Finite Group Databasebeta

← #38754224

Subgroups — #38754224

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×#20263355C211yesmaximal, characteristic, complemented
2C2×SD16×D197C211yesmaximal, characteristic, complemented
3Q8×D788C211yesmaximal, characteristic, complemented
4#27161724C211yesmaximal, characteristic
5#27161473C211yesmaximal, characteristic, complemented
6#27161326C211yesmaximal, characteristic, complemented
7#27161321C211yesmaximal, characteristic, complemented
8#27161301C211yesmaximal, characteristic, complemented
9#27161284C211yesmaximal, characteristic, complemented
10#27161242C211yesmaximal, characteristic, complemented
11#27161236C211yesmaximal, characteristic, complemented
12#27161231C211yesmaximal, characteristic, complemented
13#27161219C211yesmaximal, characteristic, complemented
14#27161152C211yesmaximal, characteristic, complemented
15#440×C197C211yesmaximal, characteristic, complemented
16C2×#15691357C2211yescharacteristic
17C2×Q8×D197C2211yescharacteristic
18C2×Q8×D197C2211yescharacteristic, complemented
19C2×D4×D197C2211yescharacteristic
20#20263355—24no—
21C2×Dic788C2211yescharacteristic, complemented
22C2×Dic788C2211yescharacteristic, complemented
23C2×#15691348C2211yescharacteristic, complemented
24C2×#15691347C2211yescharacteristic
25C2×#15691347C2211yescharacteristic
26C2×#15691346C2211yescharacteristic
27C2×#15691346C2211yescharacteristic
28C2×#15691345C2211yescharacteristic
29SD16×D197—24no—
30C2×#15691341C2211yescharacteristic, complemented
31C2×C8×D197C2211yescharacteristic
32#20263326—21no—
33#20263326C2211yescharacteristic, complemented
34#20263314C2211yescharacteristic
35#20263310—21no—
36#20263306—21no—
37#20263306—21no—
38#20263306C2211yescharacteristic
39#20263306C2211yescharacteristic
40#20263306C2211yescharacteristic, complemented
41#20263301C2211yescharacteristic
42#20263300—21no—
43#20263296—21no—
44C4×D788—21no—
45C4×D788C2211yescharacteristic
46C2×#15691335—21no—
47C4⋊#13096802C2211yescharacteristic, complemented
48#20263249C2211yescharacteristic, complemented
49#20263249C2211yescharacteristic, complemented
50#20263249C2211yescharacteristic, complemented
51#20263246C2211yescharacteristic
52#20263238C2211yescharacteristic
53#20263238C2211yescharacteristic, complemented
54C2×SD16×C197C2211yescharacteristic
55C4×Q8×C197C2211yescharacteristic, complemented
56C4×D4×C197C2211yescharacteristic
57#107×C197C2211yescharacteristic, complemented
58#103×C197C2211yescharacteristic, complemented
59#102×C197C2211yescharacteristic
60#97×C197C2211yescharacteristic, complemented
61C23×D197—21no—
62#15691357—22no—
63#15691357—41no—
64C2×Dic394—21no—
65C2×Dic394—21no—
66C2×Dic394C2311yescharacteristic
67C2×Dic394C2311yescharacteristic
68C2×Dic394C2311yescharacteristic
69Q8×D197—22no—
70Q8×D197—22no—
71Q8×D197—41no—
72Q8×D197—41no—
73C2×D788—21no—
74C2×D788D411yescharacteristic
75C2×D788C2311yescharacteristic
76C2×C22⋊D197—21no—
77C2×C22⋊D197—21no—
78D4×D197—22no—
79D4×D197—41no—
80C22×Dic197—21no—
81C2×C4×D197—21no—
82C2×C4×D197—21no—
83C2×C4×D197C2311yescharacteristic
84C2×C4×D197C2311yescharacteristic
85Dic788—22no—
86Dic788—41no—
87#15691348—22no—
88#15691347—22no—
89#15691347—41no—
90#15691346—22no—
91#15691346—22no—
92#15691345—22no—
93#15691341—22no—
94C2×#13096802C2311yescharacteristic
95C2×#13096802C2311yescharacteristic
96C8×D197—22no—
97C4⋊Dic197—21no—
98C4⋊Dic197—21no—
99C4⋊Dic197—21no—
100C4⋊Dic197—21no—
101C4⋊Dic197—21no—
102C4⋊Dic197D411yescharacteristic
