Finite Group Databasebeta

← #27151085

Subgroups — #27151085

Order 12416.

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

normal subgroups only

This group has 724 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
1#20261533C212yesmaximal, complemented
2#20261281C211yesmaximal, characteristic, complemented
3#20260918C212yesmaximal, complemented
4#20260913C212yesmaximal, complemented
5#20260908C212yesmaximal, complemented
6#20260903C212yesmaximal, complemented
7C2×#15690737C212yesmaximal, complemented
8#20260719C211yesmaximal, characteristic, complemented
9#585×C97C211yesmaximal, characteristic, complemented
10C2×#13095511C2211yescharacteristic, complemented
11#15690841—22no—
12#15690840C2214yescomplemented
13#15690836—22no—
14#15690835C2214yescomplemented
15#15690830—24no—
16C2×#13095495C2212yescomplemented
17#15690808C2211yescharacteristic, complemented
18#15690806C2211yescharacteristic, complemented
19#15690799C2211yescharacteristic, complemented
20#15690797C2211yescharacteristic, complemented
21#15690782—22no—
22C4×C22⋊D97—22no—
23C2×C4×Dic97C2211yescharacteristic, complemented
24#15690737C2214yescomplemented
25#15690737C2214yescomplemented
26#15690729C2212yescomplemented
27#15690727C2212yescomplemented
282−1+4×C97—22no—
292−1+4×C97C2212yes—
30H(5,𝔽2)×C97—22no—
31H(5,𝔽2)×C97—22no—
32H(5,𝔽2)×C97C2212yes—
33C2×#41×C97—24no—
34C2×#41×C97—22no—
35C2×#41×C97C2212yes—
36C2×#41×C97C2211yescharacteristic
37#13095511—22no—
38#13095511—22no—
39C2×Dic194C2311yescharacteristic
40C2×D388C2311yescharacteristic
41C2×C22⋊D97—22no—
42C2×C22⋊D97—21no—
43C22×Dic97—21no—
44C2×C4×D97—21no—
45#13095501—24no—
46#13095499—24no—
47#13095498—24no—
48#13095496—24no—
49#13095495—22no—
50#13095495C2314yes—
51C2×#11715895—22no—
52#13095490—22no—
53C22⋊Dic97—22no—
54#13095488—22no—
55C4×Dic97—21no—
56C4×Dic97C2311yescharacteristic
57C4×Dic97C2311yescharacteristic
58#41×C97—24no—
59#41×C97—28no—
60#41×C97—24no—
61#41×C97—24no—
62#41×C97—24no—
63#41×C97—22no—
64#41×C97—22no—
65#41×C97—24no—
66#41×C97—22no—
67#41×C97—22no—
68#41×C97D414yescomplemented
69#41×C97C2314yes—
70C2×Q8×C97—22no—
71C2×Q8×C97—24no—
72C2×Q8×C97D412yescomplemented
73C2×Q8×C97C2311yescharacteristic
74C2×D4×C97—28no—
75C2×D4×C97—22no—
76C2×D4×C97—24no—
77C2×D4×C97—22no—
78C2×D4×C97—24no—
79C2×D4×C97D412yes—
80C2×D4×C97D412yescomplemented
81C2×D4×C97C2311yescharacteristic
82C22×C4×C97—24no—
83C22×C4×C97—22no—
84C22×C4×C97D412yes—
85C22×C4×C97C2311yescharacteristic
86C23×C97—24no—
87C23×C97—22no—
88C23×C97C2×D412yes—
89C23×C97C2×D411yescharacteristic
90C22×D97—21no—
91Q8×C97—24no—
92Q8×C97—22no—
93Q8×C97—22no—
94Q8×C97C2×D414yes—
95D4×C97—28no—
96D4×C97—24no—
97D4×C97—24no—
98D4×C97—24no—
99D4×C97—24no—
100D4×C97—22no—
101D4×C97—22no—
102D4×C97C2×D414yes—
103C2×C4×C97—24no—
104C2×C4×C97—28no—
105C2×C4×C97—22no—
106C2×C4×C97—24no—
107C2×C4×C97—22no—
108C2×C4×C97—22no—
109C2×C4×C97—24no—
110C2×C4×C97C2×D412yes—
111C2×C4×C97C2×D411yescharacteristic
112C2×C4×C97C2×D412yes—
113C2×C4×C97C2×D412yes—
114C2×C4×C97C2411yescharacteristic
115C22⋊D97—44no—
116C22⋊D97—42no—
117C2×Dic97—21no—
118C2×Dic97—21no—
119C2×Dic97—21no—
120C2×Dic97—42no—
121D388—22no—
122C4×D97—41no—
123C4×D97—41no—
124Dic194—22no—
125#11715895—24no—
126C22×C97—22no—
127C22×C97—24no—
128C22×C97—41no—
129C22×C97—42no—
130C22×C97—82no—
131C22×C97—84no—
132C22×C97#12012yes—
133C22×C97C22×D411yescharacteristic
134C2×D97—42no—
135C2×D97—41no—
136C4×C97—24no—
137C4×C97—22no—
138C4×C97#12011yescharacteristic
139C4×C97#12011yescharacteristic
140C4×C97C22×D412yes—
141Dic97—22no—
142Dic97—41no—
143Dic97—41no—
144C2×C97—21no—
145C2×C97—22no—
146C2×C97—42no—
147C2×C97—44no—
148C2×C97C2×#12011yescharacteristic
149D97—81no—
150#3018—971nomaximal, Sylow
151C97#301811yescharacteristic, complemented, Sylow
