Finite Group Databasebeta

← #38754209

Subgroups — #38754209

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×#20263353C211yesmaximal, characteristic, complemented
2C2×SD16×D197C211yesmaximal, characteristic, complemented
3#27161741C211yesmaximal, characteristic, complemented
4D4×D788C211yesmaximal, characteristic, complemented
5#27161468C211yesmaximal, characteristic, complemented
6#27161326C211yesmaximal, characteristic, complemented
7#27161316C211yesmaximal, characteristic, complemented
8#27161294C211yesmaximal, characteristic, complemented
9#27161279C211yesmaximal, characteristic, complemented
10#27161231C211yesmaximal, characteristic, complemented
11#27161227C211yesmaximal, characteristic, complemented
12#27161219C211yesmaximal, characteristic, complemented
13#27161214C211yesmaximal, characteristic, complemented
14#27161152C211yesmaximal, characteristic, complemented
15#440×C197C211yesmaximal, characteristic, complemented
16C2×#15691359C2211yescharacteristic, complemented
17C2×Q8×D197C2211yescharacteristic, complemented
18C22×D788—21no—
19C2×D4×D197C2211yescharacteristic
20C2×D4×D197C2211yescharacteristic
21#20263353—24no—
22C2×#15691347C2211yescharacteristic, complemented
23C2×#15691347C2211yescharacteristic, complemented
24C2×#15691346C2211yescharacteristic
25C2×#15691345C2211yescharacteristic, complemented
26C2×#15691345C2211yescharacteristic, complemented
27SD16×D197—24no—
28C2×D1576C2211yescharacteristic, complemented
29C2×D1576C2211yescharacteristic, complemented
30C2×#15691343C2211yescharacteristic
31C2×#15691341C2211yescharacteristic, complemented
32C2×C8×D197C2211yescharacteristic, complemented
33#20263323C2211yescharacteristic
34#20263314—21no—
35#20263314C2211yescharacteristic, complemented
36#20263306—21no—
37#20263306C2211yescharacteristic, complemented
38#20263301—21no—
39#20263301C2211yescharacteristic
40#20263301C2211yescharacteristic
41#20263301C2211yescharacteristic, complemented
42#20263300—21no—
43#20263296—21no—
44#20263296—21no—
45C4×D788—21no—
46C4×D788C2211yescharacteristic
47C4⋊#13096802C2211yescharacteristic, complemented
48#20263249C2211yescharacteristic, complemented
49#20263246C2211yescharacteristic, complemented
50#20263246C2211yescharacteristic, complemented
51#20263246C2211yescharacteristic, complemented
52#20263238C2211yescharacteristic, complemented
53#20263238C2211yescharacteristic, complemented
54C2×SD16×C197C2211yescharacteristic
55C4×Q8×C197C2211yescharacteristic, complemented
56C4×D4×C197C2211yescharacteristic
57#107×C197C2211yescharacteristic, complemented
58#103×C197C2211yescharacteristic, complemented
59#102×C197C2211yescharacteristic
60#97×C197C2211yescharacteristic, complemented
61C23×D197—21no—
62C23×D197—21no—
63#15691359—22no—
64#15691359—41no—
65C2×Dic394—21no—
66C2×Dic394C2311yescharacteristic
67Q8×D197—22no—
68Q8×D197—41no—
69C2×D788—21no—
70C2×D788—21no—
71C2×D788—21no—
72C2×D788—21no—
73C2×D788—21no—
74C2×D788—21no—
75C2×D788—44no—
76C2×D788D411yescharacteristic
77C2×D788C2311yescharacteristic
78C2×D788C2311yescharacteristic
79C2×D788C2311yescharacteristic
80C2×C22⋊D197—21no—
81C2×C22⋊D197—21no—
82D4×D197—22no—
83D4×D197—22no—
84D4×D197—41no—
85D4×D197—41no—
86C2×C4×D197—21no—
87C2×C4×D197—21no—
88C2×C4×D197C2311yescharacteristic
89C2×C4×D197C2311yescharacteristic
90#15691347—22no—
91#15691347—41no—
92#15691346—22no—
93#15691345—22no—
94#15691345—22no—
95D1576—22no—
96D1576—41no—
97#15691343—22no—
98#15691341—22no—
99C2×#13096802C2311yescharacteristic
100C2×#13096802C2311yescharacteristic
101C8×D197—22no—
102C4⋊Dic197—21no—
103C4⋊Dic197D411yescharacteristic
104#15691335—21no—
105#15691335—21no—
106#15691335—21no—
107#15691335—21no—
108#15691335—21no—
109#15691335—21no—
