Finite Group Databasebeta

← C2×#72215806 #72341384

Subgroups — C2×#72215806 #72341384

Order 3072.

At least 6640 conjugacy classes of subgroups; only the 208 normal subgroups are shown.

normal subgroups only

This group has at least 6640 conjugacy classes of subgroups — too many to draw. Only its 208 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
1C2×#11711331C212yesmaximal, complemented
2C2×#14679⋊C3C211yesmaximal, characteristic, complemented
3C22×#72991C212yesmaximal, complemented
4C22×#87524C211yesmaximal, characteristic, complemented
5#72215806C218yesmaximal, direct product factor
6C24×#1020C211yesmaximal, characteristic, complemented
7C23×#1020C2212yes—
8C23×#1020C2214yescomplemented
9C23×#1020C2211yescharacteristic, complemented
10C2×#87524C2214yescomplemented
11C22×#5936C2211yescharacteristic, complemented
12C2×#72991C2218yescomplemented
13#11711331C2218yescomplemented
14C24×C22⋊A4C2211yescharacteristic
15C22×#379⋊C3C2212yescomplemented
16#14679⋊C3C2214yescomplemented
17C2×#14679D311yescharacteristic, complemented
18C23×C22⋊A4D414yescomplemented
19C23×C22⋊A4C2312yes—
20C23×C22⋊A4C2311yescharacteristic
21C22×#1020D418yescomplemented
22C22×#1020C2314yes—
23C22×#1020C2312yescomplemented
24C2×#5936C2312yescomplemented
25C2×#379⋊C3C2314yescomplemented
26C28C2×D311yescharacteristic
27C22×#379C2×D312yescomplemented
28#14679C2×D314yescomplemented
29C22×C22⋊A4C2×D412yes—
30C22×C22⋊A4C2×D414yes—
31C22×C22⋊A4C2×D414yescomplemented
32C22×C22⋊A4C2411yescharacteristic
33C2×#1020C2×D418yescomplemented
34C27D1214yescomplemented
35C27C22×D312yes—
36C27C22×D311yescharacteristic
37C23×C22⋊C4S411yescharacteristic, complemented
38C2×#379C22×D314yescomplemented
39C2×C22⋊A4#12014yescomplemented
40C2×C22⋊A4C22×D411yescharacteristic
41C2×C22⋊A4C22×D412yes—
42C26C2×D1212yes—
43C26C2×D1214yescomplemented
44C26D3×D414yes—
45C26C2×S411yescharacteristic
46C26C23×D311yescharacteristic
47C24×C4C2×S412yescomplemented
48C22×C22⋊C4C2×S414yescomplemented
49C22⋊A4C2×#12011yescharacteristic, complemented
50C25#88214yescomplemented
51C25C4⋊S414yes—
52C25C22×D1211yescharacteristic
53C25C2×D3×D412yes—
54C25C22×S412yes—
55C25C22×S411yescharacteristic
56C23×C4C22×S414yescomplemented
57C24#1020⋊C211yescharacteristic
58C24C2×#88211yescharacteristic, complemented
59C24C2×C4⋊S414yes—
60C24C2×C4⋊S412yes—
61C24D4×S414yes—
62C24C23×S411yescharacteristic
63C23#7299114yescomplemented
64C23#8529714yes—
65C23C2×#1020⋊C212yes—
66C23C2×#1020⋊C211yescharacteristic
67C23C22×C4⋊S411yescharacteristic
68C23C2×D4×S412yes—
69C22C2×#7299112yes—
70C22C2×#7299114yescomplemented
71C22#1171320014yes—
72C22C2×#8529711yescharacteristic
73C22C22×#1020⋊C211yescharacteristic
74C2C2×#1171320012yes—
75C2#7221580614yesdirect product factor
76C2C22×#7299111yescharacteristic
subgroupG/H
centerC23C2×#1020⋊C2
commutator, cosocleC22×C22⋊A4C24
FrattiniC24C23×S4
FittingC2×#14679D3
socleC25C22×S4
hypercenterC24#1020⋊C2
hypocenterC22⋊A4C2×#120
LCSG ▷ C22×C22⋊A4 ▷ C22⋊A4
UCSC1 ◁ C23 ◁ C24

Complements

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

NH ≅ G/Ntype
1. C2×#11711331C2semidirect
2. C2×#14679⋊C3C2semidirect
3. C22×#72991C2semidirect
4. C22×#87524C2semidirect
5. #72215806C2direct (N × H)
6. C24×#1020C2semidirect
8. C23×#1020C22semidirect
9. C23×#1020C22semidirect
10. C2×#87524C22semidirect
11. C22×#5936C22semidirect
12. C2×#72991C22semidirect
13. #11711331C22semidirect
15. C22×#379⋊C3C22semidirect
16. #14679⋊C3C22semidirect
17. C2×#14679D3semidirect
18. C23×C22⋊A4D4semidirect
21. C22×#1020D4semidirect
23. C22×#1020C23semidirect
24. C2×#5936C23semidirect
25. C2×#379⋊C3C23semidirect
27. C22×#379C2×D3semidirect
28. #14679C2×D3semidirect
31. C22×C22⋊A4C2×D4semidirect
33. C2×#1020C2×D4semidirect
34. C27D12semidirect
37. C23×C22⋊C4S4semidirect
38. C2×#379C22×D3semidirect
39. C2×C22⋊A4#120semidirect
43. C26C2×D12semidirect
47. C24×C4C2×S4semidirect
48. C22×C22⋊C4C2×S4semidirect
49. C22⋊A4C2×#120semidirect
50. C25#882semidirect
56. C23×C4C22×S4semidirect
58. C24C2×#882semidirect
63. C23#72991semidirect
70. C22C2×#72991semidirect
75. C2#72215806direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#117113312
C2×#14679⋊C32
C22×#729912
C22×#875242
C2×#3940583
#722158062
C24×#10202
C2×#852974