Finite Group Databasebeta

← #11929231

Subgroups — #11929231

Order 1248.

332 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 108 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
1#10597794C211yesmaximal, characteristic, complemented
2C13×#239C211yesmaximal, characteristic, complemented
3C3×#6131C211yesmaximal, characteristic, complemented
4#10597743C213yesmaximal, complemented
5D3×D52C213yesmaximal, complemented
6D12×D13C213yesmaximal, complemented
7#10597737C213yesmaximal, complemented
8#88256—31nomaximal, Hall
9C2×D3×D13C2216yescomplemented
10C3×Q8×C13C2211yescharacteristic, complemented
11D156C2213yescomplemented
12C4×D39C2213yescomplemented
13C13×D12C2213yescomplemented
14C4×D3×C13C2213yescomplemented
15C3×D52C2213yescomplemented
16C3×C4×D13C2213yescomplemented
17#64605C2211yescharacteristic, complemented
18#64604C2213yescomplemented
19#64603C2213yescomplemented
20#64602C2213yescomplemented
21C13×#41—31no—
22#6131—31no—
23#6131D311yescharacteristic, complemented
24D4×D13—33no—
25D4×D13—33no—
26#6127—33no—
27C2×D52—33no—
28C2×D39C2313yes—
29C2×D3×C13C2313yes—
30C2×C3×D13C2313yes—
31D3×D13—46no—
32C3×C4×C13C2313yes—
33Dic39C2311yescharacteristic
34C3×Dic13C2311yescharacteristic
35Dic3×C13C2311yescharacteristic
36C22×D13—36no—
37Q8×C13C2×D311yescharacteristic, complemented
38D4×C13—33no—
39C2×C4×C13—33no—
40C22⋊D13—33no—
41C22⋊D13—33no—
42D52—33no—
43D52—33no—
44D52C2×D313yescomplemented
45C4×D13—33no—
46C4×D13C2×D313yescomplemented
47Dic26—31no—
48#1009—131nomaximal, Hall
49C2×C3×C13C2411yescharacteristic
50D39—23no—
51C3×D13—23no—
52D3×C13—23no—
53C22×C13—33no—
54C2×D13—33no—
55C2×D13—126no—
56C2×D13C22×D313yes—
57C4×C13—31no—
58C4×C13C22×D313yes—
59Dic13—31no—
60Dic13C22×D311yescharacteristic
61C3×#41—131no—
62#239—131no—
63#239D1311yescharacteristic, complemented
64D3×D4—133no—
65D3×D4—133no—
66#235—133no—
67C2×D12—133no—
68C3×C13H(5,𝔽2)11yescharacteristic, complemented, Hall
69H(5,𝔽2)—391noSylow
70C2×C13—63no—
71C2×C13C23×D311yescharacteristic
72D13—23no—
73D13—63no—
74C22×D3—136no—
75C3×Q8C2×D1311yescharacteristic, complemented
76C3×D4—133no—
77C2×C3×C4—133no—
78#67—133no—
79#67—133no—
80D12—133no—
81D12—133no—
82D12C2×D1313yescomplemented
83C4×D3—133no—
84C4×D3C2×D1313yescomplemented
85Dic6—131no—
86#41—31no—
87#41—131no—
88#41—393no—
89#41—391no—
90C2×D4—393no—
91C2×D4—393no—
92C2×D4—393no—
93C13#100911yescharacteristic, complemented, Sylow
94C22×C3—133no—
95C2×D3—133no—
96C2×D3—526no—
97C2×D3C22×D1313yes—
98C3×C4—131no—
99C3×C4C22×D1313yes—
100Dic3—131no—
101Dic3C22×D1311yescharacteristic
102C23—396no—
103Q8—391no—
104Q8D3×D1311yescharacteristic, complemented
105D4—33no—
106D4—133no—
107D4—393no—
108D4—393no—
109D4—393no—
110D4—393no—
111C2×C4—33no—
112C2×C4—133no—
113C2×C4—393no—
114C2×C3—263no—
115C2×C3C23×D1311yescharacteristic
116D3—23no—
117D3—263no—
118C22—33no—
119C22—133no—
120C22—393no—
121C22—1566no—
122C4—31no—
123C4—131no—
124C4—391no—
125C4C2×D3×D1313yes—
126C3#8825611yescharacteristic, complemented, Sylow
127C2—63no—
128C2—263no—
129C2—783no—
130C2C22×D3×D1311yescharacteristic
subgroupG/H
center, FrattiniC2C22×D3×D13
commutator, socle, cosocleC2×C3×C13C24
FittingC3×Q8×C13C22
hypercenterQ8D3×D13
hypocenterC3×C13H(5,𝔽2)
LCSG ▷ C2×C3×C13 ▷ C3×C13
UCSC1 ◁ C2 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. #10597794C2semidirect
2. C13×#239C2semidirect
3. C3×#6131C2semidirect
4. #10597743C2semidirect
5. D3×D52C2semidirect
6. D12×D13C2semidirect
7. #10597737C2semidirect
9. C2×D3×D13C22semidirect
10. C3×Q8×C13C22semidirect
11. D156C22semidirect
12. C4×D39C22semidirect
13. C13×D12C22semidirect
14. C4×D3×C13C22semidirect
15. C3×D52C22semidirect
16. C3×C4×D13C22semidirect
17. #64605C22semidirect
18. #64604C22semidirect
19. #64603C22semidirect
20. #64602C22semidirect
23. #6131D3semidirect
37. Q8×C13C2×D3semidirect
44. D52C2×D3semidirect
46. C4×D13C2×D3semidirect
63. #239D13semidirect
68. C3×C13H(5,𝔽2)semidirect
75. C3×Q8C2×D13semidirect
82. D12C2×D13semidirect
84. C4×D3C2×D13semidirect
93. C13#1009semidirect
104. Q8D3×D13semidirect
126. C3#88256semidirect

Maximal subgroups

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

subgroupindex
#105977942
C13×#2392
C3×#61312
#105977432
D3×D522
D12×D132
#105977372
#882563
#100913