Finite Group Databasebeta

← #562×D97 #27155858

Subgroups — #562×D97 #27155858

Order 12416.

1152 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261483C211yesmaximal, characteristic, complemented
2#20261482C214yesmaximal, complemented
3#20261481C214yesmaximal, complemented
4#20261480C214yesmaximal, complemented
5#20261479C212yesmaximal, complemented
6#125×D97C211yesmaximal, characteristic, complemented
7#124×D97C212yesmaximal, complemented
8#123×D97C214yesmaximal, complemented
9C22⋊Q8×D97C214yesmaximal, complemented
10#121×D97C214yesmaximal, complemented
11#562×C97C211yesmaximal, characteristic, complemented
12C2×Q8×D97C2212yes—
13C2×D4×D97C2214yes—
14C2×D4×D97C2212yes—
15C22×C4×D97C2214yes—
16#15690802C2211yescharacteristic, complemented
17#15690801C2212yescomplemented
18#15690800C2214yescomplemented
19#15690799C2212yescomplemented
20#15690798C2212yescomplemented
21#15690797C2212yescomplemented
22#15690796C2214yescomplemented
23#15690796C2214yescomplemented
24#15690795C2214yescomplemented
25#15690795C2214yescomplemented
26#15690794C2214yescomplemented
27#15690793C2214yescomplemented
28#15690792C2214yescomplemented
29#15690792C2214yescomplemented
30#15690791C2214yescomplemented
31#15690790C2214yescomplemented
32#15690789C2214yescomplemented
33#15690788C2214yescomplemented
34#15690788C2214yescomplemented
35#15690787C2214yescomplemented
36#15690787C2214yescomplemented
37#15690786C2214yescomplemented
38#15690785C2214yescomplemented
39#15690784C2214yescomplemented
40#15690783C2214yescomplemented
41#15690783C2214yescomplemented
42#15690782C2214yescomplemented
43#15690782C2214yescomplemented
44#15690781C2214yescomplemented
45C4⋊C4×D97C2214yes—
46C4⋊C4×D97C2214yes—
47C4⋊C4×D97C2212yes—
48C22⋊C4×D97C2214yes—
49C22⋊C4×D97C2214yes—
50C22⋊C4×D97C2214yes—
51C42×D97C2211yescharacteristic
52#125×C97C2211yescharacteristic, complemented
53#124×C97C2212yescomplemented
54#123×C97C2214yescomplemented
55C22⋊Q8×C97C2214yescomplemented
56#121×C97C2214yescomplemented
57C23×D97C2314yes—
58C2×Dic194C2312yes—
59C2×Dic194C2314yes—
60Q8×D97—44no—
61C2×D388C2312yes—
62C2×D388C2314yes—
63C2×C22⋊D97C2314yes—
64C2×C22⋊D97C2314yes—
65C2×C22⋊D97C2314yes—
66D4×D97—48no—
67D4×D97—44no—
68C22×Dic97C2314yes—
69C2×C4×D97—48no—
70C2×C4×D97C2311yescharacteristic
71C2×C4×D97C2312yes—
72C2×C4×D97C2314yes—
73C2×C4×D97C2314yes—
74C2×C4×D97C2314yes—
75C2×C4×D97C2314yes—
76C4⋊Dic97C2314yes—
77C4⋊Dic97C2314yes—
78C4⋊Dic97C2312yes—
79#13095490C2314yes—
80#13095490C2314yes—
81#13095490C2314yes—
82#13095490C2314yes—
83#13095490C2314yes—
84C22⋊Dic97C2314yes—
85C22⋊Dic97C2314yes—
86C22⋊Dic97C2314yes—
87#13095488C2314yes—
88#13095488C2314yes—
89#13095488C2314yes—
90#13095488C2314yes—
91#13095488C2314yes—
92#13095488C2314yes—
93C4×Dic97C2311yescharacteristic
94C4×Dic97C2312yes—
95C2×Q8×C97C2312yes—
96C2×D4×C97C2314yes—
97C2×D4×C97C2312yes—
98C22×C4×C97C2314yes—
99C4⋊C4×C97C2314yes—
100C4⋊C4×C97C2314yes—
101C4⋊C4×C97C2312yes—
102C22⋊C4×C97C2314yes—
103C22⋊C4×C97C2314yes—
104C22⋊C4×C97C2314yes—
105C42×C97C2311yescharacteristic
106C23×C97C2414yes—
107C22×D97—28no—
108C22×D97—28no—
109C22×D97—28no—
110C22×D97C2414yes—
111C22×D97C2411yescharacteristic
112Q8×C97—42no—
113D4×C97—42no—
114D4×C97—44no—
115C2×C4×C97—44no—
116C2×C4×C97C2411yescharacteristic
117C2×C4×C97C2412yes—
118C2×C4×C97C2414yes—
119C2×C4×C97C2414yes—
120C22⋊D97—44no—
121C22⋊D97—44no—
122C22⋊D97—44no—
123C2×Dic97—44no—
124C2×Dic97C2411yescharacteristic
125C2×Dic97C2412yes—
126C2×Dic97C2414yes—
127C2×Dic97C2414yes—
128D388—42no—
129D388—44no—
130C4×D97—28no—
131C4×D97—28no—
132C4×D97—24no—
133C4×D97—22no—
