Finite Group Databasebeta

← C2×#72241384 #72344138

Subgroups — C2×#72241384 #72344138

Order 3072.

At least 6406 conjugacy classes of subgroups; only the 272 normal subgroups are shown.

The subgroup lattice has 102 nodes (normal subgroups only) — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20609⋊C3C211yesmaximal, characteristic, complemented
2C2×#32899⋊C3C212yesmaximal, complemented
3C23×B(4,𝔽2)⋊C3C211yesmaximal, characteristic, complemented
4C2×#32898⋊C3C212yesmaximal, complemented
5C22×#87547C211yesmaximal, characteristic, complemented
6#72241384C218yesmaximal, direct product factor
7C2×#6628896C311yesmaximal, characteristic, complemented, Sylow
8C2×#87547C2214yescomplemented
9C22×#5950C2211yescharacteristic, complemented
10C23×#997C2211yescharacteristic
11C23×#997C2211yescharacteristic, complemented
12C22×B(4,𝔽2)⋊C3C2212yescomplemented
13C22×B(4,𝔽2)⋊C3C2214yescomplemented
14#32899⋊C3C2218yescomplemented
15#32898⋊C3C2218yescomplemented
16#20609⋊C3C2214yescomplemented
17C2×#1790⋊C3C2212yescomplemented
18C22×H(7,𝔽2)C2×C311yescharacteristic, complemented
19C23×B(4,𝔽2)C2×C311yescharacteristic, complemented
20#6628896C2×C318yescomplemented
21C2×#32899C2×C312yescomplemented
22C2×#32898C2×C312yescomplemented
23C2×#20609C2×C311yescharacteristic, complemented
24C2×#5950C2312yescomplemented
25C22×#997D414yescomplemented
26C22×#997C2311yescharacteristic
27C22×#997C2312yes—
28C22×#997C2312yescomplemented
29C2×B(4,𝔽2)⋊C3C2314yescomplemented
30#1790⋊C3C2314yescomplemented
31C2×H(7,𝔽2)C22×C314yescomplemented
32C22×#585C22×C311yescharacteristic, complemented
33C23×H(5,𝔽2)C22×C311yescharacteristic
34C23×H(5,𝔽2)C22×C311yescharacteristic, complemented
35C23×#120A411yescharacteristic, complemented
36C22×B(4,𝔽2)C22×C312yes—
37C22×B(4,𝔽2)C22×C314yescomplemented
38#32899C22×C318yescomplemented
39#32898C22×C318yescomplemented
40#20609C22×C314yescomplemented
41C2×#1790C22×C312yescomplemented
42C2×#997C2×D412yescomplemented
43C2×#997C2×D414yescomplemented
44C2×#997C2411yescharacteristic
45C27C2×A411yescharacteristic, complemented
46C2×#585C23×C312yescomplemented
47C22×H(5,𝔽2)C3×D414yescomplemented
48C22×H(5,𝔽2)C23×C311yescharacteristic
49C22×H(5,𝔽2)C23×C312yes—
50C22×H(5,𝔽2)C23×C312yescomplemented
51C24×D4C2×A411yescharacteristic
52C24×D4C2×A412yescomplemented
53C22×#120C2×A418yescomplemented
54C23×C22⋊C4C2×A412yescomplemented
55C23×C22⋊C4C2×A411yescharacteristic, complemented
56C2×B(4,𝔽2)C23×C314yes—
57#1790C23×C314yescomplemented
58#997C22×D411yescharacteristic, complemented
59C26C22×A411yescharacteristic
60C26C22×A412yes—
61C26C22×A411yescharacteristic
62C26C22×A414yescomplemented
63C2×H(5,𝔽2)C2×C3×D412yes—
64C2×H(5,𝔽2)C2×C3×D414yescomplemented
65C2×H(5,𝔽2)C24×C311yescharacteristic
66C23×D4C22×A414yes—
67C23×D4C22×A418yescomplemented
68C24×C4C22×A411yescharacteristic
