Finite Group Databasebeta

← #27154869

Subgroups — #27154869

Order 12416.

364 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 113 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#135×D97C212yesmaximal, complemented
2#20261177C212yesmaximal, complemented
3#20261174C212yesmaximal, complemented
4#20261172C212yesmaximal, complemented
5#20261169C212yesmaximal, complemented
6#20261167C211yesmaximal, characteristic, complemented
7#20261165C212yesmaximal, complemented
8C2×#15690749C211yesmaximal, characteristic, complemented
9#508×C97C211yesmaximal, characteristic, complemented
10#41×D97—22no—
11#15690833C2212yes—
12#15690832C2212yescomplemented
13#15690831—22no—
14#15690830—22no—
15#15690829C2212yes—
16Q16×D97C2212yescomplemented
17SD16×D97—22no—
18D8×D97C2212yescomplemented
19C2×C8×D97C2211yescharacteristic
20Dic776C2211yescharacteristic, complemented
21#15690756C2212yescomplemented
22#15690755C2212yescomplemented
23#15690754C2212yescomplemented
24#15690753C2212yescomplemented
25D1552C2211yescharacteristic, complemented
26#15690751C2212yescomplemented
27#15690749C2212yescomplemented
28#15690749C2212yescomplemented
29C2×#13095486C2211yescharacteristic, complemented
30#135×C97C2212yescomplemented
31Q32×C97C2211yescharacteristic, complemented
32SD32×C97C2212yescomplemented
33D16×C97C2211yescharacteristic, complemented
34C2×C16×C97C2211yescharacteristic, complemented
35#13095512—22no—
36#13095511—22no—
37#13095511—42no—
38#13095510—22no—
39#13095510—42no—
40Q8×D97—22no—
41D4×D97—22no—
42D4×D97—42no—
43C2×C4×D97—42no—
44C2×C4×D97D411yescharacteristic
45Dic388C2312yes—
46#13095501—22no—
47#13095500—22no—
48#13095499—22no—
49#13095498—22no—
50D776C2312yes—
51#13095496—22no—
52C2×#11715895D411yescharacteristic
53C8×D97D412yes—
54C8×D97C2312yes—
55#13095486C2312yes—
56#41×C97—22no—
57Q16×C97C2312yes—
58SD16×C97—22no—
59D8×C97C2312yes—
60C2×C8×C97C2311yescharacteristic
61C16×C97C2311yescharacteristic
62C16×C97C2311yescharacteristic
63C22×D97—42no—
64C22×D97D811yescharacteristic
65Q8×C97—22no—
66D4×C97—22no—
67D4×C97—42no—
68C2×C4×C97—42no—
69C2×C4×C97C2×D411yescharacteristic
70C22⋊D97—42no—
71C22⋊D97—42no—
72C22⋊D97—42no—
73C2×Dic97—42no—
74C2×Dic97D811yescharacteristic
75D388—22no—
76D388—42no—
77C4×D97—42no—
78C4×D97—42no—
79C4×D97—42no—
80C4×D97D812yes—
81C4×D97C2×D412yes—
82Dic194—22no—
83Dic194—42no—
84C8×C97C2×D411yescharacteristic
85C8×C97C2411yescharacteristic
86#11715895C2×D412yes—
87C22×C97—42no—
88C22×C97C2×D811yescharacteristic
89C2×D97—41no—
90C2×D97—42no—
91C2×D97—84no—
92C2×D97C2×D812yes—
93C4×C97—42no—
94C4×C97C2×D811yescharacteristic
95C4×C97C22×D411yescharacteristic
96Dic97—42no—
97Dic97C2×D812yes—
98C2×C97—21no—
99C2×C97—82no—
100C2×C97C22×D811yescharacteristic
101D97—22no—
102D97—82no—
103#3414—971nomaximal, Sylow
104C97#341411yescharacteristic, complemented, Sylow
105C2×#135—972no—
106#510—972no—
107#510—972no—
108#509—972no—
109#509—972no—
110#508—972no—
111#508—971no—
112#508D9711yescharacteristic, complemented
113C2×M32—971no—
