Finite Group Databasebeta

← #38754227

Subgroups — #38754227

Order 25216.

494 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263359C211yesmaximal, characteristic, complemented
2C22×#15691347C211yesmaximal, characteristic, complemented
3D4×Dic394C211yesmaximal, characteristic, complemented
4#27161725C211yesmaximal, characteristic
5#27161473C211yesmaximal, characteristic, complemented
6#27161325C211yesmaximal, characteristic, complemented
7#27161320C211yesmaximal, characteristic, complemented
8#27161302C211yesmaximal, characteristic, complemented
9#27161285C211yesmaximal, characteristic, complemented
10#27161243C211yesmaximal, characteristic, complemented
11#27161237C211yesmaximal, characteristic, complemented
12#27161232C211yesmaximal, characteristic, complemented
13#27161220C211yesmaximal, characteristic, complemented
14#27161181C211yesmaximal, characteristic, complemented
15#435×C197C211yesmaximal, characteristic, complemented
16C2×#15691358C2211yescharacteristic
17C22×Dic394C2211yescharacteristic
18C22×Dic394C2211yescharacteristic, complemented
19C22×D788C2211yescharacteristic
20#20263359—24no—
21C2×Dic788C2211yescharacteristic, complemented
22C2×Dic788C2211yescharacteristic, complemented
23C2×Dic788C2211yescharacteristic, complemented
24C2×#15691347—24no—
25C2×#15691347C2211yescharacteristic
26C2×#15691347C2211yescharacteristic
27C2×#15691347C2211yescharacteristic
28C2×#15691347C2211yescharacteristic
29C2×#15691347C2211yescharacteristic
30C2×#15691347C2211yescharacteristic
31#20263326C2211yescharacteristic, complemented
32#20263325—21no—
33#20263314C2211yescharacteristic
34#20263311—21no—
35#20263305C2211yescharacteristic
36#20263305C2211yescharacteristic
37#20263305C2211yescharacteristic, complemented
38#20263303—21no—
39#20263303—21no—
40#20263300C2211yescharacteristic
41#20263299—21no—
42#20263294—21no—
43C4×Dic394C2211yescharacteristic, complemented
44C4×D788C2211yescharacteristic
45D4×Dic197—21no—
46C22⋊C4×D197—21no—
47#20263260C2211yescharacteristic, complemented
48#20263251C2211yescharacteristic, complemented
49#20263251C2211yescharacteristic, complemented
50#20263251C2211yescharacteristic, complemented
51#20263251C2211yescharacteristic, complemented
52#20263248C2211yescharacteristic
53#20263248C2211yescharacteristic
54C2×M16×C197C2211yescharacteristic, complemented
55C22×C8×C197C2211yescharacteristic
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—41no—
64C2×Dic394—22no—
65C2×Dic394—22no—
66C2×Dic394—21no—
67C2×Dic394—21no—
68C2×Dic394—41no—
69C2×Dic394—41no—
70C2×Dic394C2311yescharacteristic
71C2×Dic394C2311yescharacteristic
72C2×Dic394C2311yescharacteristic
73C2×Dic394C2311yescharacteristic
74C2×D788—22no—
75C2×D788—21no—
76C2×D788—41no—
77C2×D788C2311yescharacteristic
78C2×D788C2311yescharacteristic
79C2×C22⋊D197—21no—
80C2×C22⋊D197—21no—
81C22×Dic197—21no—
82C22×Dic197—21no—
83C2×C4×D197—21no—
84C2×C4×D197—21no—
85Dic788—22no—
86Dic788—22no—
87Dic788—41no—
88#15691347—22no—
89#15691347—22no—
90#15691347—22no—
91#15691347—22no—
92#15691347—41no—
93#15691347D414yescomplemented
94C4⋊Dic197—21no—
95C4⋊Dic197—21no—
96C4⋊Dic197C2311yescharacteristic
97C4⋊Dic197C2311yescharacteristic
98#15691337—21no—
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—
109C4×Dic197—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—42no—
126C22×D197—42no—
127C22×D197—81no—
128Dic394—21no—
129Dic394—21no—
130Dic394—21no—
131Dic394—21no—
132Dic394—21no—
133Dic394—21no—
134Dic394—21no—
135Dic394—42no—