110#15691335D411yescharacteristic
111#15691335D411yescharacteristic
112C2×Q8×C197C2311yescharacteristic
113C2×D4×C197C2311yescharacteristic
114C22×C4×C197—21no—
115SD16×C197D414yescomplemented
116C2×C8×C197C2311yescharacteristic
117C2×C8×C197C2311yescharacteristic
118C4⋊C4×C197—21no—
119C4⋊C4×C197C2311yescharacteristic
120C4⋊C4×C197C2311yescharacteristic
121C22⋊C4×C197—21no—
122C42×C197—21no—
123C42×C197C2311yescharacteristic
124C22×D197—21no—
125C22×D197—21no—
126C22×D197—21no—
127C22×D197—42no—
128C22×D197—42no—
129C22×D197—42no—
130C22×D197—42no—
131C22×D197—81no—
132C22×D197—81no—
133C22×D197C2×D411yescharacteristic
134C22×D197C2×D411yescharacteristic
135Dic394—21no—
136Dic394—21no—
137Dic394—42no—
138D788—22no—
139D788—21no—
140D788—21no—
141D788—21no—
142D788—21no—
143D788—21no—
144D788—21no—
145D788—42no—
146D788—42no—
147D788—42no—
148D788—42no—
149D788—42no—
150D788—81no—
151C22⋊D197—42no—
152C22⋊D197—42no—
153C2×Dic197—21no—
154C2×Dic197C2×D411yescharacteristic
155C2×Dic197C2×D411yescharacteristic
156C4×D197—21no—
157C4×D197—21no—
158C4×D197—21no—
159C4×D197—21no—
160C4×D197—42no—
161C4×D197—42no—
162#13096802—21no—
163#13096802—21no—
164C23×C197—21no—
165Q8×C197—21no—
166Q8×C197C2×D412yescomplemented
167D4×C197—21no—
168D4×C197C2×D412yes—
169C2×C4×C197—21no—
170C2×C4×C197—21no—
171C2×C4×C197—21no—
172C2×C4×C197—42no—
173C2×C4×C197C2×D411yescharacteristic
174C2×C4×C197C2×D411yescharacteristic
175C2×C4×C197C2411yescharacteristic
176C8×C197—21no—
177C8×C197C2×D412yescomplemented
178C22×C197—22no—
179C22×C197—41no—
180C22×C197—41no—
181C22×C197C22×D411yescharacteristic
182C2×D197—21no—
183C2×D197—21no—
184C2×D197—21no—
185C2×D197—21no—
186C2×D197—21no—
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—84no—
197C2×D197—84no—
198C2×D197AGL(1,ℤ/8)12yescomplemented
199C4×C197—22no—
200C4×C197—21no—
201C4×C197—21no—
202C4×C197—41no—
203C4×C197—41no—
204C4×C197C22×D411yescharacteristic
205C4×C197H(5,𝔽2)11yescharacteristic
206Dic197—21no—
207Dic197—21no—
208Dic197—41no—
209C2×C197—42no—
210C2×C197D4211yescharacteristic
211C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
212C2×C197#57711yescharacteristic
213D197—22no—
214D197—41no—
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#440D19711yescharacteristic, complemented
234#435—1971no—
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×SD16C2×D19711yescharacteristic
248C2×D8—1971no—
249C2×D8—1971no—
250C2×D8—1971no—
251C2×M16—1971no—
252C22×C8—1971no—
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×Q8C2×D19711yescharacteristic, complemented
266C4×D4—3941no—
267C4×D4—1971no—
268C4×D4C2×D19711yescharacteristic
269#107C2×D19711yescharacteristic, complemented
270C4⋊C8—1971no—
271#103—1971no—
272#103C2×D19711yescharacteristic, complemented
273#102—1971no—
274#102—1971no—
275#102—1971no—
276#102C2×D19711yescharacteristic
277C22⋊C8—1971no—
278C22⋊C8—1971no—
279#97C2×D19711yescharacteristic, complemented
280C24—3941no—
281C24—3941no—
282#41—7881no—
283#41—3942no—
284C2×Q8—7881no—
285C2×Q8—3941no—
286C2×Q8—3942no—
287C2×Q8—1971no—
288C2×Q8C22×D19711yescharacteristic
289C2×D4—7884no—
290C2×D4—7881no—
291C2×D4—7881no—
292C2×D4—3941no—
293C2×D4—3941no—
294C2×D4—3941no—
295C2×D4—3941no—
296C2×D4—3942no—
297C2×D4—3942no—
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×D4C22×D19711yescharacteristic
307C22×C4—21no—
308C22×C4—3941no—
309C22×C4—3941no—
310C22×C4—1971no—
311C22×C4—1971no—
312SD16—7881no—
313SD16—3942no—
314SD16—3942no—
315SD16—3942no—
316SD16—3942no—
317SD16D78814yescomplemented
318D8—7881no—
319D8—3942no—
320D8—3942no—
321M16—3942no—
322C2×C8—3942no—
323C2×C8—1971no—
324C2×C8—1971no—
325C2×C8C22×D19711yescharacteristic
326C2×C8C22×D19711yescharacteristic
327C4⋊C4—21no—
328C4⋊C4—3941no—
