Finite Group Databasebeta

← #38745885

Subgroups — #38745885

Order 25216.

520 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263312C211yesmaximal, characteristic
2#27160984C211yesmaximal, characteristic, complemented
3#27160983C211yesmaximal, characteristic, complemented
4#27160973C211yesmaximal, characteristic, complemented
5#27160972C211yesmaximal, characteristic, complemented
6#27160971C211yesmaximal, characteristic, complemented
7#27160968C211yesmaximal, characteristic, complemented
8#27160958C211yesmaximal, characteristic
9#27160955C211yesmaximal, characteristic
10#27160952C211yesmaximal, characteristic, complemented
11#27160950C211yesmaximal, characteristic, complemented
12#27160950C211yesmaximal, characteristic, complemented
13#27160944C211yesmaximal, characteristic, complemented
14#27160943C211yesmaximal, characteristic, complemented
15#385×C197C211yesmaximal, characteristic, complemented
16C22×Dic394C2211yescharacteristic
17C22×C22⋊D197C2211yescharacteristic
18C22×C4×D197C2211yescharacteristic
19#20263312C418yescomplemented
20C2×C4⋊Dic197C2211yescharacteristic
21C2×#15691337C2211yescharacteristic
22C2×#15691337C2211yescharacteristic
23C2×#15691337C2211yescharacteristic
24C2×C22⋊Dic197C2211yescharacteristic
25C2×C22⋊Dic197C2211yescharacteristic
26C2×C22⋊Dic197C2211yescharacteristic
27C2×#15691335C2211yescharacteristic
28C2×#15691335C2211yescharacteristic
29C2×#15691335C2211yescharacteristic
30C2×#15691335C2211yescharacteristic
31C2×#15691335C2211yescharacteristic
32C2×C4×Dic197C2211yescharacteristic
33C2×C4×Dic197C2211yescharacteristic
34C2×C4×Dic197C2211yescharacteristic
35#20263233C2211yescharacteristic
36#20263233C2211yescharacteristic
37#20263233C2211yescharacteristic
38#20263233C2211yescharacteristic
39#20263233C2211yescharacteristic
40#20263233C2211yescharacteristic
41#20263233C2211yescharacteristic
42#20263233C2211yescharacteristic
43#20263233C2211yescharacteristic
44#20263233C2211yescharacteristic
45C23×C4×C197C2211yescharacteristic
46C2×C4⋊C4×C197C2211yescharacteristic
47C2×C4⋊C4×C197C2211yescharacteristic
48C2×C22⋊C4×C197C2211yescharacteristic
49C2×C22⋊C4×C197C2211yescharacteristic
50#95×C197C2211yescharacteristic
51#95×C197C2211yescharacteristic
52C23×D197C2311yescharacteristic
53C2×Dic394—41no—
54C2×Dic394—41no—
55C2×Dic394C2×C414yescomplemented
56C2×C22⋊D197—41no—
57C2×C22⋊D197—41no—
58C2×C22⋊D197C2×C414yes—
59C22×Dic197C2311yescharacteristic
60C22×Dic197C2311yescharacteristic
61C22×Dic197C2311yescharacteristic
62C22×Dic197C2311yescharacteristic
63C22×Dic197C2311yescharacteristic
64C22×Dic197C2311yescharacteristic
65C22×Dic197C2311yescharacteristic
66C2×C4×D197—41no—
67C2×C4×D197—41no—
68C2×C4×D197—41no—
69C4⋊Dic197—41no—
70#15691337—41no—
71#15691337—41no—
72#15691337—41no—
73C22⋊Dic197—41no—
74C22⋊Dic197—41no—
75C22⋊Dic197C2×C414yescomplemented
76#15691335—41no—
77#15691335—41no—
78#15691335—41no—
79#15691335C2×C414yescomplemented
80#15691335C2×C414yescomplemented
81C4×Dic197—41no—
82C4×Dic197—41no—
83C4×Dic197C2×C414yescomplemented
84C24×C197C2311yescharacteristic
85C22×C4×C197—41no—
86C22×C4×C197—41no—
87C22×C4×C197—41no—
88C22×C4×C197C2311yescharacteristic
89C22×C4×C197C2311yescharacteristic
90C22×C4×C197C2311yescharacteristic
91C22×C4×C197C2311yescharacteristic
92C22×C4×C197C2311yescharacteristic
93C22×C4×C197C2311yescharacteristic
94C4⋊C4×C197—41no—
95C4⋊C4×C197—41no—
96C22⋊C4×C197—41no—
