Finite Group Databasebeta

← #38751445

Subgroups — #38751445

Order 25216.

544 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#15691341C211yesmaximal, characteristic, complemented
2C2×M16×D197C211yesmaximal, characteristic, complemented
3C4×D4×D197C211yesmaximal, characteristic
4#27161155C211yesmaximal, characteristic, complemented
5#27161154C211yesmaximal, characteristic, complemented
6#27161152C211yesmaximal, characteristic, complemented
7#27161151C211yesmaximal, characteristic, complemented
8#27161149C211yesmaximal, characteristic, complemented
9C8×C22⋊D197C211yesmaximal, characteristic, complemented
10#27161142C211yesmaximal, characteristic, complemented
11#27161101C211yesmaximal, characteristic, complemented
12#27161041C211yesmaximal, characteristic, complemented
13#27161041C211yesmaximal, characteristic, complemented
14#27161031C211yesmaximal, characteristic, complemented
15#435×C197C211yesmaximal, characteristic, complemented
16C2×D4×D197C411yescharacteristic
17C22×C4×D197C2211yescharacteristic
18C22×C4×D197C2211yescharacteristic
19C2×#15691342C2211yescharacteristic, complemented
20C2×#15691341—24no—
21C2×#15691341C2211yescharacteristic
22C2×#15691341C2211yescharacteristic
23C2×#15691341C2211yescharacteristic
24C2×#15691341C2211yescharacteristic
25C2×#15691341C2211yescharacteristic, complemented
26C2×#15691341C2211yescharacteristic, complemented
27M16×D197—24no—
28C22×#13096802C2211yescharacteristic, complemented
29C2×C8×D197C2211yescharacteristic, complemented
30C2×C8×D197C2211yescharacteristic, complemented
31C4×D788C2211yescharacteristic
32D4×Dic197C411yescharacteristic
33#20263288C411yescharacteristic
34C4×C22⋊D197C2211yescharacteristic
35C4×C22⋊D197C2211yescharacteristic
36#20263286C411yescharacteristic
37#20263286C411yescharacteristic
38C4⋊C4×D197C411yescharacteristic
39C22⋊C4×D197C411yescharacteristic
40C22⋊C4×D197C411yescharacteristic
41C42×D197C2211yescharacteristic
42C4⋊#13096802C2211yescharacteristic, complemented
43#20263255C2211yescharacteristic, complemented
44#20263255C2211yescharacteristic, complemented
45C22⋊#13096802C2211yescharacteristic, complemented
46C22⋊#13096802C2211yescharacteristic, complemented
47#20263238C2211yescharacteristic, complemented
48#20263238C2211yescharacteristic, complemented
49#20263238C2211yescharacteristic, complemented
50#20263238C2211yescharacteristic, complemented
51#20263237C2211yescharacteristic, complemented
52C8×Dic197C2211yescharacteristic, complemented
53C8×Dic197C2211yescharacteristic, complemented
54C2×M16×C197C2211yescharacteristic, complemented
55C22×C8×C197C2211yescharacteristic, complemented
56C4×D4×C197C2211yescharacteristic
57C4⋊C8×C197C2211yescharacteristic, complemented
58C22⋊C8×C197C2211yescharacteristic, complemented
59C22⋊C8×C197C2211yescharacteristic, complemented
60#97×C197C2211yescharacteristic, complemented
61C23×D197C2×C411yescharacteristic
62C23×D197C2×C411yescharacteristic
63C2×D788C2×C411yescharacteristic
64C2×C22⋊D197C2×C411yescharacteristic
65C2×C22⋊D197C2×C411yescharacteristic
66D4×D197—24no—
67C22×Dic197C2×C411yescharacteristic
68C22×Dic197C2×C411yescharacteristic
69C2×C4×D197—22no—
70C2×C4×D197—22no—
71C2×C4×D197—22no—
72C2×C4×D197—22no—
73C2×C4×D197C2×C411yescharacteristic
74C2×C4×D197C2×C411yescharacteristic
