Finite Group Databasebeta

← #38754405

Subgroups — #38754405

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×#15691347C211yesmaximal, characteristic, complemented
2C2×SD16×D197C211yesmaximal, characteristic, complemented
3#27161742C211yesmaximal, characteristic, complemented
4#27161713C211yesmaximal, characteristic
5#27161450C211yesmaximal, characteristic, complemented
6#27161309C211yesmaximal, characteristic, complemented
7#27161283C211yesmaximal, characteristic, complemented
8#27161278C211yesmaximal, characteristic, complemented
9#27161231C211yesmaximal, characteristic, complemented
10#27161229C211yesmaximal, characteristic, complemented
11#27161219C211yesmaximal, characteristic, complemented
12#27161217C211yesmaximal, characteristic, complemented
13#27161166C211yesmaximal, characteristic, complemented
14C8×C22⋊D197C211yesmaximal, characteristic, complemented
15#465×C197C211yesmaximal, characteristic, complemented
16C22×Dic394C2211yescharacteristic, complemented
17C2×Q8×D197C2211yescharacteristic, complemented
18C22×D788C2211yescharacteristic
19C2×D4×D197—21no—
20C2×D4×D197C2211yescharacteristic
21C2×#15691347—24no—
22C2×#15691347C2211yescharacteristic
23C2×#15691347C2211yescharacteristic
24C2×#15691347C2211yescharacteristic, complemented
25C2×#15691347C2211yescharacteristic, complemented
26C2×#15691346C2211yescharacteristic, complemented
27C2×#15691346C2211yescharacteristic, complemented
28C2×#15691345C2211yescharacteristic
29C2×#15691345C2211yescharacteristic, complemented
30SD16×D197—24no—
31C2×C8×D197C2211yescharacteristic, complemented
32#20263325—21no—
33#20263324C2211yescharacteristic, complemented
34#20263322C2211yescharacteristic
35#20263306C2211yescharacteristic, complemented
36#20263304—21no—
37#20263303—21no—
38#20263301C2211yescharacteristic
39#20263299—21no—
40#20263296—21no—
41#20263294—21no—
42#20263291C2211yescharacteristic, complemented
43#20263288C2211yescharacteristic
44C4×C22⋊D197C2211yescharacteristic
45#20263286—21no—
46#20263255C2211yescharacteristic, complemented
47#20263249C2211yescharacteristic, complemented
48#20263249C2211yescharacteristic, complemented
49#20263246C2211yescharacteristic
50#20263246C2211yescharacteristic, complemented
51C22⋊#13096802C2211yescharacteristic, complemented
52#20263238C2211yescharacteristic, complemented
53C8×Dic197C2211yescharacteristic, complemented
54C2×SD16×C197C2211yescharacteristic, complemented
55C22×C8×C197C2211yescharacteristic, complemented
56C22⋊Q8×C197C2211yescharacteristic, complemented
57#121×C197C2211yescharacteristic
58#106×C197C2211yescharacteristic, complemented
59#103×C197C2211yescharacteristic, complemented
60#102×C197C2211yescharacteristic, complemented
61C23×D197—21no—
62C23×D197—21no—
63C2×Dic394—21no—
64C2×Dic394—22no—
65C2×Dic394—21no—
66C2×Dic394—41no—
67C2×Dic394C2311yescharacteristic
68C2×Dic394C2311yescharacteristic
69Q8×D197—22no—
70Q8×D197—41no—
71C2×D788—22no—
72C2×D788—21no—
73C2×D788—21no—
74C2×D788—41no—
75C2×D788C2311yescharacteristic
76C2×D788C2311yescharacteristic
77C2×C22⋊D197—21no—
78C2×C22⋊D197—21no—
79C2×C22⋊D197—21no—
80C2×C22⋊D197D411yescharacteristic
81D4×D197—22no—
82D4×D197—41no—
83D4×D197—44no—
84C22×Dic197—21no—
85C2×C4×D197—21no—
86C2×C4×D197—21no—
87C2×C4×D197C2311yescharacteristic
88#15691347—22no—
89#15691347—22no—
90#15691347—22no—
91#15691347D414yescomplemented
92#15691346—22no—
93#15691346—41no—
94#15691345—22no—
95#15691345—41no—
96C2×#13096802C2311yescharacteristic
97C2×#13096802C2311yescharacteristic
98C8×D197—22no—
99C4⋊Dic197—21no—
100#15691337—21no—
101#15691337—21no—
102#15691337—21no—