136Dic394—42no—
137Dic394C2×D412yes—
138D788—21no—
139D788—21no—
140D788—21no—
141D788—42no—
142D788C2×D412yes—
143C22⋊D197—42no—
144C22⋊D197—81no—
145C2×Dic197—21no—
146C2×Dic197—21no—
147C2×Dic197—21no—
148C2×Dic197—21no—
149C2×Dic197—21no—
150C2×Dic197—21no—
151C2×Dic197—42no—
152C2×Dic197—42no—
153C4×D197—42no—
154C4×D197—81no—
155C23×C197C2×D411yescharacteristic
156C23×C197C2×D411yescharacteristic
157D4×C197—22no—
158C2×C4×C197—21no—
159C2×C4×C197—21no—
160C2×C4×C197—21no—
161C2×C4×C197—21no—
162C2×C4×C197C2×D411yescharacteristic
163C2×C4×C197C2×D411yescharacteristic
164C2×C4×C197C2×D411yescharacteristic
165C2×C4×C197C2×D411yescharacteristic
166C2×C4×C197C2411yescharacteristic
167C8×C197—21no—
168C8×C197—21no—
169C8×C197—21no—
170C8×C197C2×D412yes—
171C22×C197—21no—
172C22×C197—21no—
173C22×C197—21no—
174C22×C197—21no—
175C22×C197—21no—
176C22×C197#13712yescomplemented
177C22×C197C22×D411yescharacteristic
178C2×D197—22no—
179C2×D197—41no—
180C2×D197—41no—
181C2×D197—41no—
182C2×D197—41no—
183C2×D197—41no—
184C2×D197—84no—
185C4×C197—21no—
186C4×C197—21no—
187C4×C197—21no—
188C4×C197—21no—
189C4×C197C22×D411yescharacteristic
190C4×C197H(5,𝔽2)11yescharacteristic
191Dic197—22no—
192Dic197—41no—
193Dic197—41no—
194Dic197—41no—
195Dic197—41no—
196Dic197—41no—
197C2×C197—22no—
198C2×C197—41no—
199C2×C197D4211yescharacteristic
200C2×C197C2×#13711yescharacteristic
201C2×C197#57711yescharacteristic
202D197—42no—
203D197—81no—
204C197#326611yescharacteristic, complemented, Sylow
205#3266—1971nomaximal, Sylow
206C2×#137—1971no—
207C22×SD16—1971no—
208D4×Q8—1971no—
209#546—1971no—
210#497—1971no—
211#468—1971no—
212#467—1971no—
213#464—1971no—
214#461—1971no—
215#452—1971no—
216#451—1971no—
217#450—1971no—
218#448—1971no—
219#440—1971no—
220#435D19711yescharacteristic, complemented
221C2×#41—1971no—
222C22×Q8—1971no—
223C22×Q8—1971no—
224C22×D4—1971no—
225#137—3944no—
226C2×Q16—1971no—
227C2×Q16—1971no—
228C2×Q16—1971no—
229C2×SD16—3944no—
230C2×SD16—1971no—
231C2×SD16—1971no—
232C2×SD16—1971no—
233C2×SD16—1971no—
234C2×SD16—1971no—
235C2×SD16—1971no—
236C2×M16C2×D19711yescharacteristic, complemented
237C22×C8C2×D19711yescharacteristic
238C4⋊Q8—3941no—
239C4⋊Q8—1971no—
240#124—1971no—
241#123—3941no—
242C22⋊Q8—3941no—
243C22⋊Q8—3941no—
244C22⋊Q8—1971no—
245C22⋊Q8—1971no—
246C22⋊Q8—1971no—
247#121—3941no—
248#121—1971no—
249#120—3941no—
250C4×Q8—1971no—
251C4×D4—3941no—
252C4×D4—1971no—
253C4×D4C2×D19711yescharacteristic
254C2×C22⋊C4—3941no—
255#107—1971no—
256C4⋊C8C2×D19711yescharacteristic, complemented
257#103—1971no—
258#103—1971no—
259#103—1971no—
260#103—1971no—
261#102—1971no—
262#102—1971no—
263C22⋊C8C2×D19711yescharacteristic
264C22⋊C8C2×D19711yescharacteristic, complemented
265#97C2×D19711yescharacteristic, complemented
266C24—3941no—
267#41—7881no—
268#41—3942no—
269C2×Q8—7881no—
270C2×Q8—7881no—
271C2×Q8—3942no—
272C2×Q8—3942no—
273C2×Q8—3941no—
274C2×Q8—3941no—
275C2×Q8—1971no—
276C2×Q8—1971no—
277C2×Q8—1971no—
278C2×Q8—1971no—
279C2×D4—7881no—
280C2×D4—3942no—
281C2×D4—3941no—
282C2×D4—3941no—
283C2×D4—3941no—
284C2×D4—1971no—
285C2×D4—1971no—
286C2×D4D78811yescharacteristic
287C22×C4—3941no—
288C22×C4—3941no—
289C22×C4—3941no—
290C22×C4—3941no—
291C22×C4C22×D19711yescharacteristic
292C22×C4C22×D19711yescharacteristic
293Q16—7881no—
294Q16—3942no—
295Q16—3942no—
296SD16—7881no—
297SD16—3942no—
298SD16—3942no—
299SD16—3942no—
300SD16—3942no—
301SD16—1974no—
302M16—22no—
303C2×C8—22no—
