Finite Group Databasebeta

← #38754212

Subgroups — #38754212

Order 25216.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263354C211yesmaximal, characteristic, complemented
2C22×#15691347C211yesmaximal, characteristic, complemented
3#27161738C211yesmaximal, characteristic, complemented
4D4×D788C211yesmaximal, characteristic, complemented
5#27161468C211yesmaximal, characteristic, complemented
6#27161325C211yesmaximal, characteristic, complemented
7#27161315C211yesmaximal, characteristic, complemented
8#27161295C211yesmaximal, characteristic, complemented
9#27161280C211yesmaximal, characteristic, complemented
10#27161232C211yesmaximal, characteristic, complemented
11#27161226C211yesmaximal, characteristic, complemented
12#27161220C211yesmaximal, characteristic, complemented
13#27161215C211yesmaximal, characteristic, complemented
14#27161181C211yesmaximal, characteristic, complemented
15#435×C197C211yesmaximal, characteristic, complemented
16C2×#15691358C2211yescharacteristic, complemented
17C22×Dic394C2211yescharacteristic, complemented
18C22×D788C2211yescharacteristic
19C22×D788C2211yescharacteristic
20C2×D4×D197—21no—
21#20263354—24no—
22C2×#15691347—24no—
23C2×#15691347C2211yescharacteristic
24C2×#15691347C2211yescharacteristic
25C2×#15691347C2211yescharacteristic, complemented
26C2×#15691347C2211yescharacteristic, complemented
27C2×#15691347C2211yescharacteristic, complemented
28C2×#15691347C2211yescharacteristic, complemented
29C2×D1576C2211yescharacteristic
30C2×D1576C2211yescharacteristic, complemented
31C2×D1576C2211yescharacteristic, complemented
32#20263323C2211yescharacteristic
33#20263314C2211yescharacteristic, complemented
34#20263312—21no—
35#20263305C2211yescharacteristic, complemented
36#20263304—21no—
37#20263301—21no—
38#20263300C2211yescharacteristic
39#20263300C2211yescharacteristic
40#20263300C2211yescharacteristic, complemented
41#20263298—21no—
42#20263296—21no—
43#20263296—21no—
44C4×Dic394C2211yescharacteristic, complemented
45C4×D788C2211yescharacteristic
46#20263286—21no—
47#20263260C2211yescharacteristic, complemented
48#20263251C2211yescharacteristic, complemented
49#20263251C2211yescharacteristic, complemented
50#20263248C2211yescharacteristic
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—
63#15691358—22no—
64#15691358—41no—
65C2×Dic394—22no—
66C2×Dic394—21no—
67C2×Dic394—41no—
68C2×Dic394C2311yescharacteristic
69C2×Dic394C2311yescharacteristic
70C2×D788—22no—
71C2×D788—22no—
72C2×D788—21no—
73C2×D788—21no—
74C2×D788—21no—
75C2×D788—21no—
76C2×D788—41no—
77C2×D788—41no—
78C2×D788C2311yescharacteristic
79C2×D788C2311yescharacteristic
80C2×D788C2311yescharacteristic
81C2×D788C2311yescharacteristic
82C2×C22⋊D197—21no—
83C2×C22⋊D197—21no—
84C2×C22⋊D197—21no—
85C2×C22⋊D197—21no—
86D4×D197—44no—
87C22×Dic197—21no—
88C2×C4×D197—21no—
89C2×C4×D197—21no—
90#15691347—22no—
91#15691347—22no—
92#15691347—22no—
93#15691347—22no—
94#15691347—41no—
95#15691347D414yescomplemented
96D1576—22no—
97D1576—22no—
98D1576—41no—
99C4⋊Dic197C2311yescharacteristic
100C4⋊Dic197C2311yescharacteristic
101#15691337—21no—
102#15691337—21no—
103C22⋊Dic197—21no—
104#15691335—21no—
105#15691335—21no—
106#15691335—21no—
107#15691335—21no—
108#15691335—21no—
109#15691335—21no—
110C4×Dic197—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—42no—
129C22×D197—42no—
130C22×D197—42no—
131C22×D197—42no—
132C22×D197—81no—
133C22×D197—81no—
134Dic394—21no—
135Dic394—21no—
136Dic394—21no—
137Dic394—42no—
138Dic394C2×D412yescomplemented
139D788—21no—
140D788—21no—
141D788—21no—
142D788—21no—
143D788—21no—
144D788—21no—
145D788—21no—
146D788—42no—
147D788—42no—
148D788—42no—
