Finite Group Databasebeta

← #38754242

Subgroups — #38754242

Order 25216.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263354C211yesmaximal, characteristic, complemented
2C2×#20263348C211yesmaximal, characteristic, complemented
3#27161734C211yesmaximal, characteristic, complemented
4#27161725C211yesmaximal, characteristic, complemented
5#27161473C211yesmaximal, characteristic, complemented
6#27161337C211yesmaximal, characteristic, complemented
7#27161308C211yesmaximal, characteristic, complemented
8#27161295C211yesmaximal, characteristic, complemented
9#27161285C211yesmaximal, characteristic, complemented
10#27161243C211yesmaximal, characteristic, complemented
11#27161243C211yesmaximal, characteristic, complemented
12#27161232C211yesmaximal, characteristic, complemented
13#27161226C211yesmaximal, characteristic, complemented
14#27161192C211yesmaximal, characteristic, complemented
15#435×C197C211yesmaximal, characteristic, complemented
16C2×#15691358C2211yescharacteristic
17C2×#15691358C2211yescharacteristic
18C2×#15691358C2211yescharacteristic, complemented
19C22×D788C2211yescharacteristic
20#20263354—24no—
21#20263348—24no—
22C2×Dic788C2211yescharacteristic, complemented
23C2×Dic788C2211yescharacteristic, complemented
24C2×#15691347C2211yescharacteristic
25C2×#15691347C2211yescharacteristic
26C2×#15691347C2211yescharacteristic
27C2×#15691347C2211yescharacteristic
28C2×#15691347C2211yescharacteristic, complemented
29C2×D1576C2211yescharacteristic
30C2×D1576C2211yescharacteristic
31#20263326C2211yescharacteristic, complemented
32#20263314C2211yescharacteristic
33#20263311—21no—
34#20263308—21no—
35#20263308—21no—
36#20263305C2211yescharacteristic
37#20263305C2211yescharacteristic
38#20263305C2211yescharacteristic, complemented
39#20263300C2211yescharacteristic
40#20263299—21no—
41#20263297—21no—
42#20263294—21no—
43C4×D788C2211yescharacteristic
44C4×D788C2211yescharacteristic, complemented
45C4⋊C4×D197—21no—
46C22⋊C4×D197—21no—
47#20263258C2211yescharacteristic, complemented
48#20263251C2211yescharacteristic, complemented
49#20263251C2211yescharacteristic, complemented
50#20263251C2211yescharacteristic, complemented
51#20263248C2211yescharacteristic
52#20263248C2211yescharacteristic
53#20263248C2211yescharacteristic, complemented
54C2×M16×C197C2211yescharacteristic
55C22×C8×C197C2211yescharacteristic, complemented
56C4×D4×C197C2211yescharacteristic
57C4⋊C8×C197C2211yescharacteristic, complemented
58C22⋊C8×C197C2211yescharacteristic
59C22⋊C8×C197C2211yescharacteristic, complemented
60#97×C197C2211yescharacteristic, complemented
61C23×D197—21no—
62#15691358—22no—
63#15691358—22no—
64#15691358—22no—
65#15691358—41no—
66#15691358—41no—
67#15691358—41no—
68C2×Dic394C2311yescharacteristic
69C2×Dic394C2311yescharacteristic
