Finite Group Databasebeta

← C22×#72209772 #72275344

Subgroups — C22×#72209772 #72275344

Order 6144.

At least 12062 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has at least 12062 conjugacy classes of subgroups — too many to draw. Only its 129 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#11710730C211yesmaximal, characteristic, complemented
2C22×#62817D311yescharacteristic, complemented
3C2×#72209772C216yesmaximal, direct product factor
4C2×#11710730C2213yescomplemented
5#72209772C2214yesdirect product factor
6#11710730C2311yescharacteristic, complemented
7C2×#62817C2×D313yescomplemented
8C28S411yescharacteristic, complemented
9C23×H(5,𝔽2)S411yescharacteristic, complemented
10C23×H(5,𝔽2)S411yescharacteristic, complemented
11#62817C22×D311yescharacteristic, complemented
12C22×H(3,𝔽4)S414yescomplemented
13C27C2×S413yescomplemented
14C22×H(5,𝔽2)C2×S413yescomplemented
15C22×H(5,𝔽2)C2×S413yescomplemented
16C2×H(3,𝔽4)C2×S4112yescomplemented
17C26C22×S411yescharacteristic, complemented
18C26#102011yescharacteristic
19C26#102012yes—
20C26#102012yescomplemented
21C2×H(5,𝔽2)C22×S411yescharacteristic, complemented
22C2×H(5,𝔽2)C22×S411yescharacteristic, complemented
23C23×Q8#102012yescomplemented
24H(3,𝔽4)C22×S414yescomplemented
25C25#593411yescharacteristic
26C25#593413yes—
27C25C2×#102013yes—
28C25C2×#102016yes—
29C25C2×#102016yescomplemented
30C22×Q8C2×#102016yescomplemented
31C24C2×#593413yes—
32C24C2×#593413yes—
33C24C22×#102011yescharacteristic
34C24C22×#102012yes—
35C24C22×#102012yescomplemented
36C24#8758811yescharacteristic
37C2×Q8C22×#102012yescomplemented
38C23C22×#593411yescharacteristic
39C23#1171585013yes—
40C23#1171585011yescharacteristic
41C23C2×#8758813yes—
42C22#7220977214yesdirect product factor
43C22C2×#1171585013yes—
44C22C2×#1171585013yes—
45C22C22×#8758811yescharacteristic
46C2C2×#7220977216yesdirect product factor
47C2C22×#1171585011yescharacteristic
subgroupG/H
center, socleC23#11715850
commutator, cosocle, hypocenter#11710730C23
FrattiniC22C22×#87588
FittingC22×#62817D3
hypercenterC24#87588
LCSG ▷ #11710730
UCSC1 ◁ C23 ◁ C24

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C22×#11710730C2semidirect
2. C22×#62817D3semidirect
3. C2×#72209772C2direct (N × H)
4. C2×#11710730C22semidirect
5. #72209772C22direct (N × H)
6. #11710730C23semidirect
7. C2×#62817C2×D3semidirect
8. C28S4semidirect
9. C23×H(5,𝔽2)S4semidirect
10. C23×H(5,𝔽2)S4semidirect
11. #62817C22×D3semidirect
12. C22×H(3,𝔽4)S4semidirect
13. C27C2×S4semidirect
14. C22×H(5,𝔽2)C2×S4semidirect
15. C22×H(5,𝔽2)C2×S4semidirect
16. C2×H(3,𝔽4)C2×S4semidirect
17. C26C22×S4semidirect
20. C26#1020semidirect
21. C2×H(5,𝔽2)C22×S4semidirect
22. C2×H(5,𝔽2)C22×S4semidirect
23. C23×Q8#1020semidirect
24. H(3,𝔽4)C22×S4semidirect
29. C25C2×#1020semidirect
30. C22×Q8C2×#1020semidirect
35. C24C22×#1020semidirect
37. C2×Q8C22×#1020semidirect
42. C22#72209772direct (N × H)
46. C2C2×#72209772direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#855574
C22×#875164
C22×#875154
C22×#117107302
C22×#875244
C22×#5117903
C2×#722097722