97C22⋊C4×C197C2×C414yescomplemented
98C22×D197—21no—
99C22×D197—21no—
100C22×D197—21no—
101C22×D197—21no—
102C22×D197—21no—
103C22×D197—21no—
104C22×D197C22×C412yes—
105Dic394—44no—
106C22⋊D197—44no—
107C2×Dic197—21no—
108C2×Dic197—21no—
109C2×Dic197—21no—
110C2×Dic197—21no—
111C2×Dic197—21no—
112C2×Dic197—21no—
113C2×Dic197—21no—
114C2×Dic197—21no—
115C2×Dic197—21no—
116C2×Dic197—21no—
117C2×Dic197—21no—
118C2×Dic197—21no—
119C2×Dic197—21no—
120C2×Dic197—21no—
121C2×Dic197—21no—
122C2×Dic197—21no—
123C2×Dic197—21no—
124C2×Dic197—21no—
125C2×Dic197C22×C412yes—
126C2×Dic197C22×C412yes—
127C2×Dic197C22×C412yes—
128C4×D197—44no—
129C23×C197—21no—
130C23×C197—21no—
131C23×C197—21no—
132C23×C197—21no—
133C23×C197—21no—
134C23×C197—21no—
135C23×C197C22×C412yes—
136C23×C197C2411yescharacteristic
137C2×C4×C197—21no—
138C2×C4×C197—21no—
139C2×C4×C197—21no—
140C2×C4×C197—21no—
141C2×C4×C197—21no—
142C2×C4×C197—21no—
143C2×C4×C197—21no—
144C2×C4×C197—21no—
145C2×C4×C197—21no—
146C2×C4×C197—21no—
147C2×C4×C197—21no—
148C2×C4×C197—21no—
149C2×C4×C197—21no—
150C2×C4×C197—21no—
151C2×C4×C197—21no—
152C2×C4×C197—21no—
153C2×C4×C197—44no—
154C2×C4×C197C22×C412yes—
155C2×C4×C197C22×C412yes—
156C22×C197—22no—
157C22×C197—22no—
158C22×C197—22no—
159C22×C197—41no—
160C22×C197—41no—
161C22×C197—41no—
162C22×C197—41no—
163C22×C197C23×C411yescharacteristic
164C22×C197H(5,𝔽2)11yescharacteristic
165C22×C197H(5,𝔽2)11yescharacteristic
166C22×C197H(5,𝔽2)11yescharacteristic
167C22×C197H(5,𝔽2)11yescharacteristic
168C22×C1972−1+411yescharacteristic
169C22×C1972−1+411yescharacteristic
170C2×D197—22no—
171C2×D197—22no—
172C2×D197—22no—
173C2×D197—41no—
174C2×D197—41no—
175C2×D197—41no—
176C2×D197—41no—
177C4×C197—22no—
178C4×C197—22no—
179C4×C197—41no—
180C4×C197—41no—
181C4×C197—41no—
182C4×C197—41no—
183Dic197—22no—
184Dic197—22no—
185Dic197—22no—
186Dic197—41no—
187Dic197—41no—
188Dic197—41no—
189Dic197—41no—
190C2×C197—42no—
191C2×C197#51811yescharacteristic
192C2×C197#52011yescharacteristic
193C2×C197#52011yescharacteristic
194C2×C197H(3,𝔽4)11yescharacteristic
195C2×C197#56211yescharacteristic
196C2×C197#56211yescharacteristic
197C2×C197#56311yescharacteristic
198D197—42no—
199C197#237911yescharacteristic, complemented, Sylow
200#2379—1971nomaximal, Sylow
201C2×#124—1971no—
202#391—1971no—
203#391—1971no—
204#388—1971no—
205#388—1971no—
206#388—1971no—
207#388—1971no—
208#386—1971no—
209#386—1971no—
210#385—1971no—
211#385—1971no—
212#385—1971no—
213#385D19711yescharacteristic, complemented
214#382—1971no—
215#382—1971no—
216C22×Q8—1971no—
217C22×D4—1971no—
218C23×C4—1971no—
219C23×C4C2×D19711yescharacteristic
220#124—1978no—
221C2×C4⋊C4—1971no—
222C2×C4⋊C4—1971no—
223C2×C4⋊C4—1971no—
224C2×C4⋊C4—1971no—
225C2×C4⋊C4C2×D19711yescharacteristic
226C2×C4⋊C4C2×D19711yescharacteristic
227C2×C22⋊C4—1971no—
228C2×C22⋊C4—1971no—
229C2×C22⋊C4—1971no—
230C2×C22⋊C4—1971no—
231C2×C22⋊C4—1971no—
232C2×C22⋊C4—1971no—
233C2×C22⋊C4—1971no—
234C2×C22⋊C4—1971no—
235C2×C22⋊C4C2×D19711yescharacteristic
236C2×C22⋊C4C2×D19711yescharacteristic
237C2×C42—1971no—
238C2×C42—1971no—
239C2×C42—1971no—
240#95—1971no—
241#95—1971no—
242#95—1971no—
243#95—1971no—
244#95—1971no—
245#95—1971no—
246#95—1971no—
247#95—1971no—
248#95—1971no—
249#95—1971no—
250#95C2×D19711yescharacteristic
251#95C2×D19711yescharacteristic
252C24—1971no—
253C24C22×D19711yescharacteristic
254C2×Q8—7881no—
255C2×Q8—7881no—
256C2×Q8—1974no—
