Finite Group Databasebeta

← #541×D97 #27155098

Subgroups — #541×D97 #27155098

Order 12416.

1004 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261348C211yesmaximal, characteristic, complemented
2#20261347C211yesmaximal, characteristic, complemented
3#20261346C211yesmaximal, characteristic, complemented
4#20261345C212yesmaximal, complemented
5#20261344C214yesmaximal, complemented
6#20261343C212yesmaximal, complemented
7#20261342C214yesmaximal, complemented
8C4⋊Q8×D97C211yesmaximal, characteristic, complemented
9#126×D97C214yesmaximal
10#125×D97C211yesmaximal, characteristic, complemented
11#125×D97C214yesmaximal, complemented
12C22⋊Q8×D97C212yesmaximal
13C4×Q8×D97C212yesmaximal, complemented
14#117×D97C211yesmaximal, characteristic
15#541×C97C211yesmaximal, characteristic, complemented
16C2×Q8×D97C2212yes—
17C22×C4×D97C2211yescharacteristic
18#15690810C2211yescharacteristic, complemented
19#15690809C2214yescomplemented
20#15690808C2212yescomplemented
21#15690805C2214yes—
22#15690805C2214yes—
23#15690805C2214yes—
24#15690804C2214yes—
25#15690803C2214yes—
26#15690803C2214yes—
27#15690803C2214yes—
28#15690802C2214yescomplemented
29#15690802C2211yescharacteristic, complemented
30#15690801C2214yescomplemented
31#15690801C2212yescomplemented
32#15690801C2214yescomplemented
33#15690800C2214yescomplemented
34#15690800C2214yescomplemented
35#15690800C2214yescomplemented
36#15690800C2214yescomplemented
37#15690800C2214yescomplemented
38#15690790C2212yes—
39#15690789C2212yes—
40#15690788C2214yes—
41#15690787C2214yes—
42#15690786C2212yes—
43Q8×Dic97C2212yescomplemented
44C4×Dic194C2214yescomplemented
45C4×Dic194C2212yescomplemented
46#15690775C2214yescomplemented
47#15690775C2212yescomplemented
48#15690769C2211yescharacteristic
49#15690768C2212yes—
50#15690767C2211yescharacteristic
51#15690767C2211yescharacteristic
52#15690766C2212yes—
53C4⋊C4×D97C2212yes—
54C4⋊C4×D97C2212yes—
55C4⋊C4×D97C2214yes—
56C4⋊C4×D97C2214yes—
57C4⋊C4×D97C2214yes—
58C4⋊C4×D97C2214yes—
59C22⋊C4×D97C2212yes—
60C22⋊C4×D97C2214yes—
61C42×D97C2211yescharacteristic
62C42×D97C2211yescharacteristic
63C42×D97C2214yes—
64C4⋊Q8×C97C2211yescharacteristic, complemented
65#126×C97C2214yes—
66#125×C97C2214yescomplemented
67#125×C97C2211yescharacteristic, complemented
68C22⋊Q8×C97C2212yes—
69C4×Q8×C97C2212yescomplemented
70#117×C97C2211yescharacteristic
71C23×D97C2311yescharacteristic
72C2×Dic194C2314yes—
73C2×Dic194C2312yes—
74Q8×D97—28no—
75C22×Dic97C2311yescharacteristic
76C2×C4×D97—24no—
77C2×C4×D97C2312yes—
78C2×C4×D97C2312yes—
79C2×C4×D97C2314yes—
80C2×C4×D97C2314yes—
81C2×C4×D97C2311yescharacteristic
82C2×C4×D97C2311yescharacteristic
83C2×C4×D97C2311yescharacteristic
84C2×C4×D97C2311yescharacteristic
85C4⋊Dic97C2312yes—
86C4⋊Dic97C2312yes—
87C4⋊Dic97C2314yes—
88C4⋊Dic97C2314yes—
89C4⋊Dic97C2314yes—
90C4⋊Dic97C2314yes—
91#13095490C2312yes—
92#13095490C2312yes—
93#13095490C2314yes—
94#13095490C2314yes—
95#13095490C2314yes—
96#13095490C2314yes—
97#13095490C2314yes—
98#13095490C2314yes—
99#13095490C2314yes—
100#13095490C2314yes—
101#13095490C2314yes—
102C22⋊Dic97C2312yes—
103C22⋊Dic97C2314yes—
104#13095488C2312yes—
105#13095488C2312yes—
106#13095488C2314yes—
107#13095488C2314yes—
108C4×Dic97C2312yes—
109C4×Dic97C2312yes—
110C4×Dic97C2311yescharacteristic
111C4×Dic97C2311yescharacteristic
112C4×Dic97C2314yes—
113C4×Dic97C2314yes—
114C4×Dic97C2314yes—
115C2×Q8×C97C2312yes—
116C22×C4×C97C2311yescharacteristic
117C4⋊C4×C97C2312yes—
118C4⋊C4×C97C2312yes—
119C4⋊C4×C97C2314yes—
120C4⋊C4×C97C2314yes—
121C4⋊C4×C97C2314yes—
122C4⋊C4×C97C2314yes—
123C22⋊C4×C97C2312yes—
124C22⋊C4×C97C2314yes—
125C42×C97C2311yescharacteristic
126C42×C97C2311yescharacteristic
127C42×C97C2314yes—
128C23×C97C2411yescharacteristic
129C22×D97—22no—
130C22×D97—24no—
131C22×D97C2411yescharacteristic
132C22×D97C2411yescharacteristic
133Q8×C97—24no—
134C2×C4×C97—22no—
135C2×C4×C97C2412yes—
136C2×C4×C97C2412yes—
137C2×C4×C97C2414yes—
138C2×C4×C97C2414yes—
139C2×C4×C97C2411yescharacteristic
140C2×C4×C97C2411yescharacteristic
141C2×Dic97—22no—
142C2×Dic97C2412yes—
143C2×Dic97C2412yes—
144C2×Dic97C2414yes—
145C2×Dic97C2414yes—
146C2×Dic97C2411yescharacteristic
147C2×Dic97C2411yescharacteristic
148C4×D97—22no—