152#585D9711yescharacteristic, complemented
153#575—972no—
154#532—971no—
155#456—972no—
156#455—972no—
157#454—972no—
158#453—972no—
159C2×#104—972no—
160#413—971no—
1612−1+4—22no—
1622−1+4C2×D9712yes—
163H(5,𝔽2)—22no—
164H(5,𝔽2)—22no—
165H(5,𝔽2)C2×D9712yes—
166C2×#41—24no—
167C2×#41—22no—
168C2×#41—971no—
169C2×#41C2×D9712yes—
170C2×#41C2×D9711yescharacteristic
171#137—974no—
172#137—1942no—
173AGL(1,ℤ/8)—974no—
174AGL(1,ℤ/8)—1942no—
175#135—1944no—
176C2×M16—972no—
177C4⋊Q8—971no—
178#127—971no—
179#124—971no—
180#124—971no—
181#121—1942no—
182C4×D4—1942no—
183C2×C42—971no—
184#104—974no—
185#104—974no—
186#101—972no—
187#100—972no—
188#41—24no—
189#41—28no—
190#41—24no—
191#41—24no—
192#41—24no—
193#41—22no—
194#41—22no—
195#41—24no—
196#41—22no—
197#41—22no—
198#41—1942no—
199#41—1942no—
200#41C22⋊D9714yescomplemented
201#41C22×D9714yes—
202C2×Q8—22no—
203C2×Q8—24no—
204C2×Q8—971no—
205C2×Q8C22⋊D9712yescomplemented
206C2×Q8C22×D9711yescharacteristic
207C2×D4—28no—
208C2×D4—22no—
209C2×D4—24no—
210C2×D4—22no—
211C2×D4—24no—
212C2×D4—971no—
213C2×D4—1942no—
214C2×D4—1941no—
215C2×D4C22⋊D9712yes—
216C2×D4C22⋊D9712yescomplemented
217C2×D4C22×D9711yescharacteristic
218C22×C4—24no—
219C22×C4—22no—
220C22×C4—1941no—
221C22×C4—1941no—
222C22×C4C22⋊D9712yes—
223C22×C4C22×D9711yescharacteristic
224Q16—1944no—
225SD16—1944no—
226SD16—1944no—
227D8—1944no—
228M16—974no—
229M16—1942no—
230C2×C8—1942no—
231C4⋊C4—1942no—
232C22⋊C4—1942no—
233C22⋊C4—1942no—
234C42—971no—
235C42—971no—
236C42—1941no—
237C23—24no—
238C23—22no—
239C23—1941no—
240C23C2×C22⋊D9712yes—
241C23C2×C22⋊D9711yescharacteristic
242Q8—24no—
243Q8—22no—
244Q8—22no—
245Q8—1942no—
246Q8C2×C22⋊D9714yes—
247D4—28no—
248D4—24no—
249D4—24no—
250D4—24no—
251D4—24no—
252D4—22no—
253D4—22no—
254D4—3882no—
255D4—3884no—
256D4—1942no—
257D4C2×C22⋊D9714yes—
258C2×C4—24no—
259C2×C4—28no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—24no—
263C2×C4—22no—
264C2×C4—24no—
265C2×C4—3882no—
266C2×C4—3881no—
267C2×C4—3881no—
268C2×C4—1941no—
269C2×C4—1941no—
270C2×C4—1941no—
271C2×C4C2×C22⋊D9712yes—
272C2×C4C2×C22⋊D9711yescharacteristic
273C2×C4C2×C22⋊D9712yes—
274C2×C4C2×C22⋊D9712yes—
275C2×C4C23×D9711yescharacteristic
276C8—1944no—
277C22—22no—
278C22—24no—
279C22—41no—
280C22—42no—
281C22—82no—
282C22—84no—
283C22—3882no—
284C22—3881no—
285C22#1569077912yes—
286C22C22×C22⋊D9711yescharacteristic
287C4—24no—
288C4—22no—
289C4—3881no—
290C4—3881no—
291C4—1942no—
292C4#1569077911yescharacteristic
293C4#1569077911yescharacteristic
294C4C22×C22⋊D9712yes—
295C2—21no—
296C2—22no—
297C2—42no—
298C2—44no—
299C2—7761no—
300C2C2×#1569077911yescharacteristic
subgroupG/H
centerC4#15690779
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#585×C97C2
socleC2×C97C2×#120
hypercenter#585D97
hypocenterC97#3018
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #20261533C2semidirect
2. #20261281C2semidirect
3. #20260918C2semidirect
4. #20260913C2semidirect
5. #20260908C2semidirect
6. #20260903C2semidirect
7. C2×#15690737C2semidirect
8. #20260719C2semidirect
9. #585×C97C2semidirect
10. C2×#13095511C22semidirect
12. #15690840C22semidirect
14. #15690835C22semidirect
16. C2×#13095495C22semidirect
17. #15690808C22semidirect
18. #15690806C22semidirect
19. #15690799C22semidirect
20. #15690797C22semidirect
23. C2×C4×Dic97C22semidirect
24. #15690737C22semidirect
25. #15690737C22semidirect
26. #15690729C22semidirect
27. #15690727C22semidirect
68. #41×C97D4semidirect
72. C2×Q8×C97D4semidirect
80. C2×D4×C97D4semidirect
151. C97#3018semidirect
152. #585D97semidirect
200. #41C22⋊D97semidirect
205. C2×Q8C22⋊D97semidirect
216. C2×D4C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
#202615332
#202612812
#202609182
#202609132
#202609082
#202609032
C2×#156907372
#202607192
#585×C972
#301897