Finite Group Databasebeta

← #38754204

Subgroups — #38754204

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×#20263350C211yesmaximal, characteristic, complemented
2C22×#15691347C211yesmaximal, characteristic, complemented
3#27161735C211yesmaximal, characteristic, complemented
4#27161713C211yesmaximal, characteristic, complemented
5#27161467C211yesmaximal, characteristic, complemented
6#27161326C211yesmaximal, characteristic, complemented
7#27161293C211yesmaximal, characteristic, complemented
8#27161278C211yesmaximal, characteristic, complemented
9#27161231C211yesmaximal, characteristic, complemented
10#27161225C211yesmaximal, characteristic, complemented
11#27161217C211yesmaximal, characteristic, complemented
12#27161212C211yesmaximal, characteristic, complemented
13#27161178C211yesmaximal, characteristic, complemented
14#27161150C211yesmaximal, characteristic, complemented
15#466×C197C211yesmaximal, characteristic, complemented
16C2×#15691357C2211yescharacteristic, complemented
17C22×Dic394C2211yescharacteristic, complemented
18C22×D788C2211yescharacteristic
19C2×D4×D197—21no—
20C2×D4×D197C2211yescharacteristic
21#20263350—24no—
22C2×#15691347—24no—
23C2×#15691347C2211yescharacteristic
24C2×#15691347C2211yescharacteristic
25C2×#15691347C2211yescharacteristic, complemented
26C2×#15691347C2211yescharacteristic, complemented
27C2×#15691346C2211yescharacteristic, complemented
28C2×#15691346C2211yescharacteristic, complemented
29C2×#15691343C2211yescharacteristic
30C2×#15691343C2211yescharacteristic, complemented
31C2×#15691341C2211yescharacteristic, complemented
32#20263322C2211yescharacteristic
33#20263313C2211yescharacteristic, complemented
34#20263312—21no—
35#20263306C2211yescharacteristic, complemented
36#20263303—21no—
37#20263301C2211yescharacteristic
38#20263299—21no—
39#20263298—21no—
40#20263296—21no—
41#20263294—21no—
42#20263291C2211yescharacteristic, complemented
43D4×Dic197—21no—
44#20263288C2211yescharacteristic
45C4×C22⋊D197C2211yescharacteristic
46#20263255C2211yescharacteristic, complemented
47#20263249C2211yescharacteristic, complemented
48#20263249C2211yescharacteristic, complemented
49#20263246C2211yescharacteristic
50#20263246C2211yescharacteristic, complemented
51C22⋊#13096802C2211yescharacteristic, complemented
52#20263238C2211yescharacteristic, complemented
53#20263236C2211yescharacteristic, complemented
54C2×D8×C197C2211yescharacteristic, complemented
55C22×C8×C197C2211yescharacteristic, complemented
56#121×C197C2211yescharacteristic
57#121×C197C2211yescharacteristic, complemented
58#107×C197C2211yescharacteristic, complemented
59#102×C197C2211yescharacteristic, complemented
60#102×C197C2211yescharacteristic, complemented
61C23×D197—21no—
62C23×D197—21no—
63#15691357—22no—
64#15691357—41no—
65C2×Dic394—22no—
66C2×Dic394—21no—
67C2×Dic394—41no—
68C2×Dic394C2311yescharacteristic
69C2×Dic394C2311yescharacteristic
70C2×D788—22no—
71C2×D788—21no—
72C2×D788—21no—
73C2×D788—41no—
74C2×D788C2311yescharacteristic
75C2×D788C2311yescharacteristic
76C2×C22⋊D197—21no—
77C2×C22⋊D197—21no—
78C2×C22⋊D197—21no—
79C2×C22⋊D197—21no—
80C2×C22⋊D197D411yescharacteristic
81D4×D197—22no—
82D4×D197—41no—
83D4×D197—44no—
84C22×Dic197—21no—
85C22×Dic197—21no—
86C2×C4×D197—21no—
87C2×C4×D197C2311yescharacteristic
88#15691347—22no—
89#15691347—22no—
90#15691347—22no—
91#15691347D414yescomplemented
92#15691346—22no—
93#15691346—41no—
94#15691343—22no—
95#15691343—41no—
96#15691341—22no—
97C2×#13096802C2311yescharacteristic
98C2×#13096802C2311yescharacteristic
99C4⋊Dic197—21no—
100#15691337—21no—
101#15691337D411yescharacteristic
102C22⋊Dic197—21no—
103C22⋊Dic197—21no—
104C22⋊Dic197D411yescharacteristic
105#15691335—21no—
106#15691335—21no—
107#15691335—21no—
108#15691335D411yescharacteristic
109C4×Dic197—21no—
110C4×Dic197C2311yescharacteristic
111C2×D4×C197—21no—
112C2×D4×C197—21no—
113C2×D4×C197C2311yescharacteristic
114C2×D4×C197C2311yescharacteristic
115C22×C4×C197C2311yescharacteristic
116D8×C197—22no—
117C2×C8×C197—22no—
118C2×C8×C197C2311yescharacteristic
119C2×C8×C197C2311yescharacteristic
120C4⋊C4×C197C2311yescharacteristic
121C4⋊C4×C197C2311yescharacteristic
122C22⋊C4×C197—21no—
123C22⋊C4×C197—21no—
124C22×D197—21no—
125C22×D197—21no—
126C22×D197—42no—
127C22×D197—42no—
128C22×D197—42no—
129C22×D197—42no—
130C22×D197—81no—
131C22×D197—81no—
132C22×D197C2×D411yescharacteristic
133Dic394—21no—
134Dic394—21no—
135Dic394—21no—
136Dic394—42no—
137Dic394C2×D412yescomplemented
138D788—21no—
139D788—21no—
140D788—21no—
141D788—42no—
142D788—42no—
143D788C2×D412yes—
144C22⋊D197—22no—
145C22⋊D197—42no—
146C22⋊D197—42no—
147C22⋊D197—42no—
148C22⋊D197—81no—
149C2×Dic197—21no—
150C2×Dic197—21no—
151C2×Dic197—42no—
