Finite Group Databasebeta

← #41×C22⋊D97 #27155221

Subgroups — #41×C22⋊D97 #27155221

Order 12416.

1648 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D97C211yesmaximal, characteristic, complemented
2#20261404C211yesmaximal, characteristic, complemented
3Q8×C22⋊D97C211yesmaximal, characteristic, complemented
4#20261396C211yesmaximal, characteristic, complemented
5#20261392C211yesmaximal, characteristic, complemented
6#20261384C213yesmaximal, complemented
7#20261379C213yesmaximal, complemented
8#20261373C213yesmaximal, complemented
9D4×C22⋊D97C213yesmaximal, complemented
10#20261281C213yesmaximal, complemented
11#20261246C213yesmaximal, complemented
12#20261242C213yesmaximal, complemented
13#41×Dic97C211yesmaximal, characteristic, complemented
14C2×C4×C22⋊D97C213yesmaximal, complemented
15C22×#41×C97C211yesmaximal, characteristic, complemented
16C2×#13095512C2211yescharacteristic, complemented
17C2×#13095511C2213yescomplemented
18C2×#13095510C2213yescomplemented
19#41×D97—28no—
20C2×Q8×D97C2211yescharacteristic, complemented
21C22×C22⋊D97C2213yes—
22C2×D4×D97C2213yescomplemented
23C22×C4×D97C2213yescomplemented
24#15690808C2213yescomplemented
25#15690806C2213yescomplemented
26#15690799C2213yescomplemented
27#15690797C2213yescomplemented
28#15690793C2216yescomplemented
29#15690791C2216yescomplemented
30#15690789C2213yescomplemented
31#15690789C2213yescomplemented
32#15690786C2213yescomplemented
33#15690786C2213yescomplemented
34#15690784C2213yescomplemented
35#15690784C2213yescomplemented
36#15690782C2216yescomplemented
37#15690782C2216yescomplemented
38#15690781C2213yescomplemented
39#15690781C2213yescomplemented
40#15690779C2216yescomplemented
41#15690778C2216yescomplemented
42Q8×Dic97C2211yescharacteristic, complemented
43Q8×Dic97C2211yescharacteristic, complemented
44D4×Dic97C2213yescomplemented
45D4×Dic97C2213yescomplemented
46C4×C22⋊D97C2213yes—
47C4×C22⋊D97C2211yescharacteristic
48C4×C22⋊D97C2213yescomplemented
49C4×C22⋊D97C2213yescomplemented
50C4×C22⋊D97C2213yescomplemented
51C4×C22⋊D97C2213yescomplemented
52C4×C22⋊D97C2213yescomplemented
53C4×C22⋊D97C2213yescomplemented
54C4×C22⋊D97C2216yescomplemented
55#15690769C2213yescomplemented
56C2×#13095490C2213yescomplemented
57C2×C22⋊Dic97C2213yescomplemented
58C2×#13095488C2213yescomplemented
59C2×C4×Dic97C2213yescomplemented
60C2×#41×C97—28no—
61C2×#41×C97C2211yescharacteristic
62C2×#41×C97C2213yescomplemented
63C2×#41×C97C2211yescharacteristic, complemented
64C2×#41×C97C2213yescomplemented
65C22×Q8×C97C2211yescharacteristic, complemented
66C22×D4×C97C2213yescomplemented
67C23×C4×C97C2213yescomplemented
68C23×D97C2313yes—
69#13095512—24no—
70#13095511—212no—
71#13095510—212no—
72C2×Dic194C2313yes—
73Q8×D97—24no—
74C2×D388C2313yes—
75C2×C22⋊D97—212no—
76C2×C22⋊D97C2313yes—
77C2×C22⋊D97C2313yes—
78C2×C22⋊D97C2313yes—
79C2×C22⋊D97C2316yes—
80C2×C22⋊D97C2316yes—
81C2×C22⋊D97C2311yescharacteristic
82D4×D97—212no—
83C22×Dic97C2313yes—
84C22×Dic97C2313yes—
85C22×Dic97C2313yes—
86C2×C4×D97—212no—
87C2×C4×D97C2311yescharacteristic
88C2×C4×D97C2313yes—
89C2×C4×D97C2313yes—
90C2×C4×D97C2313yes—
91C4⋊Dic97C2313yes—