304C2×C8C22×D19711yescharacteristic
305C2×C8C22×D19711yescharacteristic
306C2×C8C22×D19711yescharacteristic
307C2×C8C22×D19711yescharacteristic
308C4⋊C4—3941no—
309C4⋊C4—3941no—
310C4⋊C4—3941no—
311C4⋊C4—3941no—
312C4⋊C4—3941no—
313C4⋊C4—3941no—
314C4⋊C4—1971no—
315C4⋊C4—1971no—
316C4⋊C4D78811yescharacteristic
317C22⋊C4—3941no—
318C22⋊C4—3941no—
319C22⋊C4—3941no—
320C22⋊C4—3941no—
321C22⋊C4—3941no—
322C22⋊C4—3941no—
323C22⋊C4—3941no—
324C22⋊C4D78811yescharacteristic
325C22⋊C4D78811yescharacteristic
326C42—3941no—
327C42C22×D19711yescharacteristic
328C23—7882no—
329C23—7882no—
330C23—15761no—
331C23—3941no—
332C23—3941no—
333C23C2×D78811yescharacteristic
334C23C2×D78811yescharacteristic
335Q8—7882no—
336Q8—7882no—
337Q8—3941no—
338Q8—3941no—
339Q8—3941no—
340Q8—3941no—
341Q8—3941no—
342Q8—3941no—
343Q8—3941no—
344Q8—1972no—
345D4—22no—
346D4—7882no—
347D4—7882no—
348D4—15761no—
349D4—3941no—
350D4—3941no—
351D4—3941no—
352D4—1972no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—7882no—
358C2×C4—7882no—
359C2×C4—7882no—
360C2×C4—15761no—
361C2×C4—3941no—
362C2×C4—3941no—
363C2×C4—3941no—
364C2×C4—3941no—
365C2×C4—3941no—
366C2×C4—3941no—
367C2×C4C2×D78811yescharacteristic
368C2×C4C2×D78811yescharacteristic
369C2×C4C2×D78811yescharacteristic
370C2×C4C2×D78811yescharacteristic
371C2×C4C23×D19711yescharacteristic
372C8—21no—
373C8—21no—
374C8—21no—
375C8D4×D19712yes—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—7881no—
382C22—7881no—
383C22—7881no—
384C22—7881no—
385C22—7881no—
386C22—15764no—
387C22—3942no—
388C22#2026335912yescomplemented
389C22C22×D78811yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—7881no—
395C4—7881no—
396C4—7881no—
397C4—7881no—
398C4—7881no—
399C4—3942no—
400C4C2×D4×D19711yescharacteristic
401C4#2026337211yescharacteristic
402C2—22no—
403C2—41no—
404C2—7882no—
405C2—15761no—
406C2D4×D78811yescharacteristic
407C2C2×#2026335911yescharacteristic
408C2#2716189111yescharacteristic
subgroupG/H
centerC22C22×D788
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#435×C197C2
socleC22×C197C22×D4
prime radical#27161725C2
hypercenter#435D197
hypocenterC197#3266
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×#20263359C2semidirect
2. C22×#15691347C2semidirect
3. D4×Dic394C2semidirect
5. #27161473C2semidirect
6. #27161325C2semidirect
7. #27161320C2semidirect
8. #27161302C2semidirect
9. #27161285C2semidirect
10. #27161243C2semidirect
11. #27161237C2semidirect
12. #27161232C2semidirect
13. #27161220C2semidirect
14. #27161181C2semidirect
15. #435×C197C2semidirect
18. C22×Dic394C22semidirect
21. C2×Dic788C22semidirect
22. C2×Dic788C22semidirect
23. C2×Dic788C22semidirect
31. #20263326C22semidirect
37. #20263305C22semidirect
43. C4×Dic394C22semidirect
47. #20263260C22semidirect
48. #20263251C22semidirect
49. #20263251C22semidirect
50. #20263251C22semidirect
51. #20263251C22semidirect
54. C2×M16×C197C22semidirect
57. C4⋊C8×C197C22semidirect
59. C22⋊C8×C197C22semidirect
60. #97×C197C22semidirect
93. #15691347D4semidirect
176. C22×C197#137semidirect
204. C197#3266semidirect
220. #435D197semidirect
236. C2×M16C2×D197semidirect
256. C4⋊C8C2×D197semidirect
264. C22⋊C8C2×D197semidirect
265. #97C2×D197semidirect
388. C22#20263359semidirect

Maximal subgroups

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

subgroupindex
C2×#202633592
C22×#156913472
D4×Dic3942
#271617252
#271614732
#271613252
#271613202
#271613022
#271612852
#271612432
#271612372
#271612322
#271612202
#271611812
#435×C1972
#3266197