Finite Group Databasebeta

← C2×C4×#87588⋊C2 #72343224

Subgroups — C2×C4×#87588⋊C2 #72343224

Order 6144.

At least 17236 conjugacy classes of subgroups; only the 277 normal subgroups are shown.

The subgroup lattice has 100 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#72233208C211yesmaximal, characteristic, complemented
2C2×C4×#87588C212yesmaximal, complemented
3C2×C4×#85645C211yesmaximal, characteristic, complemented
4C2×C4×C22⋊(C22⋊A4)C2211yescharacteristic, complemented
5C23×C4×#120D311yescharacteristic, complemented
6C4×#87588⋊C2C218yesmaximal, direct product factor
7C4×#87588C2218yescomplemented
8C4×#85645C2214yescomplemented
9C2×#72234981C212yesmaximal, complemented
10C2×#11709759C2211yescharacteristic, complemented
11C22×#87588⋊C2C211yesmaximal, characteristic
12#72234981C2218yescomplemented
13#72233208C2214yescomplemented
14C22×#85645C2211yescharacteristic
15C22×#87588C2212yes—
16C2×#11715843C2212yescomplemented
17C2×#87588⋊C2C418yesdirect product factor
18C2×#87588⋊C2C2214yes—
19C2×#87588C2×C418yescomplemented
20C2×#87588C2314yes—
21#11715843C2314yescomplemented
22#87588⋊C2C2×C4116yesdirect product factor
23C22×C22⋊(C22⋊A4)C2311yescharacteristic
24C4×C22⋊(C22⋊A4)C2312yescomplemented
25C2×#85645C2×C414yescomplemented
26C2×#85645C2312yes—
27#11709759C2312yescomplemented
28C27×C4C2×D311yescharacteristic, complemented
29C24×#120C2×D311yescharacteristic
30C22×C4×#120C2×D314yescomplemented
31C23×#379C2×D311yescharacteristic, complemented
32C2×C22⋊(C22⋊A4)C22×C412yescomplemented
33C2×C22⋊(C22⋊A4)C2411yescharacteristic
34#87588C22×C418yescomplemented
35#85645C22×C414yescomplemented
36C28C22×D311yescharacteristic
37C26×C4S411yescharacteristic, complemented
38C26×C4C22×D312yescomplemented
39C23×#120C4×D314yescomplemented
40C23×#120C22×D312yes—
41C2×C4×#120S412yescomplemented
42C22×#379C22×D312yescomplemented
43C22⋊(C22⋊A4)C23×C411yescharacteristic, complemented
44C27C2×C4×D312yescomplemented
45C27C2×S411yescharacteristic
46C27C23×D311yescharacteristic
47C25×C4C2×S411yescharacteristic
48C25×C4C2×S411yescharacteristic
49C25×C4C2×S412yescomplemented
50C25×C4C2×S414yescomplemented
51C22×#120C2×C4×D314yescomplemented
52C22×#120C2×S412yes—
53C4×#120C2×S418yescomplemented
54C2×#379C2×S412yescomplemented
55C26C4×S414yescomplemented
56C26C22×C4×D311yescharacteristic, complemented
57C26C22×S411yescharacteristic
58C26C22×S412yes—
59C26C22×S412yes—
60C24×C4C22×S412yes—
61C24×C4C22×S412yes—
62C24×C4C22×S414yescomplemented
63C24×C4#102011yescharacteristic, complemented
64C2×#120C4×S418yescomplemented
65C2×#120C22×S414yes—
66#379C22×S414yescomplemented
67C25C2×C4×S412yes—
68C25C2×C4×S414yescomplemented
69C25C2×C4×S414yescomplemented
70C25C23×S411yescharacteristic
71C25C23×S412yes—
72C25C2×#102011yescharacteristic
73C23×C4#1020⋊C212yescomplemented
74C23×C4C2×#102011yescharacteristic
75C23×C4C2×#102011yescharacteristic