92C4⋊Dic97C2313yes—
93#13095490C2311yescharacteristic
94#13095490C2313yes—
95#13095490C2313yes—
96#13095490C2313yes—
97#13095490C2316yes—
98C22⋊Dic97C2311yescharacteristic
99C22⋊Dic97C2313yes—
100C22⋊Dic97C2313yes—
101C22⋊Dic97C2313yes—
102C22⋊Dic97C2316yes—
103C22⋊Dic97C2316yes—
104#13095488C2311yescharacteristic
105#13095488C2313yes—
106#13095488C2313yes—
107#13095488C2313yes—
108#13095488C2316yes—
109C4×Dic97C2311yescharacteristic
110C4×Dic97C2313yes—
111C4×Dic97C2313yes—
112C4×Dic97C2313yes—
113C24×C97C2313yes—
114#41×C97—24no—
115#41×C97—212no—
116#41×C97—212no—
117#41×C97D418yescomplemented
118C2×Q8×C97—24no—
119C2×Q8×C97C2313yes—
120C2×Q8×C97C2311yescharacteristic
121C2×D4×C97—212no—
122C2×D4×C97C2316yes—
123C2×D4×C97C2313yes—
124C2×D4×C97C2313yes—
125C22×C4×C97—212no—
126C22×C4×C97C2313yes—
127C22×C4×C97C2311yescharacteristic
128C22×C4×C97C2313yes—
129C22×C4×C97C2313yes—
130C22×C4×C97C2313yes—
131C22×C4×C97C2313yes—
132C23×C97—26no—
133C23×C97—26no—
134C23×C97—43no—
135C23×C97C2413yes—
136C23×C97C2413yes—
137C23×C97C2411yescharacteristic
138C22×D97—26no—
139C22×D97—26no—
140C22×D97—43no—
141C22×D97C2413yes—
142C22×D97C2411yescharacteristic
143Q8×C97—26no—
144Q8×C97C2×D414yescomplemented
145D4×C97—212no—
146D4×C97—26no—
147D4×C97C2×D4112yescomplemented
148C2×C4×C97—26no—
149C2×C4×C97—22no—
150C2×C4×C97—26no—
151C2×C4×C97—26no—
152C2×C4×C97—26no—
153C2×C4×C97C2×D4112yescomplemented
154C2×C4×C97C2411yescharacteristic
155C2×C4×C97C2413yes—
156C2×C4×C97C2413yes—
157C2×C4×C97C2411yescharacteristic
158C22⋊D97—212no—
159C22⋊D97—26no—
160C22⋊D97—26no—
161C22⋊D97—26no—
162C22⋊D97—46no—
163C22⋊D97#4114yesdirect product factor
164C2×Dic97—26no—
165C2×Dic97—26no—
166C2×Dic97—43no—
167C2×Dic97C2411yescharacteristic
168C2×Dic97C2413yes—
169C2×Dic97C2413yes—
170C2×Dic97C2411yescharacteristic
171C2×Dic97C2413yes—
172C2×Dic97C2411yescharacteristic
173D388—26no—
174C4×D97—22no—
175C4×D97—26no—
176C4×D97—26no—
177C4×D97—26no—
178Dic194—26no—
179C22×C97—21no—
180C22×C97—23no—
181C22×C97—23no—
182C22×C97—21no—
183C22×C97—23no—
184C22×C97—23no—
185C22×C97—23no—
186C22×C97—412no—
187C22×C97C22×D416yes—
188C22×C97C2×#4112yescomplemented
189C22×C97C2511yescharacteristic
190C2×D97—21no—
191C2×D97—21no—
192C2×D97—23no—
193C2×D97—23no—
194C2×D97—23no—
195C2×D97—412no—
196C2×D97C2×#4112yescomplemented
197C4×C97—21no—
198C4×C97—23no—
199C4×C97C22×D416yes—
200C4×C97C22×D412yes—
201Dic97—23no—
202Dic97—21no—
203Dic97—23no—
204Dic97—21no—
205Dic97—23no—
206Dic97C2×#4112yescomplemented
207C2×C97—26no—
208C2×C97—22no—
209C2×C97—43no—
210C2×C97C23×D411yescharacteristic
211C2×C97C22×#4111yescharacteristic
212C2×C97#58511yescharacteristic
213D97—22no—
214D97—43no—
215D4×#41—971nomaximal, Sylow
216C97D4×#4111yescharacteristic, complemented, Sylow
217C22×#41—971no—
218C22×#41D9711yescharacteristic, complemented
219#550—971no—
220D4×Q8—971no—
221#548—971no—
222#548—971no—
223#547—973no—
224#546—973no—
225#546—973no—
226D42—973no—
227#532—973no—
228#525—973no—
229#525—973no—
230C4×#41—971no—
231C2×C4×D4—973no—
232C2×#41—28no—
233C2×#41—971no—
234C2×#41—973no—
235C2×#41—973no—
