Finite Group Databasebeta

← #38754203

Subgroups — #38754203

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×#20263352C211yesmaximal, characteristic, complemented
2C22×#15691346C211yesmaximal, characteristic, complemented
3#27161738C211yesmaximal, characteristic, complemented
4#27161466C211yesmaximal, characteristic, complemented
5#27161323C211yesmaximal, characteristic, complemented
6#27161313C211yesmaximal, characteristic, complemented
7#27161294C211yesmaximal, characteristic, complemented
8#27161276C211yesmaximal, characteristic, complemented
9#27161230C211yesmaximal, characteristic, complemented
10#27161222C211yesmaximal, characteristic, complemented
11#27161217C211yesmaximal, characteristic, complemented
12#27161213C211yesmaximal, characteristic, complemented
13#27161179C211yesmaximal, characteristic, complemented
14#27161151C211yesmaximal, characteristic, complemented
15D42×C197C211yesmaximal, characteristic, complemented
16C2×#15691358C2211yescharacteristic, complemented
17C22×Dic394C2211yescharacteristic, complemented
18#20263352—24no—
19C2×#15691346—24no—
20C2×#15691346C2211yescharacteristic
21C2×#15691346C2211yescharacteristic
22C2×#15691346C2211yescharacteristic, complemented
23C2×#15691346C2211yescharacteristic, complemented
24C2×#15691346C2211yescharacteristic, complemented
25C2×#15691346C2211yescharacteristic, complemented
26C2×#15691343C2211yescharacteristic
27C2×#15691343C2211yescharacteristic, complemented
28C2×#15691343C2211yescharacteristic, complemented
29C2×#15691342C2211yescharacteristic, complemented
30C22×#13096802C2211yescharacteristic, complemented
31#20263314C2211yescharacteristic, complemented
32#20263312—21no—
33#20263305C2211yescharacteristic, complemented
34#20263304—21no—
35#20263300C2211yescharacteristic, complemented
36#20263298—21no—
37C4×Dic394C2211yescharacteristic, complemented
38#20263286—21no—
39#20263259C2211yescharacteristic, complemented
40C4⋊#13096802C2211yescharacteristic, complemented
41#20263249C2211yescharacteristic, complemented
42#20263249C2211yescharacteristic, complemented
43#20263247C2211yescharacteristic
44#20263247C2211yescharacteristic, complemented
45#20263247C2211yescharacteristic, complemented
46#20263246C2211yescharacteristic, complemented
47C22⋊#13096802C2211yescharacteristic, complemented
48C22⋊#13096802C2211yescharacteristic, complemented
49#20263237C2211yescharacteristic, complemented
50C22×D4×C197—21no—
51C22×D4×C197C2211yescharacteristic
52C22×D4×C197C2211yescharacteristic
53#127×C197C2211yescharacteristic
54#121×C197—21no—
55#121×C197C2211yescharacteristic
56#121×C197C2211yescharacteristic
57#120×C197—21no—
58#120×C197—21no—
59C4×D4×C197C2211yescharacteristic
60C4×D4×C197C2211yescharacteristic
61#15691358—22no—
62#15691358—41no—
63C2×Dic394—22no—
64C2×Dic394—21no—
65C2×Dic394—41no—
66C2×Dic394C2311yescharacteristic
67C2×Dic394C2311yescharacteristic
68C2×D788C2311yescharacteristic
69C2×C22⋊D197—21no—
70C2×C22⋊D197—21no—
71C22×Dic197—21no—
72C2×C4×D197—21no—
73#15691346—22no—
74#15691346—22no—
75#15691346—22no—
76#15691346—22no—
77#15691346—41no—
78#15691346D414yescomplemented
79#15691343—22no—
80#15691343—22no—
81#15691343—41no—
82#15691342—22no—
83C2×#13096802—22no—
84C2×#13096802C2311yescharacteristic
85C2×#13096802C2311yescharacteristic
86C2×#13096802C2311yescharacteristic
87C2×#13096802C2311yescharacteristic
88C4⋊Dic197C2311yescharacteristic
89#15691337—21no—
90#15691337—21no—
91C22⋊Dic197—21no—
92#15691335—21no—
93#15691335—21no—
94#15691335—21no—
95C4×Dic197—21no—
96C24×C197—21no—
97C24×C197—21no—
98C2×D4×C197—22no—
99C2×D4×C197—22no—
100C2×D4×C197—21no—
101C2×D4×C197—21no—
102C2×D4×C197—21no—
103C2×D4×C197—21no—
104C2×D4×C197—21no—
105C2×D4×C197—21no—
106C2×D4×C197—41no—
107C2×D4×C197—41no—
