Finite Group Databasebeta

← #38754534

Subgroups — #38754534

Order 25216.

462 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161784C211yesmaximal, characteristic, complemented
2#27161713C211yesmaximal, characteristic
3#27161424C211yesmaximal, characteristic, complemented
4#27161399C211yesmaximal, characteristic, complemented
5#27161368C211yesmaximal, characteristic, complemented
6#27161309C211yesmaximal, characteristic, complemented
7#27161306C211yesmaximal, characteristic, complemented
8#27161278C211yesmaximal, characteristic, complemented
9#27161232C211yesmaximal, characteristic, complemented
10#27161231C211yesmaximal, characteristic, complemented
11#27161166C211yesmaximal, characteristic, complemented
12#27161142C211yesmaximal, characteristic, complemented
13C2×#20263246C211yesmaximal, characteristic, complemented
14#102×D197C211yesmaximal, characteristic, complemented
15#481×C197C211yesmaximal, characteristic, complemented
16C22×D788C2211yescharacteristic
17C2×D4×D197—21no—
18C2×D4×D197C2211yescharacteristic
19C2×#15691347C2211yescharacteristic
20C2×#15691347C2211yescharacteristic, complemented
21C2×#15691346C2211yescharacteristic, complemented
22C22×#13096802C2211yescharacteristic, complemented
23C2×C8×D197C2211yescharacteristic, complemented
24#20263325C2211yescharacteristic, complemented
25#20263322C2211yescharacteristic
26#20263316—21no—
27#20263306C2211yescharacteristic, complemented
28#20263305—21no—
29#20263305C2211yescharacteristic, complemented
30#20263304—21no—
31#20263301C2211yescharacteristic
32#20263299—21no—
33#20263296—21no—
34#20263294—21no—
35#20263291C2211yescharacteristic, complemented
36#20263288C2211yescharacteristic
37C4×C22⋊D197—21no—
38C4×C22⋊D197C2211yescharacteristic
39C4⋊C4×D197C2211yescharacteristic, complemented
40#20263257C2211yescharacteristic, complemented
41#20263255C2211yescharacteristic, complemented
42#20263249C2211yescharacteristic, complemented
43#20263248C2211yescharacteristic
44#20263248C2211yescharacteristic, complemented
45#20263247C2211yescharacteristic, complemented
46#20263246C2211yescharacteristic
47#20263246C2211yescharacteristic
48#20263246C2211yescharacteristic, complemented
49#20263246C2211yescharacteristic, complemented
50#20263238C2211yescharacteristic, complemented
51C8×Dic197C2211yescharacteristic, complemented
52#121×C197C2211yescharacteristic
53C2×C4⋊C4×C197C2211yescharacteristic, complemented
54#106×C197C2211yescharacteristic, complemented
55#106×C197C2211yescharacteristic, complemented
56#102×C197C2211yescharacteristic, complemented
57#102×C197C2211yescharacteristic, complemented
58C22⋊C8×C197C2211yescharacteristic, complemented
59C23×D197—21no—
60C23×D197—21no—
61C2×Dic394—21no—
62C2×Dic394C2311yescharacteristic
63C2×D788—22no—
64C2×D788—21no—
65C2×D788—21no—
66C2×D788—41no—
67C2×D788C2311yescharacteristic
68C2×D788C2311yescharacteristic
69C2×C22⋊D197—21no—
70C2×C22⋊D197—21no—
71C2×C22⋊D197—21no—
72C2×C22⋊D197D411yescharacteristic
73D4×D197—22no—
74D4×D197—41no—
75D4×D197—44no—
76C2×C4×D197—21no—
77C2×C4×D197—21no—
78C2×C4×D197C2311yescharacteristic
79#15691347—22no—
80#15691347—41no—
81#15691346—41no—
82C2×#13096802—41no—
83C2×#13096802C2311yescharacteristic
84C2×#13096802C2311yescharacteristic
85C8×D197—41no—
86C4⋊Dic197—21no—
87C4⋊Dic197C2311yescharacteristic
88#15691337—21no—
89#15691337—21no—
90#15691337—21no—
91#15691337—21no—
92#15691337D411yescharacteristic
93C22⋊Dic197—21no—
94C22⋊Dic197D411yescharacteristic
95#15691335—21no—
96#15691335—21no—
97#15691335—21no—
98#15691335D411yescharacteristic
99C4×Dic197—21no—
100C4×Dic197C2311yescharacteristic
101C2×D4×C197—21no—
102C2×D4×C197C2311yescharacteristic
103C22×C4×C197—21no—
