Finite Group Databasebeta

← #38754218

Subgroups — #38754218

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×#20263356C211yesmaximal, characteristic, complemented
2C22×#15691346C211yesmaximal, characteristic, complemented
3D4×Dic394C211yesmaximal, characteristic, complemented
4#27161474C211yesmaximal, characteristic, complemented
5#27161323C211yesmaximal, characteristic, complemented
6#27161318C211yesmaximal, characteristic, complemented
7#27161298C211yesmaximal, characteristic, complemented
8#27161284C211yesmaximal, characteristic, complemented
9#27161241C211yesmaximal, characteristic, complemented
10#27161234C211yesmaximal, characteristic, complemented
11#27161230C211yesmaximal, characteristic, complemented
12#27161217C211yesmaximal, characteristic, complemented
13#27161179C211yesmaximal, characteristic, complemented
14#27161151C211yesmaximal, characteristic, complemented
15#546×C197C211yesmaximal, characteristic
16C22×Dic394C2211yescharacteristic
17C22×Dic394C2211yescharacteristic, complemented
18#20263356—24no—
19C2×#15691348C2211yescharacteristic, complemented
20C2×#15691348C2211yescharacteristic, complemented
21C2×#15691348C2211yescharacteristic, complemented
22C2×#15691346—24no—
23C2×#15691346C2211yescharacteristic
24C2×#15691346C2211yescharacteristic
25C2×#15691346C2211yescharacteristic
26C2×#15691346C2211yescharacteristic
27C2×#15691346C2211yescharacteristic
28C2×#15691346C2211yescharacteristic
29C2×#15691342C2211yescharacteristic, complemented
30C22×#13096802C2211yescharacteristic
31#20263326C2211yescharacteristic, complemented
32#20263325—21no—
33#20263305C2211yescharacteristic
34#20263305C2211yescharacteristic, complemented
35#20263303—21no—
36#20263303—21no—
37C4×Dic394C2211yescharacteristic, complemented
38D4×Dic197—21no—
39#20263259C2211yescharacteristic, complemented
40C4⋊#13096802C2211yescharacteristic, complemented
41#20263250C2211yescharacteristic, complemented
42#20263249C2211yescharacteristic, complemented
43#20263249C2211yescharacteristic, complemented
44#20263249C2211yescharacteristic, complemented
45#20263247C2211yescharacteristic
46#20263247C2211yescharacteristic
47C22⋊#13096802C2211yescharacteristic
48C22⋊#13096802C2211yescharacteristic, complemented
49#20263237C2211yescharacteristic, complemented
50C2×#41×C197C2211yescharacteristic
51C22×D4×C197C2211yescharacteristic
52#124×C197C2211yescharacteristic
53#123×C197—21no—
54C22⋊Q8×C197C2211yescharacteristic
55#121×C197—21no—
56#121×C197C2211yescharacteristic
57#120×C197—21no—
58C4×D4×C197C2211yescharacteristic
59C4×D4×C197C2211yescharacteristic
60C2×C22⋊C4×C197—21no—
61C2×Dic394—22no—
62C2×Dic394—22no—
63C2×Dic394—21no—
64C2×Dic394—21no—
65C2×Dic394—41no—
66C2×Dic394—41no—
67C2×Dic394C2311yescharacteristic
68C2×Dic394C2311yescharacteristic
69C2×Dic394C2311yescharacteristic
70C22×Dic197—21no—
71C22×Dic197—21no—
72#15691348—22no—
73#15691348—22no—
74#15691348—41no—
75#15691346—22no—
76#15691346—22no—
77#15691346—22no—
78#15691346—22no—
79#15691346—41no—
80#15691346D414yescomplemented
81#15691342—22no—
82C2×#13096802—22no—
83C2×#13096802C2311yescharacteristic
84C2×#13096802C2311yescharacteristic
85C2×#13096802C2311yescharacteristic
86C2×#13096802C2311yescharacteristic
87C4⋊Dic197—21no—
88C4⋊Dic197—21no—
89C4⋊Dic197C2311yescharacteristic
90#15691337—21no—
91#15691337—21no—
92#15691337—21no—
93C22⋊Dic197—21no—
94C22⋊Dic197—21no—
95C4×Dic197—21no—
96C24×C197—21no—
97#41×C197—22no—