75C2×C4×D197C2×C411yescharacteristic
76C2×C4×D197C2×C411yescharacteristic
77C2×C4×D197C2×C411yescharacteristic
78C2×C4×D197C2×C411yescharacteristic
79C2×C4×D197C2311yescharacteristic
80C2×C4×D197C2311yescharacteristic
81C2×C4×D197C2311yescharacteristic
82#15691342—22no—
83#15691341—22no—
84#15691341—22no—
85#15691341—22no—
86#15691341—22no—
87#15691341—22no—
88#15691341D414yescomplemented
89C2×#13096802—22no—
90C2×#13096802C2311yescharacteristic
91C2×#13096802C2311yescharacteristic
92C2×#13096802C2311yescharacteristic
93C2×#13096802C2311yescharacteristic
94C8×D197—22no—
95C8×D197—22no—
96C4⋊Dic197C2×C411yescharacteristic
97#15691337C2×C411yescharacteristic
98#15691337C2×C411yescharacteristic
99C22⋊Dic197C2×C411yescharacteristic
100C22⋊Dic197C2×C411yescharacteristic
101#15691335C2×C411yescharacteristic
102#15691335C2×C411yescharacteristic
103#15691335C2×C411yescharacteristic
104#15691335C2×C411yescharacteristic
105C4×Dic197C2×C411yescharacteristic
106C4×Dic197C2×C411yescharacteristic
107C4×Dic197C2311yescharacteristic
108C2×D4×C197C2×C411yescharacteristic
109C22×C4×C197C2311yescharacteristic
110C22×C4×C197C2311yescharacteristic
111M16×C197—22no—
112C2×C8×C197—22no—
113C2×C8×C197C2311yescharacteristic
114C2×C8×C197C2311yescharacteristic
115C2×C8×C197C2311yescharacteristic
116C2×C8×C197C2311yescharacteristic
117C4⋊C4×C197C2×C411yescharacteristic
118C22⋊C4×C197C2×C411yescharacteristic
119C22⋊C4×C197C2×C411yescharacteristic
120C42×C197C2311yescharacteristic
121C22×D197—22no—
122C22×D197—22no—
123C22×D197—22no—
124C22×D197—22no—
125C22×D197—41no—
126C22×D197—41no—
127C22×D197C22×C411yescharacteristic
128C22×D197C22×C411yescharacteristic
129C22×D197C22×C411yescharacteristic
130D788—22no—
131C22⋊D197—22no—
132C22⋊D197M1614yescomplemented
133C2×Dic197—22no—
134C2×Dic197—22no—
135C2×Dic197C22×C411yescharacteristic
136C2×Dic197C22×C411yescharacteristic
137C2×Dic197C22×C411yescharacteristic
138C2×Dic197C22×C411yescharacteristic
139C2×Dic197C22×C411yescharacteristic
140C4×D197—22no—
141C4×D197—22no—
142C4×D197—21no—
143C4×D197—21no—
144C4×D197—21no—
145C4×D197—21no—
146C4×D197—22no—
147C4×D197—22no—
148C4×D197—22no—
149C4×D197—41no—
150C4×D197C2×D412yes—
151C4×D197#4112yes—
152#13096802—21no—
153#13096802—21no—
154#13096802—21no—
155#13096802C2×D412yescomplemented
156C23×C197C22×C411yescharacteristic
157C23×C197C22×C411yescharacteristic
158D4×C197—22no—
159C2×C4×C197—21no—
160C2×C4×C197—21no—
161C2×C4×C197—21no—
162C2×C4×C197—21no—
163C2×C4×C197C22×C411yescharacteristic
164C2×C4×C197C22×C411yescharacteristic
165C2×C4×C197C22×C411yescharacteristic
166C2×C4×C197C22×C411yescharacteristic
167C2×C4×C197C2411yescharacteristic
168C8×C197—21no—
169C8×C197—21no—
170C8×C197—21no—
171C8×C197C2×D412yescomplemented
172C22×C197—21no—
173C22×C197—21no—
174C22×C197—21no—
175C22×C197—21no—
176C22×C197—21no—
177C22×C197C2×M1612yescomplemented
178C22×C197C23×C411yescharacteristic
179C2×D197—21no—
180C2×D197—21no—
181C2×D197—21no—
182C2×D197—21no—
183C2×D197—21no—
184C2×D197—21no—
185C2×D197—21no—
186C2×D197—44no—
187C2×D197—44no—
188C2×D197C4×D412yes—
189C2×D197C2×M1612yescomplemented
190C4×C197—21no—
191C4×C197—21no—
192C4×C197—21no—
193C4×C197—21no—
194C4×C197C22×D411yescharacteristic
195C4×C197C2×#4111yescharacteristic
