Finite Group Databasebeta

← #38754673

Subgroups — #38754673

Order 25216.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161733C211yesmaximal, characteristic, complemented
2#27161438C211yesmaximal, characteristic, complemented
3#27161407C211yesmaximal, characteristic, complemented
4#27161395C211yesmaximal, characteristic, complemented
5#27161369C211yesmaximal, characteristic, complemented
6#27161330C211yesmaximal, characteristic, complemented
7#27161310C211yesmaximal, characteristic, complemented
8#27161291C211yesmaximal, characteristic, complemented
9#27161224C211yesmaximal, characteristic, complemented
10#27161213C211yesmaximal, characteristic, complemented
11#27161189C211yesmaximal, characteristic, complemented
12#27161151C211yesmaximal, characteristic, complemented
13#27161080C211yesmaximal, characteristic, complemented
14C2×#20263247C211yesmaximal, characteristic, complemented
15#546×C197C211yesmaximal, characteristic, complemented
16C2×#15691357—21no—
17C2×#15691345C2211yescharacteristic, complemented
18C2×#15691343C2211yescharacteristic
19C2×#15691343C2211yescharacteristic
20C2×#15691342C2211yescharacteristic, complemented
21C22×#13096802C2211yescharacteristic
22#20263326C2211yescharacteristic, complemented
23#20263310—21no—
24#20263310—21no—
25#20263306—21no—
26#20263300C2211yescharacteristic
27#20263300C2211yescharacteristic, complemented
28C4×D788C2211yescharacteristic, complemented
29C2×C4⋊Dic197C2211yescharacteristic
30C2×C4⋊Dic197C2211yescharacteristic, complemented
31#20263257C2211yescharacteristic, complemented
32#20263257C2211yescharacteristic, complemented
33#20263257C2211yescharacteristic, complemented
34C4⋊#13096802C2211yescharacteristic, complemented
35#20263250C2211yescharacteristic, complemented
36#20263250C2211yescharacteristic, complemented
37#20263250C2211yescharacteristic, complemented
38#20263247C2211yescharacteristic
39#20263247C2211yescharacteristic
40#20263247C2211yescharacteristic
41#20263247C2211yescharacteristic
42#20263247C2211yescharacteristic
43#20263247C2211yescharacteristic
44#20263246C2211yescharacteristic, complemented
45C22⋊#13096802C2211yescharacteristic
46C22⋊#13096802C2211yescharacteristic, complemented
47#20263237C2211yescharacteristic, complemented
48C2×#41×C197C2211yescharacteristic
49C22×D4×C197C2211yescharacteristic
50#124×C197C2211yescharacteristic
51#123×C197—21no—
52C22⋊Q8×C197C2211yescharacteristic
53#121×C197—21no—
54#121×C197C2211yescharacteristic
55#120×C197—21no—
56C4×D4×C197C2211yescharacteristic
57C4×D4×C197C2211yescharacteristic
58C2×C22⋊C4×C197—21no—
59#15691357—44no—
60C2×Dic394—21no—
61C2×D788C2311yescharacteristic
62C2×C22⋊D197—21no—
63C2×C22⋊D197—21no—
64C22×Dic197—21no—
65C22×Dic197—21no—
66C2×C4×D197—21no—
67#15691345—41no—
68#15691343—22no—
69#15691343—41no—
70#15691342—41no—
71C2×#13096802—41no—
72C2×#13096802C2311yescharacteristic
73C2×#13096802C2311yescharacteristic
74C2×#13096802C2311yescharacteristic
75C2×#13096802C2311yescharacteristic
76C4⋊Dic197—21no—
77C4⋊Dic197—21no—
78C4⋊Dic197—21no—
79C4⋊Dic197C2311yescharacteristic
80C4⋊Dic197C2311yescharacteristic
81C4⋊Dic197C2311yescharacteristic
82#15691335—21no—
83#15691335—21no—
84#15691335—21no—
85C24×C197—21no—
86#41×C197—22no—
87#41×C197—41no—
88C2×Q8×C197C2311yescharacteristic
89C2×D4×C197—22no—
90C2×D4×C197—21no—
91C2×D4×C197—21no—
92C2×D4×C197—21no—
93C2×D4×C197—41no—
94C2×D4×C197D411yescharacteristic
95C2×D4×C197C2311yescharacteristic
96C2×D4×C197C2311yescharacteristic