149D788—81no—
150D788C2×D412yes—
151C22⋊D197—42no—
152C22⋊D197—42no—
153C22⋊D197—42no—
154C22⋊D197—42no—
155C2×Dic197—21no—
156C2×Dic197—21no—
157C2×Dic197—21no—
158C2×Dic197—21no—
159C2×Dic197—42no—
160C4×D197—42no—
161C4×D197—42no—
162C23×C197C2×D411yescharacteristic
163C23×C197C2×D411yescharacteristic
164D4×C197—22no—
165C2×C4×C197—21no—
166C2×C4×C197—21no—
167C2×C4×C197—21no—
168C2×C4×C197—21no—
169C2×C4×C197C2×D411yescharacteristic
170C2×C4×C197C2×D411yescharacteristic
171C2×C4×C197C2×D411yescharacteristic
172C2×C4×C197C2×D411yescharacteristic
173C2×C4×C197C2411yescharacteristic
174C8×C197—21no—
175C8×C197—21no—
176C8×C197—21no—
177C8×C197C2×D412yescomplemented
178C22×C197—21no—
179C22×C197—21no—
180C22×C197—21no—
181C22×C197—21no—
182C22×C197—21no—
183C22×C197AGL(1,ℤ/8)12yescomplemented
184C22×C197C22×D411yescharacteristic
185C2×D197—22no—
186C2×D197—41no—
187C2×D197—41no—
188C2×D197—41no—
189C2×D197—41no—
190C2×D197—41no—
191C2×D197—41no—
192C2×D197—41no—
193C2×D197—41no—
194C2×D197—41no—
195C2×D197—41no—
196C2×D197—41no—
197C2×D197—84no—
198C2×D197—84no—
199C4×C197—21no—
200C4×C197—21no—
201C4×C197—21no—
202C4×C197—21no—
203C4×C197C22×D411yescharacteristic
204C4×C197H(5,𝔽2)11yescharacteristic
205Dic197—22no—
206Dic197—41no—
207Dic197—41no—
208Dic197—41no—
209C2×C197—22no—
210C2×C197—41no—
211C2×C197D4211yescharacteristic
212C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C197#57711yescharacteristic
214D197—42no—
215D197—81no—
216D197—81no—
217D197—81no—
218C197#326511yescharacteristic, complemented, Sylow
219#3265—1971nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1971no—
221C22×SD16—1971no—
222#548—1971no—
223D42—1971no—
224#496—1971no—
225#468—1971no—
226#466—1971no—
227#463—1971no—
228#460—1971no—
229#450—1971no—
230#449—1971no—
231#448—1971no—
232#447—1971no—
233#440—1971no—
234#435D19711yescharacteristic, complemented
235C2×#41—1971no—
236C22×Q8—1971no—
237C22×D4—3941no—
238C22×D4—1971no—
239C22×D4—1971no—
240AGL(1,ℤ/8)—3944no—
241C2×SD16—3944no—
242C2×SD16—1971no—
243C2×SD16—1971no—
244C2×SD16—1971no—
245C2×SD16—1971no—
246C2×SD16—1971no—
247C2×SD16—1971no—
248C2×D8—1971no—
249C2×D8—1971no—
250C2×D8—1971no—
251C2×M16C2×D19711yescharacteristic, complemented
252C22×C8C2×D19711yescharacteristic, complemented
253#127—1971no—
254#124—3941no—
255#124—1971no—
256C22⋊Q8—3941no—
257C22⋊Q8—1971no—
258#121—3941no—
259#121—3941no—
260#121—1971no—
261#121—1971no—
262#121—1971no—
263#120—3941no—
264#120—3941no—
265C4×Q8—1971no—
266C4×D4—3941no—
267C4×D4—1971no—
268C4×D4C2×D19711yescharacteristic
269#107—1971no—
270C4⋊C8C2×D19711yescharacteristic, complemented
271#103—1971no—
272#103—1971no—
273#102—1971no—
274#102—1971no—
275#102—1971no—
276#102—1971no—
277C22⋊C8C2×D19711yescharacteristic, complemented
278C22⋊C8C2×D19711yescharacteristic, complemented
279#97C2×D19711yescharacteristic, complemented
280C24—3941no—
281C24—3941no—
282#41—7881no—
283#41—3942no—
284C2×Q8—7881no—
285C2×Q8—3942no—
286C2×Q8—3941no—
287C2×Q8—1971no—
288C2×Q8—1971no—
289C2×D4—7881no—
290C2×D4—7881no—
291C2×D4—7884no—
292C2×D4—3942no—
293C2×D4—3942no—
294C2×D4—3941no—
295C2×D4—3941no—
296C2×D4—3941no—
297C2×D4—3941no—
298C2×D4—3941no—
299C2×D4—3941no—
300C2×D4—3941no—
301C2×D4—3941no—
302C2×D4—1971no—
303C2×D4—1971no—
304C2×D4—1971no—
305C2×D4—1971no—
306C2×D4D78811yescharacteristic
307C22×C4—3941no—
308C22×C4—3941no—
309C22×C4—3941no—
310C22×C4C22×D19711yescharacteristic
311C22×C4C22×D19711yescharacteristic
312SD16—7881no—
313SD16—3942no—
314SD16—3942no—
315SD16—3942no—
316SD16—3942no—
317SD16—1974no—
318D8—7881no—
319D8—3942no—
320D8—3942no—
321M16—22no—
322C2×C8—22no—
323C2×C8C22×D19711yescharacteristic
