Finite Group Databasebeta

← #38745697

Subgroups — #38745697

Order 25216.

652 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263311C211yesmaximal, characteristic
2C2×#20263288C211yesmaximal, characteristic
3C2×#20263283C211yesmaximal, characteristic, complemented
4#27160974C211yesmaximal, characteristic, complemented
5#27160968C211yesmaximal, characteristic, complemented
6#27160968C211yesmaximal, characteristic, complemented
7#27160961C211yesmaximal, characteristic, complemented
8#27160957C211yesmaximal, characteristic
9#27160953C211yesmaximal, characteristic, complemented
10#27160953C211yesmaximal, characteristic, complemented
11C4⋊C4×Dic197C211yesmaximal, characteristic, complemented
12C4×#15691335C211yesmaximal, characteristic, complemented
13#27160926C211yesmaximal, characteristic, complemented
14#95×D197C211yesmaximal, characteristic, complemented
15#387×C197C211yesmaximal, characteristic, complemented
16C22×D788C2211yescharacteristic
17C22×C4×D197C2211yescharacteristic
18C22×C4×D197C2211yescharacteristic
19C22×C4×D197C2211yescharacteristic
20C22×C4×D197C2211yescharacteristic
21#20263311C418yescomplemented
22#20263288—24no—
23#20263283—24no—
24C2×C4⋊Dic197C2211yescharacteristic
25C2×#15691337C2211yescharacteristic
26C2×#15691337C2211yescharacteristic
27C2×#15691337C2211yescharacteristic
28C2×#15691337C2211yescharacteristic
29C2×#15691335C2211yescharacteristic
30C2×#15691335C2211yescharacteristic
31C2×#15691335C2211yescharacteristic
32C2×#15691335C2211yescharacteristic
33C2×#15691335C2211yescharacteristic
34C2×#15691335C2211yescharacteristic
35C2×#15691335C2211yescharacteristic
36C2×#15691335C2211yescharacteristic
37C2×C4×Dic197C2211yescharacteristic
38C2×C4×Dic197C2211yescharacteristic
39C2×C4×Dic197C2211yescharacteristic
40C2×C4×Dic197C2211yescharacteristic
41#20263233C2211yescharacteristic
42#20263233C2211yescharacteristic
43#20263233C2211yescharacteristic
44#20263233C2211yescharacteristic
45#20263233C2211yescharacteristic
46#20263233C2211yescharacteristic
47C2×C4⋊C4×C197C2211yescharacteristic
48C2×C4⋊C4×C197C2211yescharacteristic
49C2×C42×C197C2211yescharacteristic
50#95×C197C2211yescharacteristic
51#95×C197C2211yescharacteristic
52#95×C197C2211yescharacteristic
53#95×C197C2211yescharacteristic
54C23×D197C2311yescharacteristic
55C23×D197C2311yescharacteristic
56C2×D788—22no—
57C2×D788—22no—
58C2×D788C2×C414yes—
59C22×Dic197C2311yescharacteristic
60C22×Dic197C2311yescharacteristic
61C22×Dic197C2311yescharacteristic
62C22×Dic197C2311yescharacteristic
63C22×Dic197C2311yescharacteristic
64C22×Dic197C2311yescharacteristic
65C2×C4×D197—22no—
66C2×C4×D197—22no—
67C2×C4×D197—22no—
68C2×C4×D197—22no—
69C2×C4×D197—22no—
70C2×C4×D197—22no—
71C2×C4×D197—22no—
72C2×C4×D197—22no—
73C2×C4×D197—41no—
74C2×C4×D197—41no—
75C2×C4×D197—41no—
76C2×C4×D197C2×C414yescomplemented
77C4⋊Dic197—22no—
78#15691337—22no—
79#15691337—22no—
80#15691337—41no—
81#15691337C2×C414yescomplemented
82#15691335—22no—
83#15691335—22no—
84#15691335—22no—
85#15691335—41no—
86#15691335C2×C414yescomplemented
87#15691335C2×C414yescomplemented
88#15691335C2×C414yescomplemented
89#15691335D414yes—
90C4×Dic197—22no—
91C4×Dic197—22no—
92C4×Dic197—22no—
93C4×Dic197—41no—
94C22×C4×C197C2311yescharacteristic
95C22×C4×C197C2311yescharacteristic
96C22×C4×C197C2311yescharacteristic
97C22×C4×C197C2311yescharacteristic
98C22×C4×C197C2311yescharacteristic
99C22×C4×C197C2311yescharacteristic
100C22×C4×C197C2311yescharacteristic