114C2×#41—1942no—
115#135—972no—
116#135—972no—
117#135—972no—
118#135—1942no—
119#135—1942no—
120#135C2×D9712yescomplemented
121C2×Q16—972no—
122C2×SD16—1942no—
123C2×D8—972no—
124C22×C8—971no—
125Q32—972no—
126Q32—971no—
127Q32C2×D9711yescharacteristic, complemented
128SD32—972no—
129SD32—972no—
130SD32—972no—
131SD32C2×D9712yescomplemented
132D16—972no—
133D16—971no—
134D16C2×D9711yescharacteristic, complemented
135M32—972no—
136M32—972no—
137C2×C16—971no—
138C2×C16C2×D9711yescharacteristic, complemented
139#41—22no—
140#41—3882no—
141#41—3882no—
142#41—1942no—
143#41—1942no—
144#41—1942no—
145C2×Q8—1942no—
146C2×D4—3882no—
147C2×D4—1942no—
148C22×C4—971no—
149C22×C4—3882no—
150Q16—972no—
151Q16—1942no—
152Q16C22×D9712yes—
153SD16—22no—
154SD16—1942no—
155SD16—1942no—
156SD16—1942no—
157D8—972no—
158D8—1942no—
159D8C22×D9712yes—
160C2×C8—971no—
161C2×C8—972no—
162C2×C8—972no—
163C2×C8C22×D9711yescharacteristic
164C16—972no—
165C16C22×D9711yescharacteristic
166C16C22×D9711yescharacteristic
167C23—971no—
168C23—3882no—
169Q8—22no—
170Q8—3882no—
171Q8—1942no—
172D4—22no—
173D4—42no—
174D4—3882no—
175D4—3882no—
176D4—3882no—
177D4—3882no—
178D4—1942no—
179C2×C4—42no—
180C2×C4—971no—
181C2×C4—972no—
182C2×C4—972no—
183C2×C4—3882no—
184C2×C4—3882no—
185C2×C4—3882no—
186C2×C4—3882no—
187C2×C4D4×D9711yescharacteristic
188C8—972no—
189C8D4×D9711yescharacteristic
190C8C23×D9711yescharacteristic
191C22—42no—
192C22—7764no—
193C22—972no—
194C22—3881no—
195C22—3882no—
196C22D8×D9711yescharacteristic
197C4—42no—
198C4—972no—
199C4—3882no—
200C4D8×D9711yescharacteristic
201C4C2×D4×D9711yescharacteristic
202C2—21no—
203C2—82no—
204C2—7762no—
205C2—1942no—
206C2C2×D8×D9711yescharacteristic
subgroupG/H
centerC4D8×D97
commutator, cosocleC8×C97C24
FrattiniC8C23×D97
Fitting#508×C97C2
socleC2×C97C22×D8
hypercenter#508D97
hypocenterC97#3414
LCSG ▷ C8×C97 ▷ C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #135×D97C2semidirect
2. #20261177C2semidirect
3. #20261174C2semidirect
4. #20261172C2semidirect
5. #20261169C2semidirect
6. #20261167C2semidirect
7. #20261165C2semidirect
8. C2×#15690749C2semidirect
9. #508×C97C2semidirect
12. #15690832C22semidirect
16. Q16×D97C22semidirect
18. D8×D97C22semidirect
20. Dic776C22semidirect
21. #15690756C22semidirect
22. #15690755C22semidirect
23. #15690754C22semidirect
24. #15690753C22semidirect
25. D1552C22semidirect
26. #15690751C22semidirect
27. #15690749C22semidirect
28. #15690749C22semidirect
29. C2×#13095486C22semidirect
30. #135×C97C22semidirect
31. Q32×C97C22semidirect
32. SD32×C97C22semidirect
33. D16×C97C22semidirect
34. C2×C16×C97C22semidirect
104. C97#3414semidirect
112. #508D97semidirect
120. #135C2×D97semidirect
127. Q32C2×D97semidirect
131. SD32C2×D97semidirect
134. D16C2×D97semidirect
138. C2×C16C2×D97semidirect

Maximal subgroups

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

subgroupindex
#135×D972
#202611772
#202611742
#202611722
#202611692
#202611672
#202611652
C2×#156907492
#508×C972
#341497