Finite Group Databasebeta

← #38754406

Subgroups — #38754406

Order 25216.

546 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#15691346C211yesmaximal, characteristic, complemented
2C2×SD16×D197C211yesmaximal, characteristic, complemented
3#27161742C211yesmaximal, characteristic, complemented
4D4×C22⋊D197C211yesmaximal, characteristic
5#27161450C211yesmaximal, characteristic, complemented
6#27161310C211yesmaximal, characteristic, complemented
7#27161307C211yesmaximal, characteristic, complemented
8#27161282C211yesmaximal, characteristic, complemented
9#27161278C211yesmaximal, characteristic, complemented
10#27161231C211yesmaximal, characteristic, complemented
11#27161220C211yesmaximal, characteristic, complemented
12#27161216C211yesmaximal, characteristic, complemented
13#27161170C211yesmaximal, characteristic, complemented
14#27161142C211yesmaximal, characteristic, complemented
15#450×C197C211yesmaximal, characteristic, complemented
16C22×Dic394C2211yescharacteristic, complemented
17C2×Q8×D197C2211yescharacteristic, complemented
18C22×C22⋊D197—21no—
19C2×D4×D197C2211yescharacteristic
20C2×#15691347C2211yescharacteristic, complemented
21C2×#15691347C2211yescharacteristic, complemented
22C2×#15691346—24no—
23C2×#15691346C2211yescharacteristic
24C2×#15691346C2211yescharacteristic
25C2×#15691346C2211yescharacteristic, complemented
26C2×#15691346C2211yescharacteristic, complemented
27C2×#15691345C2211yescharacteristic, complemented
28SD16×D197—24no—
29C22×#13096802C2211yescharacteristic, complemented
30C2×C8×D197C2211yescharacteristic, complemented
31#20263325—21no—
32#20263324C2211yescharacteristic, complemented
33#20263322C2211yescharacteristic
34#20263306C2211yescharacteristic, complemented
35#20263304—21no—
36#20263303—21no—
37#20263300C2211yescharacteristic
38#20263298—21no—
39#20263297C2211yescharacteristic
40#20263295—21no—
41#20263294—21no—
42#20263291C2211yescharacteristic, complemented
43D4×Dic197C2211yescharacteristic
44C4×C22⋊D197C2211yescharacteristic
45#20263286—21no—
46#20263257C2211yescharacteristic, complemented
47#20263255C2211yescharacteristic, complemented
48#20263251C2211yescharacteristic, complemented
49#20263251C2211yescharacteristic, complemented
50#20263250C2211yescharacteristic, complemented
51#20263246C2211yescharacteristic, complemented
52#20263238C2211yescharacteristic, complemented
53C8×Dic197C2211yescharacteristic, complemented
54C22×D4×C197C2211yescharacteristic
55C2×SD16×C197C2211yescharacteristic
56C2×SD16×C197C2211yescharacteristic, complemented
57C22⋊Q8×C197C2211yescharacteristic, complemented
58#102×C197C2211yescharacteristic
59#102×C197C2211yescharacteristic, complemented
60C22⋊C8×C197C2211yescharacteristic, complemented
61C23×D197—21no—
62C2×Dic394—21no—
63C2×Dic394—22no—
64C2×Dic394—21no—
65C2×Dic394—41no—
66C2×Dic394C2311yescharacteristic
67C2×Dic394C2311yescharacteristic
68Q8×D197—22no—
69Q8×D197—41no—
70C2×D788C2311yescharacteristic
71C2×C22⋊D197—21no—
72C2×C22⋊D197—21no—
73C2×C22⋊D197—21no—