101C4⋊C4×C197—22no—
102C4⋊C4×C197C2×C414yescomplemented
103C42×C197—22no—
104C22×D197—21no—
105C22×D197—21no—
106C22×D197—21no—
107C22×D197—21no—
108C22×D197—21no—
109C22×D197—21no—
110C22×D197—21no—
111C22×D197—21no—
112C22×D197—21no—
113C22×D197—21no—
114C22×D197C22×C412yes—
115C22×D197C22×C412yes—
116C22×D197C2×D412yes—
117C22×D197#4112yes—
118D788—28no—
119C2×Dic197—21no—
120C2×Dic197—21no—
121C2×Dic197—21no—
122C2×Dic197—21no—
123C2×Dic197—21no—
124C2×Dic197—21no—
125C2×Dic197—21no—
126C2×Dic197—21no—
127C2×Dic197—21no—
128C2×Dic197—21no—
129C2×Dic197—21no—
130C2×Dic197—21no—
131C2×Dic197—21no—
132C2×Dic197C22×C412yes—
133C2×Dic197C22×C412yes—
134C2×Dic197C2×D412yes—
135C2×Dic197#4112yes—
136C2×Dic197#4112yes—
137C4×D197—28no—
138C4×D197—44no—
139C4×D197—44no—
140C4×D197—44no—
141C23×C197C2411yescharacteristic
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—21no—
154C2×C4×C197—21no—
155C2×C4×C197—21no—
156C2×C4×C197—21no—
157C2×C4×C197—21no—
158C2×C4×C197—21no—
159C2×C4×C197C22×C412yes—
160C2×C4×C197C22×C412yes—
161C2×C4×C197C22×C412yes—
162C2×C4×C197C2×D412yes—
163C22×C197C23×C411yescharacteristic
164C22×C197C22×D411yescharacteristic
165C22×C197C2×#4111yescharacteristic
166C22×C197C2×#4111yescharacteristic
167C22×C197C2×#4111yescharacteristic
168C22×C197H(5,𝔽2)11yescharacteristic
169C22×C1972−1+411yescharacteristic
170C2×D197—22no—
171C2×D197—22no—
172C2×D197—22no—
173C2×D197—22no—
174C2×D197—22no—
175C2×D197—22no—
176C2×D197—22no—
177C2×D197—22no—
178C2×D197—22no—
179C2×D197—41no—
180C2×D197—41no—
181C2×D197—41no—
182C2×D197—41no—
183C2×D197C4×D414yes—
184C4×C197—22no—
185C4×C197—22no—
186C4×C197—22no—
187C4×C197—22no—
188C4×C197—41no—
189C4×C197—41no—
190C4×C197—41no—
191Dic197—22no—
192Dic197—22no—
193Dic197—22no—
194Dic197—22no—
195Dic197—22no—
196Dic197—41no—
197C2×C197C2×C4×D411yescharacteristic
198C2×C197C4×#4111yescharacteristic
199C2×C197#52011yescharacteristic
200C2×C197#54611yescharacteristic
201C2×C197#54711yescharacteristic
202C2×C197#55111yescharacteristic
203C2×C197#55211yescharacteristic
204D197—24no—
205D197—42no—
206C197#235911yescharacteristic, complemented, Sylow
207#2359—1971nomaximal, Sylow
208C2×#123—1971no—
209C2×C4×D4—1971no—
210C2×#117—1971no—
211#389—1971no—
212#388—1971no—
213#388—1971no—
214#387—1971no—
215#387D19711yescharacteristic, complemented
216#386—1971no—
217#385—1971no—
218#385—1971no—
219C4×C4⋊C4—1971no—
220C4×C22⋊C4—1971no—
221C4×C22⋊C4—1971no—
222C2×#95—1971no—
223C22×D4—1971no—
224C23×C4—1971no—
225C23×C4—1971no—
226C23×C4—1971no—
227C23×C4—1971no—
228#123—1978no—
229C4×D4—3944no—
230#117—3944no—
231C2×C4⋊C4—1971no—
232C2×C4⋊C4—1971no—
233C2×C4⋊C4—1971no—
234C2×C4⋊C4—1971no—
235C2×C4⋊C4—1971no—
236C2×C4⋊C4C2×D19711yescharacteristic
237C2×C4⋊C4C2×D19711yescharacteristic
238C2×C22⋊C4—1971no—
239C2×C22⋊C4—1971no—
240C2×C22⋊C4—1971no—
241C2×C22⋊C4—1971no—
242C2×C22⋊C4—1971no—
243C2×C22⋊C4—1971no—
244C2×C22⋊C4—1971no—
245C2×C22⋊C4—1971no—
246C2×C42—1971no—
247C2×C42—1971no—
248C2×C42—1971no—
249C2×C42—1971no—
250C2×C42C2×D19711yescharacteristic
251#95—1971no—
252#95—1971no—
253#95—1971no—
254#95—1971no—
255#95—1971no—
256#95—1971no—
257#95C2×D19711yescharacteristic
258#95C2×D19711yescharacteristic
259#95C2×D19711yescharacteristic
260#95C2×D19711yescharacteristic
261C24—1971no—
262C24—1971no—
263C2×D4—3942no—
264C2×D4—3942no—
265C2×D4—1974no—
266C22×C4—7881no—
267C22×C4—7881no—
268C22×C4—7881no—
269C22×C4—3942no—
270C22×C4—3942no—
271C22×C4—3942no—