108C2×D4×C197—44no—
109C2×D4×C197D411yescharacteristic
110C2×D4×C197C2311yescharacteristic
111C2×D4×C197C2311yescharacteristic
112C2×D4×C197C2311yescharacteristic
113C22×C4×C197—21no—
114C22×C4×C197C2311yescharacteristic
115C22×C4×C197C2311yescharacteristic
116C4⋊C4×C197D411yescharacteristic
117C4⋊C4×C197C2311yescharacteristic
118C22⋊C4×C197—21no—
119C22⋊C4×C197—21no—
120C22⋊C4×C197—21no—
121C22⋊C4×C197D411yescharacteristic
122C22⋊C4×C197D411yescharacteristic
123C42×C197C2311yescharacteristic
124C22×D197—21no—
125Dic394—21no—
126Dic394—21no—
127Dic394—21no—
128Dic394—42no—
129Dic394C2×D412yescomplemented
130D788—21no—
131D788—21no—
132C22⋊D197—42no—
133C22⋊D197—42no—
134C2×Dic197—21no—
135C2×Dic197—21no—
136C2×Dic197—21no—
137C2×Dic197—42no—
138C4×D197—42no—
139#13096802—21no—
140#13096802—21no—
141#13096802—21no—
142#13096802C2×D412yescomplemented
143C23×C197—21no—
144C23×C197—21no—
145C23×C197—21no—
146C23×C197—42no—
147C23×C197—42no—
148C23×C197—42no—
149C23×C197—42no—
150C23×C197—81no—
151C23×C197—81no—
152C23×C197C2×D411yescharacteristic
153C23×C197C2×D411yescharacteristic
154D4×C197—22no—
155D4×C197—21no—
156D4×C197—21no—
157D4×C197—21no—
158D4×C197—21no—
159D4×C197—21no—
160D4×C197—42no—
161D4×C197—42no—
162D4×C197—42no—
163D4×C197—42no—
164D4×C197—42no—
165D4×C197—81no—
166D4×C197C2×D412yes—
167C2×C4×C197—21no—
168C2×C4×C197—21no—
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×C197C2×D411yescharacteristic
176C2×C4×C197C2×D411yescharacteristic
177C2×C4×C197C2411yescharacteristic
178C22×C197—21no—
179C22×C197—21no—
180C22×C197—21no—
181C22×C197—22no—
182C22×C197—21no—
183C22×C197—21no—
184C22×C197—41no—
185C22×C197—41no—
186C22×C197—41no—
187C22×C197—41no—
188C22×C197—41no—
189C22×C197—41no—
190C22×C197—41no—
191C22×C197—41no—
192C22×C197—84no—
193C22×C197—84no—
194C22×C197AGL(1,ℤ/8)12yescomplemented
195C22×C197C22×D411yescharacteristic
196C2×D197—41no—
197C2×D197—41no—
198C2×D197—41no—
199C4×C197—21no—
200C4×C197—21no—
201C4×C197—21no—
202C4×C197—21no—
203C4×C197—41no—
204C4×C197C22×D411yescharacteristic
205C4×C197H(5,𝔽2)11yescharacteristic
206Dic197—22no—
207Dic197—41no—
208Dic197—41no—
209C2×C197—22no—
210C2×C197—41no—
211C2×C197—42no—
212C2×C197—81no—
213C2×C197—81no—
214C2×C197D4211yescharacteristic
215C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
216C2×C197#57711yescharacteristic
217D197—81no—
218C197#326511yescharacteristic, complemented, Sylow
219#3265—1971nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1971no—
221C22×SD16—1971no—
222#548—1971no—
223D42D19711yescharacteristic, complemented
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#435—1971no—
235C2×#41—1971no—
236C22×Q8—1971no—
237C22×D4—21no—
238C22×D4C2×D19711yescharacteristic
239C22×D4C2×D19711yescharacteristic
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×M16—1971no—
252C22×C8—1971no—
253#127C2×D19711yescharacteristic
254#124—3941no—
255#124—1971no—
256C22⋊Q8—3941no—
257C22⋊Q8—1971no—
258#121—21no—
259#121—3941no—
260#121—1971no—
261#121C2×D19711yescharacteristic
262#121C2×D19711yescharacteristic
263#120—21no—
264#120—21no—
265C4×Q8—1971no—
266C4×D4—3941no—
267C4×D4C2×D19711yescharacteristic
268C4×D4C2×D19711yescharacteristic
269#107—1971no—
270C4⋊C8—1971no—
271#103—1971no—
272#103—1971no—
273#102—1971no—
274#102—1971no—
275#102—1971no—
276#102—1971no—
277C22⋊C8—1971no—
278C22⋊C8—1971no—
279#97—1971no—
280C24—21no—
281C24—21no—
282#41—7881no—
283#41—3942no—
284C2×Q8—7881no—
285C2×Q8—3942no—
286C2×Q8—3941no—
287C2×Q8—1971no—
288C2×Q8—1971no—
289C2×D4—22no—
290C2×D4—22no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—44no—
300C2×D4—3941no—
301C2×D4—3941no—
302C2×D4—1971no—
303C2×D4C22⋊D19711yescharacteristic
304C2×D4C22×D19711yescharacteristic
