Finite Group Databasebeta

← #38754541

Subgroups — #38754541

Order 25216.

408 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161725C211yesmaximal, characteristic
2#27161436C211yesmaximal, characteristic, complemented
3#27161409C211yesmaximal, characteristic, complemented
4#27161403C211yesmaximal, characteristic, complemented
5#27161376C211yesmaximal, characteristic, complemented
6#27161318C211yesmaximal, characteristic, complemented
7#27161310C211yesmaximal, characteristic, complemented
8#27161301C211yesmaximal, characteristic, complemented
9#27161239C211yesmaximal, characteristic, complemented
10#27161229C211yesmaximal, characteristic, complemented
11C4×#15691345C211yesmaximal, characteristic, complemented
12D4×#13096802C211yesmaximal, characteristic, complemented
13#27161073C211yesmaximal, characteristic, complemented
14C2×#20263246C211yesmaximal, characteristic, complemented
15#552×C197C211yesmaximal, characteristic, complemented
16C2×#15691358C2211yescharacteristic
17C22×D788C2211yescharacteristic
18C2×#15691348C2211yescharacteristic, complemented
19C2×#15691345C2211yescharacteristic
20C2×#15691345C2211yescharacteristic
21C22×#13096802C2211yescharacteristic
22C22×#13096802C2211yescharacteristic, complemented
23#20263314C2211yescharacteristic
24#20263311—21no—
25#20263305C2211yescharacteristic
26#20263300C2211yescharacteristic
27#20263299—21no—
28#20263294—21no—
29C4×D788C2211yescharacteristic
30C22⋊C4×D197—21no—
31#20263259C2211yescharacteristic, complemented
32#20263257C2211yescharacteristic, complemented
33#20263257C2211yescharacteristic, complemented
34C4⋊#13096802C2211yescharacteristic, complemented
35#20263250C2211yescharacteristic, complemented
36#20263249C2211yescharacteristic, complemented
37#20263249C2211yescharacteristic, complemented
38#20263249C2211yescharacteristic, complemented
39#20263246C2211yescharacteristic
40#20263246C2211yescharacteristic
41#20263246C2211yescharacteristic
42#20263246C2211yescharacteristic
43#20263246C2211yescharacteristic
44#20263246C2211yescharacteristic
45C22⋊#13096802C2211yescharacteristic
46C22⋊#13096802C2211yescharacteristic, complemented
47C4×#13096802C2211yescharacteristic, complemented
48#126×C197—21no—
49#125×C197—21no—
50#125×C197C2211yescharacteristic, complemented
51#123×C197—21no—
52C22⋊Q8×C197C2211yescharacteristic
53C22⋊Q8×C197C2211yescharacteristic, complemented
54C4×Q8×C197C2211yescharacteristic, complemented
55C4×D4×C197C2211yescharacteristic
56#117×C197—21no—
57#117×C197C2211yescharacteristic, complemented
58C2×C4⋊C4×C197C2211yescharacteristic
59C23×D197—21no—
60#15691358—22no—
61#15691358—41no—
62C2×Dic394C2311yescharacteristic
63C2×D788—22no—
64C2×D788—21no—
65C2×D788—41no—
66C2×D788C2311yescharacteristic
67C2×D788C2311yescharacteristic
68C2×C22⋊D197—21no—
69C2×C22⋊D197—21no—
70C2×C4×D197—21no—
71C2×C4×D197—21no—
72#15691348—41no—
73#15691345—22no—
74#15691345—41no—
75C2×#13096802—41no—
76C2×#13096802—41no—
77C2×#13096802C2311yescharacteristic
78C2×#13096802C2311yescharacteristic
79C2×#13096802C2311yescharacteristic
80C2×#13096802C2311yescharacteristic
81C4⋊Dic197C2311yescharacteristic
82#15691337—21no—
83C22⋊Dic197—21no—
84#15691335—21no—
85#15691335—21no—
86#15691335—21no—
87#15691335—21no—
88C2×Q8×C197C2311yescharacteristic
89C2×D4×C197D411yescharacteristic
90C22×C4×C197—21no—
91C22×C4×C197C2311yescharacteristic
92C22×C4×C197C2311yescharacteristic
93C4⋊C4×C197—21no—
94C4⋊C4×C197—21no—
95C4⋊C4×C197—21no—
96C4⋊C4×C197—21no—
97C4⋊C4×C197—21no—
98C4⋊C4×C197—21no—
99C4⋊C4×C197D411yescharacteristic
100C4⋊C4×C197C2311yescharacteristic
101C4⋊C4×C197C2311yescharacteristic
102C4⋊C4×C197C2311yescharacteristic
103C22⋊C4×C197—21no—
104C22⋊C4×C197—21no—
105C22⋊C4×C197—21no—
106C22⋊C4×C197D411yescharacteristic
107C22⋊C4×C197D411yescharacteristic
108C42×C197—21no—
109C42×C197—21no—
110C42×C197C2311yescharacteristic
111C22×D197—21no—
112C22×D197—21no—
113C22×D197—42no—
114C22×D197—42no—
115C22×D197—81no—
116Dic394—21no—
117Dic394—21no—
118D788—21no—
119D788—21no—
120D788—21no—
121D788—42no—
122D788#4112yes—
123C22⋊D197—42no—
124C22⋊D197—81no—
125C2×Dic197—21no—
126C2×Dic197—21no—
127C4×D197—42no—
128C4×D197—81no—
129#13096802—21no—
130#13096802—21no—
131#13096802—21no—
132#13096802—21no—
133C23×C197C2×D411yescharacteristic
134C23×C197C2×D411yescharacteristic
135Q8×C197—21no—
136Q8×C197—21no—
137D4×C197—22no—
138C2×C4×C197—21no—
139C2×C4×C197—21no—
140C2×C4×C197—21no—
141C2×C4×C197—21no—
142C2×C4×C197—21no—
143C2×C4×C197—21no—
144C2×C4×C197—21no—
145C2×C4×C197—21no—
146C2×C4×C197—42no—