134C4×D97—44no—
135C4×D97—44no—
136Dic194—42no—
137Dic194—44no—
138C22×C97—24no—
139C22×C97—24no—
140C22×C97—24no—
141C22×C97C2511yescharacteristic
142C2×D97—24no—
143C2×D97—24no—
144C2×D97—24no—
145C2×D97—416no—
146C2×D97H(5,𝔽2)14yes—
147C2×D972−1+412yes—
148C4×C97—24no—
149C4×C97—22no—
150C4×C97—24no—
151C4×C97—21no—
152Dic97—24no—
153Dic97—22no—
154Dic97—24no—
155Dic97—21no—
156C2×C97—44no—
157C2×C97C2×H(5,𝔽2)12yes—
158C2×C97C2×2−1+411yescharacteristic
159D97—44no—
160D97#56214yesdirect product factor
161C2×#562—971nomaximal, Sylow
162C97C2×#56211yescharacteristic, complemented, Sylow
163#562—971no—
164#562—972no—
165#562—974no—
166#562—974no—
167#562—974no—
168#562D9711yescharacteristic, direct product factor
169C2×#125—971no—
170C2×#124—972no—
171C2×#123—974no—
172C2×C22⋊Q8—974no—
173C2×#121—974no—
174C22×Q8—972no—
175C22×D4—974no—
176C22×D4—972no—
177C23×C4—974no—
178#125—974no—
179#125—972no—
180#125—971no—
181#125C2×D9711yescharacteristic, complemented
182#124—972no—
183#124—972no—
184#124—974no—
185#124—974no—
186#124—972no—
187#124C2×D9712yescomplemented
188#123—974no—
189#123—974no—
190#123—974no—
191#123—974no—
192#123—974no—
193#123—974no—
194#123—974no—
195#123C2×D9714yescomplemented
196C22⋊Q8—974no—
197C22⋊Q8—974no—
198C22⋊Q8—974no—
199C22⋊Q8—974no—
200C22⋊Q8—974no—
201C22⋊Q8—974no—
202C22⋊Q8—974no—
203C22⋊Q8C2×D9714yescomplemented
204#121—974no—
205#121—974no—
206#121—974no—
207#121—974no—
208#121—974no—
209#121—974no—
210#121—974no—
211#121C2×D9714yescomplemented
212C2×C4⋊C4—974no—
213C2×C4⋊C4—974no—
214C2×C4⋊C4—972no—
215C2×C22⋊C4—974no—
216C2×C22⋊C4—974no—
217C2×C22⋊C4—974no—
218C2×C42—971no—
219C24—974no—
220C2×Q8—974no—
221C2×Q8—972no—
222C2×Q8—3884no—
223C2×Q8C22×D9712yes—
224C2×D4—974no—
225C2×D4—974no—
226C2×D4—974no—
227C2×D4—974no—
228C2×D4—972no—
229C2×D4—3888no—
230C2×D4—3884no—
231C2×D4C22×D9714yes—
232C2×D4C22×D9712yes—
233C22×C4—974no—
234C22×C4—974no—
235C22×C4—974no—
236C22×C4—971no—
237C22×C4—972no—
238C22×C4—974no—
239C22×C4—974no—
240C22×C4—3888no—
241C22×C4C22×D9714yes—
242C4⋊C4—974no—
243C4⋊C4—974no—
244C4⋊C4—974no—
245C4⋊C4—974no—
246C4⋊C4—974no—
247C4⋊C4—974no—
248C4⋊C4—974no—
249C4⋊C4—972no—
250C4⋊C4C22×D9714yes—
251C4⋊C4C22×D9714yes—
252C4⋊C4C22×D9712yes—
253C22⋊C4—974no—
254C22⋊C4—974no—
255C22⋊C4—974no—
256C22⋊C4—974no—
257C22⋊C4—974no—
258C22⋊C4—974no—
259C22⋊C4—974no—
260C22⋊C4—974no—
261C22⋊C4—974no—
262C22⋊C4C22×D9714yes—
263C22⋊C4C22×D9714yes—
264C22⋊C4C22×D9714yes—
265C42—972no—
266C42—971no—
267C42C22×D9711yescharacteristic
268C23—971no—
269C23—974no—
270C23—1948no—
271C23—1948no—
272C23—1948no—
273C23C23×D9714yes—
274Q8—42no—
275Q8—3884no—
276Q8—3882no—
277D4—44no—
278D4—42no—
279D4—3884no—
280D4—3882no—
281D4—3884no—
282D4—3884no—
283D4—3884no—
284C2×C4—44no—
285C2×C4—971no—
286C2×C4—972no—
287C2×C4—974no—
288C2×C4—974no—
289C2×C4—3884no—
290C2×C4—3884no—
291C2×C4—3884no—
292C2×C4—1948no—
293C2×C4—1948no—
294C2×C4—1944no—
295C2×C4—1942no—
296C2×C4C23×D9711yescharacteristic
297C2×C4C23×D9712yes—
298C2×C4C23×D9714yes—
299C2×C4C23×D9714yes—
300C22—24no—
301C22—24no—
302C22—24no—
303C22—974no—
304C22—972no—
305C22—38816no—
306C22—1944no—
307C22—1944no—
308C22—1944no—
309C22C24×D9711yescharacteristic
310C4—24no—
311C4—24no—
312C4—22no—
313C4—21no—
314C4—1944no—
315C4—1944no—
316C4—1942no—
317C4—1941no—
318C2—44no—
319C2—974no—
320C2—3884no—
321C2H(5,𝔽2)×D9712yes—
322C22−1+4×D9711yescharacteristic
subgroupG/H
center, FrattiniC22C24×D97
commutator, socle, cosocleC22×C97C25
Fitting#562×C97C2
hypercenter#562D97
hypocenterC97C2×#562
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ #562