69C24×C4C22×A412yes—
70C22×C22⋊C4C22×A414yescomplemented
71C22×C22⋊C4C22×A418yescomplemented
72C25D4×A414yes—
73C25C23×A412yes—
74C25C23×A414yes—
75C25C23×A411yescharacteristic
76C25C23×A412yes—
77H(5,𝔽2)C22×C3×D411yescharacteristic
78C22×D4#120⋊C311yescharacteristic
79C23×C4C23×A412yes—
80C23×C4C23×A414yes—
81C24C2×#120⋊C311yescharacteristic
82C24C2×#120⋊C311yescharacteristic
83C24C2×D4×A412yes—
84C24C2×D4×A414yes—
85C24C24×A411yescharacteristic
86C2×D4C2×#120⋊C314yes—
87C22×C4C2×#120⋊C311yescharacteristic
88C23AΣL(1,𝔽64)14yes—
89C23C22×#120⋊C312yes—
90C23C22×#120⋊C312yes—
91C23C22×#120⋊C311yescharacteristic
92C23C22×D4×A411yescharacteristic
93C2×C4C22×#120⋊C312yes—
94C22C2×AΣL(1,𝔽64)14yes—
95C22C2×AΣL(1,𝔽64)12yes—
96C22#60399⋊C314yes—
97C22C23×#120⋊C311yescharacteristic
98C2C2×#60399⋊C312yes—
99C2#7224138414yesdirect product factor
100C2C22×AΣL(1,𝔽64)11yescharacteristic
subgroupG/H
center, socleC23C22×#120⋊C3
commutator, cosocleC2×H(5,𝔽2)C24×C3
FrattiniC24C24×A4
FittingC2×#6628896C3
hypercenterC22×D4#120⋊C3
hypocenterH(5,𝔽2)C22×C3×D4
LCSG ▷ C2×H(5,𝔽2) ▷ H(5,𝔽2)
UCSC1 ◁ C23 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C2×#20609⋊C3C2semidirect
2. C2×#32899⋊C3C2semidirect
3. C23×B(4,𝔽2)⋊C3C2semidirect
4. C2×#32898⋊C3C2semidirect
5. C22×#87547C2semidirect
6. #72241384C2direct (N × H)
7. C2×#6628896C3semidirect
8. C2×#87547C22semidirect
9. C22×#5950C22semidirect
11. C23×#997C22semidirect
12. C22×B(4,𝔽2)⋊C3C22semidirect
13. C22×B(4,𝔽2)⋊C3C22semidirect
14. #32899⋊C3C22semidirect
15. #32898⋊C3C22semidirect
16. #20609⋊C3C22semidirect
17. C2×#1790⋊C3C22semidirect
18. C22×H(7,𝔽2)C2×C3semidirect
19. C23×B(4,𝔽2)C2×C3semidirect
20. #6628896C2×C3semidirect
21. C2×#32899C2×C3semidirect
22. C2×#32898C2×C3semidirect
23. C2×#20609C2×C3semidirect
24. C2×#5950C23semidirect
25. C22×#997D4semidirect
28. C22×#997C23semidirect
29. C2×B(4,𝔽2)⋊C3C23semidirect
30. #1790⋊C3C23semidirect
31. C2×H(7,𝔽2)C22×C3semidirect
32. C22×#585C22×C3semidirect
34. C23×H(5,𝔽2)C22×C3semidirect
35. C23×#120A4semidirect
37. C22×B(4,𝔽2)C22×C3semidirect
38. #32899C22×C3semidirect
39. #32898C22×C3semidirect
40. #20609C22×C3semidirect
41. C2×#1790C22×C3semidirect
42. C2×#997C2×D4semidirect
43. C2×#997C2×D4semidirect
45. C27C2×A4semidirect
46. C2×#585C23×C3semidirect
47. C22×H(5,𝔽2)C3×D4semidirect
50. C22×H(5,𝔽2)C23×C3semidirect
52. C24×D4C2×A4semidirect
53. C22×#120C2×A4semidirect
54. C23×C22⋊C4C2×A4semidirect
55. C23×C22⋊C4C2×A4semidirect
57. #1790C23×C3semidirect
58. #997C22×D4semidirect
62. C26C22×A4semidirect
64. C2×H(5,𝔽2)C2×C3×D4semidirect
67. C23×D4C22×A4semidirect
70. C22×C22⋊C4C22×A4semidirect
71. C22×C22⋊C4C22×A4semidirect
99. C2#72241384direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#20609⋊C32
C2×#32899⋊C32
C23×B(4,𝔽2)⋊C32
C2×#32898⋊C32
C22×#875472
#722413842
C2×#66288963
C2×A4×#1204