Finite Group Databasebeta

← #38754208

Subgroups — #38754208

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×#20263351C211yesmaximal, characteristic, complemented
2C22×#15691345C211yesmaximal, characteristic, complemented
3D4×D788C211yesmaximal, characteristic, complemented
4#27161469C211yesmaximal, characteristic, complemented
5#27161327C211yesmaximal, characteristic, complemented
6#27161313C211yesmaximal, characteristic, complemented
7#27161291C211yesmaximal, characteristic, complemented
8#27161279C211yesmaximal, characteristic, complemented
9#27161229C211yesmaximal, characteristic, complemented
10#27161224C211yesmaximal, characteristic, complemented
11#27161218C211yesmaximal, characteristic, complemented
12#27161212C211yesmaximal, characteristic, complemented
13#27161177C211yesmaximal, characteristic, complemented
14#27161151C211yesmaximal, characteristic, complemented
15#548×C197C211yesmaximal, characteristic, complemented
16C22×D788C2211yescharacteristic
17C22×D788C2211yescharacteristic
18C2×D4×D197—21no—
19#20263351—24no—
20C2×#15691345—24no—
21C2×#15691345C2211yescharacteristic
22C2×#15691345C2211yescharacteristic
23C2×#15691345C2211yescharacteristic, complemented
24C2×#15691345C2211yescharacteristic, complemented
25C2×#15691345C2211yescharacteristic, complemented
26C2×#15691345C2211yescharacteristic, complemented
27C2×#15691343C2211yescharacteristic
28C2×#15691343C2211yescharacteristic, complemented
29C2×#15691343C2211yescharacteristic, complemented
30C2×#15691342C2211yescharacteristic, complemented
31C22×#13096802C2211yescharacteristic, complemented
32#20263323C2211yescharacteristic
33#20263301—21no—
34#20263300C2211yescharacteristic
35#20263300C2211yescharacteristic
36#20263296—21no—
37#20263296—21no—
38C4×D788C2211yescharacteristic
39#20263259C2211yescharacteristic, complemented
40C4⋊#13096802C2211yescharacteristic, complemented
41#20263250C2211yescharacteristic, complemented
42#20263250C2211yescharacteristic, complemented
43#20263247C2211yescharacteristic, complemented
44#20263246C2211yescharacteristic
45#20263246C2211yescharacteristic, complemented
46#20263246C2211yescharacteristic, complemented
47C22⋊#13096802C2211yescharacteristic, complemented
48C22⋊#13096802C2211yescharacteristic, complemented
49#20263237C2211yescharacteristic, complemented
50C2×#41×C197C2211yescharacteristic, complemented
51C22×Q8×C197C2211yescharacteristic, complemented
52#124×C197—21no—
53#124×C197C2211yescharacteristic, complemented
54C22⋊Q8×C197—21no—
55C22⋊Q8×C197C2211yescharacteristic, complemented
56#121×C197—21no—
57#121×C197C2211yescharacteristic, complemented
58C4×Q8×C197C2211yescharacteristic, complemented
59C4×D4×C197—21no—
60C4×D4×C197C2211yescharacteristic
61C23×D197—21no—
62C23×D197—21no—
63C2×D788—22no—
64C2×D788—22no—
65C2×D788—21no—
66C2×D788—21no—
67C2×D788—21no—
68C2×D788—21no—
69C2×D788—41no—
70C2×D788—41no—
71C2×D788C2311yescharacteristic
72C2×D788C2311yescharacteristic
73C2×D788C2311yescharacteristic
74C2×C22⋊D197—21no—
75C2×C22⋊D197—21no—
76D4×D197—44no—
77C2×C4×D197—21no—
78#15691345—22no—
79#15691345—22no—
80#15691345—22no—
81#15691345—22no—
82#15691345—41no—
83#15691345D414yescomplemented
84#15691343—22no—
85#15691343—22no—
86#15691343—41no—
87#15691342—22no—
88C2×#13096802—22no—
89C2×#13096802C2311yescharacteristic
90C2×#13096802C2311yescharacteristic
91C2×#13096802C2311yescharacteristic
92C2×#13096802C2311yescharacteristic
93C4⋊Dic197C2311yescharacteristic
94#15691335—21no—
95#15691335—21no—