257C2×D4—7881no—
258C2×D4—7881no—
259C2×D4—1974no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—7881no—
264C22×C4—7881no—
265C22×C4—7881no—
266C22×C4—1971no—
267C22×C4—1971no—
268C22×C4—1971no—
269C22×C4—1971no—
270C22×C4—1971no—
271C22×C4—1971no—
272C22×C4—1971no—
273C22×C4C22×D19711yescharacteristic
274C22×C4C22×D19711yescharacteristic
275C22×C4C22×D19711yescharacteristic
276C22×C4C22×D19711yescharacteristic
277C22×C4C22×D19711yescharacteristic
278C22×C4C22×D19711yescharacteristic
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—7881no—
282C4⋊C4—7881no—
283C4⋊C4—7881no—
284C4⋊C4—7881no—
285C22⋊C4—41no—
286C22⋊C4—7881no—
287C22⋊C4—7881no—
288C22⋊C4—7881no—
289C22⋊C4—7881no—
290C22⋊C4—7881no—
291C22⋊C4—1974no—
292C22⋊C4—1974no—
293C22⋊C4—1974no—
294C22⋊C4C4×D19714yescomplemented
295C42—7881no—
296C42—7881no—
297C42—1974no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—3941no—
305C23—3941no—
306C23—3941no—
307C23—3941no—
308C23—3941no—
309C23—3941no—
310C23—1972no—
311C23C2×C4×D19712yes—
312C23C23×D19711yescharacteristic
313Q8—7884no—
314D4—7884no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—44no—
332C2×C4—7884no—
333C2×C4—3941no—
334C2×C4—3941no—
335C2×C4—3941no—
336C2×C4—3941no—
337C2×C4—3941no—
338C2×C4—3941no—
339C2×C4—3941no—
340C2×C4—3941no—
341C2×C4—3941no—
342C2×C4—3941no—
343C2×C4—3941no—
344C2×C4—3941no—
345C2×C4—3941no—
346C2×C4—3941no—
347C2×C4—3941no—
348C2×C4—3941no—
349C2×C4—3941no—
350C2×C4—3941no—
351C2×C4—1972no—
352C2×C4—1972no—
353C2×C4—1972no—
354C2×C4C2×C4×D19712yes—
355C2×C4C2×C4×D19712yes—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—7881no—
364C22—7881no—
365C22—7881no—
366C22—7881no—
367C22—3942no—
368C22—3942no—
369C22—3942no—
370C22C22×C4×D19711yescharacteristic
371C22#2026337111yescharacteristic
372C22#2026337111yescharacteristic
373C22#2026337111yescharacteristic
374C22#2026337211yescharacteristic
375C22#2026337311yescharacteristic
376C22#2026337411yescharacteristic
377C4—22no—
378C4—22no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—7881no—
384C4—7881no—
385C4—7881no—
386C4—7881no—
387C4—3942no—
388C4—3942no—
389C4—3942no—
390C2—42no—
391C2—7882no—
392C2#2716155111yescharacteristic
393C2#2716155811yescharacteristic
394C2#2716156011yescharacteristic
395C2#2716182211yescharacteristic
396C2#2716182511yescharacteristic
397C2#2716182611yescharacteristic
398C2#2716183011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D197
commutatorC22×C197C23×C4
Fitting#385×C197C2
socle, cosocleC23×C197C24
prime radicalC22×C22⋊D197C22
hypercenter#385D197
hypocenterC197#2379
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. #27160984C2semidirect
3. #27160983C2semidirect
4. #27160973C2semidirect
5. #27160972C2semidirect
6. #27160971C2semidirect
7. #27160968C2semidirect
10. #27160952C2semidirect
11. #27160950C2semidirect
12. #27160950C2semidirect
13. #27160944C2semidirect
14. #27160943C2semidirect
15. #385×C197C2semidirect
19. #20263312C4semidirect
55. C2×Dic394C2×C4semidirect
75. C22⋊Dic197C2×C4semidirect
79. #15691335C2×C4semidirect
80. #15691335C2×C4semidirect
83. C4×Dic197C2×C4semidirect
97. C22⋊C4×C197C2×C4semidirect
199. C197#2379semidirect
213. #385D197semidirect
294. C22⋊C4C4×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633122
#271609842
#271609832
#271609732
#271609722
#271609712
#271609682
#271609582
#271609552
#271609522
#271609502
#271609442
#271609432
#385×C1972
#2379197