Finite Group DatabasebetaFinite Group DatabasebetaOrder 128.
83 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #410 | C2 | 1 | 2 | yes | maximal |
| 2 | #408 | C2 | 1 | 1 | yes | maximal, characteristic |
| 3 | C2×#41 | C22 | 1 | 1 | yes | characteristic |
| 4 | C2×M16 | C4 | 1 | 1 | yes | characteristic, complemented |
| 5 | C22×C8 | C4 | 1 | 1 | yes | characteristic, complemented |
| 6 | #117 | — | 2 | 2 | no | — |
| 7 | #99 | — | 2 | 2 | no | — |
| 8 | #99 | C4 | 1 | 4 | yes | complemented |
| 9 | C22⋊C8 | — | 2 | 2 | no | — |
| 10 | #41 | — | 4 | 1 | no | — |
| 11 | #41 | — | 4 | 1 | no | — |
| 12 | C2×Q8 | Q8 | 1 | 1 | yes | characteristic |
| 13 | C2×D4 | C2×C4 | 1 | 2 | yes | — |
| 14 | C2×D4 | D4 | 1 | 1 | yes | characteristic |
| 15 | C22×C4 | C2×C4 | 1 | 1 | yes | characteristic |
| 16 | C22×C4 | D4 | 1 | 2 | yes | — |
| 17 | M16 | — | 4 | 1 | no | — |
| 18 | C2×C8 | — | 2 | 1 | no | — |
| 19 | C2×C8 | — | 2 | 1 | no | — |
| 20 | C2×C8 | — | 4 | 1 | no | — |
| 21 | C4⋊C4 | — | 4 | 2 | no | — |
| 22 | C22⋊C4 | — | 2 | 4 | no | — |
| 23 | C42 | — | 4 | 2 | no | — |
| 24 | C23 | C42 | 1 | 1 | yes | characteristic |
| 25 | C23 | C22⋊C4 | 1 | 2 | yes | — |
| 26 | Q8 | — | 4 | 1 | no | — |
| 27 | D4 | — | 4 | 2 | no | — |
| 28 | D4 | — | 4 | 1 | no | — |
| 29 | C2×C4 | — | 2 | 2 | no | — |
| 30 | C2×C4 | — | 2 | 1 | no | — |
| 31 | C2×C4 | — | 2 | 2 | no | — |
| 32 | C2×C4 | — | 2 | 1 | no | — |
| 33 | C2×C4 | — | 4 | 4 | no | — |
| 34 | C2×C4 | C22⋊C4 | 1 | 1 | yes | characteristic |
| 35 | C2×C4 | C4⋊C4 | 1 | 1 | yes | characteristic |
| 36 | C2×C4 | C4⋊C4 | 1 | 2 | yes | — |
| 37 | C8 | — | 4 | 1 | no | — |
| 38 | C8 | — | 4 | 1 | no | — |
| 39 | C22 | — | 2 | 2 | no | — |
| 40 | C22 | — | 2 | 1 | no | — |
| 41 | C22 | — | 4 | 1 | no | — |
| 42 | C22 | — | 4 | 2 | no | — |
| 43 | C22 | #95 | 1 | 1 | yes | characteristic |
| 44 | C4 | — | 2 | 2 | no | — |
| 45 | C4 | — | 2 | 1 | no | — |
| 46 | C4 | — | 8 | 4 | no | — |
| 47 | C4 | #99 | 1 | 1 | yes | characteristic |
| 48 | C4 | #99 | 1 | 1 | yes | characteristic |
| 49 | C2 | — | 2 | 1 | no | — |
| 50 | C2 | — | 4 | 2 | no | — |
| 51 | C2 | — | 4 | 1 | no | — |
| 52 | C2 | #342 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center | C4 | #99 |
| commutator | C23 | C42 |
| Frattini, prime radical, cosocle | C2×#41 | C22 |
| socle | C2 | #342 |
| LCS | G ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. C2×M16 | C4 | semidirect |
| 5. C22×C8 | C4 | semidirect |
| 8. #99 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #410 | 2 |
| #408 | 2 |