Finite Group Databasebeta

← #27151092

Subgroups — #27151092

Order 12416.

762 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C2×#15690855C211yesmaximal, characteristic, complemented
2#20261533C212yesmaximal, complemented
3#20260912C214yesmaximal, complemented
4#20260902C214yesmaximal, complemented
5#20260753C212yesmaximal, complemented
6C2×#15690727C211yesmaximal, characteristic, complemented
7#535×C97C211yesmaximal, characteristic, complemented
8#15690855—24no—
9#15690855—22no—
10#15690855C2214yes—
11C2×#13095511C2212yes—
12C2×#13095510—22no—
13C22×C22⋊D97—22no—
14C2×D4×D97—22no—
15#15690841C2214yescomplemented
16#15690840—22no—
17#15690836C2214yescomplemented
18#15690835—22no—
19#15690830—24no—
20C2×#13095495C2212yescomplemented
21#15690738C2218yescomplemented
22#15690727C2214yescomplemented
23C2×#41×C97C2211yescharacteristic, complemented
24C22×D4×C97C2211yescharacteristic
25#127×C97C2212yescomplemented
26#124×C97C2212yescomplemented
27#121×C97—22no—
28#120×C97—22no—
29#117×C97C2211yescharacteristic, complemented
30C23×D97—22no—
31#13095511—22no—
32#13095511—22no—
33#13095511D414yescomplemented
34#13095511C2314yes—
35#13095510—28no—
36#13095510—28no—
37C2×Dic194D412yescomplemented
38C2×D388D412yescomplemented
39C2×C22⋊D97—24no—
40C2×C22⋊D97—28no—
41C2×C22⋊D97—22no—
42C2×C22⋊D97—28no—
43D4×D97—28no—
44D4×D97—28no—
45C22×Dic97—22no—
46C2×C4×D97—22no—
47#13095501—24no—
48#13095499—24no—
49#13095498—24no—
50#13095496—24no—
51#13095495—22no—
52#13095495C2314yescomplemented
53C2×#11715895—22no—
54C24×C97D412yes—
55#41×C97—22no—
56#41×C97—22no—
57C2×Q8×C97C2311yescharacteristic, complemented
58C2×D4×C97—21no—
59C2×D4×C97—24no—
60C2×D4×C97—22no—
61C2×D4×C97—22no—
62C2×D4×C97D412yes—
63C2×D4×C97C2312yes—
64C2×D4×C97C2311yescharacteristic, complemented
65C22×C4×C97—21no—
66C22×C4×C97C2311yescharacteristic
67C4⋊C4×C97—21no—
68C22⋊C4×C97—21no—
69C22⋊C4×C97—24no—
70C42×C97C2312yescomplemented
71C23×C97—22no—
72C23×C97—22no—
73C23×C97—21no—
74C23×C97—82no—
75C23×C97C2×D414yes—
76C23×C97C2×D411yescharacteristic
77C22×D97—22no—
78C22×D97—28no—
79C22×D97—84no—
80Q8×C97—22no—
81D4×C97—24no—
82D4×C97—24no—
83D4×C97—22no—
84D4×C97—41no—
85D4×C97—41no—
86D4×C97—84no—
87C2×C4×C97—22no—
88C2×C4×C97—21no—
89C2×C4×C97—42no—
90C2×C4×C97C2×D412yes—
91C2×C4×C97C2×D412yes—
92C2×C4×C97C2×D411yescharacteristic
93C2×C4×C97C2411yescharacteristic
94C22⋊D97—216no—
95C22⋊D97—24no—
96C22⋊D97—24no—
97C2×Dic97—28no—
98C2×Dic97—22no—
99D388—22no—
100D388C2×D414yes—
101C4×D97—24no—
102C4×D97—24no—
103Dic194—22no—
104Dic194C2×D414yes—
105#11715895—24no—
106C22×C97—24no—
107C22×C97—41no—
108C22×C97—44no—
109C22×C97—44no—
110C22×C97—42no—
111C22×C97—41no—
112C22×C97—82no—
113C22×C97#12011yescharacteristic
114C22×C97#12011yescharacteristic
115C22×C97C22×D411yescharacteristic
116C2×D97—24no—
117C2×D97—44no—
118C2×D97—88no—
119C4×C97—42no—
120C4×C97—41no—
121C4×C97#12012yes—
122C4×C97C22×D412yes—
123Dic97—24no—
124C2×C97—21no—
125C2×C97—21no—
126C2×C97—21no—
127C2×C97—44no—
128C2×C97—81no—
129C2×C97C2×#12011yescharacteristic
130D97—44no—
131#3019—971nomaximal, Sylow
132C97#301911yescharacteristic, complemented, Sylow
133C2×H(5,𝔽2)—971no—
134#575—972no—
135#535D9711yescharacteristic, complemented
136#455—974no—
137#453—974no—
138#421—972no—
139C2×#100—971no—
140H(5,𝔽2)—974no—
141H(5,𝔽2)—1944no—
142H(5,𝔽2)—1942no—
143C2×#41—972no—
