Finite Group Databasebeta

← #38754187

Subgroups — #38754187

Order 25216.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263350C211yesmaximal, characteristic, complemented
2C2×D8×D197C211yesmaximal, characteristic, complemented
3#27161724C211yesmaximal, characteristic, complemented
4#27161714C211yesmaximal, characteristic, complemented
5#27161466C211yesmaximal, characteristic, complemented
6#27161331C211yesmaximal, characteristic, complemented
7#27161307C211yesmaximal, characteristic, complemented
8#27161295C211yesmaximal, characteristic, complemented
9#27161231C211yesmaximal, characteristic, complemented
10#27161221C211yesmaximal, characteristic, complemented
11#27161214C211yesmaximal, characteristic, complemented
12#27161211C211yesmaximal, characteristic, complemented
13#27161189C211yesmaximal, characteristic, complemented
14#27161149C211yesmaximal, characteristic, complemented
15#459×C197C211yesmaximal, characteristic, complemented
16C2×#15691357C2211yescharacteristic, complemented
17C2×D4×D197—21no—
18C2×D4×D197C2211yescharacteristic
19C2×D4×D197C2211yescharacteristic, complemented
20C2×D4×D197C2211yescharacteristic, complemented
21#20263350—24no—
22C2×#15691347C2211yescharacteristic, complemented
23C2×#15691346C2211yescharacteristic, complemented
24C2×#15691346C2211yescharacteristic, complemented
25C2×D1576C2211yescharacteristic
26C2×#15691343C2211yescharacteristic
27C2×#15691343C2211yescharacteristic, complemented
28C2×#15691343C2211yescharacteristic, complemented
29D8×D197—24no—
30C2×#15691341C2211yescharacteristic, complemented
31C2×C8×D197C2211yescharacteristic, complemented
32#20263314C2211yescharacteristic, complemented
33#20263310—21no—
34#20263306C2211yescharacteristic, complemented
35#20263301C2211yescharacteristic, complemented
36#20263300—21no—
37#20263297—21no—
38#20263297C2211yescharacteristic
39#20263297C2211yescharacteristic, complemented
40#20263296—21no—
41#20263294—21no—
42#20263294—21no—
43C4×D788C2211yescharacteristic
44C4×D788C2211yescharacteristic, complemented
45C2×#15691335—21no—
46#20263257C2211yescharacteristic, complemented
47#20263251C2211yescharacteristic, complemented
48#20263248C2211yescharacteristic, complemented
49#20263247C2211yescharacteristic, complemented
50#20263246C2211yescharacteristic, complemented
51#20263238C2211yescharacteristic, complemented
52#20263238C2211yescharacteristic, complemented
53#20263237C2211yescharacteristic, complemented
54C2×D8×C197C2211yescharacteristic, complemented
55C2×D8×C197C2211yescharacteristic, complemented
56#127×C197C2211yescharacteristic, complemented
57C4×D4×C197C2211yescharacteristic, complemented
58C4⋊C8×C197C2211yescharacteristic, complemented
59#102×C197C2211yescharacteristic, complemented
60#102×C197C2211yescharacteristic, complemented
61C23×D197—21no—
62C23×D197—21no—
63C23×D197—21no—
64#15691357—22no—
65#15691357—41no—
66C2×Dic394C2311yescharacteristic
67C2×D788—21no—
68C2×D788D411yescharacteristic
69C2×D788C2311yescharacteristic
70C2×D788C2311yescharacteristic
71C2×C22⋊D197—21no—
72C2×C22⋊D197—21no—
73C2×C22⋊D197—21no—
74C2×C22⋊D197—21no—
75C2×C22⋊D197—21no—