152C2×Dic197—42no—
153C2×Dic197C2×D411yescharacteristic
154C2×Dic197C2×D411yescharacteristic
155C2×Dic197C2×D411yescharacteristic
156C4×D197—21no—
157C4×D197—21no—
158C4×D197—42no—
159#13096802—21no—
160#13096802—21no—
161C23×C197—21no—
162C23×C197—21no—
163C23×C197C2×D411yescharacteristic
164D4×C197—21no—
165D4×C197—21no—
166D4×C197—21no—
167D4×C197—21no—
168D4×C197—42no—
169D4×C197—42no—
170C2×C4×C197—21no—
171C2×C4×C197—21no—
172C2×C4×C197—21no—
173C2×C4×C197—21no—
174C2×C4×C197C2×D411yescharacteristic
175C2×C4×C197C2411yescharacteristic
176C8×C197—21no—
177C8×C197C2×D412yescomplemented
178C22×C197—21no—
179C22×C197—21no—
180C22×C197—41no—
181C22×C197—41no—
182C22×C197—41no—
183C22×C197—41no—
184C22×C197—41no—
185C22×C197—41no—
186C22×C197AGL(1,ℤ/8)12yescomplemented
187C22×C197C22×D411yescharacteristic
188C2×D197—21no—
189C2×D197—22no—
190C2×D197—21no—
191C2×D197—21no—
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—41no—
201C4×C197—41no—
202C4×C197C22×D411yescharacteristic
203C4×C197H(5,𝔽2)11yescharacteristic
204Dic197—21no—
205Dic197—22no—
206Dic197—21no—
207Dic197—21no—
208Dic197—41no—
209C2×C197—22no—
210C2×C197—81no—
211C2×C197—81no—
212C2×C197D4211yescharacteristic
213C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C197#57711yescharacteristic
215D197—41no—
216D197—42no—
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#466D19711yescharacteristic, complemented
227#463—1971no—
228#460—1971no—
229#450—1971no—
230#449—1971no—
231#448—1971no—
232#447—1971no—
233#440—1971no—
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×SD16—1971no—
248C2×D8—1971no—
249C2×D8—1971no—
250C2×D8C2×D19711yescharacteristic, complemented
251C2×M16—1971no—
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#121C2×D19711yescharacteristic
262#121C2×D19711yescharacteristic, complemented
263#120—3941no—
264#120—3941no—
265C4×Q8—1971no—
266C4×D4—3941no—
267C4×D4—1971no—
268C4×D4—1971no—
269#107C2×D19711yescharacteristic, complemented
270C4⋊C8—1971no—
271#103—1971no—
272#103—1971no—
273#102—1971no—
274#102—1971no—
275#102C2×D19711yescharacteristic, complemented
276#102C2×D19711yescharacteristic, complemented
277C22⋊C8—1971no—
278C22⋊C8—1971no—
279#97—1971no—
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—21no—
290C2×D4—21no—
291C2×D4—7881no—
292C2×D4—7884no—
293C2×D4—7881no—
294C2×D4—3941no—
295C2×D4—3942no—
296C2×D4—3942no—
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×D4C22×D19711yescharacteristic
306C2×D4C22×D19711yescharacteristic
307C22×C4—3941no—
308C22×C4—3941no—
309C22×C4—3941no—
310C22×C4—1971no—
311C22×C4C22×D19711yescharacteristic
312SD16—7881no—
313SD16—3942no—
314SD16—3942no—
315SD16—3942no—
316SD16—3942no—
317SD16—1974no—
318D8—22no—
319D8—7881no—
320D8—3942no—
321M16—3942no—
322C2×C8—22no—
323C2×C8—1971no—
324C2×C8—1971no—
325C2×C8C22×D19711yescharacteristic
326C2×C8C22×D19711yescharacteristic
327C4⋊C4—3941no—
328C4⋊C4—3941no—
329C4⋊C4—1971no—
330C4⋊C4C22×D19711yescharacteristic
331C4⋊C4C22×D19711yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—3941no—
335C22⋊C4—3941no—
336C22⋊C4—3941no—
337C22⋊C4—3941no—
338C22⋊C4—3941no—
339C22⋊C4—1971no—
340C22⋊C4—1971no—
341C42—3941no—
342C42—1971no—
343C23—21no—
344C23—21no—
345C23—7882no—
346C23—7882no—
347C23—7882no—
348C23—7882no—
349C23—15761no—
350C23—15761no—
351C23—3941no—
352C23—3941no—
353C23—1971no—
354C23D4×D19711yescharacteristic
355Q8—7882no—
356Q8—3941no—
357Q8—3941no—
358Q8—3941no—
359Q8—1972no—
360D4—21no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—42no—
365D4—42no—
366D4—7882no—
367D4—7882no—
368D4—7882no—
369D4—7882no—
370D4—7882no—
371D4—15761no—
372D4—3942no—
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×C4—1971no—
389C2×C4—1971no—
390C2×C4—1971no—
391C2×C4D4×D19711yescharacteristic
392C2×C4C23×D19711yescharacteristic
393C8—21no—
394C8—3941no—
395C8—3941no—
396C8D4×D19712yescomplemented
397C22—21no—
398C22—21no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—7881no—
406C22—7881no—
407C22—7881no—
408C22—7881no—
409C22—7881no—
410C22—15764no—
411C22—15764no—
412C22—3941no—
413C22—3941no—
414C22—3941no—
415C22—3942no—
416C22#2026335012yescomplemented
417C22C2×D4×D19711yescharacteristic
418C4—21no—
419C4—41no—
420C4—41no—
421C4—7881no—
422C4—3941no—
423C4—3941no—
424C4—3941no—
425C4—3942no—
426C4C2×D4×D19711yescharacteristic
427C4#2026337211yescharacteristic
428C2—22no—
429C2—81no—
430C2—81no—
431C2—7882no—
432C2—7881no—
433C2—15761no—
434C2#2716171311yescharacteristic
435C2C2×#2026335011yescharacteristic
436C2#2716188611yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#466×C197C2
socleC22×C197C22×D4
hypercenter#466D197
hypocenterC197#3265
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #466