74C2×C22⋊D197—21no—
75C2×C22⋊D197—21no—
76C2×C22⋊D197—44no—
77C2×C22⋊D197D411yescharacteristic
78D4×D197—22no—
79D4×D197—41no—
80C22×Dic197—21no—
81C22×Dic197—21no—
82C2×C4×D197—21no—
83C2×C4×D197C2311yescharacteristic
84#15691347—22no—
85#15691347—41no—
86#15691346—22no—
87#15691346—22no—
88#15691346—22no—
89#15691346D414yescomplemented
90#15691345—22no—
91C2×#13096802—22no—
92C2×#13096802C2311yescharacteristic
93C2×#13096802C2311yescharacteristic
94C8×D197—22no—
95C4⋊Dic197—21no—
96C4⋊Dic197C2311yescharacteristic
97#15691337—21no—
98#15691337—21no—
99#15691337—21no—
100#15691337D411yescharacteristic
101C22⋊Dic197—21no—
102C22⋊Dic197—21no—
103C22⋊Dic197D411yescharacteristic
104#15691335—21no—
105#15691335—21no—
106#15691335D411yescharacteristic
107C4×Dic197—21no—
108C4×Dic197C2311yescharacteristic
109C24×C197—21no—
110C2×Q8×C197C2311yescharacteristic
111C2×D4×C197—22no—
112C2×D4×C197—21no—
113C2×D4×C197—41no—
114C2×D4×C197C2311yescharacteristic
115C2×D4×C197C2311yescharacteristic
116C22×C4×C197C2311yescharacteristic
117SD16×C197—22no—
118SD16×C197—41no—
119C2×C8×C197C2311yescharacteristic
120C2×C8×C197C2311yescharacteristic
121C4⋊C4×C197—21no—
122C4⋊C4×C197C2311yescharacteristic
123C22⋊C4×C197—21no—
124C22×D197—21no—
125C22×D197—42no—
126C22×D197—42no—
127C22×D197—81no—
128C22×D197C2×D411yescharacteristic
129Dic394—21no—
130Dic394—21no—
131Dic394—21no—
132Dic394—42no—
133Dic394—42no—
134Dic394C2×D412yescomplemented
135D788—21no—
136D788—21no—
137C22⋊D197—42no—
138C22⋊D197—42no—
139C22⋊D197—42no—
140C22⋊D197—42no—
141C22⋊D197—81no—
142C22⋊D197SD1614yescomplemented
143C2×Dic197—21no—
144C2×Dic197—21no—
145C2×Dic197—21no—
146C2×Dic197—42no—
147C2×Dic197—42no—
148C2×Dic197C2×D411yescharacteristic
149C2×Dic197C2×D411yescharacteristic
150C2×Dic197C2×D411yescharacteristic
151C4×D197—21no—
152C4×D197—21no—
153C4×D197—42no—
154#13096802—21no—
155#13096802C2×D412yescomplemented
156C23×C197—21no—
157C23×C197—21no—
158C23×C197—42no—
159C23×C197—42no—
160C23×C197—81no—
161C23×C197C2×D411yescharacteristic
162Q8×C197—21no—
163Q8×C197—21no—
164D4×C197—21no—
165D4×C197—21no—
166D4×C197—21no—
167D4×C197—42no—
168D4×C197C2×D412yes—
169C2×C4×C197—21no—
170C2×C4×C197—21no—
171C2×C4×C197—21no—
172C2×C4×C197—21no—
173C2×C4×C197C2×D411yescharacteristic
174C2×C4×C197C2411yescharacteristic
175C8×C197—21no—
176C8×C197—21no—
177C22×C197—21no—
178C22×C197—21no—
179C22×C197—22no—
180C22×C197—41no—
181C22×C197—41no—
182C22×C197—41no—
183C22×C197—41no—
184C22×C197—41no—
185C22×C197—84no—
186C22×C197C2×SD1612yescomplemented
187C22×C197C22×D411yescharacteristic
188C2×D197—21no—
189C2×D197—21no—
190C2×D197—41no—
191C2×D197—41no—
192C2×D197—41no—
193C2×D197—84no—
194C2×D197C2×SD1612yescomplemented
195C4×C197—21no—
196C4×C197—41no—
197C4×C197—41no—
198C4×C197C22×D411yescharacteristic
199C4×C197H(5,𝔽2)11yescharacteristic
200Dic197—21no—
201Dic197—22no—
202Dic197—21no—
203Dic197—41no—
204Dic197—41no—
205Dic197C2×SD1612yescomplemented
206C2×C197—22no—
207C2×C197—42no—
208C2×C197—81no—