76C2×C22⋊D197—21no—
77D4×D197—22no—
78D4×D197—22no—
79D4×D197—22no—
80D4×D197—41no—
81D4×D197—41no—
82D4×D197—41no—
83D4×D197—44no—
84C22×Dic197—21no—
85C2×C4×D197—21no—
86C2×C4×D197C2311yescharacteristic
87C2×C4×D197C2311yescharacteristic
88#15691347—22no—
89#15691346—22no—
90#15691346—41no—
91D1576—22no—
92#15691343—22no—
93#15691343—41no—
94#15691343D414yescomplemented
95#15691341—22no—
96C2×#13096802C2311yescharacteristic
97C2×#13096802C2311yescharacteristic
98C8×D197—22no—
99C4⋊Dic197—21no—
100C4⋊Dic197D411yescharacteristic
101C4⋊Dic197C2311yescharacteristic
102C22⋊Dic197—21no—
103C22⋊Dic197—21no—
104#15691335—21no—
105#15691335—21no—
106#15691335—21no—
107#15691335—21no—
108#15691335—21no—
109#15691335D411yescharacteristic
110#15691335D411yescharacteristic
111C2×D4×C197—21no—
112C2×D4×C197—21no—
113C2×D4×C197C2311yescharacteristic
114C2×D4×C197C2311yescharacteristic
115C2×D4×C197C2311yescharacteristic
116C22×C4×C197—21no—
117D8×C197—22no—
118D8×C197—22no—
119C2×C8×C197C2311yescharacteristic
120C2×C8×C197C2311yescharacteristic
121C4⋊C4×C197C2311yescharacteristic
122C22⋊C4×C197—21no—
123C42×C197C2311yescharacteristic
124C22×D197—21no—
125C22×D197—21no—
126C22×D197—42no—
127C22×D197—42no—
128C22×D197—42no—
129C22×D197—42no—
130C22×D197—42no—
131C22×D197—42no—
132C22×D197—81no—
133C22×D197—81no—
134C22×D197—81no—
135C22×D197C2×D411yescharacteristic
136C22×D197C2×D411yescharacteristic
137Dic394—21no—
138Dic394—21no—
139D788—22no—
140D788—21no—
141D788—21no—
142D788—21no—
143D788—81no—
144D788C2×D412yes—
145C22⋊D197—42no—
146C22⋊D197—42no—
147C22⋊D197—42no—
148C22⋊D197—42no—
149C22⋊D197—42no—
150C22⋊D197—42no—
151C2×Dic197—21no—
152C2×Dic197—21no—
153C2×Dic197—42no—
154C2×Dic197C2×D411yescharacteristic
155C2×Dic197C2×D411yescharacteristic
156C4×D197—21no—
157C4×D197—21no—
158C4×D197—21no—
159C4×D197—21no—
160C4×D197—42no—
161#13096802—21no—
162#13096802C2×D412yescomplemented
163C23×C197—21no—
164C23×C197—21no—
165C23×C197—21no—
166D4×C197—21no—
167D4×C197—21no—
168D4×C197—21no—
169D4×C197—21no—
170D4×C197—21no—
171D4×C197—42no—
172D4×C197—81no—
173D4×C197C2×D412yescomplemented
174C2×C4×C197—21no—
175C2×C4×C197—81no—
176C2×C4×C197C2×D411yescharacteristic
177C2×C4×C197C2×D411yescharacteristic
178C2×C4×C197C2411yescharacteristic
179C8×C197—21no—
180C8×C197—21no—
181C22×C197—22no—
182C22×C197—41no—
183C22×C197—41no—
184C22×C197—41no—
185C22×C197—41no—
186C22×C197—41no—
187C22×C197—41no—
188C22×C197—41no—
189C22×C197—41no—
190C22×C197C22×D411yescharacteristic
191C2×D197—21no—
192C2×D197—21no—
193C2×D197—21no—
194C2×D197—22no—
195C2×D197—21no—
196C2×D197—21no—
197C2×D197—41no—
198C2×D197—41no—
199C2×D197—41no—
200C2×D197—41no—
201C2×D197—41no—
202C2×D197—84no—
203C2×D197—84no—
204C2×D197—84no—
205C2×D197AGL(1,ℤ/8)12yescomplemented
206C4×C197—21no—
207C4×C197—21no—
208C4×C197—41no—
209C4×C197C22×D411yescharacteristic
210C4×C197H(5,𝔽2)11yescharacteristic
211Dic197—21no—
212Dic197—21no—
213Dic197—41no—
214Dic197—41no—
215C2×C197—42no—
216C2×C197—81no—
217C2×C197—81no—
218C2×C197D4211yescharacteristic
219C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C197#57611yescharacteristic
