Finite Group Databasebeta

← #38754197

Subgroups — #38754197

Order 25216.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263354C211yesmaximal, characteristic, complemented
2C22×D1576C211yesmaximal, characteristic, complemented
3#27161725C211yesmaximal, characteristic, complemented
4D4×D788C211yesmaximal, characteristic, complemented
5#27161468C211yesmaximal, characteristic, complemented
6#27161332C211yesmaximal, characteristic, complemented
7#27161308C211yesmaximal, characteristic, complemented
8#27161295C211yesmaximal, characteristic, complemented
9#27161275C211yesmaximal, characteristic, complemented
10#27161232C211yesmaximal, characteristic, complemented
11#27161226C211yesmaximal, characteristic, complemented
12#27161215C211yesmaximal, characteristic, complemented
13#27161215C211yesmaximal, characteristic, complemented
14#27161192C211yesmaximal, characteristic, complemented
15#435×C197C211yesmaximal, characteristic, complemented
16C2×#15691358C2211yescharacteristic, complemented
17C22×D788C2211yescharacteristic
18C22×D788C2211yescharacteristic, complemented
19C22×D788C2211yescharacteristic, complemented
20C2×D4×D197—21no—
21#20263354—24no—
22C2×#15691347C2211yescharacteristic, complemented
23C2×#15691347C2211yescharacteristic, complemented
24C2×#15691347C2211yescharacteristic, complemented
25C2×D1576—24no—
26C2×D1576C2211yescharacteristic
27C2×D1576C2211yescharacteristic
28C2×D1576C2211yescharacteristic, complemented
29C2×D1576C2211yescharacteristic, complemented
30C2×D1576C2211yescharacteristic, complemented
31C2×D1576C2211yescharacteristic, complemented
32#20263323C2211yescharacteristic, complemented
33#20263314C2211yescharacteristic, complemented
34#20263311—21no—
35#20263305C2211yescharacteristic, complemented
36#20263301—21no—
37#20263300C2211yescharacteristic
38#20263300C2211yescharacteristic, complemented
39#20263300C2211yescharacteristic, complemented
40#20263299—21no—
41#20263296—21no—
42#20263296—21no—
43#20263294—21no—
44C4×D788C2211yescharacteristic, complemented
45C4×D788C2211yescharacteristic, complemented
46C22⋊C4×D197—21no—
47#20263258C2211yescharacteristic, complemented
48#20263251C2211yescharacteristic, complemented
49#20263248C2211yescharacteristic, complemented
50#20263248C2211yescharacteristic, complemented
51#20263248C2211yescharacteristic, complemented
52#20263248C2211yescharacteristic, complemented
53#20263248C2211yescharacteristic, complemented
54C2×M16×C197C2211yescharacteristic, complemented
55C22×C8×C197C2211yescharacteristic, complemented
56C4×D4×C197C2211yescharacteristic
57C4⋊C8×C197C2211yescharacteristic, complemented
58C22⋊C8×C197C2211yescharacteristic, complemented
59C22⋊C8×C197C2211yescharacteristic, complemented
60#97×C197C2211yescharacteristic, complemented
61C23×D197—21no—
62C23×D197—21no—
63C23×D197—21no—
64#15691358—22no—
65#15691358—41no—
66C2×Dic394C2311yescharacteristic
67C2×D788—22no—
68C2×D788—22no—
69C2×D788—22no—
70C2×D788—21no—
71C2×D788—21no—
72C2×D788—21no—
73C2×D788—21no—
74C2×D788—21no—
75C2×D788—41no—
76C2×D788—41no—
77C2×D788—41no—
78C2×D788C2311yescharacteristic
79C2×D788C2311yescharacteristic
80C2×D788C2311yescharacteristic