98#41×C197—41no—
99C2×Q8×C197C2311yescharacteristic
100C2×D4×C197—22no—
101C2×D4×C197—21no—
102C2×D4×C197—21no—
103C2×D4×C197—21no—
104C2×D4×C197—41no—
105C2×D4×C197D411yescharacteristic
106C2×D4×C197C2311yescharacteristic
107C2×D4×C197C2311yescharacteristic
108C22×C4×C197—21no—
109C22×C4×C197—21no—
110C22×C4×C197C2311yescharacteristic
111C22×C4×C197C2311yescharacteristic
112C4⋊C4×C197—21no—
113C4⋊C4×C197D411yescharacteristic
114C4⋊C4×C197C2311yescharacteristic
115C22⋊C4×C197—21no—
116C22⋊C4×C197—21no—
117C22⋊C4×C197—21no—
118C22⋊C4×C197—21no—
119C22⋊C4×C197—21no—
120C22⋊C4×C197D411yescharacteristic
121C22⋊C4×C197D411yescharacteristic
122C42×C197C2311yescharacteristic
123Dic394—21no—
124Dic394—21no—
125Dic394—21no—
126Dic394—21no—
127Dic394—21no—
128Dic394—42no—
129Dic394—42no—
130Dic394C2×D412yes—
131C2×Dic197—21no—
132C2×Dic197—21no—
133C2×Dic197—21no—
134C2×Dic197—21no—
135C2×Dic197—42no—
136C2×Dic197—42no—
137#13096802—21no—
138#13096802—21no—
139#13096802—21no—
140#13096802C2×D412yes—
141C23×C197—21no—
142C23×C197—21no—
143C23×C197—42no—
144C23×C197—42no—
145C23×C197—81no—
146C23×C197C2×D411yescharacteristic
147C23×C197C2×D411yescharacteristic
148Q8×C197—21no—
149Q8×C197—21no—
150D4×C197—22no—
151D4×C197—21no—
152D4×C197—21no—
153D4×C197—21no—
154D4×C197—42no—
155D4×C197—42no—
156D4×C197—81no—
157D4×C197C2×D412yes—
158C2×C4×C197—21no—
159C2×C4×C197—21no—
160C2×C4×C197—21no—
161C2×C4×C197—21no—
162C2×C4×C197—21no—
163C2×C4×C197—21no—
164C2×C4×C197—42no—
165C2×C4×C197—81no—
166C2×C4×C197C2×D411yescharacteristic
167C2×C4×C197C2×D411yescharacteristic
168C2×C4×C197C2×D411yescharacteristic
169C2×C4×C197C2×D411yescharacteristic
170C2×C4×C197C2411yescharacteristic
171C22×C197—21no—
172C22×C197—21no—
173C22×C197—21no—
174C22×C197—22no—
175C22×C197—21no—
176C22×C197—21no—
177C22×C197—41no—
178C22×C197—41no—
179C22×C197—41no—
180C22×C197—41no—
181C22×C197—41no—
182C22×C197—84no—
183C22×C197#13712yescomplemented
184C22×C197C22×D411yescharacteristic
185C4×C197—21no—
186C4×C197—21no—
187C4×C197—21no—
188C4×C197—21no—
189C4×C197—41no—
190C4×C197—41no—
191C4×C197C22×D411yescharacteristic
192C4×C197H(5,𝔽2)11yescharacteristic
193Dic197—22no—
194Dic197—41no—
195Dic197—41no—
196Dic197—41no—
197C2×C197—22no—
198C2×C197—41no—
199C2×C197—42no—
200C2×C197—81no—
201C2×C197D4211yescharacteristic
202C2×C197C2×#13711yescharacteristic
203C2×C197#57711yescharacteristic
204C197#326611yescharacteristic, complemented, Sylow
205#3266—1971nomaximal, Sylow
206C2×#137—1971no—
207C22×SD16—1971no—
208D4×Q8—1971no—
209#546D19711yescharacteristic
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#435—1971no—
221C2×#41C2×D19711yescharacteristic
222C22×Q8—1971no—
223C22×Q8—1971no—
224C22×D4C2×D19711yescharacteristic
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×M16—1971no—
237C22×C8—1971no—
238C4⋊Q8—3941no—
239C4⋊Q8—1971no—
240#124C2×D19711yescharacteristic
241#123—21no—
242C22⋊Q8—3941no—
243C22⋊Q8—3941no—
244C22⋊Q8—1971no—
245C22⋊Q8—1971no—
246C22⋊Q8C2×D19711yescharacteristic
247#121—21no—
248#121C2×D19711yescharacteristic
249#120—21no—
250C4×Q8—1971no—
251C4×D4—3941no—
252C4×D4C2×D19711yescharacteristic
253C4×D4C2×D19711yescharacteristic
254C2×C22⋊C4—21no—
255#107—1971no—
256C4⋊C8—1971no—
257#103—1971no—
258#103—1971no—
259#103—1971no—
260#103—1971no—
261#102—1971no—
262#102—1971no—
263C22⋊C8—1971no—
264C22⋊C8—1971no—
265#97—1971no—
266C24—21no—
267#41—22no—
268#41—41no—