149C4×D97—28no—
150C4×D97—24no—
151C4×D97—24no—
152C4×D97—28no—
153C4×D97—22no—
154C4×D97—22no—
155C4×D97#4114yes—
156Dic194—24no—
157Dic194—28no—
158C22×C97—21no—
159C22×C97—22no—
160C22×C97C2511yescharacteristic
161C2×D97—21no—
162C2×D97—22no—
163C2×D97—44no—
164C2×D97C2×#4112yes—
165C2×D972−1+414yes—
166C4×C97—24no—
167C4×C97—21no—
168C4×C97—22no—
169C4×C97—22no—
170C4×C97—24no—
171C4×C97C2×#4112yes—
172Dic97—24no—
173Dic97—21no—
174Dic97—22no—
175Dic97—22no—
176Dic97—24no—
177Dic97C2×#4112yes—
178C2×C97—41no—
179C2×C97C22×#4111yescharacteristic
180C2×C97C2×2−1+412yes—
181D97—41no—
182D97#54114yesdirect product factor
183C2×#541—971nomaximal, Sylow
184C97C2×#54111yescharacteristic, complemented, Sylow
185#541—971no—
186#541—974no—
187#541—974no—
188#541—971no—
189#541—972no—
190#541—972no—
191#541—971no—
192#541D9711yescharacteristic, direct product factor
193C2×C4⋊Q8—971no—
194C2×#126—974no—
195C2×#125—971no—
196C2×#125—974no—
197C2×C22⋊Q8—972no—
198C2×C4×Q8—972no—
199C2×#117—971no—
200C22×Q8—972no—
201C23×C4—971no—
202C4⋊Q8—972no—
203C4⋊Q8—974no—
204C4⋊Q8—971no—
205C4⋊Q8C2×D9711yescharacteristic, complemented
206#126—974no—
207#126—974no—
208#126—974no—
209#126—974no—
210#126—974no—
211#126—974no—
212#126—974no—
213#126C2×D9714yes—
214#125—974no—
215#125—974no—
216#125—974no—
217#125—974no—
218#125—974no—
219#125—972no—
220#125—974no—
221#125—974no—
222#125—971no—
223#125—974no—
224#125C2×D9711yescharacteristic, complemented
225#125C2×D9714yescomplemented
226C22⋊Q8—972no—
227C22⋊Q8—972no—
228C22⋊Q8—974no—
229C22⋊Q8—972no—
230C22⋊Q8—974no—
231C22⋊Q8C2×D9712yes—
232C4×Q8—972no—
233C4×Q8—974no—
234C4×Q8—972no—
235C4×Q8—974no—
236C4×Q8—972no—
237C4×Q8C2×D9712yescomplemented
238#117—971no—
239#117—972no—
240#117—972no—
241#117—971no—
242#117—971no—
243#117C2×D9711yescharacteristic
244C2×C4⋊C4—972no—
245C2×C4⋊C4—972no—
246C2×C4⋊C4—974no—
247C2×C4⋊C4—974no—
248C2×C4⋊C4—974no—
249C2×C4⋊C4—974no—
250C2×C22⋊C4—972no—
251C2×C22⋊C4—974no—
252C2×C42—971no—
253C2×C42—971no—
254C2×C42—974no—
255C24—971no—
256C2×Q8—974no—
257C2×Q8—972no—
258C2×Q8—1948no—
259C2×Q8C22×D9712yes—
260C22×C4—971no—
261C22×C4—971no—
262C22×C4—972no—
263C22×C4—972no—
264C22×C4—974no—
265C22×C4—974no—
266C22×C4—971no—
267C22×C4—971no—
268C22×C4—971no—
269C22×C4—1944no—
270C22×C4C22×D9711yescharacteristic
271C4⋊C4—972no—
272C4⋊C4—972no—
273C4⋊C4—974no—
274C4⋊C4—974no—
275C4⋊C4—974no—
276C4⋊C4—974no—
277C4⋊C4—974no—
278C4⋊C4—974no—
279C4⋊C4—974no—
280C4⋊C4—974no—
281C4⋊C4—974no—
282C4⋊C4—972no—
283C4⋊C4—972no—
284C4⋊C4—974no—
285C4⋊C4—974no—
286C4⋊C4—974no—
287C4⋊C4—974no—
288C4⋊C4C22×D9712yes—
289C4⋊C4C22×D9712yes—
290C4⋊C4C22×D9714yes—
291C4⋊C4C22×D9714yes—
292C4⋊C4C22×D9714yes—
293C4⋊C4C22×D9714yes—
294C22⋊C4—972no—
295C22⋊C4—974no—
296C22⋊C4—972no—
297C22⋊C4—972no—
298C22⋊C4—974no—
299C22⋊C4—974no—
300C22⋊C4C22×D9712yes—
301C22⋊C4C22×D9714yes—
302C42—972no—
303C42—971no—
304C42—972no—
305C42—971no—
306C42—974no—
307C42—974no—
308C42—974no—
309C42C22×D9711yescharacteristic
310C42C22×D9711yescharacteristic
311C42C22×D9714yes—
312C23—971no—
313C23—971no—
314C23—1942no—
315C23—1944no—
316C23C23×D9711yescharacteristic
317Q8—24no—
318Q8—1948no—
319Q8—1944no—
320C2×C4—22no—
321C2×C4—971no—
322C2×C4—974no—
323C2×C4—972no—
324C2×C4—972no—
325C2×C4—974no—
326C2×C4—974no—
327C2×C4—971no—
328C2×C4—1942no—
329C2×C4—1942no—
330C2×C4—1948no—
331C2×C4—1944no—
332C2×C4—1944no—
333C2×C4—1948no—
334C2×C4—1942no—
335C2×C4—1942no—
336C2×C4C23×D9712yes—
337C2×C4C23×D9712yes—
338C2×C4C23×D9714yes—
339C2×C4C23×D9714yes—
340C2×C4C23×D9711yescharacteristic
341C2×C4C23×D9711yescharacteristic
342C22—21no—
343C22—22no—
344C22—972no—
345C22—974no—
346C22—3884no—
347C22—1941no—
348C22—1942no—
349C22C24×D9711yescharacteristic
350C4—21no—
351C4—24no—
352C4—22no—
353C4—22no—
354C4—24no—
355C4—972no—
356C4—1941no—
357C4—1944no—
358C4—1942no—
359C4—1942no—
360C4—1944no—
361C4#41×D9712yes—
362C2—41no—
363C2—974no—
364C2—3881no—
365C2C2×#41×D9711yescharacteristic
366C22−1+4×D9712yes—
subgroupG/H
center, FrattiniC22C24×D97
commutator, socle, cosocleC22×C97C25
Fitting#541×C97C2
prime radicalC23×D97C23
hypercenter#541D97
hypocenterC97C2×#541
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ #541