209C2×C197D4211yescharacteristic
210C2×C197C22×SD1611yescharacteristic
211C2×C197#57711yescharacteristic
212D197—22no—
213D197—81no—
214C197D4×SD1611yescharacteristic, complemented, Sylow
215D4×SD16—1971nomaximal, Sylow
216C22×SD16—1971no—
217C22×SD16—1971no—
218D4×Q8—1971no—
219D42—1971no—
220#492—1971no—
221#465—1971no—
222#465—1971no—
223#461—1971no—
224#460—1971no—
225#450—1971no—
226#450D19711yescharacteristic, complemented
227#448—1971no—
228#448—1971no—
229C4×SD16—1971no—
230C8×D4—1971no—
231C22×Q8—1971no—
232C22×Q8—1971no—
233C22×D4—3941no—
234C22×D4—1971no—
235C22×D4C2×D19711yescharacteristic
236C2×SD16—3944no—
237C2×SD16—3944no—
238C2×SD16—1971no—
239C2×SD16—1971no—
240C2×SD16—1971no—
241C2×SD16—1971no—
242C2×SD16—1971no—
243C2×SD16—1971no—
244C2×SD16—1971no—
245C2×SD16C2×D19711yescharacteristic
246C2×SD16C2×D19711yescharacteristic, complemented
247C22×C8—1971no—
248C22×C8—1971no—
249C4⋊Q8—3941no—
250C4⋊Q8—1971no—
251#127—1971no—
252C22⋊Q8—3941no—
253C22⋊Q8—3941no—
254C22⋊Q8—1971no—
255C22⋊Q8C2×D19711yescharacteristic, complemented
256#121—3941no—
257#121—1971no—
258#121—1971no—
259#120—3941no—
260#120—3941no—
261C4×Q8—1971no—
262C4×D4—3941no—
263C4×D4—1971no—
264C4×D4—1971no—
265#106—1971no—
266C4⋊C8—1971no—
267#103—1971no—
268#103—1971no—
269#103—1971no—
270#102—1971no—
271#102C2×D19711yescharacteristic
272#102C2×D19711yescharacteristic, complemented
273C22⋊C8—1971no—
274C22⋊C8C2×D19711yescharacteristic, complemented
275C4×C8—1971no—
276C24—21no—
277C24—3941no—
278C2×Q8—7881no—
279C2×Q8—7881no—
280C2×Q8—3942no—
281C2×Q8—3941no—
282C2×Q8—3942no—
283C2×Q8—3941no—
284C2×Q8—1971no—
285C2×Q8—1971no—
286C2×Q8C22×D19711yescharacteristic
287C2×D4—22no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—7884no—
291C2×D4—7881no—
292C2×D4—3941no—
293C2×D4—3941no—
294C2×D4—3942no—
295C2×D4—3941no—
296C2×D4—3941no—
297C2×D4—3941no—
298C2×D4—1971no—
299C2×D4—1971no—
300C2×D4C22×D19711yescharacteristic
301C2×D4C22×D19711yescharacteristic
302C22×C4—3941no—
303C22×C4—3941no—
304C22×C4—3941no—
305C22×C4—1971no—
306C22×C4C22×D19711yescharacteristic
307SD16—22no—
308SD16—41no—
309SD16—7881no—
310SD16—3942no—
311SD16—3942no—
312SD16—3942no—
313SD16—3942no—
314SD16—3942no—
315SD16—1974no—
316C2×C8—3942no—
317C2×C8—3942no—
318C2×C8—1971no—
319C2×C8—1971no—
320C2×C8C22×D19711yescharacteristic
321C2×C8C22×D19711yescharacteristic
322C4⋊C4—21no—
323C4⋊C4—3941no—
324C4⋊C4—3941no—
325C4⋊C4—3941no—
326C4⋊C4—3941no—
327C4⋊C4—1971no—
328C4⋊C4—1971no—
329C4⋊C4C22×D19711yescharacteristic
330C22⋊C4—21no—
331C22⋊C4—3941no—
332C22⋊C4—3941no—
333C22⋊C4—3941no—
334C22⋊C4—3941no—
335C22⋊C4—1971no—
336C22⋊C4—1971no—
337C42—3941no—
338C42—1971no—
339C23—21no—
340C23—21no—
341C23—42no—
342C23—42no—
343C23—81no—
344C23—7882no—
345C23—7882no—
346C23—15761no—
347C23—3941no—
348C23—1971no—
349C23D4×D19711yescharacteristic
350Q8—21no—
351Q8—21no—
352Q8—7882no—
353Q8—7882no—
354Q8—3941no—
355Q8—3941no—
356Q8—3941no—
357Q8—1972no—
358D4—21no—
359D4—21no—
360D4—21no—
361D4—42no—
362D4—7882no—