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×Q8C22×D19711yescharacteristic
279C2×D4—22no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—41no—
284C2×D4C22⋊D19711yescharacteristic
285C2×D4C22×D19711yescharacteristic
286C2×D4C22×D19711yescharacteristic
287C22×C4—21no—
288C22×C4—21no—
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—3942no—
303C2×C8—3942no—
304C2×C8—1971no—
305C2×C8—1971no—
306C2×C8—1971no—
307C2×C8—1971no—
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—21no—
319C22⋊C4—21no—
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—3941no—
323C22⋊C4—3941no—
324C22⋊C4C22⋊D19711yescharacteristic
325C22⋊C4C22⋊D19711yescharacteristic
326C42—3941no—
327C42C22×D19711yescharacteristic
328C23—21no—
329C23—21no—
330C23—42no—
331C23—42no—
332C23—81no—
333C23C2×C22⋊D19711yescharacteristic
334C23C2×C22⋊D19711yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—7882no—
338Q8—7882no—
339Q8—3941no—
340Q8—3941no—
341Q8—3941no—
342Q8—3941no—
343Q8—3941no—
344Q8—1972no—
345D4—22no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—42no—
350D4—42no—
351D4—81no—
352D4D4×D19712yes—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—42no—
360C2×C4—81no—
361C2×C4—7882no—
362C2×C4—7882no—
363C2×C4—3941no—
364C2×C4—3941no—
365C2×C4—3941no—
366C2×C4—3941no—
367C2×C4C2×C22⋊D19711yescharacteristic
368C2×C4C2×C22⋊D19711yescharacteristic
369C2×C4C2×C22⋊D19711yescharacteristic
370C2×C4C2×C22⋊D19711yescharacteristic
371C2×C4C23×D19711yescharacteristic
372C8—3941no—
373C8—3941no—
374C8—3941no—
375C8—1972no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—22no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—84no—
388C22#2026335612yescomplemented
389C22C22×C22⋊D19711yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—41no—
395C4—41no—
396C4—7881no—
397C4—7881no—
398C4—7881no—
399C4—3942no—
400C4C2×D4×D19711yescharacteristic
401C4#2026337111yescharacteristic
402C2—22no—
403C2—41no—
404C2—42no—
405C2—81no—
406C2D4×C22⋊D19711yescharacteristic
407C2C2×#2026335611yescharacteristic
408C2#2716189011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting, prime radical#546×C197C2
socleC22×C197C22×D4
hypercenter#546D197
hypocenterC197#3266
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C2×#20263356C2semidirect
2. C22×#15691346C2semidirect
3. D4×Dic394C2semidirect
4. #27161474C2semidirect
5. #27161323C2semidirect
6. #27161318C2semidirect
7. #27161298C2semidirect
8. #27161284C2semidirect
9. #27161241C2semidirect
10. #27161234C2semidirect
11. #27161230C2semidirect
12. #27161217C2semidirect
13. #27161179C2semidirect
14. #27161151C2semidirect
17. C22×Dic394C22semidirect
19. C2×#15691348C22semidirect
20. C2×#15691348C22semidirect
21. C2×#15691348C22semidirect
29. C2×#15691342C22semidirect
31. #20263326C22semidirect
34. #20263305C22semidirect
37. C4×Dic394C22semidirect
39. #20263259C22semidirect
40. C4⋊#13096802C22semidirect
41. #20263250C22semidirect
42. #20263249C22semidirect
43. #20263249C22semidirect
44. #20263249C22semidirect
48. C22⋊#13096802C22semidirect
49. #20263237C22semidirect
80. #15691346D4semidirect
183. C22×C197#137semidirect
204. C197#3266semidirect
388. C22#20263356semidirect

Maximal subgroups

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

subgroupindex
C2×#202633562
C22×#156913462
D4×Dic3942
#271614742
#271613232
#271613182
#271612982
#271612842
#271612412
#271612342
#271612302
#271612172
#271611792
#271611512
#546×C1972
#3266197