76C23×C4C2×#102014yescomplemented
77#120C2×C4×S418yescomplemented
78C24C2×#1020⋊C212yes—
79C24C22×C4×S411yescharacteristic
80C24C22×C4×S412yescomplemented
81C24C4×#102014yescomplemented
82C24C22×#102011yescharacteristic
83C24C22×#102012yes—
84C22×C4C2×#1020⋊C214yescomplemented
85C22×C4C22×#102012yes—
86C22×C4C22×#102012yes—
87C23C4×#1020⋊C214yescomplemented
88C23C22×#1020⋊C212yes—
89C23C2×C4×#102012yes—
90C23C2×C4×#102014yescomplemented
91C23C23×#102011yescharacteristic
92C2×C4#87588⋊C211yescharacteristic, direct product factor
93C22C2×#87588⋊C211yescharacteristic
94C22C2×C4×#1020⋊C212yescomplemented
95C22C22×C4×#102011yescharacteristic
96C4C2×#87588⋊C212yesdirect product factor
97C2C22×#87588⋊C211yescharacteristic
98C2C4×#87588⋊C212yesdirect product factor
subgroupG/H
center, hypercenterC2×C4#87588⋊C2
commutator, hypocenterC22⋊(C22⋊A4)C23×C4
FrattiniC23C23×#1020
FittingC23×C4×#120D3
socleC26C22×S4
prime radicalC22×#87588⋊C2C2
cosocleC2×C22⋊(C22⋊A4)C24
LCSG ▷ C22⋊(C22⋊A4)
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
1. C2×#72233208C2semidirect
2. C2×C4×#87588C2semidirect
3. C2×C4×#85645C2semidirect
4. C2×C4×C22⋊(C22⋊A4)C22semidirect
5. C23×C4×#120D3semidirect
6. C4×#87588⋊C2C2direct (N × H)
7. C4×#87588C22semidirect
8. C4×#85645C22semidirect
9. C2×#72234981C2semidirect
10. C2×#11709759C22semidirect
12. #72234981C22semidirect
13. #72233208C22semidirect
16. C2×#11715843C22semidirect
17. C2×#87588⋊C2C4direct (N × H)
19. C2×#87588C2×C4semidirect
21. #11715843C23semidirect
22. #87588⋊C2C2×C4direct (N × H)
24. C4×C22⋊(C22⋊A4)C23semidirect
25. C2×#85645C2×C4semidirect
27. #11709759C23semidirect
28. C27×C4C2×D3semidirect
30. C22×C4×#120C2×D3semidirect
31. C23×#379C2×D3semidirect
32. C2×C22⋊(C22⋊A4)C22×C4semidirect
34. #87588C22×C4semidirect
35. #85645C22×C4semidirect
37. C26×C4S4semidirect
38. C26×C4C22×D3semidirect
39. C23×#120C4×D3semidirect
41. C2×C4×#120S4semidirect
42. C22×#379C22×D3semidirect
43. C22⋊(C22⋊A4)C23×C4semidirect
44. C27C2×C4×D3semidirect
49. C25×C4C2×S4semidirect
50. C25×C4C2×S4semidirect
51. C22×#120C2×C4×D3semidirect
53. C4×#120C2×S4semidirect
54. C2×#379C2×S4semidirect
55. C26C4×S4semidirect
56. C26C22×C4×D3semidirect
62. C24×C4C22×S4semidirect
63. C24×C4#1020semidirect
64. C2×#120C4×S4semidirect
66. #379C22×S4semidirect
68. C25C2×C4×S4semidirect
69. C25C2×C4×S4semidirect
73. C23×C4#1020⋊C2semidirect
76. C23×C4C2×#1020semidirect
77. #120C2×C4×S4semidirect
80. C24C22×C4×S4semidirect
81. C24C4×#1020semidirect
84. C22×C4C2×#1020⋊C2semidirect
87. C23C4×#1020⋊C2semidirect
90. C23C2×C4×#1020semidirect
92. C2×C4#87588⋊C2direct (N × H)
94. C22C2×C4×#1020⋊C2semidirect
96. C4C2×#87588⋊C2direct (N × H)
98. C2C4×#87588⋊C2direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#722332082
C2×C4×#875882
C2×C4×#856452
C2×C4×#239003
C4×#87588⋊C22
C2×#722349812
C22×C4×#10204
C2×C4×#1020⋊C24
C22×#87588⋊C22