Finite Group Databasebeta

← C3×#1009 #64351

Subgroups — C3×#1009 #64351

Order 288.

352 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 170 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
1C32×#41C211yesmaximal, characteristic, complemented
2C3×#239C212yesmaximal, complemented
3C3×D3×D4C216yesmaximal, complemented
4C3×#235C213yesmaximal, complemented
5C2×C3×D12C213yesmaximal, complemented
6C3×H(5,𝔽2)—31nomaximal
7#1009C311yesmaximal, characteristic, direct product factor
8C22×C3×D3C2216yescomplemented
9C32×Q8C2211yescharacteristic, complemented
10C32×D4C2213yescomplemented
11C2×C32×C4C2213yescomplemented
12C3×#67C2216yescomplemented
13C3×D12C2213yescomplemented
14C3×D12C2216yescomplemented
15C3×C4×D3C2216yescomplemented
16C3×Dic6C2211yescharacteristic, complemented
17C3×#41—21no—
18C3×#41—32no—
19C3×#41—33no—
20C3×#41D311yescharacteristic, complemented
21C3×#41C2×C311yescharacteristic, complemented
22C2×C3×D4—36no—
23C2×C3×D4—33no—
24#239C2×C312yescomplemented
25D3×D4C2×C316yescomplemented
26#235C2×C313yescomplemented
27C2×D12C2×C313yescomplemented
28C22×C32C2313yes—
29C2×C3×D3—46no—
30C2×C3×D3C2316yes—
31C32×C4C2311yescharacteristic
32C32×C4C2313yes—
33C3×Dic3C2312yes—
34H(5,𝔽2)—31noSylow
35C23×C3—36no—
36C22×D3C22×C316yescomplemented
37C3×Q8—21no—
38C3×Q8—31no—
39C3×Q8C2×D311yescharacteristic, complemented
40C3×Q8C22×C311yescharacteristic, complemented
41C3×D4—23no—
42C3×D4—36no—
43C3×D4—36no—
44C3×D4—33no—
45C3×D4C2×D313yescomplemented
46C3×D4C22×C313yescomplemented
47C2×C3×C4—23no—
48C2×C3×C4—36no—
49C2×C3×C4C2×D313yescomplemented
50C2×C3×C4C22×C313yescomplemented
51#67C22×C316yescomplemented
52D12C22×C313yescomplemented
53D12C22×C316yescomplemented
54C4×D3C22×C316yescomplemented
55Dic6C22×C311yescharacteristic, complemented
56C2×C32—23no—
57C2×C32C2411yescharacteristic
58C3×D3—26no—
59#41—32no—
60#41—33no—
61#41C3×D311yescharacteristic, complemented
62C2×D4—36no—
63C2×D4—33no—
64C22×C3—23no—
65C22×C3—36no—
66C22×C3—126no—
67C22×C3C22×D313yes—
68C22×C3C23×C313yes—
69C2×D3—46no—
70C2×D3C23×C316yes—
71C3×C4—21no—
72C3×C4—23no—
73C3×C4—32no—
74C3×C4C22×D313yes—
75C3×C4C22×D311yescharacteristic
76C3×C4C23×C311yescharacteristic
77C3×C4C23×C313yes—
78Dic3C23×C312yes—
79C32H(5,𝔽2)11yescharacteristic, complemented, Sylow
80C23—36no—
81Q8—31no—
82Q8C2×C3×D311yescharacteristic, complemented
83D4—36no—
84D4—36no—
85D4—33no—
86D4C2×C3×D313yescomplemented
87C2×C4—36no—
88C2×C4C2×C3×D313yescomplemented
89C2×C3—23no—
90C2×C3—23no—
91C2×C3—21no—
92C2×C3—43no—
93C2×C3—66no—
94C2×C3C23×D311yescharacteristic
95C2×C3C24×C311yescharacteristic
96D3—26no—
97C22—36no—
98C22—126no—
99C22C22×C3×D313yes—
100C4—32no—
101C4C22×C3×D313yes—
102C4C22×C3×D311yescharacteristic
103C3—21no—
104C3#100911yescharacteristic, direct product factor
105C3C3×H(5,𝔽2)11yescharacteristic, complemented
106C2—23no—
107C2—66no—
108C2C23×C3×D311yescharacteristic
subgroupG/H
centerC2×C3C23×D3
commutator, cosocleC2×C3C24×C3
FrattiniC2C23×C3×D3
FittingC32×#41C2
socleC2×C32C24
hypercenterC3×#41D3
hypocenterC3C3×H(5,𝔽2)
LCSG ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C3 ◁ C3×#41

Complements

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

NH ≅ G/Ntype
1. C32×#41C2semidirect
2. C3×#239C2semidirect
3. C3×D3×D4C2semidirect
4. C3×#235C2semidirect
5. C2×C3×D12C2semidirect
7. #1009C3direct (N × H)
8. C22×C3×D3C22semidirect
9. C32×Q8C22semidirect
10. C32×D4C22semidirect
11. C2×C32×C4C22semidirect
12. C3×#67C22semidirect
13. C3×D12C22semidirect
14. C3×D12C22semidirect
15. C3×C4×D3C22semidirect
16. C3×Dic6C22semidirect
20. C3×#41D3semidirect
21. C3×#41C2×C3semidirect
24. #239C2×C3semidirect
25. D3×D4C2×C3semidirect
26. #235C2×C3semidirect
27. C2×D12C2×C3semidirect
36. C22×D3C22×C3semidirect
39. C3×Q8C2×D3semidirect
40. C3×Q8C22×C3semidirect
45. C3×D4C2×D3semidirect
46. C3×D4C22×C3semidirect
49. C2×C3×C4C2×D3semidirect
50. C2×C3×C4C22×C3semidirect
51. #67C22×C3semidirect
52. D12C22×C3semidirect
53. D12C22×C3semidirect
54. C4×D3C22×C3semidirect
55. Dic6C22×C3semidirect
61. #41C3×D3semidirect
79. C32H(5,𝔽2)semidirect
82. Q8C2×C3×D3semidirect
86. D4C2×C3×D3semidirect
88. C2×C4C2×C3×D3semidirect
104. C3#1009direct (N × H)
105. C3C3×H(5,𝔽2)semidirect

Maximal subgroups

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

subgroupindex
C32×#412
C3×#2392
C3×D3×D42
C3×#2352
C2×C3×D122
C3×H(5,𝔽2)3
#10093