Finite Group DatabasebetaFinite Group DatabasebetaOrder 12416.
122 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 122 conjugacy classes of subgroups — too many to draw. Only its 20 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 | #20261639 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 2 | #20261593 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 3 | #20260905 | C2 | 1 | 1 | yes | maximal, characteristic |
| 4 | #15690877 | — | 2 | 1 | no | — |
| 5 | #15690869 | — | 2 | 1 | no | — |
| 6 | #15690855 | C4 | 1 | 1 | yes | characteristic, complemented |
| 7 | #15690838 | — | 2 | 1 | no | — |
| 8 | #15690806 | C22 | 1 | 1 | yes | characteristic |
| 9 | #15690736 | — | 2 | 1 | no | — |
| 10 | #100×C97 | C4 | 1 | 1 | yes | characteristic, complemented |
| 11 | C2×#11715897 | — | 4 | 1 | no | — |
| 12 | C22⋊#87604 | — | 4 | 1 | no | — |
| 13 | #13095511 | — | 2 | 1 | no | — |
| 14 | #13095510 | — | 4 | 1 | no | — |
| 15 | C2×D388 | D4 | 1 | 1 | yes | characteristic |
| 16 | C2×C22⋊D97 | — | 4 | 1 | no | — |
| 17 | C2×C22⋊D97 | — | 4 | 1 | no | — |
| 18 | D4×D97 | — | 4 | 1 | no | — |
| 19 | #13095500 | — | 4 | 1 | no | — |
| 20 | D776 | — | 4 | 1 | no | — |
| 21 | C4×Dic97 | D4 | 1 | 1 | yes | characteristic |
| 22 | C2×D4×C97 | C2×C4 | 1 | 1 | yes | characteristic |
| 23 | M16×C97 | — | 2 | 1 | no | — |
| 24 | C23×C97 | — | 2 | 1 | no | — |
| 25 | C22×D97 | — | 2 | 1 | no | — |
| 26 | C22×D97 | — | 4 | 1 | no | — |
| 27 | C2×#87604 | — | 8 | 1 | no | — |
| 28 | D4×C97 | — | 4 | 1 | no | — |
| 29 | C2×C4×C97 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 30 | C22⋊D97 | — | 4 | 2 | no | — |
| 31 | C22⋊D97 | — | 4 | 1 | no | — |
| 32 | C22⋊D97 | — | 8 | 1 | no | — |
| 33 | C22⋊D97 | — | 8 | 1 | no | — |
| 34 | C2×Dic97 | — | 2 | 1 | no | — |
| 35 | C2×Dic97 | — | 4 | 1 | no | — |
| 36 | D388 | — | 2 | 1 | no | — |
| 37 | D388 | — | 4 | 1 | no | — |
| 38 | C4×D97 | — | 4 | 1 | no | — |
| 39 | Dic194 | — | 2 | 1 | no | — |
| 40 | #11715897 | — | 8 | 1 | no | — |
| 41 | C8×C97 | — | 4 | 1 | no | — |
| 42 | C22×C97 | — | 4 | 1 | no | — |
| 43 | C22×C97 | — | 8 | 1 | no | — |
| 44 | C22×C97 | #99 | 1 | 1 | yes | characteristic |
| 45 | C2×D97 | — | 4 | 1 | no | — |
| 46 | C2×D97 | — | 4 | 1 | no | — |
| 47 | C2×D97 | — | 8 | 1 | no | — |
| 48 | C2×D97 | — | 16 | 1 | no | — |
| 49 | #87604 | — | 16 | 1 | no | — |
| 50 | C4×C97 | — | 2 | 1 | no | — |
| 51 | Dic97 | — | 2 | 1 | no | — |
| 52 | Dic97 | — | 2 | 1 | no | — |
| 53 | Dic97 | — | 4 | 1 | no | — |
| 54 | C2×C97 | — | 2 | 1 | no | — |
| 55 | C2×C97 | — | 8 | 1 | no | — |
| 56 | C2×C97 | AΣL(1,𝔽16) | 1 | 1 | yes | characteristic |
| 57 | D97 | — | 8 | 1 | no | — |
| 58 | D97 | — | 8 | 1 | no | — |