Complements

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

NH ≅ G/Ntype
1. #20261348C2semidirect
2. #20261347C2semidirect
3. #20261346C2semidirect
4. #20261345C2semidirect
5. #20261344C2semidirect
6. #20261343C2semidirect
7. #20261342C2semidirect
8. C4⋊Q8×D97C2semidirect
10. #125×D97C2semidirect
11. #125×D97C2semidirect
13. C4×Q8×D97C2semidirect
15. #541×C97C2semidirect
18. #15690810C22semidirect
19. #15690809C22semidirect
20. #15690808C22semidirect
28. #15690802C22semidirect
29. #15690802C22semidirect
30. #15690801C22semidirect
31. #15690801C22semidirect
32. #15690801C22semidirect
33. #15690800C22semidirect
34. #15690800C22semidirect
35. #15690800C22semidirect
36. #15690800C22semidirect
37. #15690800C22semidirect
43. Q8×Dic97C22semidirect
44. C4×Dic194C22semidirect
45. C4×Dic194C22semidirect
46. #15690775C22semidirect
47. #15690775C22semidirect
64. C4⋊Q8×C97C22semidirect
66. #125×C97C22semidirect
67. #125×C97C22semidirect
69. C4×Q8×C97C22semidirect
182. D97#541direct (N × H)
184. C97C2×#541semidirect
192. #541D97direct (N × H)
205. C4⋊Q8C2×D97semidirect
224. #125C2×D97semidirect
225. #125C2×D97semidirect
237. C4×Q8C2×D97semidirect

Maximal subgroups

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

subgroupindex
#202613482
#202613472
#202613462
#202613452
#202613442
#202613432
#202613422
C4⋊Q8×D972
#126×D972
#125×D972
C22⋊Q8×D972
C4×Q8×D972
#117×D972
#541×C972
C2×#54197