305C2×D4C22×D19711yescharacteristic
306C2×D4C22×D19711yescharacteristic
307C22×C4—21no—
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—3942no—
322C2×C8—3942no—
323C2×C8—1971no—
324C2×C8—1971no—
325C2×C8—1971no—
326C2×C8—1971no—
327C4⋊C4—3941no—
328C4⋊C4—3941no—
329C4⋊C4—1971no—
330C4⋊C4C22⋊D19711yescharacteristic
331C4⋊C4C22×D19711yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—3941no—
336C22⋊C4—3941no—
337C22⋊C4—3941no—
338C22⋊C4—3941no—
339C22⋊C4C22⋊D19711yescharacteristic
340C22⋊C4C22⋊D19711yescharacteristic
341C42—3941no—
342C42C22×D19711yescharacteristic
343C23—21no—
344C23—21no—
345C23—21no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—81no—
351C23—81no—
352C23—3941no—
353C23C2×C22⋊D19711yescharacteristic
354C23C2×C22⋊D19711yescharacteristic
355Q8—7882no—
356Q8—3941no—
357Q8—3941no—
358Q8—3941no—
359Q8—1972no—
360D4—22no—
361D4—21no—
362D4—21no—
363D4—21no—
364D4—21no—
365D4—21no—
366D4—42no—
367D4—42no—
368D4—42no—
369D4—42no—
370D4—42no—
371D4—81no—
372D4—7882no—
373D4—7882no—
374D4—3941no—
375D4—3941no—
376D4D4×D19712yes—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—42no—
383C2×C4—7882no—
384C2×C4—7882no—
385C2×C4—3941no—
386C2×C4—3941no—
387C2×C4—3941no—
388C2×C4C2×C22⋊D19711yescharacteristic
389C2×C4C2×C22⋊D19711yescharacteristic
390C2×C4C2×C22⋊D19711yescharacteristic
391C2×C4C2×C22⋊D19711yescharacteristic
392C2×C4C23×D19711yescharacteristic
393C8—3941no—
394C8—3941no—
395C8—3941no—
396C8—1972no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—22no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—84no—
412C22—84no—
413C22—7881no—
414C22—7881no—
415C22—7881no—
416C22#2026335212yescomplemented
417C22C22×C22⋊D19711yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—41no—
423C4—7881no—
424C4—7881no—
425C4—3942no—
426C4C2×D4×D19711yescharacteristic
427C4#2026337111yescharacteristic
428C2—22no—
429C2—41no—
430C2—42no—
431C2—81no—
432C2—81no—
433C2—15761no—
434C2D4×C22⋊D19711yescharacteristic
435C2C2×#2026335211yescharacteristic
436C2#2716189011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingD42×C197C2
socleC22×C197C22×D4
hypercenterD42D197
hypocenterC197#3265
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#20263352C2semidirect
2. C22×#15691346C2semidirect
3. #27161738C2semidirect
4. #27161466C2semidirect
5. #27161323C2semidirect
6. #27161313C2semidirect
7. #27161294C2semidirect
8. #27161276C2semidirect
9. #27161230C2semidirect
10. #27161222C2semidirect
11. #27161217C2semidirect
12. #27161213C2semidirect
13. #27161179C2semidirect
14. #27161151C2semidirect
15. D42×C197C2semidirect
16. C2×#15691358C22semidirect
17. C22×Dic394C22semidirect
22. C2×#15691346C22semidirect
23. C2×#15691346C22semidirect
24. C2×#15691346C22semidirect
25. C2×#15691346C22semidirect
27. C2×#15691343C22semidirect
28. C2×#15691343C22semidirect
29. C2×#15691342C22semidirect
30. C22×#13096802C22semidirect
31. #20263314C22semidirect
33. #20263305C22semidirect
35. #20263300C22semidirect
37. C4×Dic394C22semidirect
39. #20263259C22semidirect
40. C4⋊#13096802C22semidirect
41. #20263249C22semidirect
42. #20263249C22semidirect
44. #20263247C22semidirect
45. #20263247C22semidirect
46. #20263246C22semidirect
47. C22⋊#13096802C22semidirect
48. C22⋊#13096802C22semidirect
49. #20263237C22semidirect
78. #15691346D4semidirect
129. Dic394C2×D4semidirect
142. #13096802C2×D4semidirect
194. C22×C197AGL(1,ℤ/8)semidirect
218. C197#3265semidirect
223. D42D197semidirect
416. C22#20263352semidirect

Maximal subgroups

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

subgroupindex
C2×#202633522
C22×#156913462
#271617382
#271614662
#271613232
#271613132
#271612942
#271612762
#271612302
#271612222
#271612172
#271612132
#271611792
#271611512
D42×C1972
#3265197