96#15691335—21no—
97#41×C197—22no—
98#41×C197—41no—
99C2×Q8×C197—21no—
100C2×Q8×C197—22no—
101C2×Q8×C197—41no—
102C2×Q8×C197C2311yescharacteristic
103C2×Q8×C197C2311yescharacteristic
104C2×D4×C197—21no—
105C2×D4×C197—21no—
106C2×D4×C197D411yescharacteristic
107C2×D4×C197C2311yescharacteristic
108C22×C4×C197—21no—
109C22×C4×C197—21no—
110C22×C4×C197C2311yescharacteristic
111C22×C4×C197C2311yescharacteristic
112C4⋊C4×C197—21no—
113C4⋊C4×C197—21no—
114C4⋊C4×C197D411yescharacteristic
115C4⋊C4×C197C2311yescharacteristic
116C22⋊C4×C197—21no—
117C22⋊C4×C197—21no—
118C22⋊C4×C197—21no—
119C22⋊C4×C197—21no—
120C22⋊C4×C197D411yescharacteristic
121C22⋊C4×C197D411yescharacteristic
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—
133D788—21no—
134D788—21no—
135D788—21no—
136D788—21no—
137D788—21no—
138D788—42no—
139D788—42no—
140D788—42no—
141D788—81no—
142D788C2×D412yes—
143C22⋊D197—42no—
144C22⋊D197—42no—
145C2×Dic197—21no—
146C4×D197—42no—
147#13096802—21no—
148#13096802—21no—
149#13096802—21no—
150#13096802C2×D412yescomplemented
151C23×C197—21no—
152C23×C197C2×D411yescharacteristic
153C23×C197C2×D411yescharacteristic
154Q8×C197—21no—
155Q8×C197—21no—
156Q8×C197—21no—
157Q8×C197—42no—
158Q8×C197C2×D412yescomplemented
159D4×C197—22no—
160D4×C197—21no—
161D4×C197—21no—
162D4×C197—42no—
163D4×C197—42no—
164C2×C4×C197—21no—
165C2×C4×C197—21no—
166C2×C4×C197—21no—
167C2×C4×C197—21no—
168C2×C4×C197—21no—
169C2×C4×C197—21no—
170C2×C4×C197—21no—
171C2×C4×C197—42no—
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—21no—
182C22×C197—21no—
183C22×C197—41no—
184C22×C197—41no—
185C22×C197—41no—
186C22×C197AGL(1,ℤ/8)12yescomplemented
187C22×C197C22×D411yescharacteristic
188C2×D197—22no—
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—22no—
202C4×C197—21no—
203C4×C197—21no—
204C4×C197—41no—
205C4×C197—41no—
206C4×C197C22×D411yescharacteristic
207C4×C197H(5,𝔽2)11yescharacteristic
208Dic197—41no—
209C2×C197—22no—
210C2×C197—41no—
211C2×C197—81no—
212C2×C197D4211yescharacteristic
213C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
214C2×C197#57711yescharacteristic
215D197—42no—
216D197—81no—
217D197—81no—
218C197#326511yescharacteristic, complemented, Sylow
219#3265—1971nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1971no—
221C22×SD16—1971no—
222#548D19711yescharacteristic, complemented
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#435—1971no—
235C2×#41C2×D19711yescharacteristic, complemented
236C22×Q8C2×D19711yescharacteristic, complemented
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×M16—1971no—
252C22×C8—1971no—
253#127—1971no—
254#124—21no—
255#124C2×D19711yescharacteristic, complemented
256C22⋊Q8—21no—
257C22⋊Q8C2×D19711yescharacteristic, complemented
258#121—21no—
259#121—3941no—
260#121—1971no—
261#121—1971no—
262#121C2×D19711yescharacteristic, complemented
263#120—3941no—
264#120—3941no—
265C4×Q8C2×D19711yescharacteristic, complemented
266C4×D4—21no—
267C4×D4—1971no—
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—3941no—
281C24—3941no—
282#41—22no—
283#41—41no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8C22×D19711yescharacteristic
288C2×Q8C22×D19711yescharacteristic
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—7881no—
292C2×D4—7881no—
293C2×D4—7884no—
294C2×D4—3942no—
295C2×D4—3942no—
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×D4C22⋊D19711yescharacteristic
306C2×D4C22×D19711yescharacteristic
307C22×C4—21no—
308C22×C4—21no—