196Dic197—21no—
197Dic197—21no—
198Dic197—21no—
199Dic197C4×D412yes—
200Dic197C2×M1612yescomplemented
201C2×C197—22no—
202C2×C197—41no—
203C2×C197C2×C4×D411yescharacteristic
204C2×C197C22×M1611yescharacteristic
205C2×C197#56811yescharacteristic
206D197—22no—
207D197—22no—
208D197—41no—
209C197D4×M1611yescharacteristic, complemented, Sylow
210D4×M16—1971nomaximal, Sylow
211C22×M16—1971no—
212C22×M16—1971no—
213C2×C4×D4—1971no—
214#436—1971no—
215#436—1971no—
216#435—1971no—
217#435—1971no—
218#435—1971no—
219#435D19711yescharacteristic, complemented
220C8×D4—1971no—
221C8×D4—1971no—
222#423—1971no—
223#407—1971no—
224#407—1971no—
225C4×M16—1971no—
226C22×D4—1971no—
227C23×C4—1971no—
228C23×C4—1971no—
229C2×M16—3944no—
230C2×M16—3944no—
231C2×M16—1971no—
232C2×M16—1971no—
233C2×M16—1971no—
234C2×M16—1971no—
235C2×M16—1971no—
236C2×M16—1971no—
237C2×M16—1971no—
238C2×M16C2×D19711yescharacteristic, complemented
239C22×C8—1971no—
240C22×C8—1971no—
241C22×C8—1971no—
242C22×C8C2×D19711yescharacteristic, complemented
243C4×D4—1971no—
244C4×D4—1971no—
245C4×D4—1971no—
246C4×D4—1971no—
247C4×D4—1971no—
248C4×D4—1971no—
249C4×D4—1971no—
250C4×D4C2×D19711yescharacteristic
251C2×C4⋊C4—1971no—
252C2×C22⋊C4—1971no—
253C2×C22⋊C4—1971no—
254C2×C42—1971no—
255C4⋊C8—1971no—
256C4⋊C8—1971no—
257C4⋊C8—1971no—
258C4⋊C8C2×D19711yescharacteristic, complemented
259C22⋊C8—1971no—
260C22⋊C8—1971no—
261C22⋊C8—1971no—
262C22⋊C8—1971no—
263C22⋊C8—1971no—
264C22⋊C8—1971no—
265C22⋊C8C2×D19711yescharacteristic, complemented
266C22⋊C8C2×D19711yescharacteristic, complemented
267#97—1971no—
268#97C2×D19711yescharacteristic, complemented
269C4×C8—1971no—
270C4×C8—1971no—
271C24—1971no—
272C24—1971no—
273C2×D4—3944no—
274C2×D4—1971no—
275C2×D4—1971no—
276C2×D4—1971no—
277C2×D4C4×D19711yescharacteristic
278C22×C4—3942no—
279C22×C4—3942no—
280C22×C4—3942no—
281C22×C4—3942no—
282C22×C4—1971no—
283C22×C4—1971no—
284C22×C4—1971no—
285C22×C4—1971no—
286C22×C4—1971no—
287C22×C4—1971no—
288C22×C4—1971no—
289C22×C4—1971no—
290C22×C4—1971no—
291C22×C4—1971no—
292C22×C4—1971no—
293C22×C4C22×D19711yescharacteristic
294C22×C4C22×D19711yescharacteristic
295M16—22no—
296M16—3942no—
297M16—3942no—
298M16—3942no—
299M16—3942no—
300M16—3942no—
301M16—3942no—
302M16—1974no—
303C2×C8—22no—
304C2×C8—3942no—
305C2×C8—3942no—
306C2×C8—3942no—
307C2×C8—1971no—
308C2×C8—1971no—
309C2×C8—1971no—
310C2×C8—1971no—
311C2×C8C22×D19711yescharacteristic
312C2×C8C22×D19711yescharacteristic
313C2×C8C22×D19711yescharacteristic
314C2×C8C22×D19711yescharacteristic
315C4⋊C4—1971no—
316C4⋊C4—1971no—
317C4⋊C4—1971no—
318C4⋊C4C4×D19711yescharacteristic
319C22⋊C4—1971no—
320C22⋊C4—1971no—
321C22⋊C4—1971no—
322C22⋊C4—1971no—
323C22⋊C4—1971no—
324C22⋊C4—1971no—
325C22⋊C4C4×D19711yescharacteristic
326C22⋊C4C4×D19711yescharacteristic
327C42—1971no—
328C42—1971no—
329C42—1971no—
330C42C22×D19711yescharacteristic
331C23—7881no—
332C23—7881no—
333C23—3942no—
334C23—3942no—
335C23—3942no—
336C23—3942no—
337C23—1971no—
338C23—1971no—
339C23—1971no—
340C23C2×C4×D19711yescharacteristic
341C23C2×C4×D19711yescharacteristic
342D4—22no—
343D4—3942no—
344D4—3942no—
345D4—1974no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—7881no—
351C2×C4—3942no—
352C2×C4—3942no—
353C2×C4—3942no—
354C2×C4—3942no—