81C2×D788C2311yescharacteristic
82C2×D788C2311yescharacteristic
83C2×C22⋊D197—21no—
84C2×C22⋊D197—21no—
85C2×C22⋊D197—21no—
86C2×C22⋊D197—21no—
87D4×D197—44no—
88C2×C4×D197—21no—
89C2×C4×D197—21no—
90C2×C4×D197—21no—
91#15691347—22no—
92#15691347—22no—
93#15691347—41no—
94D1576—22no—
95D1576—22no—
96D1576—22no—
97D1576—22no—
98D1576—41no—
99D1576D414yescomplemented
100C4⋊Dic197C2311yescharacteristic
101C4⋊Dic197C2311yescharacteristic
102#15691337—21no—
103C22⋊Dic197—21no—
104#15691335—21no—
105#15691335—21no—
106#15691335—21no—
107#15691335—21no—
108#15691335—21no—
109#15691335—21no—
110#15691335—21no—
111C2×D4×C197D411yescharacteristic
112C22×C4×C197C2311yescharacteristic
113C22×C4×C197C2311yescharacteristic
114M16×C197—22no—
115C2×C8×C197—22no—
116C2×C8×C197C2311yescharacteristic
117C2×C8×C197C2311yescharacteristic
118C2×C8×C197C2311yescharacteristic
119C2×C8×C197C2311yescharacteristic
120C4⋊C4×C197D411yescharacteristic
121C22⋊C4×C197D411yescharacteristic
122C22⋊C4×C197D411yescharacteristic
123C42×C197C2311yescharacteristic
124C22×D197—21no—
125C22×D197—21no—
126C22×D197—21no—
127C22×D197—21no—
128C22×D197—21no—
129C22×D197—42no—
130C22×D197—42no—
131C22×D197—42no—
132C22×D197—42no—
133C22×D197—42no—
134C22×D197—42no—
135C22×D197—81no—
136C22×D197—81no—
137C22×D197—81no—
138Dic394—21no—
139Dic394—21no—
140D788—21no—
141D788—21no—
142D788—21no—
143D788—21no—
144D788—21no—
145D788—21no—
146D788—21no—
147D788—21no—
148D788—42no—
149D788—42no—
150D788—42no—
151D788—42no—
152D788—81no—
153D788C2×D412yes—
154D788C2×D412yescomplemented
155C22⋊D197—42no—
156C22⋊D197—42no—
157C22⋊D197—42no—
158C22⋊D197—81no—
159C2×Dic197—21no—
160C2×Dic197—21no—
161C2×Dic197—21no—
162C4×D197—42no—
163C4×D197—42no—
164C4×D197—81no—
165C23×C197C2×D411yescharacteristic
166C23×C197C2×D411yescharacteristic
167D4×C197—22no—
168C2×C4×C197—21no—
169C2×C4×C197—21no—
170C2×C4×C197—21no—
171C2×C4×C197—21no—
172C2×C4×C197C2×D411yescharacteristic
173C2×C4×C197C2×D411yescharacteristic
174C2×C4×C197C2×D411yescharacteristic
175C2×C4×C197C2×D411yescharacteristic
176C2×C4×C197C2411yescharacteristic
177C8×C197—21no—
178C8×C197—21no—
179C8×C197—21no—
180C8×C197C2×D412yescomplemented
181C22×C197—21no—
182C22×C197—21no—
183C22×C197—21no—
184C22×C197—21no—
185C22×C197—21no—
186C22×C197AGL(1,ℤ/8)12yescomplemented
187C22×C197C22×D411yescharacteristic
188C2×D197—22no—
189C2×D197—22no—
190C2×D197—41no—
191C2×D197—41no—
192C2×D197—41no—
193C2×D197—41no—
194C2×D197—41no—
195C2×D197—41no—
196C2×D197—41no—
197C2×D197—41no—
198C2×D197—41no—
199C2×D197—41no—
200C2×D197—41no—
201C2×D197—41no—
202C2×D197—41no—
203C2×D197—84no—
204C2×D197—84no—
205C2×D197—84no—
206C4×C197—21no—
207C4×C197—21no—
208C4×C197—21no—
209C4×C197—21no—
210C4×C197C22×D411yescharacteristic
211C4×C197H(5,𝔽2)11yescharacteristic
212Dic197—41no—
213Dic197—41no—
214Dic197—41no—
215C2×C197—22no—
216C2×C197—41no—
217C2×C197D4211yescharacteristic
218C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C197#57611yescharacteristic
220D197—42no—
221D197—42no—
222D197—81no—
223D197—81no—
224D197—81no—
225C197#326411yescharacteristic, complemented, Sylow