104C22×C4×C197C2311yescharacteristic
105C2×C8×C197C2311yescharacteristic
106C2×C8×C197C2311yescharacteristic
107C4⋊C4×C197—21no—
108C4⋊C4×C197C2311yescharacteristic
109C4⋊C4×C197C2311yescharacteristic
110C4⋊C4×C197C2311yescharacteristic
111C22⋊C4×C197—21no—
112C22×D197—21no—
113C22×D197—21no—
114C22×D197—42no—
115C22×D197—42no—
116C22×D197—42no—
117C22×D197—42no—
118C22×D197—81no—
119C22×D197—81no—
120C22×D197C2×D411yescharacteristic
121Dic394—21no—
122Dic394—21no—
123Dic394—81no—
124D788—21no—
125D788—21no—
126D788—21no—
127D788—42no—
128D788—42no—
129D788#4112yes—
130C22⋊D197—42no—
131C22⋊D197—42no—
132C22⋊D197—81no—
133C22⋊D197SD1614yescomplemented
134C2×Dic197—21no—
135C2×Dic197—21no—
136C2×Dic197C2×D411yescharacteristic
137C2×Dic197C2×D411yescharacteristic
138C2×Dic197C2×D411yescharacteristic
139C4×D197—21no—
140C4×D197—21no—
141C4×D197—42no—
142C4×D197—81no—
143#13096802—21no—
144#13096802—21no—
145C23×C197—21no—
146C23×C197C2×D411yescharacteristic
147D4×C197—21no—
148D4×C197—21no—
149D4×C197—42no—
150C2×C4×C197—21no—
151C2×C4×C197—21no—
152C2×C4×C197—21no—
153C2×C4×C197—21no—
154C2×C4×C197—21no—
155C2×C4×C197—81no—
156C2×C4×C197C2×D411yescharacteristic
157C2×C4×C197C2411yescharacteristic
158C8×C197—21no—
159C8×C197—21no—
160C22×C197—21no—
161C22×C197—21no—
162C22×C197—41no—
163C22×C197—41no—
164C22×C197—41no—
165C22×C197C2×SD1612yes—
166C22×C197C22×D411yescharacteristic
167C2×D197—21no—
168C2×D197—21no—
169C2×D197—22no—
170C2×D197—41no—
171C2×D197—41no—
172C2×D197—41no—
173C2×D197—41no—
174C2×D197—41no—
175C2×D197—84no—
176C2×D197—84no—
177C2×D197C2×SD1612yes—
178C4×C197—21no—
179C4×C197—41no—
180C4×C197—41no—
181C4×C197—41no—
182C4×C197C2×#4111yescharacteristic
183C4×C197H(5,𝔽2)11yescharacteristic
184Dic197—21no—
185Dic197—21no—
186Dic197—41no—
187Dic197—41no—
188Dic197C2×SD1612yes—
189C2×C197—22no—
190C2×C197—81no—
191C2×C197#54611yescharacteristic
192C2×C197C22×SD1611yescharacteristic
193C2×C197#57611yescharacteristic
194D197—22no—
195D197—42no—
196D197—81no—
197C197#329511yescharacteristic, complemented, Sylow
198#3295—1971nomaximal, Sylow
199#554—1971no—
200D42—1971no—
201#486—1971no—
202#481—1971no—
203#481D19711yescharacteristic, complemented
204#476—1971no—
205#465—1971no—
206#465—1971no—
207#460—1971no—
208#450—1971no—
209#450—1971no—
210C4×SD16—1971no—
211C8×D4—1971no—
212C2×#102—1971no—
213C2×#102—1971no—
214C22×D4—3941no—
215C22×D4—1971no—
216C22×D4—1971no—
217C2×SD16—1971no—
218C2×SD16—1971no—
219C2×SD16—1971no—
220C22×C8—1971no—
221C22×C8—1971no—
222C4⋊Q8—1971no—
223#127—1971no—
224#125—3941no—
225C22⋊Q8—3941no—
226C22⋊Q8—3941no—
227C22⋊Q8—1971no—
228C22⋊Q8—1971no—
229#121—3941no—
230#121—1971no—
231#121C2×D19711yescharacteristic
232#120—3941no—
233#120—3941no—
234C4×Q8—1971no—
235C4×D4—3941no—
236C4×D4—1971no—
237C4×D4—1971no—
238C2×C4⋊C4—1971no—
239C2×C4⋊C4C2×D19711yescharacteristic, complemented
240#106—1971no—
241#106C2×D19711yescharacteristic, complemented
242#106C2×D19711yescharacteristic, complemented
243C4⋊C8—1971no—
244#103—1971no—
245#102—1971no—
246#102—1971no—
247#102—1971no—
248#102—1971no—
249#102—1971no—
250#102—1971no—
251#102—1971no—
252#102C2×D19711yescharacteristic, complemented
253#102C2×D19711yescharacteristic, complemented
254C22⋊C8—1971no—
255C22⋊C8C2×D19711yescharacteristic, complemented
256C4×C8—1971no—
257C24—3941no—
258C24—3941no—
259C2×Q8—3941no—
260C2×Q8—1971no—
261C2×D4—21no—
262C2×D4—7881no—
263C2×D4—7884no—
264C2×D4—7881no—
265C2×D4—3941no—
266C2×D4—3942no—
267C2×D4—3942no—
268C2×D4—3941no—
269C2×D4—3941no—
270C2×D4—3941no—
271C2×D4—3941no—
272C2×D4—1971no—
273C2×D4—1971no—
274C2×D4—1971no—
275C2×D4C22×D19711yescharacteristic
276C22×C4—21no—
277C22×C4—3941no—
278C22×C4—3941no—