355C2×C4—3942no—
356C2×C4—3942no—
357C2×C4—3942no—
358C2×C4—3941no—
359C2×C4—3941no—
360C2×C4—3941no—
361C2×C4—3941no—
362C2×C4—1971no—
363C2×C4—1971no—
364C2×C4—1971no—
365C2×C4—1971no—
366C2×C4—1971no—
367C2×C4—1972no—
368C2×C4—1972no—
369C2×C4C2×C4×D19711yescharacteristic
370C2×C4C2×C4×D19711yescharacteristic
371C2×C4C2×C4×D19711yescharacteristic
372C2×C4C2×C4×D19711yescharacteristic
373C2×C4C23×D19711yescharacteristic
374C8—21no—
375C8—21no—
376C8—21no—
377C8—3941no—
378C8—3941no—
379C8—3941no—
380C8—1972no—
381C8D4×D19712yescomplemented
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—7884no—
388C22—7884no—
389C22—3941no—
390C22—3941no—
391C22—3941no—
392C22—3941no—
393C22—3941no—
394C22—3941no—
395C22—3941no—
396C22—1972no—
397C22—1972no—
398C22M16×D19712yescomplemented
399C22C22×C4×D19711yescharacteristic
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—3941no—
405C4—3941no—
406C4—3941no—
407C4—1972no—
408C4—1972no—
409C4C2×D4×D19711yescharacteristic
410C4#41×D19711yescharacteristic
411C2—22no—
412C2—41no—
413C2—7881no—
414C2—3942no—
415C2—3942no—
416C2C4×D4×D19711yescharacteristic
417C2C2×M16×D19711yescharacteristic
418C2#2716184411yescharacteristic
subgroupG/H
center, FrattiniC2×C4C23×D197
commutator, socleC22×C197C23×C4
Fitting#435×C197C2
prime radicalC2×D4×D197C4
cosocleC2×C4×C197C24
hypercenter#435D197
hypocenterC197D4×M16
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C2×C4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. C22×#15691341C2semidirect
2. C2×M16×D197C2semidirect
4. #27161155C2semidirect
5. #27161154C2semidirect
6. #27161152C2semidirect
7. #27161151C2semidirect
8. #27161149C2semidirect
9. C8×C22⋊D197C2semidirect
10. #27161142C2semidirect
11. #27161101C2semidirect
12. #27161041C2semidirect
13. #27161041C2semidirect
14. #27161031C2semidirect
15. #435×C197C2semidirect
19. C2×#15691342C22semidirect
25. C2×#15691341C22semidirect
26. C2×#15691341C22semidirect
28. C22×#13096802C22semidirect
29. C2×C8×D197C22semidirect
30. C2×C8×D197C22semidirect
42. C4⋊#13096802C22semidirect
43. #20263255C22semidirect
44. #20263255C22semidirect
45. C22⋊#13096802C22semidirect
46. C22⋊#13096802C22semidirect
47. #20263238C22semidirect
48. #20263238C22semidirect
49. #20263238C22semidirect
50. #20263238C22semidirect
51. #20263237C22semidirect
52. C8×Dic197C22semidirect
53. C8×Dic197C22semidirect
54. C2×M16×C197C22semidirect
55. C22×C8×C197C22semidirect
57. C4⋊C8×C197C22semidirect
58. C22⋊C8×C197C22semidirect
59. C22⋊C8×C197C22semidirect
60. #97×C197C22semidirect
88. #15691341D4semidirect
132. C22⋊D197M16semidirect
155. #13096802C2×D4semidirect
171. C8×C197C2×D4semidirect
177. C22×C197C2×M16semidirect
189. C2×D197C2×M16semidirect
200. Dic197C2×M16semidirect
209. C197D4×M16semidirect
219. #435D197semidirect
238. C2×M16C2×D197semidirect
242. C22×C8C2×D197semidirect
258. C4⋊C8C2×D197semidirect
265. C22⋊C8C2×D197semidirect
266. C22⋊C8C2×D197semidirect
268. #97C2×D197semidirect
381. C8D4×D197semidirect
398. C22M16×D197semidirect

Maximal subgroups

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

subgroupindex
C22×#156913412
C2×M16×D1972
C4×D4×D1972
#271611552
#271611542
#271611522
#271611512
#271611492
C8×C22⋊D1972
#271611422
#271611012
#271610412
#271610312
#435×C1972
D4×M16197