103#15691337D411yescharacteristic
104C22⋊Dic197D411yescharacteristic
105#15691335—21no—
106#15691335—21no—
107#15691335—21no—
108#15691335D411yescharacteristic
109C4×Dic197—21no—
110C4×Dic197C2311yescharacteristic
111C2×Q8×C197C2311yescharacteristic
112C2×D4×C197—21no—
113C2×D4×C197C2311yescharacteristic
114C22×C4×C197C2311yescharacteristic
115SD16×C197—22no—
116C2×C8×C197—22no—
117C2×C8×C197C2311yescharacteristic
118C2×C8×C197C2311yescharacteristic
119C4⋊C4×C197—21no—
120C4⋊C4×C197C2311yescharacteristic
121C4⋊C4×C197C2311yescharacteristic
122C22⋊C4×C197—21no—
123C22⋊C4×C197—21no—
124C22×D197—21no—
125C22×D197—21no—
126C22×D197—42no—
127C22×D197—42no—
128C22×D197—42no—
129C22×D197—42no—
130C22×D197—81no—
131C22×D197—81no—
132C22×D197C2×D411yescharacteristic
133Dic394—21no—
134Dic394—21no—
135Dic394—21no—
136Dic394—42no—
137Dic394—42no—
138Dic394C2×D412yescomplemented
139D788—21no—
140D788—21no—
141D788—21no—
142D788—42no—
143D788—42no—
144D788C2×D412yes—
145C22⋊D197—42no—
146C22⋊D197—42no—
147C22⋊D197—81no—
148C22⋊D197SD1614yescomplemented
149C2×Dic197—21no—
150C2×Dic197—21no—
151C2×Dic197—42no—
152C2×Dic197C2×D411yescharacteristic
153C2×Dic197C2×D411yescharacteristic
154C2×Dic197C2×D411yescharacteristic
155C4×D197—21no—
156C4×D197—21no—
157C4×D197—42no—
158C4×D197—42no—
159#13096802—21no—
160#13096802—21no—
161C23×C197—21no—
162C23×C197C2×D411yescharacteristic
163Q8×C197—21no—
164Q8×C197—21no—
165D4×C197—21no—
166D4×C197—21no—
167D4×C197—42no—
168C2×C4×C197—21no—
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×C197C2×D412yescomplemented
177C22×C197—21no—
178C22×C197—21no—
179C22×C197—41no—
180C22×C197—41no—
181C22×C197—41no—
182C22×C197C2×SD1612yescomplemented
183C22×C197C22×D411yescharacteristic
184C2×D197—21no—
185C2×D197—21no—
186C2×D197—22no—
187C2×D197—41no—
188C2×D197—41no—
189C2×D197—41no—
190C2×D197—41no—
191C2×D197—41no—
192C2×D197—84no—
193C2×D197—84no—
194C2×D197C2×SD1612yescomplemented
195C4×C197—21no—
196C4×C197—41no—
197C4×C197—41no—
198C4×C197—41no—
199C4×C197C22×D411yescharacteristic
200C4×C197H(5,𝔽2)11yescharacteristic
201Dic197—21no—
202Dic197—22no—
203Dic197—21no—
204Dic197—41no—
205Dic197C2×SD1612yescomplemented
206C2×C197—22no—
207C2×C197—81no—
208C2×C197D4211yescharacteristic
209C2×C197C22×SD1611yescharacteristic
210C2×C197#57711yescharacteristic
211D197—22no—
212D197—42no—
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#465D19711yescharacteristic, complemented
223#461—1971no—
224#460—1971no—
225#450—1971no—
226#450—1971no—
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×D4—1971no—
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×SD16—1971no—
246C2×SD16C2×D19711yescharacteristic, complemented
247C22×C8—1971no—
248C22×C8C2×D19711yescharacteristic, complemented
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#121C2×D19711yescharacteristic
259#120—3941no—
260#120—3941no—
261C4×Q8—1971no—
262C4×D4—3941no—
263C4×D4—1971no—
264C4×D4—1971no—
265#106C2×D19711yescharacteristic, complemented
266C4⋊C8—1971no—
267#103—1971no—
268#103—1971no—
269#103C2×D19711yescharacteristic, complemented
270#102—1971no—
271#102—1971no—
272#102C2×D19711yescharacteristic, complemented
273C22⋊C8—1971no—
274C22⋊C8—1971no—
275C4×C8—1971no—
276C24—3941no—
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—21no—
288C2×D4—7881no—
289C2×D4—7884no—
290C2×D4—7881no—
291C2×D4—3941no—
292C2×D4—3942no—
293C2×D4—3942no—
294C2×D4—3941no—
295C2×D4—3941no—
296C2×D4—3941no—
297C2×D4—3941no—
298C2×D4—1971no—
299C2×D4—1971no—
300C2×D4—1971no—
301C2×D4C22×D19711yescharacteristic
302C22×C4—3941no—
303C22×C4—3941no—