329C4⋊C4—1971no—
330C4⋊C4C22×D19711yescharacteristic
331C4⋊C4C22×D19711yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—3941no—
334C22⋊C4—3941no—
335C22⋊C4—3941no—
336C22⋊C4—3941no—
337C22⋊C4—3941no—
338C22⋊C4—3941no—
339C22⋊C4—1971no—
340C22⋊C4—1971no—
341C42—21no—
342C42C22×D19711yescharacteristic
343C23—21no—
344C23—7882no—
345C23—7882no—
346C23—7882no—
347C23—7882no—
348C23—15761no—
349C23—15761no—
350C23—3941no—
351C23—3941no—
352C23—3941no—
353C23—1971no—
354C23—1971no—
355Q8—21no—
356Q8—7882no—
357Q8—3941no—
358Q8—3941no—
359Q8C2×D78812yescomplemented
360D4—21no—
361D4—7882no—
362D4—7882no—
363D4—7882no—
364D4—7882no—
365D4—7882no—
366D4—7882no—
367D4—7882no—
368D4—15761no—
369D4—3942no—
370D4—3941no—
371D4—3941no—
372D4—3941no—
373D4—3941no—
374D4—3941no—
375D4—3941no—
376D4C2×D78812yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—42no—
381C2×C4—7882no—
382C2×C4—7882no—
383C2×C4—3941no—
384C2×C4—3941no—
385C2×C4—3941no—
386C2×C4—3941no—
387C2×C4—3941no—
388C2×C4—1971no—
389C2×C4—1971no—
390C2×C4D4×D19711yescharacteristic
391C2×C4D4×D19711yescharacteristic
392C2×C4C23×D19711yescharacteristic
393C8—21no—
394C8—3941no—
395C8—3941no—
396C8C2×D78812yescomplemented
397C22—22no—
398C22—41no—
399C22—41no—
400C22—7881no—
401C22—7881no—
402C22—7881no—
403C22—7881no—
404C22—7881no—
405C22—7881no—
406C22—7881no—
407C22—7881no—
408C22—7881no—
409C22—15764no—
410C22—15764no—
411C22—3941no—
412C22—3941no—
413C22—3941no—
414C22—3941no—
415C22—3941no—
416C22—1972no—
417C22C2×D4×D19711yescharacteristic
418C4—21no—
419C4—21no—
420C4—22no—
421C4—41no—
422C4—41no—
423C4—7881no—
424C4—3941no—
425C4—3941no—
426C4C22×D78811yescharacteristic
427C4#2026337211yescharacteristic
428C2—42no—
429C2—7881no—
430C2—15761no—
431C2—15761no—
432C2—15761no—
433C2—3942no—
434C2D4×D78811yescharacteristic
435C2AGL(1,ℤ/8)×D19711yescharacteristic
436C2#2716188811yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#440×C197C2
socleC22×C197C22×D4
hypercenter#440D197
hypocenterC197#3265
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. C2×#20263353C2semidirect
2. C2×SD16×D197C2semidirect
3. #27161741C2semidirect
4. D4×D788C2semidirect
5. #27161468C2semidirect
6. #27161326C2semidirect
7. #27161316C2semidirect
8. #27161294C2semidirect
9. #27161279C2semidirect
10. #27161231C2semidirect
11. #27161227C2semidirect
12. #27161219C2semidirect
13. #27161214C2semidirect
14. #27161152C2semidirect
15. #440×C197C2semidirect
16. C2×#15691359C22semidirect
17. C2×Q8×D197C22semidirect
22. C2×#15691347C22semidirect
23. C2×#15691347C22semidirect
25. C2×#15691345C22semidirect
26. C2×#15691345C22semidirect
28. C2×D1576C22semidirect
29. C2×D1576C22semidirect
31. C2×#15691341C22semidirect
32. C2×C8×D197C22semidirect
35. #20263314C22semidirect
37. #20263306C22semidirect
41. #20263301C22semidirect
47. C4⋊#13096802C22semidirect
48. #20263249C22semidirect
49. #20263246C22semidirect
50. #20263246C22semidirect
51. #20263246C22semidirect
52. #20263238C22semidirect
53. #20263238C22semidirect
55. C4×Q8×C197C22semidirect
57. #107×C197C22semidirect
58. #103×C197C22semidirect
60. #97×C197C22semidirect
115. SD16×C197D4semidirect
166. Q8×C197C2×D4semidirect
177. C8×C197C2×D4semidirect
198. C2×D197AGL(1,ℤ/8)semidirect
218. C197#3265semidirect
233. #440D197semidirect
265. C4×Q8C2×D197semidirect
269. #107C2×D197semidirect
272. #103C2×D197semidirect
279. #97C2×D197semidirect
317. SD16D788semidirect
359. Q8C2×D788semidirect
396. C8C2×D788semidirect

Maximal subgroups

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

subgroupindex
C2×#202633532
C2×SD16×D1972
#271617412
D4×D7882
#271614682
#271613262
#271613162
#271612942
#271612792
#271612312
#271612272
#271612192
#271612142
#271611522
#440×C1972
#3265197