324C2×C8C22×D19711yescharacteristic
325C2×C8C22×D19711yescharacteristic
326C2×C8C22×D19711yescharacteristic
327C4⋊C4—3941no—
328C4⋊C4—3941no—
329C4⋊C4—1971no—
330C4⋊C4—1971no—
331C4⋊C4D78811yescharacteristic
332C22⋊C4—3941no—
333C22⋊C4—3941no—
334C22⋊C4—3941no—
335C22⋊C4—3941no—
336C22⋊C4—3941no—
337C22⋊C4—3941no—
338C22⋊C4—3941no—
339C22⋊C4D78811yescharacteristic
340C22⋊C4D78811yescharacteristic
341C42—3941no—
342C42C22×D19711yescharacteristic
343C23—7882no—
344C23—7882no—
345C23—7882no—
346C23—7882no—
347C23—15761no—
348C23—15761no—
349C23—3941no—
350C23—3941no—
351C23—3941no—
352C23—3941no—
353C23C2×D78811yescharacteristic
354C23C2×D78811yescharacteristic
355Q8—7882no—
356Q8—3941no—
357Q8—3941no—
358Q8—3941no—
359Q8—1972no—
360D4—22no—
361D4—7882no—
362D4—7882no—
363D4—7882no—
364D4—7882no—
365D4—7882no—
366D4—7882no—
367D4—7882no—
368D4—15761no—
369D4—3941no—
370D4—3941no—
371D4—3941no—
372D4—3941no—
373D4—3941no—
374D4—3941no—
375D4—3941no—
376D4—1972no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—7882no—
382C2×C4—7882no—
383C2×C4—7882no—
384C2×C4—3941no—
385C2×C4—3941no—
386C2×C4—3941no—
387C2×C4—3941no—
388C2×C4C2×D78811yescharacteristic
389C2×C4C2×D78811yescharacteristic
390C2×C4C2×D78811yescharacteristic
391C2×C4C2×D78811yescharacteristic
392C2×C4C23×D19711yescharacteristic
393C8—21no—
394C8—21no—
395C8—21no—
396C8D4×D19712yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—7881no—
403C22—7881no—
404C22—7881no—
405C22—7881no—
406C22—7881no—
407C22—7881no—
408C22—7881no—
409C22—7881no—
410C22—7881no—
411C22—7881no—
412C22—7881no—
413C22—15764no—
414C22—15764no—
415C22—3942no—
416C22#2026335412yescomplemented
417C22C22×D78811yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—7881no—
423C4—7881no—
424C4—7881no—
425C4—3942no—
426C4C2×D4×D19711yescharacteristic
427C4#2026337211yescharacteristic
428C2—22no—
429C2—41no—
430C2—7882no—
431C2—15761no—
432C2—15761no—
433C2—15761no—
434C2D4×D78811yescharacteristic
435C2C2×#2026335411yescharacteristic
436C2#2716189111yescharacteristic
subgroupG/H
centerC22C22×D788
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#435×C197C2
socleC22×C197C22×D4
hypercenter#435D197
hypocenterC197#3265
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×#15691347C2semidirect
3. #27161738C2semidirect
4. D4×D788C2semidirect
5. #27161468C2semidirect
6. #27161325C2semidirect
7. #27161315C2semidirect
8. #27161295C2semidirect
9. #27161280C2semidirect
10. #27161232C2semidirect
11. #27161226C2semidirect
12. #27161220C2semidirect
13. #27161215C2semidirect
14. #27161181C2semidirect
15. #435×C197C2semidirect
16. C2×#15691358C22semidirect
17. C22×Dic394C22semidirect
25. C2×#15691347C22semidirect
26. C2×#15691347C22semidirect
27. C2×#15691347C22semidirect
28. C2×#15691347C22semidirect
30. C2×D1576C22semidirect
31. C2×D1576C22semidirect
33. #20263314C22semidirect
35. #20263305C22semidirect
40. #20263300C22semidirect
44. C4×Dic394C22semidirect
47. #20263260C22semidirect
48. #20263251C22semidirect
49. #20263251C22semidirect
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
95. #15691347D4semidirect
138. Dic394C2×D4semidirect
177. C8×C197C2×D4semidirect
183. C22×C197AGL(1,ℤ/8)semidirect
218. C197#3265semidirect
234. #435D197semidirect
251. C2×M16C2×D197semidirect
252. C22×C8C2×D197semidirect
270. C4⋊C8C2×D197semidirect
277. C22⋊C8C2×D197semidirect
278. C22⋊C8C2×D197semidirect
279. #97C2×D197semidirect
396. C8D4×D197semidirect
416. 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×#156913472
#271617382
D4×D7882
#271614682
#271613252
#271613152
#271612952
#271612802
#271612322
#271612262
#271612202
#271612152
#271611812
#435×C1972
#3265197