226#3264—1971nomaximal, Sylow
227C2×AGL(1,ℤ/8)—1971no—
228C22×D8—1971no—
229#546—1971no—
230D42—1971no—
231#496—1971no—
232#469—1971no—
233#465—1971no—
234#463—1971no—
235#459—1971no—
236#450—1971no—
237#449—1971no—
238#447—1971no—
239#447—1971no—
240#442—1971no—
241#435D19711yescharacteristic, complemented
242C2×#41—1971no—
243C22×D4—3941no—
244C22×D4—1971no—
245C22×D4—1971no—
246C22×D4—1971no—
247AGL(1,ℤ/8)—3944no—
248C2×SD16—1971no—
249C2×SD16—1971no—
250C2×SD16—1971no—
251C2×D8—3944no—
252C2×D8—1971no—
253C2×D8—1971no—
254C2×D8—1971no—
255C2×D8—1971no—
256C2×D8—1971no—
257C2×D8—1971no—
258C2×M16C2×D19711yescharacteristic, complemented
259C22×C8C2×D19711yescharacteristic, complemented
260#127—1971no—
261#124—1971no—
262#123—3941no—
263C22⋊Q8—1971no—
264#121—3941no—
265#121—3941no—
266#121—1971no—
267#121—1971no—
268#121—1971no—
269#120—3941no—
270#120—3941no—
271#120—3941no—
272C4×D4—1971no—
273C4×D4—1971no—
274C4×D4C2×D19711yescharacteristic
275C2×C22⋊C4—3941no—
276#106—1971no—
277C4⋊C8C2×D19711yescharacteristic, complemented
278#103—1971no—
279#102—1971no—
280#102—1971no—
281#102—1971no—
282#102—1971no—
283#102—1971no—
284C22⋊C8C2×D19711yescharacteristic, complemented
285C22⋊C8C2×D19711yescharacteristic, complemented
286#97C2×D19711yescharacteristic, complemented
287C24—3941no—
288C24—3941no—
289C24—3941no—
290#41—7881no—
291#41—3942no—
292C2×Q8—1971no—
293C2×D4—7881no—
294C2×D4—7881no—
295C2×D4—7881no—
296C2×D4—7884no—
297C2×D4—3942no—
298C2×D4—3942no—
299C2×D4—3942no—
300C2×D4—3941no—
301C2×D4—3941no—
302C2×D4—3941no—
303C2×D4—3941no—
304C2×D4—3941no—
305C2×D4—3941no—
306C2×D4—3941no—
307C2×D4—3941no—
308C2×D4—3941no—
309C2×D4—1971no—
310C2×D4—1971no—
311C2×D4—1971no—
312C2×D4—1971no—
313C2×D4—1971no—
314C2×D4D78811yescharacteristic
315C22×C4—3941no—
316C22×C4—3941no—
317C22×C4—3941no—
318C22×C4C22×D19711yescharacteristic
319C22×C4C22×D19711yescharacteristic
320SD16—7881no—
321SD16—3942no—
322SD16—3942no—
323D8—7881no—
324D8—3942no—
325D8—3942no—
326D8—3942no—
327D8—3942no—
328D8—1974no—
329M16—22no—
330C2×C8—22no—
331C2×C8C22×D19711yescharacteristic
332C2×C8C22×D19711yescharacteristic
333C2×C8C22×D19711yescharacteristic
334C2×C8C22×D19711yescharacteristic
335C4⋊C4—3941no—
336C4⋊C4—1971no—
337C4⋊C4—1971no—
338C4⋊C4D78811yescharacteristic
339C22⋊C4—3941no—
340C22⋊C4—3941no—
341C22⋊C4—3941no—
342C22⋊C4—3941no—
343C22⋊C4—3941no—
344C22⋊C4—3941no—
345C22⋊C4—3941no—
346C22⋊C4—3941no—
347C22⋊C4D78811yescharacteristic
348C22⋊C4D78811yescharacteristic
349C42C22×D19711yescharacteristic
350C23—7882no—
351C23—7882no—
352C23—7882no—
353C23—7882no—
354C23—7882no—
355C23—7882no—
356C23—15761no—
357C23—15761no—
358C23—15761no—
359C23—3941no—
360C23—3941no—
361C23—3941no—
362C23—3941no—
363C23—3941no—
364C23C2×D78811yescharacteristic
365C23C2×D78811yescharacteristic
366Q8—3941no—
367Q8—3941no—
368D4—22no—
369D4—7882no—
370D4—7882no—
371D4—7882no—
372D4—7882no—
373D4—7882no—
374D4—7882no—
375D4—7882no—
376D4—15761no—
377D4—15761no—
378D4—3941no—
379D4—3941no—
380D4—3941no—
381D4—3941no—
382D4—3941no—
383D4—3941no—
384D4—3941no—
385D4—3941no—
386D4—1972no—
