Finite Group Databasebeta

← #27154880

Subgroups — #27154880

Order 12416.

360 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 129 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#20261540C211yesmaximal, characteristic, complemented
2#20261176C213yesmaximal, complemented
3#20261170C213yesmaximal, complemented
4#20261165C213yesmaximal, complemented
5C2×#15690753C213yesmaximal, complemented
6#20261152C211yesmaximal, characteristic, complemented
7#578×C97C211yesmaximal, characteristic, complemented
8#15690856—21no—
9#15690838—23no—
10#15690832C2213yes—
11C2×D776C2213yes—
12#15690756C2213yescomplemented
13#15690754C2211yescharacteristic, complemented
14#15690753C2213yescomplemented
15#15690753C2216yescomplemented
16#15690751C2213yescomplemented
17#15690750C2213yescomplemented
18C2×#13095486C2213yescomplemented
192−1+4×C97—21no—
20#137×C97—23no—
21#135×C97C2213yescomplemented
22C2×Q16×C97C2213yescomplemented
23#131×C97C2211yescharacteristic
24#13095512—41no—
25#13095511—23no—
26C2×D388—23no—
27D4×D97—43no—
28Dic388C2311yescharacteristic
29#13095500—23no—
30D776—23no—
31D776C2313yes—
32#13095486C2311yescharacteristic
33#13095486C2313yes—
34#41×C97—23no—
35#41×C97—43no—
36#41×C97D411yescharacteristic
37C2×Q8×C97—23no—
38C2×Q8×C97—41no—
39Q16×C97—23no—
40Q16×C97C2313yes—
41SD16×C97—23no—
42D8×C97C2311yescharacteristic
43M16×C97D413yes—
44C2×C8×C97C2313yes—
45C22×D97—43no—
46Q8×C97—23no—
47Q8×C97—43no—
48Q8×C97D811yescharacteristic
49D4×C97—21no—
50D4×C97—43no—
51D4×C97D813yes—
52C2×C4×C97—43no—
53C2×C4×C97—43no—
54C2×C4×C97C2×D413yes—
55C22⋊D97—43no—
56D388—23no—
57D388—43no—
58C4×D97—43no—
59Dic194—21no—
60C8×C97C2×D413yes—
61C8×C97C2411yescharacteristic
62C22×C97—41no—
63C22×C97C2×D813yes—
64C2×D97—43no—
65C2×D97—163no—
66C4×C97—43no—
67C4×C97C2×D813yes—
68C4×C97C22×D411yescharacteristic
69Dic97—41no—
70C2×C97—23no—
71C2×C97—81no—
72C2×C97C22×D811yescharacteristic
73D97—83no—
74#3416—971nomaximal, Sylow
75C97#341611yescharacteristic, complemented, Sylow
76#578D9711yescharacteristic, complemented
77#576—971no—
78#510—973no—
79#509—973no—
80#508—973no—
81C2×SD32—973no—
82#504—971no—
832−1+4—21no—
84H(5,𝔽2)—1941no—
85#137—23no—
86AGL(1,ℤ/8)—1943no—
87#135—973no—
88#135C2×D9713yescomplemented
89C2×Q16C2×D9713yescomplemented
90C2×D8—973no—
91#131C2×D9711yescharacteristic
92Q32—973no—
93SD32—973no—
94SD32—976no—
95SD32—971no—
96D16—973no—
97M32—973no—
98C2×C16—973no—
99#41—23no—
100#41—43no—
101#41—3881no—
102#41—1943no—
103#41C22⋊D9711yescharacteristic
104C2×Q8—23no—
105C2×Q8—41no—
106C2×D4—3883no—
107C2×D4—1943no—
108Q16—23no—
109Q16—971no—
110Q16C22×D9713yes—
111SD16—23no—
112SD16—1943no—
113D8—973no—
114D8—1943no—
115D8C22×D9711yescharacteristic
116M16C22⋊D9713yes—
117C2×C8C22×D9713yes—
118C16—971no—
119C16—973no—
120C23—3883no—
121Q8—23no—
122Q8—43no—
123Q8—1941no—
124Q8#1309549611yescharacteristic
125D4—21no—
126D4—43no—
127D4—3883no—
128D4—3883no—
129D4—1943no—
130D4#1309549613yes—
131C2×C4—43no—
132C2×C4—43no—
133C2×C4—3883no—
134C2×C4C2×C22⋊D9713yes—
135C8C2×C22⋊D9713yes—
136C8C23×D9711yescharacteristic
137C22—41no—
138C22—15523no—
139C22—3883no—
140C22C2×#1309549613yes—
141C4—43no—
142C4—3881no—
143C4C2×#1309549613yes—
144C4C22×C22⋊D9711yescharacteristic
145C2—23no—
146C2—81no—
147C2—7763no—
148C2C22×#1309549611yescharacteristic
subgroupG/H
centerC2C22×#13095496
commutator, cosocleC8×C97C24
FrattiniC8C23×D97
Fitting#578×C97C2
socleC2×C97C22×D8
hypercenter#578D97
hypocenterC97#3416
LCSG ▷ C8×C97 ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ #41 ◁ #131 ◁ #578

Complements

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

NH ≅ G/Ntype
1. #20261540C2semidirect
2. #20261176C2semidirect
3. #20261170C2semidirect
4. #20261165C2semidirect
5. C2×#15690753C2semidirect
6. #20261152C2semidirect
7. #578×C97C2semidirect
12. #15690756C22semidirect
13. #15690754C22semidirect
14. #15690753C22semidirect
15. #15690753C22semidirect
16. #15690751C22semidirect
17. #15690750C22semidirect
18. C2×#13095486C22semidirect
21. #135×C97C22semidirect
22. C2×Q16×C97C22semidirect
75. C97#3416semidirect
76. #578D97semidirect
88. #135C2×D97semidirect
89. C2×Q16C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202615402
#202611762
#202611702
#202611652
C2×#156907532
#202611522
#578×C972
#341697