279C22×C4—1971no—
280C22×C4C22×D19711yescharacteristic
281SD16—7881no—
282SD16—7881no—
283SD16—3942no—
284C2×C8—7881no—
285C2×C8—7881no—
286C2×C8—1971no—
287C2×C8—1971no—
288C2×C8C22×D19711yescharacteristic
289C2×C8C22×D19711yescharacteristic
290C4⋊C4—21no—
291C4⋊C4—3941no—
292C4⋊C4—3941no—
293C4⋊C4—3941no—
294C4⋊C4—3941no—
295C4⋊C4—3941no—
296C4⋊C4—1971no—
297C4⋊C4—1971no—
298C4⋊C4C22×D19711yescharacteristic
299C4⋊C4C22×D19711yescharacteristic
300C4⋊C4C22×D19711yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—3941no—
303C22⋊C4—3941no—
304C22⋊C4—3941no—
305C22⋊C4—3941no—
306C22⋊C4—1971no—
307C22⋊C4—1971no—
308C42—3941no—
309C42—1971no—
310C23—21no—
311C23—7882no—
312C23—7882no—
313C23—7882no—
314C23—7882no—
315C23—15761no—
316C23—15761no—
317C23—3941no—
318C23—3941no—
319C23—1971no—
320C23D4×D19711yescharacteristic
321Q8—15761no—
322Q8—3941no—
323Q8—3941no—
324D4—21no—
325D4—21no—
326D4—42no—
327D4—7882no—
328D4—7882no—
329D4—7882no—
330D4—7882no—
331D4—15761no—
332D4—3941no—
333D4—3941no—
334D4—3941no—
335D4—1974no—
336D4—1972no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—81no—
343C2×C4—7882no—
344C2×C4—15761no—
345C2×C4—3941no—
346C2×C4—3941no—
347C2×C4—3941no—
348C2×C4—3941no—
349C2×C4—1971no—
350C2×C4—1971no—
351C2×C4—1971no—
352C2×C4D4×D19711yescharacteristic
353C2×C4C23×D19711yescharacteristic
354C8—21no—
355C8—21no—
356C8—3941no—
357C8—3941no—
358C22—21no—
359C22—21no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—7881no—
364C22—7881no—
365C22—7881no—
366C22—7881no—
367C22—7881no—
368C22—15764no—
369C22—15764no—
370C22—3942no—
371C22—3941no—
372C22—3941no—
373C22—1972no—
374C22SD16×D19712yes—
375C22C2×D4×D19711yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—7881no—
381C4—7881no—
382C4—3941no—
383C4—3941no—
384C4—1972no—
385C4#41×D19711yescharacteristic
386C4#2026337211yescharacteristic
387C2—22no—
388C2—81no—
389C2—7882no—
390C2—15761no—
391C2—3942no—
392C2#2716172111yescharacteristic
393C2C2×SD16×D19711yescharacteristic
394C2#2716188311yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#481×C197C2
socleC22×C197C22×D4
prime radical#27161713C2
hypercenter#481D197
hypocenterC197#3295
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
1. #27161784C2semidirect
3. #27161424C2semidirect
4. #27161399C2semidirect
5. #27161368C2semidirect
6. #27161309C2semidirect
7. #27161306C2semidirect
8. #27161278C2semidirect
9. #27161232C2semidirect
10. #27161231C2semidirect
11. #27161166C2semidirect
12. #27161142C2semidirect
13. C2×#20263246C2semidirect
14. #102×D197C2semidirect
15. #481×C197C2semidirect
20. C2×#15691347C22semidirect
21. C2×#15691346C22semidirect
22. C22×#13096802C22semidirect
23. C2×C8×D197C22semidirect
24. #20263325C22semidirect
27. #20263306C22semidirect
29. #20263305C22semidirect
35. #20263291C22semidirect
39. C4⋊C4×D197C22semidirect
40. #20263257C22semidirect
41. #20263255C22semidirect
42. #20263249C22semidirect
44. #20263248C22semidirect
45. #20263247C22semidirect
48. #20263246C22semidirect
49. #20263246C22semidirect
50. #20263238C22semidirect
51. C8×Dic197C22semidirect
53. C2×C4⋊C4×C197C22semidirect
54. #106×C197C22semidirect
55. #106×C197C22semidirect
56. #102×C197C22semidirect
57. #102×C197C22semidirect
58. C22⋊C8×C197C22semidirect
133. C22⋊D197SD16semidirect
197. C197#3295semidirect
203. #481D197semidirect
239. C2×C4⋊C4C2×D197semidirect
241. #106C2×D197semidirect
242. #106C2×D197semidirect
252. #102C2×D197semidirect
253. #102C2×D197semidirect
255. C22⋊C8C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271617842
#271617132
#271614242
#271613992
#271613682
#271613092
#271613062
#271612782
#271612322
#271612312
#271611662
#271611422
C2×#202632462
#102×D1972
#481×C1972
#3295197