70C2×Dic394C2311yescharacteristic
71C2×D788—22no—
72C2×D788—21no—
73C2×D788—41no—
74C2×D788C2311yescharacteristic
75C2×D788C2311yescharacteristic
76C2×D788C2311yescharacteristic
77C2×C22⋊D197—21no—
78C2×C22⋊D197—21no—
79C2×C22⋊D197—21no—
80C2×C22⋊D197—21no—
81C2×C4×D197—21no—
82C2×C4×D197—21no—
83C2×C4×D197—21no—
84C2×C4×D197—21no—
85C2×C4×D197—21no—
86Dic788—22no—
87Dic788—41no—
88#15691347—22no—
89#15691347—22no—
90#15691347—22no—
91#15691347—22no—
92#15691347—41no—
93D1576—22no—
94D1576D414yescomplemented
95C4⋊Dic197—21no—
96C4⋊Dic197—21no—
97C4⋊Dic197C2311yescharacteristic
98C4⋊Dic197C2311yescharacteristic
99#15691337—21no—
100#15691337—21no—
101#15691337—21no—
102C22⋊Dic197—21no—
103C22⋊Dic197—21no—
104C22⋊Dic197—21no—
105#15691335—21no—
106#15691335—21no—
107#15691335—21no—
108#15691335—21no—
109#15691335—21no—
110C2×D4×C197D411yescharacteristic
111C22×C4×C197C2311yescharacteristic
112C22×C4×C197C2311yescharacteristic
113M16×C197—22no—
114C2×C8×C197—22no—
115C2×C8×C197C2311yescharacteristic
116C2×C8×C197C2311yescharacteristic
117C2×C8×C197C2311yescharacteristic
118C2×C8×C197C2311yescharacteristic
119C4⋊C4×C197D411yescharacteristic
120C22⋊C4×C197D411yescharacteristic
121C22⋊C4×C197D411yescharacteristic
122C42×C197C2311yescharacteristic
123C22×D197—21no—
124C22×D197—21no—
125C22×D197—21no—
126C22×D197—42no—
127C22×D197—42no—
128C22×D197—81no—
129Dic394—21no—
130Dic394—21no—
131Dic394—21no—
132Dic394—21no—
133Dic394—21no—
134Dic394—21no—
135D788—21no—
136D788—21no—
137D788—21no—
138D788—21no—
139D788—42no—
140D788C2×D412yes—
141D788C2×D412yes—
142C22⋊D197—42no—
143C22⋊D197—42no—
144C22⋊D197—42no—
145C22⋊D197—81no—
146C2×Dic197—21no—
147C2×Dic197—21no—
148C2×Dic197—21no—
149C2×Dic197—21no—
150C2×Dic197—21no—
151C4×D197—42no—
152C4×D197—42no—
153C4×D197—42no—
154C4×D197—42no—
155C4×D197—81no—
156C23×C197C2×D411yescharacteristic
157C23×C197C2×D411yescharacteristic
158D4×C197—41no—
159C2×C4×C197—21no—
160C2×C4×C197—21no—
161C2×C4×C197—21no—
162C2×C4×C197—21no—
163C2×C4×C197C2×D411yescharacteristic
164C2×C4×C197C2×D411yescharacteristic
165C2×C4×C197C2×D411yescharacteristic
166C2×C4×C197C2×D411yescharacteristic
167C2×C4×C197C2411yescharacteristic
168C8×C197—21no—
169C8×C197—21no—
170C8×C197—21no—
171C8×C197C2×D412yes—
172C22×C197—21no—
173C22×C197—21no—
174C22×C197—21no—
175C22×C197—21no—
176C22×C197—21no—
177C22×C197—21no—
178C22×C197C22×D411yescharacteristic
179C2×D197—22no—
180C2×D197—22no—
181C2×D197—41no—
182C2×D197—41no—
183C2×D197—41no—
184C2×D197—41no—
185C2×D197—41no—
186C2×D197—41no—
187C2×D197—41no—
188C2×D197—84no—
189C4×C197—21no—
190C4×C197—21no—
191C4×C197—21no—
192C4×C197—21no—
193C4×C197C22×D411yescharacteristic
194C4×C197H(5,𝔽2)11yescharacteristic
195Dic197—41no—
196Dic197—41no—
197Dic197—41no—
198Dic197—41no—
199Dic197—41no—
200C2×C197—41no—
