Finite Group DatabasebetaFinite Group DatabasebetaOrder 3712.
262 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 262 conjugacy classes of subgroups — too many to draw. Only its 45 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 | #14271905 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #14271553 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #14271552 | C2 | 1 | 2 | yes | maximal, complemented |
| 4 | #14270902 | C2 | 1 | 2 | yes | maximal, complemented |
| 5 | C29×#329 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 6 | C29×#124 | C22 | 1 | 1 | yes | characteristic |
| 7 | #11745309 | — | 2 | 1 | no | — |
| 8 | C2×#90784 | — | 2 | 1 | no | — |
| 9 | #11745299 | — | 2 | 1 | no | — |
| 10 | C2×#90782 | — | 2 | 1 | no | — |
| 11 | C2×D232 | — | 2 | 2 | no | — |
| 12 | C2×#90773 | — | 2 | 2 | no | — |
| 13 | #11745247 | — | 2 | 2 | no | — |
| 14 | #11745244 | — | 2 | 2 | no | — |
| 15 | #11745237 | C22 | 1 | 2 | yes | — |
| 16 | #11745236 | C22 | 1 | 1 | yes | characteristic, complemented |
| 17 | C29×#97 | C22 | 1 | 2 | yes | complemented |
| 18 | #11745151 | C22 | 1 | 1 | yes | characteristic, complemented |
| 19 | C23×D29 | — | 2 | 1 | no | — |
| 20 | C2×Q8×C29 | D4 | 1 | 1 | yes | characteristic |
| 21 | C2×D4×C29 | D4 | 1 | 1 | yes | characteristic |
| 22 | C2×C22⋊D29 | — | 2 | 2 | no | — |
| 23 | C2×D116 | — | 2 | 1 | no | — |
| 24 | C2×D116 | — | 2 | 2 | no | — |
| 25 | C2×D116 | — | 2 | 1 | no | — |
| 26 | C2×D116 | D4 | 1 | 2 | yes | — |
| 27 | C2×Dic58 | D4 | 1 | 2 | yes | — |
| 28 | C2×C8×C29 | — | 2 | 2 | no | — |
| 29 | C22⋊C4×C29 | — | 2 | 2 | no | — |
| 30 | C42×C29 | C23 | 1 | 1 | yes | characteristic |
| 31 | C22⋊Dic29 | — | 2 | 1 | no | — |
| 32 | #90784 | — | 4 | 2 | no | — |
| 33 | #90782 | — | 4 | 2 | no | — |
| 34 | #90781 | — | 2 | 2 | no | — |
| 35 | #90781 | — | 2 | 2 | no | — |
| 36 | #90781 | — | 2 | 2 | no | — |
| 37 | C4×Dic29 | — | 2 | 1 | no | — |
| 38 | C2×#6597 | — | 2 | 1 | no | — |
| 39 | D232 | — | 4 | 4 | no | — |
| 40 | #90773 | — | 4 | 4 | no | — |
| 41 | C23×C29 | — | 2 | 1 | no | — |
| 42 | C22×D29 | — | 2 | 2 | no | — |
| 43 | C22×D29 | — | 4 | 1 | no | — |
| 44 | C22×D29 | — | 8 | 2 | no | — |
| 45 | C22×D29 | — | 8 | 1 | no | — |
| 46 | Q8×C29 | — | 4 | 1 | no | — |
| 47 | D4×C29 | — | 4 | 1 | no | — |
| 48 | C2×C4×C29 | — | 2 | 1 | no | — |
| 49 | C2×C4×C29 | C2×D4 | 1 | 1 | yes | characteristic |
| 50 | C2×C4×C29 | C2×D4 | 1 | 2 | yes | — |
| 51 | C22⋊D29 | — | 8 | 2 | no | — |
| 52 | C2×Dic29 | — | 2 | 2 | no | — |