Complements

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

NH ≅ G/Ntype
1. C2×#20263350C2semidirect
2. C22×#15691347C2semidirect
3. #27161735C2semidirect
4. #27161713C2semidirect
5. #27161467C2semidirect
6. #27161326C2semidirect
7. #27161293C2semidirect
8. #27161278C2semidirect
9. #27161231C2semidirect
10. #27161225C2semidirect
11. #27161217C2semidirect
12. #27161212C2semidirect
13. #27161178C2semidirect
14. #27161150C2semidirect
15. #466×C197C2semidirect
16. C2×#15691357C22semidirect
17. C22×Dic394C22semidirect
25. C2×#15691347C22semidirect
26. C2×#15691347C22semidirect
27. C2×#15691346C22semidirect
28. C2×#15691346C22semidirect
30. C2×#15691343C22semidirect
31. C2×#15691341C22semidirect
33. #20263313C22semidirect
35. #20263306C22semidirect
42. #20263291C22semidirect
46. #20263255C22semidirect
47. #20263249C22semidirect
48. #20263249C22semidirect
50. #20263246C22semidirect
51. C22⋊#13096802C22semidirect
52. #20263238C22semidirect
53. #20263236C22semidirect
54. C2×D8×C197C22semidirect
55. C22×C8×C197C22semidirect
57. #121×C197C22semidirect
58. #107×C197C22semidirect
59. #102×C197C22semidirect
60. #102×C197C22semidirect
91. #15691347D4semidirect
137. Dic394C2×D4semidirect
177. C8×C197C2×D4semidirect
186. C22×C197AGL(1,ℤ/8)semidirect
218. C197#3265semidirect
226. #466D197semidirect
250. C2×D8C2×D197semidirect
252. C22×C8C2×D197semidirect
262. #121C2×D197semidirect
269. #107C2×D197semidirect
275. #102C2×D197semidirect
276. #102C2×D197semidirect
396. C8D4×D197semidirect
416. C22#20263350semidirect

Maximal subgroups

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

subgroupindex
C2×#202633502
C22×#156913472
#271617352
#271617132
#271614672
#271613262
#271612932
#271612782
#271612312
#271612252
#271612172
#271612122
#271611782
#271611502
#466×C1972
#3265197