Complements

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

NH ≅ G/Ntype
1. #20261483C2semidirect
2. #20261482C2semidirect
3. #20261481C2semidirect
4. #20261480C2semidirect
5. #20261479C2semidirect
6. #125×D97C2semidirect
7. #124×D97C2semidirect
8. #123×D97C2semidirect
9. C22⋊Q8×D97C2semidirect
10. #121×D97C2semidirect
11. #562×C97C2semidirect
16. #15690802C22semidirect
17. #15690801C22semidirect
18. #15690800C22semidirect
19. #15690799C22semidirect
20. #15690798C22semidirect
21. #15690797C22semidirect
22. #15690796C22semidirect
23. #15690796C22semidirect
24. #15690795C22semidirect
25. #15690795C22semidirect
26. #15690794C22semidirect
27. #15690793C22semidirect
28. #15690792C22semidirect
29. #15690792C22semidirect
30. #15690791C22semidirect
31. #15690790C22semidirect
32. #15690789C22semidirect
33. #15690788C22semidirect
34. #15690788C22semidirect
35. #15690787C22semidirect
36. #15690787C22semidirect
37. #15690786C22semidirect
38. #15690785C22semidirect
39. #15690784C22semidirect
40. #15690783C22semidirect
41. #15690783C22semidirect
42. #15690782C22semidirect
43. #15690782C22semidirect
44. #15690781C22semidirect
52. #125×C97C22semidirect
53. #124×C97C22semidirect
54. #123×C97C22semidirect
55. C22⋊Q8×C97C22semidirect
56. #121×C97C22semidirect
160. D97#562direct (N × H)
162. C97C2×#562semidirect
168. #562D97direct (N × H)
181. #125C2×D97semidirect
187. #124C2×D97semidirect
195. #123C2×D97semidirect
203. C22⋊Q8C2×D97semidirect
211. #121C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202614832
#202614822
#202614812
#202614802
#202614792
#125×D972
#124×D972
#123×D972
C22⋊Q8×D972
#121×D972
#562×C972
C2×#56297