103#15691335—21no—
104#15691335—21no—
105#15691335—21no—
106#15691335—21no—
107#15691335—21no—
108#15691335—21no—
109#15691335D411yescharacteristic
110#15691335D411yescharacteristic
111C2×Q8×C197C2311yescharacteristic
112C2×D4×C197C2311yescharacteristic
113C22×C4×C197—21no—
114SD16×C197D414yescomplemented
115C2×C8×C197C2311yescharacteristic
116C2×C8×C197C2311yescharacteristic
117C4⋊C4×C197—21no—
118C4⋊C4×C197C2311yescharacteristic
119C4⋊C4×C197C2311yescharacteristic
120C22⋊C4×C197—21no—
121C42×C197—21no—
122C42×C197C2311yescharacteristic
123C22×D197—21no—
124C22×D197—42no—
125C22×D197—42no—
126C22×D197—81no—
127C22×D197C2×D411yescharacteristic
128C22×D197C2×D411yescharacteristic
129Dic394—21no—
130Dic394—21no—
131Dic394—21no—
132Dic394—21no—
133Dic394—21no—
134Dic394—21no—
135Dic394—42no—
136Dic394—42no—
137D788—22no—
138D788—21no—
139D788—21no—
140D788—81no—
141C22⋊D197—42no—
142C22⋊D197—42no—
143C2×Dic197—21no—
144C2×Dic197—21no—
145C2×Dic197—21no—
146C2×Dic197—42no—
147C2×Dic197C2×D411yescharacteristic
148C2×Dic197C2×D411yescharacteristic
149C4×D197—21no—
150C4×D197—21no—
151C4×D197—21no—
152C4×D197—21no—
153C4×D197—42no—
154C4×D197—42no—
155#13096802—21no—
156#13096802—21no—
157C23×C197—21no—
158Q8×C197—21no—
159Q8×C197C2×D412yes—
160D4×C197—21no—
161D4×C197C2×D412yes—
162C2×C4×C197—21no—
163C2×C4×C197—21no—
164C2×C4×C197—21no—
165C2×C4×C197—81no—
166C2×C4×C197C2×D411yescharacteristic
167C2×C4×C197C2×D411yescharacteristic
168C2×C4×C197C2411yescharacteristic
169C8×C197—21no—
170C8×C197C2×D412yes—
171C22×C197—22no—
172C22×C197—41no—
173C22×C197—41no—
174C22×C197C22×D411yescharacteristic
175C2×D197—21no—
176C2×D197—21no—
177C2×D197—21no—
178C2×D197—21no—
179C2×D197—21no—
180C2×D197—41no—
181C2×D197—41no—
182C2×D197—41no—
183C2×D197—84no—
184C2×D197#13712yescomplemented
185C4×C197—22no—
186C4×C197—21no—
187C4×C197—21no—
188C4×C197—41no—
189C4×C197—41no—
190C4×C197C22×D411yescharacteristic
191C4×C197H(5,𝔽2)11yescharacteristic
192Dic197—21no—
193Dic197—21no—
194Dic197—41no—
195Dic197—41no—
196Dic197—41no—
197C2×C197—42no—
198C2×C197D4211yescharacteristic
199C2×C197C2×#13711yescharacteristic
200C2×C197#57711yescharacteristic
201D197—22no—
202D197—41no—
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#440D19711yescharacteristic, complemented
220#435—1971no—
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×SD16C2×D19711yescharacteristic
236C2×M16—1971no—
237C22×C8—1971no—
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×Q8C2×D19711yescharacteristic, complemented
251C4×D4—3941no—
252C4×D4—1971no—
253C4×D4C2×D19711yescharacteristic
254C2×C22⋊C4—3941no—
255#107C2×D19711yescharacteristic, complemented
256C4⋊C8—1971no—
257#103—1971no—
258#103—1971no—
259#103—1971no—
260#103C2×D19711yescharacteristic, complemented
261#102—1971no—
262#102C2×D19711yescharacteristic
263C22⋊C8—1971no—
264C22⋊C8—1971no—
265#97C2×D19711yescharacteristic, complemented
266C24—3941no—
267#41—7881no—
268#41—3942no—
269C2×Q8—7881no—
270C2×Q8—7881no—
271C2×Q8—3941no—
272C2×Q8—3941no—
273C2×Q8—3942no—
274C2×Q8—3942no—
275C2×Q8—1971no—
276C2×Q8—1971no—
277C2×Q8—1971no—
278C2×Q8C22×D19711yescharacteristic
279C2×D4—7881no—
280C2×D4—3941no—
281C2×D4—3941no—
282C2×D4—3942no—
283C2×D4—3941no—
284C2×D4—1971no—
285C2×D4—1971no—
286C2×D4C22×D19711yescharacteristic
287C22×C4—21no—
288C22×C4—3941no—
289C22×C4—3941no—
290C22×C4—3941no—
291C22×C4—1971no—
292C22×C4—1971no—
293Q16—7881no—
294Q16—3942no—
295Q16—3942no—
296SD16—7881no—
297SD16—3942no—
