Finite Group DatabasebetaFinite Group DatabasebetaOrder 1024.
300501 conjugacy classes of subgroups; only the 66933 normal subgroups are shown.
This group has 300501 conjugacy classes of subgroups — too many to draw. Only its 66933 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 | C25×C4⋊C4 | C2 | 1 | 3 | yes | maximal, complemented |
| 2 | C25×C22⋊C4 | C2 | 1 | 3 | yes | maximal |
| 3 | C23×#399 | C2 | 1 | 120 | yes | maximal, direct product factor |
| 4 | C24×#95 | C2 | 1 | 1 | yes | maximal, characteristic, complemented |
| 5 | C28 | C22 | 1 | 1 | yes | characteristic |
| 6 | C26×C4 | C22 | 1 | 3 | yes | — |
| 7 | C26×C4 | C22 | 1 | 3 | yes | — |
| 8 | C24×C4⋊C4 | — | 2 | 96 | no | — |
| 9 | C24×C4⋊C4 | C22 | 1 | 180 | yes | complemented |
| 10 | C24×C22⋊C4 | — | 2 | 96 | no | — |
| 11 | C24×C22⋊C4 | C22 | 1 | 180 | yes | — |
| 12 | C22×#399 | C22 | 1 | 2240 | yes | direct product factor |
| 13 | C23×#95 | C22 | 1 | 60 | yes | complemented |
| 14 | C27 | — | 2 | 48 | no | — |
| 15 | C27 | — | 2 | 48 | no | — |
| 16 | C27 | Q8 | 1 | 32 | yes | — |
| 17 | C27 | C23 | 1 | 30 | yes | — |
| 18 | C27 | C23 | 1 | 1 | yes | characteristic |
| 19 | C25×C4 | — | 2 | 96 | no | — |
| 20 | C25×C4 | — | 2 | 48 | no | — |
| 21 | C25×C4 | — | 2 | 96 | no | — |
| 22 | C25×C4 | D4 | 1 | 96 | yes | — |
| 23 | C25×C4 | C23 | 1 | 90 | yes | — |
| 24 | C25×C4 | C23 | 1 | 90 | yes | — |
| 25 | C23×C4⋊C4 | — | 2 | 2880 | no | — |
| 26 | C23×C4⋊C4 | C23 | 1 | 1680 | yes | complemented |
| 27 | C23×C22⋊C4 | — | 2 | 2880 | no | — |
| 28 | C23×C22⋊C4 | C23 | 1 | 1680 | yes | — |
| 29 | C2×#399 | C23 | 1 | 7680 | yes | direct product factor |
| 30 | C22×#95 | C23 | 1 | 560 | yes | complemented |
| 31 | C26 | — | 2 | 1536 | no | — |
| 32 | C26 | — | 2 | 720 | no | — |
| 33 | C26 | — | 2 | 720 | no | — |
| 34 | C26 | — | 4 | 768 | no | — |
| 35 | C26 | — | 4 | 256 | no | — |
| 36 | C26 | C2×D4 | 1 | 48 | yes | — |
| 37 | C26 | C2×Q8 | 1 | 480 | yes | — |
| 38 | C26 | C2×Q8 | 1 | 16 | yes | — |
| 39 | C26 | #41 | 1 | 48 | yes | — |
| 40 | C26 | C24 | 1 | 140 | yes | — |
| 41 | C26 | C24 | 1 | 15 | yes | — |
| 42 | C24×C4 | — | 2 | 1536 | no | — |
| 43 | C24×C4 | — | 2 | 1440 | no | — |
| 44 | C24×C4 | — | 2 | 720 | no | — |
| 45 | C24×C4 | — | 2 | 1440 | no | — |
| 46 | C24×C4 | — | 4 | 768 | no | — |
| 47 | C24×C4 | C2×D4 | 1 | 1440 | yes | — |
| 48 | C24×C4 | C24 | 1 | 420 | yes | — |
| 49 | C24×C4 | C24 | 1 | 420 | yes | — |
| 50 | C22×C4⋊C4 | — | 2 | 13440 | no | — |
| 51 | C22×C4⋊C4 | C24 | 1 | 2880 | yes | complemented |