| 53 | D116 | — | 4 | 2 | no | — |
| 54 | D116 | — | 4 | 2 | no | — |
| 55 | D116 | — | 4 | 2 | no | — |
| 56 | D116 | — | 4 | 1 | no | — |
| 57 | D116 | — | 4 | 1 | no | — |
| 58 | D116 | — | 8 | 1 | no | — |
| 59 | Dic58 | — | 4 | 2 | no | — |
| 60 | C8×C29 | — | 2 | 4 | no | — |
| 61 | #6597 | — | 2 | 2 | no | — |
| 62 | #1655 | — | 29 | 1 | no | maximal, Sylow |
| 63 | C22×C29 | — | 4 | 2 | no | — |
| 64 | C22×C29 | — | 4 | 1 | no | — |
| 65 | C22×C29 | #120 | 1 | 1 | yes | characteristic |
| 66 | C2×D29 | — | 4 | 2 | no | — |
| 67 | C2×D29 | — | 4 | 2 | no | — |
| 68 | C2×D29 | — | 4 | 2 | no | — |
| 69 | C2×D29 | — | 8 | 4 | no | — |
| 70 | C2×D29 | — | 8 | 2 | no | — |
| 71 | C2×D29 | — | 8 | 1 | no | — |
| 72 | C4×C29 | — | 2 | 1 | no | — |
| 73 | C4×C29 | — | 2 | 2 | no | — |
| 74 | C4×C29 | — | 4 | 1 | no | — |
| 75 | Dic29 | — | 4 | 2 | no | — |
| 76 | H(3,𝔽4) | — | 29 | 1 | no | — |
| 77 | #496 | — | 29 | 1 | no | — |
| 78 | #496 | — | 29 | 2 | no | — |
| 79 | #329 | — | 29 | 2 | no | — |
| 80 | #329 | D29 | 1 | 1 | yes | characteristic, complemented |
| 81 | C2×C29 | — | 8 | 1 | no | — |
| 82 | C2×C29 | #453 | 1 | 2 | yes | — |
| 83 | C2×C29 | #453 | 1 | 1 | yes | characteristic |
| 84 | D29 | — | 8 | 2 | no | — |
| 85 | D29 | — | 16 | 1 | no | — |
| 86 | C2×SD16 | — | 58 | 2 | no | — |
| 87 | C2×SD16 | — | 58 | 1 | no | — |
| 88 | C2×D8 | — | 58 | 2 | no | — |
| 89 | C2×D8 | — | 58 | 1 | no | — |
| 90 | #127 | — | 29 | 1 | no | — |
| 91 | #124 | — | 29 | 2 | no | — |
| 92 | #124 | — | 58 | 2 | no | — |
| 93 | #124 | — | 58 | 1 | no | — |
| 94 | #124 | C2×D29 | 1 | 1 | yes | characteristic |
| 95 | #120 | — | 58 | 2 | no | — |
| 96 | #120 | — | 58 | 1 | no | — |
| 97 | #97 | — | 29 | 1 | no | — |
| 98 | #97 | C2×D29 | 1 | 2 | yes | complemented |
| 99 | C29 | #1655 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 100 | C24 | — | 58 | 1 | no | — |
| 101 | C2×Q8 | — | 29 | 2 | no | — |
| 102 | C2×Q8 | D116 | 1 | 1 | yes | characteristic |
| 103 | C2×D4 | — | 29 | 2 | no | — |
| 104 | C2×D4 | — | 58 | 1 | no | — |
| 105 | C2×D4 | — | 58 | 2 | no | — |
| 106 | C2×D4 | — | 58 | 1 | no | — |
| 107 | C2×D4 | — | 58 | 2 | no | — |
| 108 | C2×D4 | D116 | 1 | 1 | yes | characteristic |
| 109 | SD16 | — | 116 | 4 | no | — |
| 110 | SD16 | — | 116 | 2 | no | — |
| 111 | D8 | — | 116 | 4 | no | — |
| 112 | D8 | — | 116 | 2 | no | — |
| 113 | C2×C8 | — | 2 | 2 | no | — |
| 114 | C2×C8 | — | 58 | 1 | no | — |
| 115 | C22⋊C4 | — | 2 | 2 | no | — |
| 116 | C22⋊C4 | — | 58 | 2 | no | — |
| 117 | C22⋊C4 | — | 58 | 1 | no | — |