236C2×#41—1948no—
237C2×#41C2×D9711yescharacteristic
238C2×#41C2×D9713yescomplemented
239C2×#41C2×D9713yescomplemented
240C2×#41C2×D9711yescharacteristic, complemented
241C22×Q8—971no—
242C22×Q8C2×D9711yescharacteristic, complemented
243C22×D4—973no—
244C22×D4—973no—
245C22×D4C2×D9713yescomplemented
246C23×C4—973no—
247C23×C4C2×D9713yescomplemented
248C4⋊Q8—973no—
249#127—973no—
250#124—973no—
251#124—973no—
252#123—976no—
253#123—976no—
254C22⋊Q8—973no—
255C22⋊Q8—973no—
256C22⋊Q8—973no—
257C22⋊Q8—973no—
258#121—973no—
259#121—973no—
260#121—976no—
261#121—976no—
262#121—973no—
263#121—973no—
264#120—976no—
265#120—976no—
266C4×Q8—971no—
267C4×Q8—971no—
268C4×D4—973no—
269C4×D4—973no—
270C4×D4—971no—
271C4×D4—973no—
272C4×D4—973no—
273C4×D4—973no—
274C4×D4—973no—
275C4×D4—973no—
276C4×D4—973no—
277C4×D4—973no—
278C4×D4—976no—
279#117—973no—
280C2×C4⋊C4—973no—
281C2×C22⋊C4—973no—
282C2×C22⋊C4—973no—
283C2×C42—973no—
284C24—973no—
285C24C22×D9713yes—
286#41—24no—
287#41—212no—
288#41—212no—
289#41—1944no—
290#41—19412no—
291#41—19412no—
292#41C22⋊D9718yesdirect product factor
293C2×Q8—24no—
294C2×Q8—973no—
295C2×Q8—1944no—
296C2×Q8C22×D9713yes—
297C2×Q8C22×D9711yescharacteristic
298C2×D4—212no—
299C2×D4—971no—
300C2×D4—976no—
301C2×D4—973no—
302C2×D4—973no—
303C2×D4—973no—
304C2×D4—976no—
305C2×D4—973no—
306C2×D4—19412no—
307C2×D4—19412no—
308C2×D4C22×D9716yes—
309C2×D4C22×D9713yes—
310C2×D4C22×D9713yes—
311C22×C4—212no—
312C22×C4—973no—
313C22×C4—973no—
314C22×C4—973no—
315C22×C4—973no—
316C22×C4—973no—
317C22×C4—971no—
318C22×C4—973no—
319C22×C4—19412no—
320C22×C4C22×D9713yes—
321C22×C4C22×D9713yes—
322C22×C4C22×D9713yes—
323C22×C4C22×D9711yescharacteristic
324C22×C4C22×D9713yes—
325C22×C4C22×D9713yes—
326C4⋊C4—971no—
327C4⋊C4—973no—
328C4⋊C4—973no—
329C4⋊C4—973no—
330C4⋊C4—976no—
331C4⋊C4—973no—
332C4⋊C4—973no—
333C22⋊C4—971no—
334C22⋊C4—973no—
335C22⋊C4—973no—
336C22⋊C4—973no—
337C22⋊C4—976no—
338C22⋊C4—976no—
339C22⋊C4—971no—
340C22⋊C4—973no—
341C22⋊C4—973no—
342C22⋊C4—973no—
343C22⋊C4—976no—
344C42—973no—
345C42—973no—
346C42—971no—
347C42—973no—
348C23—26no—
349C23—26no—
350C23—43no—
351C23—971no—
352C23—973no—
353C23—3883no—
354C23—1946no—
355C23—1946no—
356C23C23×D9713yes—
357C23C23×D9713yes—
358C23C23×D9711yescharacteristic
359Q8—26no—
360Q8—1946no—
361Q8C2×C22⋊D9714yescomplemented
362D4—212no—
363D4—26no—
364D4—974no—
365D4—3886no—
366D4—1946no—
367D4—1946no—
368D4—19412no—
369D4—1946no—
370D4—1946no—
371D4C2×C22⋊D97112yescomplemented
372C2×C4—26no—
373C2×C4—22no—
374C2×C4—26no—
375C2×C4—26no—
376C2×C4—26no—
377C2×C4—971no—
378C2×C4—971no—
379C2×C4—973no—
380C2×C4—973no—
381C2×C4—971no—
382C2×C4—973no—
383C2×C4—3883no—
384C2×C4—1946no—
385C2×C4—1946no—
386C2×C4—1942no—
387C2×C4—1946no—
388C2×C4—1946no—
389C2×C4—1946no—
390C2×C4C2×C22⋊D97112yescomplemented
391C2×C4C23×D9711yescharacteristic
392C2×C4C23×D9713yes—
393C2×C4C23×D9713yes—
394C2×C4C23×D9711yescharacteristic
395C22—21no—
396C22—23no—
397C22—23no—
398C22—21no—
399C22—23no—
400C22—23no—
401C22—23no—
402C22—412no—
403C22—972no—
404C22—38812no—
405C22—1943no—
406C22—1943no—
407C22—1943no—
408C22—1941no—