221D197—22no—
222D197—41no—
223D197—42no—
224D197—81no—
225C197#326411yescharacteristic, complemented, Sylow
226#3264—1971nomaximal, Sylow
227C2×AGL(1,ℤ/8)—1971no—
228C22×D8—1971no—
229#546—1971no—
230D42—1971no—
231#496—1971no—
232#469—1971no—
233#465—1971no—
234#463—1971no—
235#459D19711yescharacteristic, complemented
236#450—1971no—
237#449—1971no—
238#447—1971no—
239#447—1971no—
240#442—1971no—
241#435—1971no—
242C2×#41—1971no—
243C22×D4—3941no—
244C22×D4—1971no—
245C22×D4—1971no—
246C22×D4—1971no—
247AGL(1,ℤ/8)—3944no—
248C2×SD16—1971no—
249C2×SD16—1971no—
250C2×SD16—1971no—
251C2×D8—3944no—
252C2×D8—1971no—
253C2×D8—1971no—
254C2×D8—1971no—
255C2×D8—1971no—
256C2×D8C2×D19711yescharacteristic, complemented
257C2×D8C2×D19711yescharacteristic, complemented
258C2×M16—1971no—
259C22×C8—1971no—
260#127C2×D19711yescharacteristic, complemented
261#124—1971no—
262#123—3941no—
263C22⋊Q8—1971no—
264#121—3941no—
265#121—3941no—
266#121—1971no—
267#121—1971no—
268#121—1971no—
269#120—3941no—
270#120—3941no—
271#120—3941no—
272C4×D4—1971no—
273C4×D4—1971no—
274C4×D4C2×D19711yescharacteristic, complemented
275C2×C22⋊C4—3941no—
276#106—1971no—
277C4⋊C8C2×D19711yescharacteristic, complemented
278#103—1971no—
279#102—1971no—
280#102—1971no—
281#102—1971no—
282#102C2×D19711yescharacteristic, complemented
283#102C2×D19711yescharacteristic, complemented
284C22⋊C8—1971no—
285C22⋊C8—1971no—
286#97—1971no—
287C24—3941no—
288C24—3941no—
289C24—3941no—
290#41—7881no—
291#41—3942no—
292C2×Q8—1971no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—7884no—
296C2×D4—7881no—
297C2×D4—7881no—
298C2×D4—7881no—
299C2×D4—3941no—
300C2×D4—3942no—
301C2×D4—3942no—
302C2×D4—3941no—
303C2×D4—3941no—
304C2×D4—3941no—
305C2×D4—3942no—
306C2×D4—3941no—
307C2×D4—3941no—
308C2×D4—3941no—
309C2×D4—1971no—
310C2×D4—1971no—
311C2×D4—1971no—
312C2×D4C22×D19711yescharacteristic
313C2×D4C22×D19711yescharacteristic
314C2×D4C22×D19711yescharacteristic
315C22×C4—21no—
316C22×C4—3941no—
317C22×C4—3941no—
318C22×C4—1971no—
319C22×C4—1971no—
320SD16—7881no—
321SD16—3942no—
322SD16—3942no—
323D8—22no—
324D8—22no—
325D8—7881no—
326D8—3942no—
327D8—3942no—
328D8—1974no—
329M16—3942no—
330C2×C8—3942no—
331C2×C8—1971no—
332C2×C8—1971no—
333C2×C8C22×D19711yescharacteristic
334C2×C8C22×D19711yescharacteristic
335C4⋊C4—3941no—
336C4⋊C4—1971no—
337C4⋊C4—1971no—
338C4⋊C4C22×D19711yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—3941no—
341C22⋊C4—3941no—
342C22⋊C4—3941no—
343C22⋊C4—3941no—
344C22⋊C4—3941no—
345C22⋊C4—3941no—
346C22⋊C4—3941no—
347C22⋊C4—1971no—
348C22⋊C4—1971no—
349C42C22×D19711yescharacteristic
350C23—21no—
351C23—21no—
352C23—21no—
353C23—7882no—
354C23—7882no—
355C23—7882no—
356C23—7882no—
357C23—7882no—
358C23—7882no—
359C23—15761no—
360C23—15761no—
361C23—15761no—
362C23—3941no—
363C23—3941no—
364C23—1971no—
365C23—1971no—
366Q8—3941no—
367Q8—3941no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—42no—
374D4—81no—
375D4—7882no—
376D4—7882no—
377D4—7882no—
378D4—7882no—
379D4—7882no—
380D4—7882no—
381D4—15761no—
382D4—3942no—