| 52 | C22×C22⋊C4 | — | 2 | 13440 | no | — |
| 53 | C22×C22⋊C4 | C24 | 1 | 2880 | yes | — |
| 54 | #399 | C24 | 1 | 4096 | yes | direct product factor |
| 55 | C2×#95 | C24 | 1 | 960 | yes | complemented |
| 56 | C25 | — | 2 | 11520 | no | — |
| 57 | C25 | — | 2 | 1680 | no | — |
| 58 | C25 | — | 2 | 1680 | no | — |
| 59 | C25 | — | 4 | 5760 | no | — |
| 60 | C25 | — | 4 | 1920 | no | — |
| 61 | C25 | — | 4 | 8192 | no | — |
| 62 | C25 | C22⋊Q8 | 1 | 768 | yes | — |
| 63 | C25 | #123 | 1 | 768 | yes | — |
| 64 | C25 | #127 | 1 | 256 | yes | — |
| 65 | C25 | C22×D4 | 1 | 360 | yes | — |
| 66 | C25 | C22×Q8 | 1 | 1120 | yes | — |
| 67 | C25 | C22×Q8 | 1 | 120 | yes | — |
| 68 | C25 | C2×#41 | 1 | 360 | yes | — |
| 69 | C25 | C25 | 1 | 120 | yes | — |
| 70 | C25 | C25 | 1 | 35 | yes | — |
| 71 | C23×C4 | — | 2 | 11520 | no | — |
| 72 | C23×C4 | — | 2 | 3360 | no | — |
| 73 | C23×C4 | — | 2 | 1680 | no | — |
| 74 | C23×C4 | — | 2 | 3360 | no | — |
| 75 | C23×C4 | — | 4 | 5760 | no | — |
| 76 | C23×C4 | C22×D4 | 1 | 3360 | yes | — |
| 77 | C23×C4 | C25 | 1 | 360 | yes | — |
| 78 | C23×C4 | C25 | 1 | 360 | yes | — |
| 79 | C2×C4⋊C4 | — | 2 | 11520 | no | — |
| 80 | C2×C4⋊C4 | C25 | 1 | 768 | yes | complemented |
| 81 | C2×C22⋊C4 | — | 2 | 11520 | no | — |
| 82 | C2×C22⋊C4 | C25 | 1 | 768 | yes | — |
| 83 | #95 | C25 | 1 | 256 | yes | complemented |
| 84 | C24 | — | 2 | 720 | no | — |
| 85 | C24 | — | 2 | 720 | no | — |
| 86 | C24 | — | 2 | 13440 | no | — |
| 87 | C24 | — | 4 | 30720 | no | — |
| 88 | C24 | — | 4 | 6720 | no | — |
| 89 | C24 | — | 4 | 2240 | no | — |
| 90 | C24 | #399 | 1 | 4096 | yes | direct product factor |
| 91 | C24 | C2×C22⋊Q8 | 1 | 2880 | yes | — |
| 92 | C24 | C2×#123 | 1 | 2880 | yes | — |
| 93 | C24 | C2×#127 | 1 | 960 | yes | — |
| 94 | C24 | C23×D4 | 1 | 420 | yes | — |
| 95 | C24 | C23×Q8 | 1 | 480 | yes | — |
| 96 | C24 | C23×Q8 | 1 | 140 | yes | — |
| 97 | C24 | C22×#41 | 1 | 420 | yes | — |
| 98 | C24 | C26 | 1 | 16 | yes | — |
| 99 | C24 | C26 | 1 | 15 | yes | — |
| 100 | C22×C4 | — | 2 | 1440 | no | — |
| 101 | C22×C4 | — | 2 | 13440 | no | — |
| 102 | C22×C4 | — | 2 | 720 | no | — |
| 103 | C22×C4 | — | 2 | 1440 | no | — |
| 104 | C22×C4 | — | 4 | 6720 | no | — |
| 105 | C22×C4 | C23×D4 | 1 | 1440 | yes | — |
| 106 | C22×C4 | C26 | 1 | 48 | yes | — |
| 107 | C22×C4 | C26 | 1 | 48 | yes | — |
| 108 | C4⋊C4 | — | 2 | 1536 | no | — |
| 109 | C22⋊C4 | — | 2 | 1536 | no | — |
| 110 | C23 | — | 2 | 48 | no | — |
| 111 | C23 | — | 2 | 48 | no | — |
| 112 | C23 | — | 2 | 2880 | no | — |