147C2×C4×C197C2×D411yescharacteristic
148C2×C4×C197C2×D411yescharacteristic
149C2×C4×C197C2×D411yescharacteristic
150C2×C4×C197C2×D411yescharacteristic
151C2×C4×C197C2411yescharacteristic
152C22×C197—21no—
153C22×C197—21no—
154C22×C197—21no—
155C22×C197—21no—
156C22×C197—21no—
157C22×C197#13512yes—
158C22×C197C22×D411yescharacteristic
159C2×D197—22no—
160C2×D197—41no—
161C2×D197—41no—
162C2×D197—41no—
163C2×D197—41no—
164C2×D197—41no—
165C2×D197—84no—
166C4×C197—21no—
167C4×C197—21no—
168C4×C197—21no—
169C4×C197—21no—
170C4×C197—41no—
171C4×C197—41no—
172C4×C197—41no—
173C4×C197—41no—
174C4×C197C2×#4111yescharacteristic
175C4×C197H(5,𝔽2)11yescharacteristic
176Dic197—41no—
177Dic197—41no—
178C2×C197—22no—
179C2×C197—41no—
180C2×C197#54611yescharacteristic
181C2×C197C2×#13511yescharacteristic
182C2×C197#57711yescharacteristic
183D197—42no—
184D197—81no—
185C197#329611yescharacteristic, complemented, Sylow
186#3296—1971nomaximal, Sylow
187#552D19711yescharacteristic, complemented
188#546—1971no—
189#488—1971no—
190#483—1971no—
191#482—1971no—
192#478—1971no—
193#467—1971no—
194#465—1971no—
195#464—1971no—
196#452—1971no—
197#450—1971no—
198C4×SD16—1971no—
199C8×D4—1971no—
200#416—1971no—
201C2×#102—1971no—
202C2×#41—1971no—
203C22×D4—1971no—
204C2×Q16—1971no—
205C2×SD16—1971no—
206C2×SD16—1971no—
207C22×C8—1971no—
208C22×C8—1971no—
209#126—21no—
210#125—21no—
211#125C2×D19711yescharacteristic, complemented
212#124—1971no—
213#123—21no—
214#123—3941no—
215C22⋊Q8—1971no—
216C22⋊Q8C2×D19711yescharacteristic
217C22⋊Q8C2×D19711yescharacteristic, complemented
218#121—3941no—
219#121—1971no—
220#120—3941no—
221C4×Q8C2×D19711yescharacteristic, complemented
222C4×D4—1971no—
223C4×D4C2×D19711yescharacteristic
224#117—21no—
225#117C2×D19711yescharacteristic, complemented
226C2×C4⋊C4C2×D19711yescharacteristic
227C2×C22⋊C4—3941no—
228#107—1971no—
229#106—1971no—
230#106—1971no—
231C4⋊C8—1971no—
232#103—1971no—
233#103—1971no—
234#103—1971no—
235#103—1971no—
236#102—1971no—
237#102—1971no—
238#102—1971no—
239#102—1971no—
240#102—1971no—
241#102—1971no—
242C22⋊C8—1971no—
243C22⋊C8—1971no—
244C4×C8—1971no—
245C24—3941no—
246#41—7881no—
247#41—3942no—
248C2×Q8—1971no—
249C2×Q8C22×D19711yescharacteristic
250C2×D4—7881no—
251C2×D4—3942no—
252C2×D4—3941no—
253C2×D4—3941no—
254C2×D4—3941no—
255C2×D4—1971no—
256C2×D4—1971no—
257C2×D4C22⋊D19711yescharacteristic
258C22×C4—21no—
259C22×C4—3941no—
260C22×C4—3941no—
261C22×C4C22×D19711yescharacteristic
262C22×C4C22×D19711yescharacteristic
263Q16—7881no—
264SD16—7881no—
265SD16—3942no—
266C2×C8—7881no—
267C2×C8—7881no—
268C2×C8—1971no—
269C2×C8—1971no—
270C2×C8—1971no—
271C2×C8—1971no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—3941no—
279C4⋊C4—1971no—
280C4⋊C4C22⋊D19711yescharacteristic
281C4⋊C4C22×D19711yescharacteristic
282C4⋊C4C22×D19711yescharacteristic
283C4⋊C4C22×D19711yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—3941no—
288C22⋊C4—3941no—
289C22⋊C4—3941no—
290C22⋊C4—3941no—
291C22⋊C4—3941no—
292C22⋊C4C22⋊D19711yescharacteristic
293C22⋊C4C22⋊D19711yescharacteristic
294C42—21no—
295C42—21no—
296C42C22×D19711yescharacteristic
297C23—7882no—
298C23—7882no—
299C23—15761no—
300C23—3941no—
301C23—3941no—
302C23C2×C22⋊D19711yescharacteristic
303C23C2×C22⋊D19711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—3941no—
307Q8—3941no—
308D4—22no—
309D4—7882no—
310D4—7882no—
311D4—15761no—
312D4—3941no—
313D4—3941no—
314D4—3941no—
315D4—1972no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—7882no—
326C2×C4—15761no—
327C2×C4—3941no—
328C2×C4—3941no—
329C2×C4C2×C22⋊D19711yescharacteristic
330C2×C4C2×C22⋊D19711yescharacteristic
331C2×C4C2×C22⋊D19711yescharacteristic
332C2×C4C2×C22⋊D19711yescharacteristic
333C2×C4C23×D19711yescharacteristic
334C8—3941no—
335C8—3941no—
336C8—3941no—
337C8—3941no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—7881no—
344C22—7881no—
345C22—7881no—
346C22—7881no—
347C22—7881no—
348C22—15764no—
349C22—3942no—
350C22#2026334612yes—
351C22C22×C22⋊D19711yescharacteristic
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—7881no—
361C4—7881no—
362C4#41×D19711yescharacteristic
363C4#2026337211yescharacteristic
364C2—22no—
365C2—41no—
366C2—7882no—
367C2—15761no—
368C2#2716171711yescharacteristic
369C2C2×#2026334611yescharacteristic
370C2#2716188711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#552×C197C2
socleC22×C197C22×D4
prime radical#27161725C2
hypercenter#552D197
hypocenterC197#3296
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ #552