304C22×C4—3941no—
305C22×C4—1971no—
306C22×C4C22×D19711yescharacteristic
307SD16—22no—
308SD16—7881no—
309SD16—7881no—
310SD16—3942no—
311SD16—3942no—
312SD16—3942no—
313SD16—3942no—
314SD16—3942no—
315SD16—1974no—
316C2×C8—22no—
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⋊C4C22×D19711yescharacteristic
329C4⋊C4C22×D19711yescharacteristic
330C22⋊C4—21no—
331C22⋊C4—21no—
332C22⋊C4—3941no—
333C22⋊C4—3941no—
334C22⋊C4—3941no—
335C22⋊C4—1971no—
336C22⋊C4—1971no—
337C42—3941no—
338C42—1971no—
339C23—21no—
340C23—7882no—
341C23—7882no—
342C23—7882no—
343C23—7882no—
344C23—15761no—
345C23—15761no—
346C23—3941no—
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—42no—
361D4—7882no—
362D4—7882no—
363D4—7882no—
364D4—7882no—
365D4—15761no—
366D4—3941no—
367D4—3941no—
368D4—3941no—
369D4—1974no—
370D4—1972no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—7882no—
377C2×C4—7882no—
378C2×C4—7882no—
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—3941no—
390C8—3941no—
391C8D4×D19712yescomplemented
392C22—21no—
393C22—21no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—7881no—
398C22—7881no—
399C22—7881no—
400C22—7881no—
401C22—7881no—
402C22—15764no—
403C22—15764no—
404C22—3942no—
405C22—3941no—
406C22—3941no—
407C22—1972no—
408C22SD16×D19712yescomplemented
409C22C2×D4×D19711yescharacteristic
410C4—21no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—7881no—
415C4—3941no—
416C4—3942no—
417C4—3941no—
418C4—1972no—
419C4C2×D4×D19711yescharacteristic
420C4#2026337211yescharacteristic
421C2—22no—
422C2—81no—
423C2—7882no—
424C2—15761no—
425C2—3942no—
426C2#2716171311yescharacteristic
427C2C2×SD16×D19711yescharacteristic
428C2#2716188611yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#465×C197C2
socleC22×C197C22×D4
prime radical#27161713C2
hypercenter#465D197
hypocenterC197D4×SD16
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. C22×#15691347C2semidirect
2. C2×SD16×D197C2semidirect
3. #27161742C2semidirect
5. #27161450C2semidirect
6. #27161309C2semidirect
7. #27161283C2semidirect
8. #27161278C2semidirect
9. #27161231C2semidirect
10. #27161229C2semidirect
11. #27161219C2semidirect
12. #27161217C2semidirect
13. #27161166C2semidirect
14. C8×C22⋊D197C2semidirect
15. #465×C197C2semidirect
16. C22×Dic394C22semidirect
17. C2×Q8×D197C22semidirect
24. C2×#15691347C22semidirect
25. C2×#15691347C22semidirect
26. C2×#15691346C22semidirect
27. C2×#15691346C22semidirect
29. C2×#15691345C22semidirect
31. C2×C8×D197C22semidirect
33. #20263324C22semidirect
35. #20263306C22semidirect
42. #20263291C22semidirect
46. #20263255C22semidirect
47. #20263249C22semidirect
48. #20263249C22semidirect
50. #20263246C22semidirect
51. C22⋊#13096802C22semidirect
52. #20263238C22semidirect
53. C8×Dic197C22semidirect
54. C2×SD16×C197C22semidirect
55. C22×C8×C197C22semidirect
56. C22⋊Q8×C197C22semidirect
58. #106×C197C22semidirect
59. #103×C197C22semidirect
60. #102×C197C22semidirect
91. #15691347D4semidirect
138. Dic394C2×D4semidirect
148. C22⋊D197SD16semidirect
176. C8×C197C2×D4semidirect
182. C22×C197C2×SD16semidirect
194. C2×D197C2×SD16semidirect
205. Dic197C2×SD16semidirect
214. C197D4×SD16semidirect
222. #465D197semidirect
246. C2×SD16C2×D197semidirect
248. C22×C8C2×D197semidirect
255. C22⋊Q8C2×D197semidirect
265. #106C2×D197semidirect
269. #103C2×D197semidirect
272. #102C2×D197semidirect
391. C8D4×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×#156913472
C2×SD16×D1972
#271617422
#271617132
#271614502
#271613092
#271612832
#271612782
#271612312
#271612292
#271612192
#271612172
#271611662
C8×C22⋊D1972
#465×C1972
D4×SD16197