383D4—3941no—
384D4—3941no—
385D4—3941no—
386D4—1972no—
387D4D4×D19712yescomplemented
388C2×C4—21no—
389C2×C4—81no—
390C2×C4—7882no—
391C2×C4—7882no—
392C2×C4—3941no—
393C2×C4—3941no—
394C2×C4—3941no—
395C2×C4—3941no—
396C2×C4—3941no—
397C2×C4—3941no—
398C2×C4—1971no—
399C2×C4—1971no—
400C2×C4D4×D19711yescharacteristic
401C2×C4D4×D19711yescharacteristic
402C2×C4C23×D19711yescharacteristic
403C8—21no—
404C8—21no—
405C8—3941no—
406C8—1972no—
407C22—22no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—7881no—
417C22—7881no—
418C22—7881no—
419C22—7881no—
420C22—7881no—
421C22—15764no—
422C22—15764no—
423C22—15764no—
424C22—3941no—
425C22—3941no—
426C22—3941no—
427C22—3942no—
428C22—3941no—
429C22—3941no—
430C22—1972no—
431C22C2×D4×D19711yescharacteristic
432C4—21no—
433C4—21no—
434C4—41no—
435C4—7881no—
436C4—7881no—
437C4—3941no—
438C4—3941no—
439C4C2×D4×D19711yescharacteristic
440C4#2026337211yescharacteristic
441C2—42no—
442C2—81no—
443C2—81no—
444C2—7882no—
445C2—7881no—
446C2—15761no—
447C2—3942no—
448C2#2716171311yescharacteristic
449C2AGL(1,ℤ/8)×D19711yescharacteristic
450C2#2716188011yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#459×C197C2
socleC22×C197C22×D4
hypercenter#459D197
hypocenterC197#3264
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. C2×#20263350C2semidirect
2. C2×D8×D197C2semidirect
3. #27161724C2semidirect
4. #27161714C2semidirect
5. #27161466C2semidirect
6. #27161331C2semidirect
7. #27161307C2semidirect
8. #27161295C2semidirect
9. #27161231C2semidirect
10. #27161221C2semidirect
11. #27161214C2semidirect
12. #27161211C2semidirect
13. #27161189C2semidirect
14. #27161149C2semidirect
15. #459×C197C2semidirect
16. C2×#15691357C22semidirect
19. C2×D4×D197C22semidirect
20. C2×D4×D197C22semidirect
22. C2×#15691347C22semidirect
23. C2×#15691346C22semidirect
24. C2×#15691346C22semidirect
27. C2×#15691343C22semidirect
28. C2×#15691343C22semidirect
30. C2×#15691341C22semidirect
31. C2×C8×D197C22semidirect
32. #20263314C22semidirect
34. #20263306C22semidirect
35. #20263301C22semidirect
39. #20263297C22semidirect
44. C4×D788C22semidirect
46. #20263257C22semidirect
47. #20263251C22semidirect
48. #20263248C22semidirect
49. #20263247C22semidirect
50. #20263246C22semidirect
51. #20263238C22semidirect
52. #20263238C22semidirect
53. #20263237C22semidirect
54. C2×D8×C197C22semidirect
55. C2×D8×C197C22semidirect
56. #127×C197C22semidirect
57. C4×D4×C197C22semidirect
58. C4⋊C8×C197C22semidirect
59. #102×C197C22semidirect
60. #102×C197C22semidirect
94. #15691343D4semidirect
162. #13096802C2×D4semidirect
173. D4×C197C2×D4semidirect
205. C2×D197AGL(1,ℤ/8)semidirect
225. C197#3264semidirect
235. #459D197semidirect
256. C2×D8C2×D197semidirect
257. C2×D8C2×D197semidirect
260. #127C2×D197semidirect
274. C4×D4C2×D197semidirect
277. C4⋊C8C2×D197semidirect
282. #102C2×D197semidirect
283. #102C2×D197semidirect
387. D4D4×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633502
C2×D8×D1972
#271617242
#271617142
#271614662
#271613312
#271613072
#271612952
#271612312
#271612212
#271612142
#271612112
#271611892
#271611492
#459×C1972
#3264197