — | 36 | 8 | no | — |
| 79 | #157 | — | 36 | 8 | no | — |
| 80 | #157 | — | 36 | 4 | no | — |
| 81 | #157 | — | 36 | 4 | no | — |
| 82 | #157 | — | 72 | 4 | no | — |
| 83 | #157 | — | 72 | 8 | no | — |
| 84 | C32×C4 | — | 9 | 2 | no | — |
| 85 | C3⋊Dic3 | C2×PSU(3,𝔽2) | 1 | 2 | yes | — |
| 86 | C3×Dic3 | — | 24 | 4 | no | — |
| 87 | C4⋊Q8 | — | 81 | 2 | no | — |
| 88 | #107 | — | 81 | 4 | no | — |
| 89 | C4×C8 | — | 81 | 1 | no | — |
| 90 | C33 | — | 4 | 2 | no | — |
| 91 | C33 | — | 16 | 2 | no | — |
| 92 | C4×D3 | — | 108 | 2 | no | — |
| 93 | C3×C8 | — | 36 | 2 | no | — |
| 94 | C3⋊C8 | — | 108 | 2 | no | — |
| 95 | C2×C32 | — | 4 | 2 | no | — |
| 96 | C2×C32 | — | 4 | 2 | no | — |
| 97 | C2×C32 | — | 8 | 2 | no | — |
| 98 | C2×C32 | — | 8 | 4 | no | — |
| 99 | C2×C32 | — | 16 | 4 | no | — |
| 100 | C2×C32 | C4⋊PSU(3,𝔽2) | 1 | 2 | yes | — |
| 101 | C3⋊D3 | — | 9 | 2 | no | — |
| 102 | C3⋊D3 | — | 9 | 2 | no | — |
| 103 | C3⋊D3 | — | 36 | 2 | no | — |
| 104 | C3⋊D3 | — | 36 | 2 | no | — |
| 105 | C3⋊D3 | — | 36 | 2 | no | — |
| 106 | C3⋊D3 | — | 36 | 2 | no | — |
| 107 | C3⋊D3 | — | 72 | 2 | no | — |
| 108 | C3⋊D3 | — | 72 | 2 | no | — |
| 109 | C3⋊D3 | — | 72 | 4 | no | — |
| 110 | C3⋊D3 | — | 72 | 4 | no | — |
| 111 | C3⋊D3 | — | 144 | 4 | no | — |
| 112 | C3⋊D3 | — | 144 | 4 | no | — |
| 113 | C2×Q8 | — | 162 | 2 | no | — |
| 114 | C2×C8 | — | 81 | 2 | no | — |
| 115 | C4⋊C4 | — | 81 | 4 | no | — |
| 116 | C4⋊C4 | — | 162 | 2 | no | — |
| 117 | C42 | — | 81 | 1 | no | — |
| 118 | C2×D3 | — | 108 | 2 | no | — |
| 119 | C2×D3 | — | 432 | 2 | no | — |
| 120 | C3×C4 | — | 36 | 2 | no | — |
| 121 | Dic3 | — | 12 | 2 | no | — |
| 122 | C32 | — | 4 | 2 | no | — |
| 123 | C32 | — | 4 | 2 | no | — |
| 124 | C32 | — | 8 | 2 | no | — |
| 125 | C32 | — | 8 | 4 | no | — |
| 126 | C32 | — | 16 | 4 | no | — |
| 127 | C32 | #10593536 | 1 | 2 | yes | complemented |
| 128 | Q8 | — | 324 | 4 | no | — |
| 129 | C2×C4 | — | 81 | 1 | no | — |
| 130 | C2×C4 | — | 81 | 1 | no | — |
| 131 | C2×C4 | — | 81 | 1 | no | — |
| 132 | C2×C4 | — | 162 | 4 | no | — |
| 133 | C8 | — | 9 | 2 | no | — |
| 134 | C8 | — | 81 | 2 | no | — |
| 135 | C2×C3 | — | 4 | 2 | no | — |
| 136 | C2×C3 | — | 16 | 2 | no | — |
| 137 | D3 | — | 108 | 2 | no | — |
| 138 | D3 | — | 108 | 2 | no | — |
| 139 | D3 | — | 432 | 2 | no | — |
| 140 | D3 | — | 432 | 2 | no | — |
| 141 | C22 | — | 81 | 1 | no | — |
| 142 | C4 | — | 9 | 2 | no | — |
| 143 | C4 | — | 324 | 4 | no | — |
| 144 | C4 | — | 81 | 2 | no | — |
| 145 | C4 | — | 81 | 2 | no | — |
| 146 | C3 | — | 4 | 2 | no | — |
| 147 | C3 | — | 16 | 2 | no | — |
| 148 | C2 | — | 81 | 1 | no | — |
| 149 | C2 | — | 81 | 1 | no | — |
| 150 | C2 | #72213277 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 14. C3⋊(C3⋊#622) | Q8 | semidirect |
| 15. C32×#622 | Q8 | semidirect |
| 45. C34 | #500 | semidirect |
| 59. #622 | PSU(3,𝔽2) | semidirect |
| 127. C32 | #10593536 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72220235 | 2 |
| #72220201 | 2 |
| #72213866 | 2 |
| #10593536 | 9 |