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—21no—
328C4⋊C4—21no—
329C4⋊C4—1971no—
330C4⋊C4C22⋊D19711yescharacteristic
331C4⋊C4C22×D19711yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—3941no—
337C22⋊C4—3941no—
338C22⋊C4—3941no—
339C22⋊C4C22⋊D19711yescharacteristic
340C22⋊C4C22⋊D19711yescharacteristic
341C42—21no—
342C42C22×D19711yescharacteristic
343C23—21no—
344C23—7882no—
345C23—7882no—
346C23—7882no—
347C23—7882no—
348C23—15761no—
349C23—15761no—
350C23—3941no—
351C23—3941no—
352C23—3941no—
353C23C2×C22⋊D19711yescharacteristic
354C23C2×C22⋊D19711yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—21no—
358Q8—42no—
359Q8D4×D19712yescomplemented
360D4—22no—
361D4—21no—
362D4—21no—
363D4—42no—
364D4—42no—
365D4—7882no—
366D4—7882no—
367D4—7882no—
368D4—7882no—
369D4—7882no—
370D4—15761no—
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—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—7882no—
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—41no—
403C22—41no—
404C22—41no—
405C22—7881no—
406C22—7881no—
407C22—7881no—
408C22—7881no—
409C22—7881no—
410C22—7881no—
411C22—7881no—
412C22—7881no—
413C22—15764no—
414C22—15764no—
415C22—3942no—
416C22#2026335112yescomplemented
417C22C22×C22⋊D19711yescharacteristic
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—22no—
423C4—41no—
424C4—41no—
425C4—7881no—
426C4C2×D4×D19711yescharacteristic
427C4#2026337111yescharacteristic
428C2—22no—
429C2—41no—
430C2—81no—
431C2—7882no—
432C2—15761no—
433C2—15761no—
434C2D4×C22⋊D19711yescharacteristic
435C2C2×#2026335111yescharacteristic
436C2#2716188711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#548×C197C2
socleC22×C197C22×D4
hypercenter#548D197
hypocenterC197#3265
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C2×#20263351C2semidirect
2. C22×#15691345C2semidirect
3. D4×D788C2semidirect
4. #27161469C2semidirect
5. #27161327C2semidirect
6. #27161313C2semidirect
7. #27161291C2semidirect
8. #27161279C2semidirect
9. #27161229C2semidirect
10. #27161224C2semidirect
11. #27161218C2semidirect
12. #27161212C2semidirect
13. #27161177C2semidirect
14. #27161151C2semidirect
15. #548×C197C2semidirect
23. C2×#15691345C22semidirect
24. C2×#15691345C22semidirect
25. C2×#15691345C22semidirect
26. C2×#15691345C22semidirect
28. C2×#15691343C22semidirect
29. C2×#15691343C22semidirect
30. C2×#15691342C22semidirect
31. C22×#13096802C22semidirect
39. #20263259C22semidirect
40. C4⋊#13096802C22semidirect
41. #20263250C22semidirect
42. #20263250C22semidirect
43. #20263247C22semidirect
45. #20263246C22semidirect
46. #20263246C22semidirect
47. C22⋊#13096802C22semidirect
48. C22⋊#13096802C22semidirect
49. #20263237C22semidirect
50. C2×#41×C197C22semidirect
51. C22×Q8×C197C22semidirect
53. #124×C197C22semidirect
55. C22⋊Q8×C197C22semidirect
57. #121×C197C22semidirect
58. C4×Q8×C197C22semidirect
83. #15691345D4semidirect
150. #13096802C2×D4semidirect
158. Q8×C197C2×D4semidirect
186. C22×C197AGL(1,ℤ/8)semidirect
218. C197#3265semidirect
222. #548D197semidirect
235. C2×#41C2×D197semidirect
236. C22×Q8C2×D197semidirect
255. #124C2×D197semidirect
257. C22⋊Q8C2×D197semidirect
262. #121C2×D197semidirect
265. C4×Q8C2×D197semidirect
359. Q8D4×D197semidirect
416. C22#20263351semidirect

Maximal subgroups

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

subgroupindex
C2×#202633512
C22×#156913452
D4×D7882
#271614692
#271613272
#271613132
#271612912
#271612792
#271612292
#271612242
#271612182
#271612122
#271611772
#271611512
#548×C1972
#3265197