97C22×C4×C197—21no—
98C22×C4×C197—21no—
99C22×C4×C197C2311yescharacteristic
100C22×C4×C197C2311yescharacteristic
101C4⋊C4×C197—21no—
102C4⋊C4×C197D411yescharacteristic
103C4⋊C4×C197C2311yescharacteristic
104C22⋊C4×C197—21no—
105C22⋊C4×C197—21no—
106C22⋊C4×C197—21no—
107C22⋊C4×C197—21no—
108C22⋊C4×C197—21no—
109C22⋊C4×C197D411yescharacteristic
110C22⋊C4×C197D411yescharacteristic
111C42×C197C2311yescharacteristic
112C22×D197—21no—
113Dic394—42no—
114D788—21no—
115D788—21no—
116C22⋊D197—42no—
117C22⋊D197—42no—
118C2×Dic197—21no—
119C2×Dic197—21no—
120C2×Dic197—21no—
121C2×Dic197—42no—
122C2×Dic197—42no—
123C4×D197—42no—
124#13096802—21no—
125#13096802—21no—
126#13096802—21no—
127#13096802—21no—
128C23×C197—21no—
129C23×C197—21no—
130C23×C197—42no—
131C23×C197—42no—
132C23×C197—81no—
133C23×C197C2×D411yescharacteristic
134C23×C197C2×D411yescharacteristic
135Q8×C197—21no—
136Q8×C197—21no—
137D4×C197—22no—
138D4×C197—21no—
139D4×C197—21no—
140D4×C197—21no—
141D4×C197—42no—
142D4×C197—42no—
143D4×C197—81no—
144D4×C197#4112yescomplemented
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—42no—
152C2×C4×C197—81no—
153C2×C4×C197C2×D411yescharacteristic
154C2×C4×C197C2×D411yescharacteristic
155C2×C4×C197C2×D411yescharacteristic
156C2×C4×C197C2×D411yescharacteristic
157C2×C4×C197C2411yescharacteristic
158C22×C197—21no—
159C22×C197—21no—
160C22×C197—21no—
161C22×C197—22no—
162C22×C197—21no—
163C22×C197—21no—
164C22×C197—41no—
165C22×C197—41no—
166C22×C197—41no—
167C22×C197—41no—
168C22×C197—41no—
169C22×C197—84no—
170C22×C197AGL(1,ℤ/8)12yes—
171C22×C197C22×D411yescharacteristic
172C2×D197—41no—
173C2×D197—41no—
174C2×D197—41no—
175C4×C197—21no—
176C4×C197—21no—
177C4×C197—21no—
178C4×C197—21no—
179C4×C197—41no—
180C4×C197—41no—
181C4×C197C2×#4111yescharacteristic
182C4×C197H(5,𝔽2)11yescharacteristic
183Dic197—41no—
184Dic197—41no—
185Dic197—41no—
186C2×C197—22no—
187C2×C197—41no—
188C2×C197—42no—
189C2×C197—81no—
190C2×C197#54611yescharacteristic
191C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
192C2×C197#57711yescharacteristic
193D197—81no—
194C197#330611yescharacteristic, complemented, Sylow
195#3306—1971nomaximal, Sylow
196#547—1971no—
197#546D19711yescharacteristic, complemented
198#489—1971no—
199#483—1971no—
200#481—1971no—
201#476—1971no—
202#469—1971no—
203#465—1971no—
204#463—1971no—
205#449—1971no—
206#447—1971no—
207#442—1971no—
208#435—1971no—
209#417—1971no—
210C2×#102—1971no—
211C2×#41—3941no—
212C2×#41C2×D19711yescharacteristic
213C22×D4C2×D19711yescharacteristic
214C2×SD16—1971no—
215C2×D8—1971no—
216C2×D8—1971no—
217C2×M16—1971no—
218C22×C8—1971no—
219C4⋊Q8—1971no—
220#124C2×D19711yescharacteristic
221#123—21no—
222#123—3941no—
223#123—3941no—
224C22⋊Q8—3941no—
225C22⋊Q8C2×D19711yescharacteristic
226#121—21no—
227#121—1971no—
228#121—1971no—
229#121C2×D19711yescharacteristic
230#120—21no—
231C4×D4—1971no—
232C4×D4C2×D19711yescharacteristic
233C4×D4C2×D19711yescharacteristic
234C2×C4⋊C4—1971no—
235C2×C4⋊C4—1971no—
236C2×C22⋊C4—21no—
237#106—1971no—
238#106—1971no—
239#106—1971no—
240C4⋊C8—1971no—
241#103—1971no—
242#103—1971no—
243#103—1971no—
244#102—1971no—
245#102—1971no—
246#102—1971no—
247#102—1971no—
248#102—1971no—
249#102—1971no—
250#102—1971no—
251C22⋊C8—1971no—
252C22⋊C8—1971no—
253#97—1971no—
254C24—21no—
255#41—22no—
256#41—41no—
257#41—7884no—
258C2×Q8—3941no—
259C2×Q8C22×D19711yescharacteristic
260C2×D4—22no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—41no—