201C2×C197—41no—
202C2×C197D4211yescharacteristic
203C2×C197#57511yescharacteristic
204C2×C197#57711yescharacteristic
205D197—42no—
206D197—42no—
207D197—81no—
208C197#326711yescharacteristic, complemented, Sylow
209#3267—1971nomaximal, Sylow
210C2×AGL(1,ℤ/8)—1971no—
211C2×#135—1971no—
212#547—1971no—
213#546—1971no—
214#497—1971no—
215#470—1971no—
216#465—1971no—
217#463—1971no—
218#461—1971no—
219#452—1971no—
220#452—1971no—
221#450—1971no—
222#449—1971no—
223#442—1971no—
224#435D19711yescharacteristic, complemented
225C2×#41—1971no—
226C2×#41—1971no—
227C2×#41—1971no—
228C22×D4—1971no—
229AGL(1,ℤ/8)—3944no—
230#135—3944no—
231C2×Q16—1971no—
232C2×Q16—1971no—
233C2×SD16—1971no—
234C2×SD16—1971no—
235C2×SD16—1971no—
236C2×SD16—1971no—
237C2×SD16—1971no—
238C2×D8—1971no—
239C2×D8—1971no—
240C2×M16C2×D19711yescharacteristic
241C22×C8C2×D19711yescharacteristic, complemented
242C4⋊Q8—1971no—
243#124—1971no—
244#123—3941no—
245#123—3941no—
246#123—3941no—
247C22⋊Q8—1971no—
248C22⋊Q8—1971no—
249C22⋊Q8—1971no—
250#121—3941no—
251#121—3941no—
252#121—1971no—
253#120—3941no—
254C4×D4—1971no—
255C4×D4—1971no—
256C4×D4C2×D19711yescharacteristic
257C2×C4⋊C4—3941no—
258C2×C22⋊C4—3941no—
259#106—1971no—
260C4⋊C8C2×D19711yescharacteristic, complemented
261#103—1971no—
262#103—1971no—
263#103—1971no—
264#102—1971no—
265#102—1971no—
266#102—1971no—
267C22⋊C8C2×D19711yescharacteristic
268C22⋊C8C2×D19711yescharacteristic, complemented
269#97C2×D19711yescharacteristic, complemented
270C24—3941no—
271#41—7881no—
272#41—7881no—
273#41—7881no—
274#41—3942no—
275#41—3942no—
276#41—3942no—
277C2×Q8—1971no—
278C2×Q8—1971no—
279C2×Q8—1971no—
280C2×D4—7881no—
281C2×D4—3942no—
282C2×D4—3941no—
283C2×D4—3941no—
284C2×D4—3941no—
285C2×D4—3941no—
286C2×D4—3941no—
287C2×D4—1971no—
288C2×D4—1971no—
289C2×D4—1971no—
290C2×D4D78811yescharacteristic
291C22×C4—3941no—
292C22×C4—3941no—
293C22×C4—3941no—
294C22×C4—3941no—
295C22×C4—3941no—
296C22×C4C22×D19711yescharacteristic
297C22×C4C22×D19711yescharacteristic
298Q16—7881no—
299Q16—3942no—
300SD16—7881no—
301SD16—3942no—
302SD16—3942no—
303SD16—3942no—
304SD16—3942no—
305D8—3942no—
306D8—1974no—
307M16—22no—
308C2×C8—22no—
309C2×C8C22×D19711yescharacteristic
310C2×C8C22×D19711yescharacteristic
311C2×C8C22×D19711yescharacteristic
312C2×C8C22×D19711yescharacteristic
313C4⋊C4—3941no—
314C4⋊C4—3941no—
315C4⋊C4—3941no—
316C4⋊C4—3941no—
317C4⋊C4—3941no—
318C4⋊C4—1971no—
319C4⋊C4—1971no—
320C4⋊C4D78811yescharacteristic
321C22⋊C4—3941no—
322C22⋊C4—3941no—
323C22⋊C4—3941no—
324C22⋊C4—3941no—
325C22⋊C4—3941no—
326C22⋊C4—3941no—
327C22⋊C4—3941no—
328C22⋊C4—3941no—
329C22⋊C4D78811yescharacteristic
330C22⋊C4D78811yescharacteristic
331C42C22×D19711yescharacteristic
332C23—7882no—
333C23—7882no—
334C23—15761no—
335C23—3941no—
