Finite Group Databasebeta

← #27151108

Subgroups — #27151108

Order 12416.

736 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 161 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×#15690835C212yesmaximal, complemented
2#20261299C211yesmaximal, characteristic, complemented
3#20260908C214yesmaximal, complemented
4#20260903C214yesmaximal, complemented
5#20260752C212yesmaximal, complemented
6#20260719C211yesmaximal, characteristic, complemented
7#585×C97C211yesmaximal, characteristic, complemented
8C22×D388C2211yescharacteristic, complemented
9#15690835—24no—
10#15690835C2218yescomplemented
11C2×#13095498—22no—
12C2×#13095496—22no—
13C2×#13095495C2212yescomplemented
14#15690806C2212yescomplemented
15#15690799C2212yescomplemented
16#15690784—22no—
17#15690778—22no—
18#15690769C2211yescharacteristic, complemented
19#15690737C2218yescomplemented
20#15690729C2212yescomplemented
21#15690727C2212yescomplemented
222−1+4×C97—22no—
232−1+4×C97C2212yescomplemented
24H(5,𝔽2)×C97—22no—
25H(5,𝔽2)×C97—22no—
26H(5,𝔽2)×C97C2212yescomplemented
27C2×#41×C97—24no—
28C2×#41×C97—22no—
29C2×#41×C97C2212yes—
30C2×#41×C97C2211yescharacteristic
31C23×D97—21no—
32C2×D388—22no—
33C2×D388—22no—
34C2×D388—21no—
35C2×D388C2312yes—
36C2×C22⋊D97—24no—
37#13095498—28no—
38#13095496—28no—
39#13095495—22no—
40#13095495C2314yes—
41C2×#11715895—22no—
42C4⋊Dic97—21no—
43C22⋊Dic97—21no—
44#13095488—24no—
45C4×Dic97C2312yes—
46#41×C97—24no—
47#41×C97—28no—
48#41×C97—24no—
49#41×C97—24no—
50#41×C97—24no—
51#41×C97—22no—
52#41×C97—22no—
53#41×C97—24no—
54#41×C97—22no—
55#41×C97—22no—
56#41×C97D414yescomplemented
57#41×C97C2314yes—
58C2×Q8×C97—22no—
59C2×Q8×C97—24no—
60C2×Q8×C97D412yescomplemented
61C2×Q8×C97C2311yescharacteristic
62C2×D4×C97—28no—
63C2×D4×C97—22no—
64C2×D4×C97—24no—
65C2×D4×C97—22no—
66C2×D4×C97—24no—
67C2×D4×C97D412yes—
68C2×D4×C97D412yescomplemented
69C2×D4×C97C2311yescharacteristic
70C22×C4×C97—24no—
71C22×C4×C97—22no—
72C22×C4×C97D412yes—
73C22×C4×C97C2311yescharacteristic
74C23×C97—24no—
75C23×C97—22no—
76C23×C97C2×D412yes—
77C23×C97C2×D411yescharacteristic
78C22×D97—22no—
79C22×D97—42no—
80C22×D97—82no—
81Q8×C97—24no—
82Q8×C97—22no—
83Q8×C97—22no—
84Q8×C97C2×D414yes—
85D4×C97—28no—
86D4×C97—24no—
87D4×C97—24no—
88D4×C97—24no—
89D4×C97—24no—
90D4×C97—22no—
91D4×C97—22no—
92D4×C97C2×D414yes—
93C2×C4×C97—24no—
94C2×C4×C97—28no—
95C2×C4×C97—22no—
96C2×C4×C97—24no—
97C2×C4×C97—22no—
98C2×C4×C97—22no—
99C2×C4×C97—24no—
100C2×C4×C97C2×D412yes—
101C2×C4×C97C2×D411yescharacteristic
102C2×C4×C97C2×D412yes—
103C2×C4×C97C2×D412yes—
104C2×C4×C97C2411yescharacteristic
105C22⋊D97—84no—
106C2×Dic97—22no—
107D388—24no—
108D388—41no—
109D388—41no—
110#11715895—24no—
111C22×C97—22no—
112C22×C97—24no—
113C22×C97—41no—
114C22×C97—42no—
115C22×C97—82no—
116C22×C97—84no—
117C22×C97#12012yes—
118C22×C97C22×D411yescharacteristic
119C2×D97—44no—
120C2×D97—42no—
121C2×D97—84no—
122C4×C97—24no—
123C4×C97—22no—
124C4×C97#12011yescharacteristic
125C4×C97#12011yescharacteristic
126C4×C97C22×D412yes—
127Dic97—42no—
128C2×C97—21no—
129C2×C97—22no—
130C2×C97—42no—
131C2×C97—44no—
132C2×C97C2×#12011yescharacteristic
133D97—82no—
134#3021—971nomaximal, Sylow
135C97#302111yescharacteristic, complemented, Sylow
136#585D9711yescharacteristic, complemented
137C2×AGL(1,ℤ/8)—972no—
138#535—971no—
139#454—974no—
140#453—974no—