| 59 | #1402 | — | 97 | 1 | no | maximal, Sylow |
| 60 | C97 | #1402 | 1 | 1 | yes | characteristic, complemented, Sylow |
| 61 | #453 | — | 97 | 1 | no | — |
| 62 | #353 | — | 97 | 1 | no | — |
| 63 | #331 | — | 97 | 1 | no | — |
| 64 | H(5,𝔽2) | — | 97 | 1 | no | — |
| 65 | AGL(1,ℤ/8) | — | 194 | 1 | no | — |
| 66 | #127 | — | 97 | 1 | no | — |
| 67 | C4⋊C8 | — | 194 | 1 | no | — |
| 68 | #104 | — | 194 | 1 | no | — |
| 69 | #100 | #87604 | 1 | 1 | yes | characteristic, complemented |
| 70 | #99 | — | 194 | 1 | no | — |
| 71 | #41 | — | 388 | 1 | no | — |
| 72 | #41 | — | 194 | 1 | no | — |
| 73 | C2×D4 | — | 97 | 1 | no | — |
| 74 | C2×D4 | — | 388 | 1 | no | — |
| 75 | C2×D4 | — | 388 | 1 | no | — |
| 76 | C2×D4 | — | 388 | 1 | no | — |
| 77 | C2×D4 | C2×#87604 | 1 | 1 | yes | characteristic |
| 78 | SD16 | — | 388 | 1 | no | — |
| 79 | D8 | — | 388 | 1 | no | — |
| 80 | M16 | — | 2 | 1 | no | — |
| 81 | C2×C8 | — | 388 | 1 | no | — |
| 82 | C22⋊C4 | — | 388 | 1 | no | — |
| 83 | C42 | — | 97 | 1 | no | — |
| 84 | C23 | — | 2 | 1 | no | — |
| 85 | C23 | — | 388 | 1 | no | — |
| 86 | C23 | — | 194 | 1 | no | — |
| 87 | Q8 | — | 194 | 1 | no | — |
| 88 | D4 | — | 4 | 1 | no | — |
| 89 | D4 | — | 776 | 1 | no | — |
| 90 | D4 | — | 776 | 1 | no | — |
| 91 | D4 | — | 388 | 1 | no | — |
| 92 | D4 | — | 388 | 2 | no | — |
| 93 | D4 | — | 388 | 1 | no | — |
| 94 | D4 | — | 194 | 1 | no | — |
| 95 | C2×C4 | — | 776 | 1 | no | — |
| 96 | C2×C4 | — | 388 | 1 | no | — |
| 97 | C2×C4 | — | 388 | 1 | no | — |
| 98 | C2×C4 | — | 194 | 1 | no | — |
| 99 | C2×C4 | C22⋊#87604 | 1 | 1 | yes | characteristic |
| 100 | C8 | — | 4 | 1 | no | — |
| 101 | C8 | — | 776 | 1 | no | — |
| 102 | C22 | — | 4 | 1 | no | — |
| 103 | C22 | — | 8 | 1 | no | — |
| 104 | C22 | — | 776 | 1 | no | — |
| 105 | C22 | — | 1552 | 1 | no | — |
| 106 | C22 | — | 388 | 1 | no | — |
| 107 | C22 | — | 388 | 1 | no | — |
| 108 | C22 | #15690869 | 1 | 1 | yes | characteristic |
| 109 | C4 | — | 2 | 1 | no | — |
| 110 | C4 | — | 1552 | 1 | no | — |
| 111 | C4 | — | 388 | 1 | no | — |
| 112 | C4 | — | 194 | 1 | no | — |
| 113 | C4 | — | 194 | 1 | no | — |
| 114 | C2 | — | 2 | 1 | no | — |
| 115 | C2 | — | 8 | 1 | no | — |
| 116 | C2 | — | 776 | 1 | no | — |
| 117 | C2 | — | 776 | 1 | no | — |
| 118 | C2 | #20261636 | 1 | 1 | yes | characteristic |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #20261639 | C2 | semidirect |
| 2. #20261593 | C2 | semidirect |
| 6. #15690855 | C4 | semidirect |
| 10. #100×C97 | C4 | semidirect |
| 60. C97 | #1402 | semidirect |
| 69. #100 | #87604 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #20261639 | 2 |
| #20261593 | 2 |
| #20260905 | 2 |
| #1402 | 97 |