272C22×C4—3942no—
273C22×C4—3942no—
274C22×C4—3942no—
275C22×C4—3942no—
276C22×C4—3942no—
277C22×C4—1971no—
278C22×C4—1971no—
279C22×C4—1971no—
280C22×C4—1971no—
281C22×C4—1971no—
282C22×C4—1971no—
283C22×C4—1974no—
284C22×C4C22×D19711yescharacteristic
285C22×C4C22×D19711yescharacteristic
286C22×C4C22×D19711yescharacteristic
287C22×C4C22×D19711yescharacteristic
288C22×C4C22×D19711yescharacteristic
289C22×C4C22×D19711yescharacteristic
290C22×C4C22×D19711yescharacteristic
291C4⋊C4—22no—
292C4⋊C4—7881no—
293C4⋊C4—3942no—
294C4⋊C4—3942no—
295C4⋊C4—3942no—
296C4⋊C4—1974no—
297C4⋊C4C4×D19714yescomplemented
298C22⋊C4—7881no—
299C22⋊C4—3942no—
300C22⋊C4—3942no—
301C22⋊C4—3942no—
302C22⋊C4—1974no—
303C22⋊C4—1974no—
304C22⋊C4—1974no—
305C22⋊C4—1974no—
306C42—22no—
307C42—7881no—
308C42—3942no—
309C42—3942no—
310C42—3942no—
311C23—3941no—
312C23—3941no—
313C23—3941no—
314C23—3941no—
315C23—3941no—
316C23—3941no—
317C23—3941no—
318C23—3941no—
319C23—3941no—
320C23—3941no—
321C23—1972no—
322C23—1972no—
323C23—1972no—
324C23—1972no—
325C23C23×D19711yescharacteristic
326D4—3948no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—7884no—
345C2×C4—7884no—
346C2×C4—7884no—
347C2×C4—3948no—
348C2×C4—3941no—
349C2×C4—3941no—
350C2×C4—3941no—
351C2×C4—3941no—
352C2×C4—3941no—
353C2×C4—3941no—
354C2×C4—3941no—
355C2×C4—3941no—
356C2×C4—3941no—
357C2×C4—3941no—
358C2×C4—3941no—
359C2×C4—3941no—
360C2×C4—3941no—
361C2×C4—1972no—
362C2×C4—1972no—
363C2×C4—1972no—
364C2×C4—1972no—
365C2×C4—1972no—
366C2×C4C2×C4×D19712yes—
367C2×C4C2×C4×D19712yes—
368C2×C4C2×C4×D19712yes—
369C2×C4D4×D19712yes—
370C22—7881no—
371C22—7881no—
372C22—7881no—
373C22—7881no—
374C22—3942no—
375C22—3942no—
376C22—3942no—
377C22—3942no—
378C22—3942no—
379C22—3942no—
380C22—3942no—
381C22—3942no—
382C22—3942no—
383C22—1974no—
384C22C22×C4×D19711yescharacteristic
385C22C2×D4×D19711yescharacteristic
386C22#41×D19711yescharacteristic
387C22#41×D19711yescharacteristic
388C22#41×D19711yescharacteristic
389C22#2026337211yescharacteristic
390C22#2026337411yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—7881no—
399C4—3942no—
400C4—3942no—
401C4—3942no—
402C4—3942no—
403C4—3942no—
404C2—7882no—
405C2—3944no—
406C2C4×D4×D19711yescharacteristic
407C2#2716154711yescharacteristic
408C2#2716156111yescharacteristic
409C2#2716172111yescharacteristic
410C2#2716173111yescharacteristic
411C2#2716175411yescharacteristic
412C2#2716176111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D197
commutatorC22×C197C23×C4
Fitting#387×C197C2
socle, cosocleC23×C197C24
prime radicalC22×D788C22
hypercenter#387D197
hypocenterC197#2359
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
3. C2×#20263283C2semidirect
4. #27160974C2semidirect
5. #27160968C2semidirect
6. #27160968C2semidirect
7. #27160961C2semidirect
9. #27160953C2semidirect
10. #27160953C2semidirect
11. C4⋊C4×Dic197C2semidirect
12. C4×#15691335C2semidirect
13. #27160926C2semidirect
14. #95×D197C2semidirect
15. #387×C197C2semidirect
21. #20263311C4semidirect
76. C2×C4×D197C2×C4semidirect
81. #15691337C2×C4semidirect
86. #15691335C2×C4semidirect
87. #15691335C2×C4semidirect
88. #15691335C2×C4semidirect
102. C4⋊C4×C197C2×C4semidirect
206. C197#2359semidirect
215. #387D197semidirect
297. C4⋊C4C4×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633112
C2×#202632882
C2×#202632832
#271609742
#271609682
#271609612
#271609572
#271609532
C4⋊C4×Dic1972
C4×#156913352
#271609262
#95×D1972
#387×C1972
#2359197