141#421—972no—
142#413—971no—
1432−1+4—22no—
1442−1+4C2×D9712yescomplemented
145H(5,𝔽2)—22no—
146H(5,𝔽2)—22no—
147H(5,𝔽2)C2×D9712yescomplemented
148C2×#41—24no—
149C2×#41—22no—
150C2×#41C2×D9712yes—
151C2×#41C2×D9711yescharacteristic
152C22×D4—971no—
153AGL(1,ℤ/8)—978no—
154AGL(1,ℤ/8)—1944no—
155C2×SD16—1942no—
156C2×D8—1942no—
157C2×M16—972no—
158#127—972no—
159#124—972no—
160#121—1942no—
161#120—1942no—
162#117—971no—
163#104—978no—
164#101—972no—
165#100—972no—
166C24—1941no—
167#41—24no—
168#41—28no—
169#41—24no—
170#41—24no—
171#41—24no—
172#41—22no—
173#41—22no—
174#41—24no—
175#41—22no—
176#41—22no—
177#41C22⋊D9714yescomplemented
178#41C22×D9714yes—
179C2×Q8—22no—
180C2×Q8—24no—
181C2×Q8C22⋊D9712yescomplemented
182C2×Q8C22×D9711yescharacteristic
183C2×D4—28no—
184C2×D4—22no—
185C2×D4—24no—
186C2×D4—22no—
187C2×D4—24no—
188C2×D4—972no—
189C2×D4—1944no—
190C2×D4—1941no—
191C2×D4—1942no—
192C2×D4—1942no—
193C2×D4C22⋊D9712yes—
194C2×D4C22⋊D9712yescomplemented
195C2×D4C22×D9711yescharacteristic
196C22×C4—24no—
197C22×C4—22no—
198C22×C4C22⋊D9712yes—
199C22×C4C22×D9711yescharacteristic
200SD16—1948no—
201D8—1948no—
202M16—974no—
203M16—1942no—
204C2×C8—1942no—
205C4⋊C4—1941no—
206C22⋊C4—1941no—
207C22⋊C4—1944no—
208C42—972no—
209C23—24no—
210C23—22no—
211C23—7762no—
212C23—3882no—
213C23—1942no—
214C23C2×C22⋊D9712yes—
215C23C2×C22⋊D9711yescharacteristic
216Q8—24no—
217Q8—22no—
218Q8—22no—
219Q8C2×C22⋊D9714yes—
220D4—28no—
221D4—24no—
222D4—24no—
223D4—24no—
224D4—24no—
225D4—22no—
226D4—22no—
227D4—7764no—
228D4—3881no—
229D4—3881no—
230D4—1944no—
231D4C2×C22⋊D9714yes—
232C2×C4—24no—
233C2×C4—28no—
234C2×C4—22no—
235C2×C4—24no—
236C2×C4—22no—
237C2×C4—22no—
238C2×C4—24no—
239C2×C4—1942no—
240C2×C4C2×C22⋊D9712yes—
241C2×C4C2×C22⋊D9711yescharacteristic
242C2×C4C2×C22⋊D9712yes—
243C2×C4C2×C22⋊D9712yes—
244C2×C4C23×D9711yescharacteristic
245C8—1944no—
246C22—22no—
247C22—24no—
248C22—41no—
249C22—42no—
250C22—82no—
251C22—84no—
252C22—7764no—
253C22—3884no—
254C22—3882no—
255C22#1569077912yes—
256C22C22×C22⋊D9711yescharacteristic
257C4—24no—
258C4—22no—
259C4—3882no—
260C4#1569077911yescharacteristic
261C4#1569077911yescharacteristic
262C4C22×C22⋊D9712yes—
263C2—21no—
264C2—22no—
265C2—42no—
266C2—44no—
267C2—7762no—
268C2C2×#1569077911yescharacteristic
subgroupG/H
centerC2C2×#15690779
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#585×C97C2
socleC2×C97C2×#120
hypercenter#585D97
hypocenterC97#3021
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. C2×#15690835C2semidirect
2. #20261299C2semidirect
3. #20260908C2semidirect
4. #20260903C2semidirect
5. #20260752C2semidirect
6. #20260719C2semidirect
7. #585×C97C2semidirect
8. C22×D388C22semidirect
10. #15690835C22semidirect
13. C2×#13095495C22semidirect
14. #15690806C22semidirect
15. #15690799C22semidirect
18. #15690769C22semidirect
19. #15690737C22semidirect
20. #15690729C22semidirect
21. #15690727C22semidirect
23. 2−1+4×C97C22semidirect
26. H(5,𝔽2)×C97C22semidirect
56. #41×C97D4semidirect
60. C2×Q8×C97D4semidirect
68. C2×D4×C97D4semidirect
135. C97#3021semidirect
136. #585D97semidirect
144. 2−1+4C2×D97semidirect
147. H(5,𝔽2)C2×D97semidirect
177. #41C22⋊D97semidirect
181. C2×Q8C22⋊D97semidirect
194. C2×D4C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908352
#202612992
#202609082
#202609032
#202607522
#202607192
#585×C972
#302197