| 118 | C22⋊C4 | — | 58 | 2 | no | — |
| 119 | C22⋊C4 | — | 58 | 2 | no | — |
| 120 | C42 | — | 58 | 1 | no | — |
| 121 | C42 | C22×D29 | 1 | 1 | yes | characteristic |
| 122 | C23 | — | 2 | 1 | no | — |
| 123 | C23 | — | 58 | 2 | no | — |
| 124 | C23 | — | 116 | 1 | no | — |
| 125 | C23 | — | 232 | 2 | no | — |
| 126 | C23 | — | 232 | 1 | no | — |
| 127 | Q8 | — | 4 | 1 | no | — |
| 128 | Q8 | — | 116 | 2 | no | — |
| 129 | D4 | — | 4 | 1 | no | — |
| 130 | D4 | — | 116 | 2 | no | — |
| 131 | D4 | — | 116 | 2 | no | — |
| 132 | D4 | — | 116 | 1 | no | — |
| 133 | D4 | — | 116 | 2 | no | — |
| 134 | D4 | — | 116 | 1 | no | — |
| 135 | D4 | — | 232 | 1 | no | — |
| 136 | D4 | — | 232 | 2 | no | — |
| 137 | C2×C4 | — | 2 | 1 | no | — |
| 138 | C2×C4 | — | 58 | 2 | no | — |
| 139 | C2×C4 | C2×D116 | 1 | 1 | yes | characteristic |
| 140 | C2×C4 | D4×D29 | 1 | 2 | yes | — |
| 141 | C8 | — | 2 | 4 | no | — |
| 142 | C8 | — | 58 | 2 | no | — |
| 143 | C22 | — | 4 | 2 | no | — |
| 144 | C22 | — | 4 | 1 | no | — |
| 145 | C22 | — | 116 | 2 | no | — |
| 146 | C22 | — | 116 | 2 | no | — |
| 147 | C22 | — | 116 | 2 | no | — |
| 148 | C22 | — | 232 | 4 | no | — |
| 149 | C22 | — | 232 | 2 | no | — |
| 150 | C22 | — | 232 | 1 | no | — |
| 151 | C22 | #11745244 | 1 | 1 | yes | characteristic |
| 152 | C4 | — | 2 | 2 | no | — |
| 153 | C4 | — | 2 | 1 | no | — |
| 154 | C4 | — | 4 | 1 | no | — |
| 155 | C4 | — | 116 | 2 | no | — |
| 156 | C2 | — | 8 | 1 | no | — |
| 157 | C2 | — | 464 | 1 | no | — |
| 158 | C2 | — | 232 | 2 | no | — |
| 159 | C2 | #14271332 | 1 | 2 | yes | — |
| 160 | C2 | #14271333 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C22 | #11745244 |
| commutator, cosocle | C42×C29 | C23 |
| Frattini | C42 | C22×D29 |
| Fitting | C29×#329 | C2 |
| socle | C22×C29 | #120 |
| hypercenter | #329 | D29 |
| hypocenter | C29 | #1655 |
| LCS | G ▷ C42×C29 ▷ C22×C29 ▷ C29 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ #329 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #14271905 | C2 | semidirect |
| 2. #14271553 | C2 | semidirect |
| 3. #14271552 | C2 | semidirect |
| 4. #14270902 | C2 | semidirect |
| 5. C29×#329 | C2 | semidirect |
| 16. #11745236 | C22 | semidirect |
| 17. C29×#97 | C22 | semidirect |
| 18. #11745151 | C22 | semidirect |
| 80. #329 | D29 | semidirect |
| 98. #97 | C2×D29 | semidirect |
| 99. C29 | #1655 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #14271905 | 2 |
| #14271553 | 2 |
| #14271552 | 2 |
| #14270902 | 2 |
| C29×#329 | 2 |
| #1655 | 29 |