387D4—1972no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—7882no—
393C2×C4—7882no—
394C2×C4—15761no—
395C2×C4—3941no—
396C2×C4—3941no—
397C2×C4—3941no—
398C2×C4C2×D78811yescharacteristic
399C2×C4C2×D78811yescharacteristic
400C2×C4C2×D78811yescharacteristic
401C2×C4C2×D78811yescharacteristic
402C2×C4C23×D19711yescharacteristic
403C8—21no—
404C8—21no—
405C8—21no—
406C8D4×D19712yescomplemented
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—7881no—
413C22—7881no—
414C22—7881no—
415C22—7881no—
416C22—7881no—
417C22—7881no—
418C22—7881no—
419C22—7881no—
420C22—7881no—
421C22—7881no—
422C22—7881no—
423C22—7881no—
424C22—7881no—
425C22—15764no—
426C22—15764no—
427C22—15764no—
428C22—3942no—
429C22—3942no—
430C22#2026335412yescomplemented
431C22C22×D78811yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—7881no—
437C4—7881no—
438C4—7881no—
439C4C2×D4×D19711yescharacteristic
440C4#2026337211yescharacteristic
441C2—22no—
442C2—41no—
443C2—7882no—
444C2—7882no—
445C2—15761no—
446C2—15761no—
447C2—15761no—
448C2D4×D78811yescharacteristic
449C2C2×#2026335411yescharacteristic
450C2#2716188411yescharacteristic
subgroupG/H
centerC22C22×D788
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#435×C197C2
socleC22×C197C22×D4
hypercenter#435D197
hypocenterC197#3264
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#20263354C2semidirect
2. C22×D1576C2semidirect
3. #27161725C2semidirect
4. D4×D788C2semidirect
5. #27161468C2semidirect
6. #27161332C2semidirect
7. #27161308C2semidirect
8. #27161295C2semidirect
9. #27161275C2semidirect
10. #27161232C2semidirect
11. #27161226C2semidirect
12. #27161215C2semidirect
13. #27161215C2semidirect
14. #27161192C2semidirect
15. #435×C197C2semidirect
16. C2×#15691358C22semidirect
18. C22×D788C22semidirect
19. C22×D788C22semidirect
22. C2×#15691347C22semidirect
23. C2×#15691347C22semidirect
24. C2×#15691347C22semidirect
28. C2×D1576C22semidirect
29. C2×D1576C22semidirect
30. C2×D1576C22semidirect
31. C2×D1576C22semidirect
32. #20263323C22semidirect
33. #20263314C22semidirect
35. #20263305C22semidirect
38. #20263300C22semidirect
39. #20263300C22semidirect
44. C4×D788C22semidirect
45. C4×D788C22semidirect
47. #20263258C22semidirect
48. #20263251C22semidirect
49. #20263248C22semidirect
50. #20263248C22semidirect
51. #20263248C22semidirect
52. #20263248C22semidirect
53. #20263248C22semidirect
54. C2×M16×C197C22semidirect
55. C22×C8×C197C22semidirect
57. C4⋊C8×C197C22semidirect
58. C22⋊C8×C197C22semidirect
59. C22⋊C8×C197C22semidirect
60. #97×C197C22semidirect
99. D1576D4semidirect
154. D788C2×D4semidirect
180. C8×C197C2×D4semidirect
186. C22×C197AGL(1,ℤ/8)semidirect
225. C197#3264semidirect
241. #435D197semidirect
258. C2×M16C2×D197semidirect
259. C22×C8C2×D197semidirect
277. C4⋊C8C2×D197semidirect
284. C22⋊C8C2×D197semidirect
285. C22⋊C8C2×D197semidirect
286. #97C2×D197semidirect
406. C8D4×D197semidirect
430. C22#20263354semidirect

Maximal subgroups

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

subgroupindex
C2×#202633542
C22×D15762
#271617252
D4×D7882
#271614682
#271613322
#271613082
#271612952
#271612752
#271612322
#271612262
#271612152
#271611922
#435×C1972
#3264197