336C23—3941no—
337C23—3941no—
338C23C2×D78811yescharacteristic
339C23C2×D78811yescharacteristic
340Q8—3941no—
341Q8—3941no—
342Q8—3941no—
343Q8—3941no—
344Q8—3941no—
345Q8—3941no—
346D4—41no—
347D4—7882no—
348D4—7882no—
349D4—7882no—
350D4—7882no—
351D4—15761no—
352D4—3941no—
353D4—3941no—
354D4—3941no—
355D4—3941no—
356D4—1972no—
357D4—1972no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—7882no—
363C2×C4—7882no—
364C2×C4—7882no—
365C2×C4—7882no—
366C2×C4—15761no—
367C2×C4—3941no—
368C2×C4—3941no—
369C2×C4—3941no—
370C2×C4—3941no—
371C2×C4—3941no—
372C2×C4C2×D78811yescharacteristic
373C2×C4C2×D78811yescharacteristic
374C2×C4C2×D78811yescharacteristic
375C2×C4C2×D78811yescharacteristic
376C2×C4C23×D19711yescharacteristic
377C8—21no—
378C8—21no—
379C8—21no—
380C8D4×D19712yes—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—7881no—
388C22—7881no—
389C22—7881no—
390C22—7881no—
391C22—7881no—
392C22—7881no—
393C22—7881no—
394C22—15764no—
395C22—3942no—
396C22—3942no—
397C22C22×D78811yescharacteristic
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—7881no—
403C4—7881no—
404C4—7881no—
405C4—7881no—
406C4—7881no—
407C4C2×D4×D19711yescharacteristic
408C4#2026337211yescharacteristic
409C2—41no—
410C2—41no—
411C2—7882no—
412C2—7882no—
413C2—15761no—
414C2D4×D78811yescharacteristic
415C2#2716187911yescharacteristic
416C2#2716189111yescharacteristic
subgroupG/H
centerC22C22×D788
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#435×C197C2
socleC22×C197C22×D4
hypercenter#435D197
hypocenterC197#3267
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C2×#20263354C2semidirect
2. C2×#20263348C2semidirect
3. #27161734C2semidirect
4. #27161725C2semidirect
5. #27161473C2semidirect
6. #27161337C2semidirect
7. #27161308C2semidirect
8. #27161295C2semidirect
9. #27161285C2semidirect
10. #27161243C2semidirect
11. #27161243C2semidirect
12. #27161232C2semidirect
13. #27161226C2semidirect
14. #27161192C2semidirect
15. #435×C197C2semidirect
18. C2×#15691358C22semidirect
22. C2×Dic788C22semidirect
23. C2×Dic788C22semidirect
28. C2×#15691347C22semidirect
31. #20263326C22semidirect
38. #20263305C22semidirect
44. C4×D788C22semidirect
47. #20263258C22semidirect
48. #20263251C22semidirect
49. #20263251C22semidirect
50. #20263251C22semidirect
53. #20263248C22semidirect
55. C22×C8×C197C22semidirect
57. C4⋊C8×C197C22semidirect
59. C22⋊C8×C197C22semidirect
60. #97×C197C22semidirect
94. D1576D4semidirect
208. C197#3267semidirect
224. #435D197semidirect
241. C22×C8C2×D197semidirect
260. C4⋊C8C2×D197semidirect
268. C22⋊C8C2×D197semidirect
269. #97C2×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633542
C2×#202633482
#271617342
#271617252
#271614732
#271613372
#271613082
#271612952
#271612852
#271612432
#271612322
#271612262
#271611922
#435×C1972
#3267197