Complements

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

NH ≅ G/Ntype
2. #27161436C2semidirect
3. #27161409C2semidirect
4. #27161403C2semidirect
5. #27161376C2semidirect
6. #27161318C2semidirect
7. #27161310C2semidirect
8. #27161301C2semidirect
9. #27161239C2semidirect
10. #27161229C2semidirect
11. C4×#15691345C2semidirect
12. D4×#13096802C2semidirect
13. #27161073C2semidirect
14. C2×#20263246C2semidirect
15. #552×C197C2semidirect
18. C2×#15691348C22semidirect
22. C22×#13096802C22semidirect
31. #20263259C22semidirect
32. #20263257C22semidirect
33. #20263257C22semidirect
34. C4⋊#13096802C22semidirect
35. #20263250C22semidirect
36. #20263249C22semidirect
37. #20263249C22semidirect
38. #20263249C22semidirect
46. C22⋊#13096802C22semidirect
47. C4×#13096802C22semidirect
50. #125×C197C22semidirect
53. C22⋊Q8×C197C22semidirect
54. C4×Q8×C197C22semidirect
57. #117×C197C22semidirect
185. C197#3296semidirect
187. #552D197semidirect
211. #125C2×D197semidirect
217. C22⋊Q8C2×D197semidirect
221. C4×Q8C2×D197semidirect
225. #117C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271617252
#271614362
#271614092
#271614032
#271613762
#271613182
#271613102
#271613012
#271612392
#271612292
C4×#156913452
D4×#130968022
#271610732
C2×#202632462
#552×C1972
#3296197