144C2×#41—1942no—
145C2×#41C2×D9711yescharacteristic, complemented
146C22×D4—1942no—
147C22×D4—1942no—
148C22×D4C2×D9711yescharacteristic
149#137—974no—
150#137—1942no—
151AGL(1,ℤ/8)—974no—
152AGL(1,ℤ/8)—1942no—
153#135—1944no—
154C2×M16—972no—
155#127C2×D9712yescomplemented
156#124C2×D9712yescomplemented
157#121—22no—
158#120—22no—
159#117C2×D9711yescharacteristic, complemented
160#104—978no—
161#100—974no—
162C24—1942no—
163C24C22⋊D9712yes—
164#41—22no—
165#41—22no—
166#41—974no—
167#41—974no—
168#41—1948no—
169#41—1948no—
170#41—1942no—
171#41—1942no—
172C2×Q8—972no—
173C2×Q8C22×D9711yescharacteristic, complemented
174C2×D4—21no—
175C2×D4—24no—
176C2×D4—22no—
177C2×D4—22no—
178C2×D4—972no—
179C2×D4—1948no—
180C2×D4—1948no—
181C2×D4—1944no—
182C2×D4—1948no—
183C2×D4—1942no—
184C2×D4—1948no—
185C2×D4C22⋊D9712yes—
186C2×D4C22×D9712yes—
187C2×D4C22×D9711yescharacteristic, complemented
188C22×C4—21no—
189C22×C4—1942no—
190C22×C4—1942no—
191C22×C4C22×D9711yescharacteristic
192Q16—1944no—
193SD16—1944no—
194SD16—1944no—
195D8—1944no—
196M16—974no—
197M16—1942no—
198C2×C8—1942no—
199C4⋊C4—21no—
200C22⋊C4—21no—
201C22⋊C4—24no—
202C42C22×D9712yescomplemented
203C23—22no—
204C23—22no—
205C23—21no—
206C23—82no—
207C23—7764no—
208C23—1948no—
209C23—1942no—
210C23C2×C22⋊D9714yes—
211C23C2×C22⋊D9711yescharacteristic
212Q8—22no—
213Q8—974no—
214Q8—1942no—
215D4—24no—
216D4—24no—
217D4—22no—
218D4—41no—
219D4—41no—
220D4—84no—
221D4—974no—
222D4—19416no—
223D4—1944no—
224D4—1944no—
225D4—1942no—
226C2×C4—22no—
227C2×C4—21no—
228C2×C4—42no—
229C2×C4—1948no—
230C2×C4—1942no—
231C2×C4—1944no—
232C2×C4—1944no—
233C2×C4D4×D9712yes—
234C2×C4D4×D9712yes—
235C2×C4C2×C22⋊D9711yescharacteristic
236C2×C4C23×D9711yescharacteristic
237C8—1944no—
238C22—24no—
239C22—41no—
240C22—44no—
241C22—44no—
242C22—42no—
243C22—41no—
244C22—82no—
245C22—7768no—
246C22—3884no—
247C22—1944no—
248C22#1569077811yescharacteristic
249C22#1569077811yescharacteristic
250C22C22×C22⋊D9711yescharacteristic
251C4—42no—
252C4—41no—
253C4—1944no—
254C4#1569077812yes—
255C4C2×D4×D9712yes—
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—81no—
261C2—3884no—
262C2C2×#1569077811yescharacteristic
subgroupG/H
centerC2C2×#15690778
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#535×C97C2
socleC2×C97C2×#120
hypercenter#535D97
hypocenterC97#3019
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#15690855C2semidirect
2. #20261533C2semidirect
3. #20260912C2semidirect
4. #20260902C2semidirect
5. #20260753C2semidirect
6. C2×#15690727C2semidirect
7. #535×C97C2semidirect
15. #15690841C22semidirect
17. #15690836C22semidirect
20. C2×#13095495C22semidirect
21. #15690738C22semidirect
22. #15690727C22semidirect
23. C2×#41×C97C22semidirect
25. #127×C97C22semidirect
26. #124×C97C22semidirect
29. #117×C97C22semidirect
33. #13095511D4semidirect
37. C2×Dic194D4semidirect
38. C2×D388D4semidirect
52. #13095495C23semidirect
57. C2×Q8×C97C23semidirect
64. C2×D4×C97C23semidirect
70. C42×C97C23semidirect
132. C97#3019semidirect
135. #535D97semidirect
145. C2×#41C2×D97semidirect
155. #127C2×D97semidirect
156. #124C2×D97semidirect
159. #117C2×D97semidirect
173. C2×Q8C22×D97semidirect
187. C2×D4C22×D97semidirect
202. C42C22×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908552
#202615332
#202609122
#202609022
#202607532
C2×#156907272
#535×C972
#301997