265C2×D4—3941no—
266C2×D4—3941no—
267C2×D4—1971no—
268C2×D4C22⋊D19711yescharacteristic
269C2×D4C22×D19711yescharacteristic
270C2×D4C22×D19711yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—3941no—
274C22×C4—3941no—
275C22×C4—3941no—
276C22×C4C22×D19711yescharacteristic
277C22×C4C22×D19711yescharacteristic
278SD16—7881no—
279D8—7881no—
280D8—3942no—
281M16—7881no—
282C2×C8—7881no—
283C2×C8—1971no—
284C2×C8—1971no—
285C2×C8—1971no—
286C2×C8—1971no—
287C4⋊C4—21no—
288C4⋊C4—3941no—
289C4⋊C4—3941no—
290C4⋊C4—3941no—
291C4⋊C4—1971no—
292C4⋊C4—1971no—
293C4⋊C4—1971no—
294C4⋊C4C22⋊D19711yescharacteristic
295C4⋊C4C22×D19711yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—3941no—
302C22⋊C4—3941no—
303C22⋊C4—3941no—
304C22⋊C4C22⋊D19711yescharacteristic
305C22⋊C4C22⋊D19711yescharacteristic
306C42C22×D19711yescharacteristic
307C23—21no—
308C23—21no—
309C23—42no—
310C23—42no—
311C23—81no—
312C23—3941no—
313C23C2×C22⋊D19711yescharacteristic
314C23C2×C22⋊D19711yescharacteristic
315Q8—21no—
316Q8—21no—
317Q8—7882no—
318D4—22no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—42no—
323D4—42no—
324D4—81no—
325D4—7882no—
326D4—7882no—
327D4—3941no—
328D4—3941no—
329D4#1569135712yescomplemented
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—42no—
337C2×C4—81no—
338C2×C4—7882no—
339C2×C4—7882no—
340C2×C4—7882no—
341C2×C4—3941no—
342C2×C4—3941no—
343C2×C4—3941no—
344C2×C4C2×C22⋊D19711yescharacteristic
345C2×C4C2×C22⋊D19711yescharacteristic
346C2×C4C2×C22⋊D19711yescharacteristic
347C2×C4C2×C22⋊D19711yescharacteristic
348C2×C4C23×D19711yescharacteristic
349C8—3941no—
350C8—3941no—
351C8—3941no—
352C8—3941no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—84no—
365C22—7881no—
366C22—7881no—
367C22—7881no—
368C22#2026335112yes—
369C22C22×C22⋊D19711yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—7881no—
377C4—7881no—
378C4—7881no—
379C4C2×#1569135711yescharacteristic
380C4#2026337111yescharacteristic
381C2—22no—
382C2—41no—
383C2—42no—
384C2—81no—
385C2—15761no—
386C2#2716172311yescharacteristic
387C2C2×#2026335111yescharacteristic
388C2#2716189011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#546×C197C2
socleC22×C197C22×D4
hypercenter#546D197
hypocenterC197#3306
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. #27161733C2semidirect
2. #27161438C2semidirect
3. #27161407C2semidirect
4. #27161395C2semidirect
5. #27161369C2semidirect
6. #27161330C2semidirect
7. #27161310C2semidirect
8. #27161291C2semidirect
9. #27161224C2semidirect
10. #27161213C2semidirect
11. #27161189C2semidirect
12. #27161151C2semidirect
13. #27161080C2semidirect
14. C2×#20263247C2semidirect
15. #546×C197C2semidirect
17. C2×#15691345C22semidirect
20. C2×#15691342C22semidirect
22. #20263326C22semidirect
27. #20263300C22semidirect
28. C4×D788C22semidirect
30. C2×C4⋊Dic197C22semidirect
31. #20263257C22semidirect
32. #20263257C22semidirect
33. #20263257C22semidirect
34. C4⋊#13096802C22semidirect
35. #20263250C22semidirect
36. #20263250C22semidirect
37. #20263250C22semidirect
44. #20263246C22semidirect
46. C22⋊#13096802C22semidirect
47. #20263237C22semidirect
144. D4×C197#41semidirect
194. C197#3306semidirect
197. #546D197semidirect
329. D4#15691357semidirect

Maximal subgroups

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

subgroupindex
#271617332
#271614382
#271614072
#271613952
#271613692
#271613302
#271613102
#271612912
#271612242
#271612132
#271611892
#271611512
#271610802
C2×#202632472
#546×C1972
#3306197