363D4—7882no—
364D4—7882no—
365D4—7882no—
366D4—15761no—
367D4—3941no—
368D4—3941no—
369D4—1974no—
370D4D4×D19712yes—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—7882no—
376C2×C4—7882no—
377C2×C4—7882no—
378C2×C4—3941no—
379C2×C4—3941no—
380C2×C4—3941no—
381C2×C4—3941no—
382C2×C4—3941no—
383C2×C4—1971no—
384C2×C4—1971no—
385C2×C4—1971no—
386C2×C4D4×D19711yescharacteristic
387C2×C4C23×D19711yescharacteristic
388C8—21no—
389C8—21no—
390C8—3941no—
391C8—1972no—
392C22—21no—
393C22—21no—
394C22—22no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—84no—
401C22—7881no—
402C22—7881no—
403C22—7881no—
404C22—15764no—
405C22—3941no—
406C22—3941no—
407C22—1972no—
408C22SD16×D19712yescomplemented
409C22C2×D4×D19711yescharacteristic
410C4—21no—
411C4—41no—
412C4—41no—
413C4—7881no—
414C4—7881no—
415C4—3941no—
416C4—3942no—
417C4—3941no—
418C4—1972no—
419C4C2×D4×D19711yescharacteristic
420C4#2026337111yescharacteristic
421C2—22no—
422C2—42no—
423C2—81no—
424C2—15761no—
425C2—3942no—
426C2#2716171111yescharacteristic
427C2C2×SD16×D19711yescharacteristic
428C2#2716188911yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#450×C197C2
socleC22×C197C22×D4
prime radicalD4×C22⋊D197C2
hypercenter#450D197
hypocenterC197D4×SD16
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C22×#15691346C2semidirect
2. C2×SD16×D197C2semidirect
3. #27161742C2semidirect
5. #27161450C2semidirect
6. #27161310C2semidirect
7. #27161307C2semidirect
8. #27161282C2semidirect
9. #27161278C2semidirect
10. #27161231C2semidirect
11. #27161220C2semidirect
12. #27161216C2semidirect
13. #27161170C2semidirect
14. #27161142C2semidirect
15. #450×C197C2semidirect
16. C22×Dic394C22semidirect
17. C2×Q8×D197C22semidirect
20. C2×#15691347C22semidirect
21. C2×#15691347C22semidirect
25. C2×#15691346C22semidirect
26. C2×#15691346C22semidirect
27. C2×#15691345C22semidirect
29. C22×#13096802C22semidirect
30. C2×C8×D197C22semidirect
32. #20263324C22semidirect
34. #20263306C22semidirect
42. #20263291C22semidirect
46. #20263257C22semidirect
47. #20263255C22semidirect
48. #20263251C22semidirect
49. #20263251C22semidirect
50. #20263250C22semidirect
51. #20263246C22semidirect
52. #20263238C22semidirect
53. C8×Dic197C22semidirect
56. C2×SD16×C197C22semidirect
57. C22⋊Q8×C197C22semidirect
59. #102×C197C22semidirect
60. C22⋊C8×C197C22semidirect
89. #15691346D4semidirect
134. Dic394C2×D4semidirect
142. C22⋊D197SD16semidirect
155. #13096802C2×D4semidirect
186. C22×C197C2×SD16semidirect
194. C2×D197C2×SD16semidirect
205. Dic197C2×SD16semidirect
214. C197D4×SD16semidirect
226. #450D197semidirect
246. C2×SD16C2×D197semidirect
255. C22⋊Q8C2×D197semidirect
272. #102C2×D197semidirect
274. C22⋊C8C2×D197semidirect
408. C22SD16×D197semidirect

Maximal subgroups

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

subgroupindex
C22×#156913462
C2×SD16×D1972
#271617422
D4×C22⋊D1972
#271614502
#271613102
#271613072
#271612822
#271612782
#271612312
#271612202
#271612162
#271611702
#271611422
#450×C1972
D4×SD16197