409C22—1941no—
410C22C22×C22⋊D9716yes—
411C22#41×D9712yescomplemented
412C22C24×D9711yescharacteristic
413C4—21no—
414C4—23no—
415C4—972no—
416C4—1941no—
417C4—1943no—
418C4—1943no—
419C4—1941no—
420C4—1943no—
421C4C22×C22⋊D9712yes—
422C4C22×C22⋊D9716yes—
423C2—22no—
424C2—26no—
425C2—43no—
426C2—3883no—
427C2—1942no—
428C2C23×C22⋊D9711yescharacteristic
429C2C2×#41×D9711yescharacteristic
430C2#2026157111yescharacteristic
subgroupG/H
centerC2×C4C23×D97
commutator, socle, cosocleC22×C97C25
FrattiniC22C24×D97
FittingC22×#41×C97C2
hypercenterC22×#41D97
hypocenterC97D4×#41
LCSG ▷ C22×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. C2×#41×D97C2semidirect
2. #20261404C2semidirect
3. Q8×C22⋊D97C2semidirect
4. #20261396C2semidirect
5. #20261392C2semidirect
6. #20261384C2semidirect
7. #20261379C2semidirect
8. #20261373C2semidirect
9. D4×C22⋊D97C2semidirect
10. #20261281C2semidirect
11. #20261246C2semidirect
12. #20261242C2semidirect
13. #41×Dic97C2semidirect
14. C2×C4×C22⋊D97C2semidirect
15. C22×#41×C97C2semidirect
16. C2×#13095512C22semidirect
17. C2×#13095511C22semidirect
18. C2×#13095510C22semidirect
20. C2×Q8×D97C22semidirect
22. C2×D4×D97C22semidirect
23. C22×C4×D97C22semidirect
24. #15690808C22semidirect
25. #15690806C22semidirect
26. #15690799C22semidirect
27. #15690797C22semidirect
28. #15690793C22semidirect
29. #15690791C22semidirect
30. #15690789C22semidirect
31. #15690789C22semidirect
32. #15690786C22semidirect
33. #15690786C22semidirect
34. #15690784C22semidirect
35. #15690784C22semidirect
36. #15690782C22semidirect
37. #15690782C22semidirect
38. #15690781C22semidirect
39. #15690781C22semidirect
40. #15690779C22semidirect
41. #15690778C22semidirect
42. Q8×Dic97C22semidirect
43. Q8×Dic97C22semidirect
44. D4×Dic97C22semidirect
45. D4×Dic97C22semidirect
48. C4×C22⋊D97C22semidirect
49. C4×C22⋊D97C22semidirect
50. C4×C22⋊D97C22semidirect
51. C4×C22⋊D97C22semidirect
52. C4×C22⋊D97C22semidirect
53. C4×C22⋊D97C22semidirect
54. C4×C22⋊D97C22semidirect
55. #15690769C22semidirect
56. C2×#13095490C22semidirect
57. C2×C22⋊Dic97C22semidirect
58. C2×#13095488C22semidirect
59. C2×C4×Dic97C22semidirect
62. C2×#41×C97C22semidirect
63. C2×#41×C97C22semidirect
64. C2×#41×C97C22semidirect
65. C22×Q8×C97C22semidirect
66. C22×D4×C97C22semidirect
67. C23×C4×C97C22semidirect
117. #41×C97D4semidirect
144. Q8×C97C2×D4semidirect
147. D4×C97C2×D4semidirect
153. C2×C4×C97C2×D4semidirect
163. C22⋊D97#41direct (N × H)
188. C22×C97C2×#41semidirect
196. C2×D97C2×#41semidirect
206. Dic97C2×#41semidirect
216. C97D4×#41semidirect
218. C22×#41D97semidirect
238. C2×#41C2×D97semidirect
239. C2×#41C2×D97semidirect
240. C2×#41C2×D97semidirect
242. C22×Q8C2×D97semidirect
245. C22×D4C2×D97semidirect
247. C23×C4C2×D97semidirect
292. #41C22⋊D97direct (N × H)
361. Q8C2×C22⋊D97semidirect
371. D4C2×C22⋊D97semidirect
390. C2×C4C2×C22⋊D97semidirect
411. C22#41×D97semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D972
#202614042
Q8×C22⋊D972
#202613962
#202613922
#202613842
#202613792
#202613732
D4×C22⋊D972
#202612812
#202612462
#202612422
#41×Dic972
C2×C4×C22⋊D972
C22×#41×C972
D4×#4197