| 113 | C23 | — | 4 | 1440 | no | — |
| 114 | C23 | — | 4 | 480 | no | — |
| 115 | C23 | — | 4 | 17920 | no | — |
| 116 | C23 | C2×#399 | 1 | 7680 | yes | direct product factor |
| 117 | C23 | C22×C22⋊Q8 | 1 | 1680 | yes | — |
| 118 | C23 | C22×#123 | 1 | 1680 | yes | — |
| 119 | C23 | C22×#127 | 1 | 560 | yes | — |
| 120 | C23 | C24×D4 | 1 | 90 | yes | — |
| 121 | C23 | C24×Q8 | 1 | 32 | yes | — |
| 122 | C23 | C24×Q8 | 1 | 30 | yes | — |
| 123 | C23 | C23×#41 | 1 | 90 | yes | — |
| 124 | C23 | C27 | 1 | 1 | yes | characteristic |
| 125 | C2×C4 | — | 2 | 96 | no | — |
| 126 | C2×C4 | — | 2 | 2880 | no | — |
| 127 | C2×C4 | — | 2 | 48 | no | — |
| 128 | C2×C4 | — | 2 | 96 | no | — |
| 129 | C2×C4 | — | 4 | 1440 | no | — |
| 130 | C2×C4 | C24×D4 | 1 | 96 | yes | — |
| 131 | C22 | — | 2 | 96 | no | — |
| 132 | C22 | — | 4 | 48 | no | — |
| 133 | C22 | — | 4 | 16 | no | — |
| 134 | C22 | — | 4 | 1920 | no | — |
| 135 | C22 | C22×#399 | 1 | 2240 | yes | direct product factor |
| 136 | C22 | C23×C22⋊Q8 | 1 | 180 | yes | — |
| 137 | C22 | C23×#123 | 1 | 180 | yes | — |
| 138 | C22 | C23×#127 | 1 | 60 | yes | — |
| 139 | C22 | C25×D4 | 1 | 3 | yes | — |
| 140 | C22 | C25×Q8 | 1 | 1 | yes | characteristic |
| 141 | C22 | C24×#41 | 1 | 3 | yes | — |
| 142 | C4 | — | 2 | 96 | no | — |
| 143 | C4 | — | 4 | 48 | no | — |
| 144 | C2 | — | 4 | 32 | no | — |
| 145 | C2 | C23×#399 | 1 | 120 | yes | direct product factor |
| 146 | C2 | C24×C22⋊Q8 | 1 | 3 | yes | — |
| 147 | C2 | C24×#123 | 1 | 3 | yes | — |
| 148 | C2 | C24×#127 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, socle | C27 | C23 |
| commutator, Frattini, cosocle | C23 | C27 |
| prime radical | C28 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C27 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C25×C4⋊C4 | C2 | semidirect |
| 3. C23×#399 | C2 | direct (N × H) |
| 4. C24×#95 | C2 | semidirect |
| 9. C24×C4⋊C4 | C22 | semidirect |
| 12. C22×#399 | C22 | direct (N × H) |
| 13. C23×#95 | C22 | semidirect |
| 26. C23×C4⋊C4 | C23 | semidirect |
| 29. C2×#399 | C23 | direct (N × H) |
| 30. C22×#95 | C23 | semidirect |
| 51. C22×C4⋊C4 | C24 | semidirect |
| 54. #399 | C24 | direct (N × H) |
| 55. C2×#95 | C24 | semidirect |
| 80. C2×C4⋊C4 | C25 | semidirect |
| 83. #95 | C25 | semidirect |
| 90. C24 | #399 | direct (N × H) |
| 116. C23 | C2×#399 | direct (N × H) |
| 135. C22 | C22×#399 | direct (N × H) |
| 145. C2 | C23×#399 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C25×C4⋊C4 | 2 |
| C25×C22⋊C4 | 2 |
| C23×#399 | 2 |
| C24×#95 | 2 |