298SD16—3942no—
299SD16—3942no—
300SD16—3942no—
301SD16D78814yescomplemented
302M16—3942no—
303C2×C8—3942no—
304C2×C8—1971no—
305C2×C8—1971no—
306C2×C8C22×D19711yescharacteristic
307C2×C8C22×D19711yescharacteristic
308C4⋊C4—21no—
309C4⋊C4—3941no—
310C4⋊C4—3941no—
311C4⋊C4—3941no—
312C4⋊C4—3941no—
313C4⋊C4—3941no—
314C4⋊C4—1971no—
315C4⋊C4C22×D19711yescharacteristic
316C4⋊C4C22×D19711yescharacteristic
317C22⋊C4—21no—
318C22⋊C4—3941no—
319C22⋊C4—3941no—
320C22⋊C4—3941no—
321C22⋊C4—3941no—
322C22⋊C4—3941no—
323C22⋊C4—3941no—
324C22⋊C4—1971no—
325C22⋊C4—1971no—
326C42—21no—
327C42C22×D19711yescharacteristic
328C23—21no—
329C23—7882no—
330C23—7882no—
331C23—15761no—
332C23—3941no—
333C23—1971no—
334C23—1971no—
335Q8—21no—
336Q8—7882no—
337Q8—7882no—
338Q8—3941no—
339Q8—3941no—
340Q8—3941no—
341Q8—3941no—
342Q8—3941no—
343Q8—3941no—
344Q8C2×D78812yes—
345D4—21no—
346D4—7882no—
347D4—7882no—
348D4—15761no—
349D4—3942no—
350D4—3941no—
351D4—3941no—
352D4C2×D78812yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—81no—
357C2×C4—7882no—
358C2×C4—7882no—
359C2×C4—7882no—
360C2×C4—3941no—
361C2×C4—3941no—
362C2×C4—3941no—
363C2×C4—3941no—
364C2×C4—3941no—
365C2×C4—3941no—
366C2×C4—3941no—
367C2×C4—1971no—
368C2×C4—1971no—
369C2×C4D4×D19711yescharacteristic
370C2×C4D4×D19711yescharacteristic
371C2×C4C23×D19711yescharacteristic
372C8—21no—
373C8—3941no—
374C8—3941no—
375C8C2×D78812yes—
376C22—22no—
377C22—41no—
378C22—41no—
379C22—7881no—
380C22—7881no—
381C22—7881no—
382C22—15764no—
383C22—3941no—
384C22—3941no—
385C22—3941no—
386C22—3941no—
387C22—3941no—
388C22—1972no—
389C22C2×D4×D19711yescharacteristic
390C4—21no—
391C4—21no—
392C4—22no—
393C4—41no—
394C4—41no—
395C4—7881no—
396C4—7881no—
397C4—7881no—
398C4—3941no—
399C4—3941no—
400C4C22×D78811yescharacteristic
401C4#2026337211yescharacteristic
402C2—42no—
403C2—7881no—
404C2—15761no—
405C2—3942no—
406C2D4×D78811yescharacteristic
407C2#137×D19711yescharacteristic
408C2#2716188911yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#440×C197C2
socleC22×C197C22×D4
prime radical#27161724C2
hypercenter#440D197
hypocenterC197#3266
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. C2×#20263355C2semidirect
2. C2×SD16×D197C2semidirect
3. Q8×D788C2semidirect
5. #27161473C2semidirect
6. #27161326C2semidirect
7. #27161321C2semidirect
8. #27161301C2semidirect
9. #27161284C2semidirect
10. #27161242C2semidirect
11. #27161236C2semidirect
12. #27161231C2semidirect
13. #27161219C2semidirect
14. #27161152C2semidirect
15. #440×C197C2semidirect
18. C2×Q8×D197C22semidirect
21. C2×Dic788C22semidirect
22. C2×Dic788C22semidirect
23. C2×#15691348C22semidirect
30. C2×#15691341C22semidirect
33. #20263326C22semidirect
40. #20263306C22semidirect
47. C4⋊#13096802C22semidirect
48. #20263249C22semidirect
49. #20263249C22semidirect
50. #20263249C22semidirect
53. #20263238C22semidirect
55. C4×Q8×C197C22semidirect
57. #107×C197C22semidirect
58. #103×C197C22semidirect
60. #97×C197C22semidirect
114. SD16×C197D4semidirect
184. C2×D197#137semidirect
204. C197#3266semidirect
219. #440D197semidirect
250. C4×Q8C2×D197semidirect
255. #107C2×D197semidirect
260. #103C2×D197semidirect
265. #97C2×D197semidirect
301. SD16D788semidirect

Maximal subgroups

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

subgroupindex
C2×#202633552
C2×SD16×D1972
Q8×D7882
#271617242
#271614732
#271613262
#271613212
#271613012
#271612842
#271612422
#271612362
#271612312
#271612192
#271611522
#440×C1972
#3266197