Finite Group Databasebeta

← #38754703

Subgroups — #38754703

Order 25216.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161733C211yesmaximal, characteristic, complemented
2#27161438C211yesmaximal, characteristic, complemented
3#27161412C211yesmaximal, characteristic, complemented
4#27161390C211yesmaximal, characteristic, complemented
5#27161374C211yesmaximal, characteristic, complemented
6#27161327C211yesmaximal, characteristic, complemented
7#27161313C211yesmaximal, characteristic, complemented
8#27161291C211yesmaximal, characteristic, complemented
9#27161224C211yesmaximal, characteristic, complemented
10#27161218C211yesmaximal, characteristic, complemented
11#27161177C211yesmaximal, characteristic, complemented
12#27161151C211yesmaximal, characteristic, complemented
13#27161080C211yesmaximal, characteristic, complemented
14C2×#20263250C211yesmaximal, characteristic, complemented
15#548×C197C211yesmaximal, characteristic, complemented
16C2×#15691357—21no—
17C2×#15691345C2211yescharacteristic
18C2×#15691345C2211yescharacteristic
19C2×#15691343C2211yescharacteristic
20C2×#15691342C2211yescharacteristic
21C22×#13096802C2211yescharacteristic
22#20263326C2211yescharacteristic
23#20263310—21no—
24#20263310—21no—
25#20263306—21no—
26#20263300C2211yescharacteristic
27#20263300C2211yescharacteristic
28C4×D788C2211yescharacteristic
29C2×C4⋊Dic197C2211yescharacteristic
30C2×C4⋊Dic197C2211yescharacteristic
31#20263259C2211yescharacteristic
32#20263259C2211yescharacteristic
33#20263259C2211yescharacteristic
34C4⋊#13096802C2211yescharacteristic
35#20263250C2211yescharacteristic
36#20263250C2211yescharacteristic
37#20263250C2211yescharacteristic
38#20263250C2211yescharacteristic
39#20263250C2211yescharacteristic
40#20263250C2211yescharacteristic
41#20263247C2211yescharacteristic
42#20263247C2211yescharacteristic
43#20263247C2211yescharacteristic
44#20263246C2211yescharacteristic
45C22⋊#13096802C2211yescharacteristic
46C22⋊#13096802C2211yescharacteristic
47#20263237C2211yescharacteristic
48C2×#41×C197C2211yescharacteristic
49C22×Q8×C197C2211yescharacteristic
50#124×C197—21no—
51#124×C197C2211yescharacteristic
52C22⋊Q8×C197—21no—
53C22⋊Q8×C197C2211yescharacteristic
54#121×C197—21no—
55#121×C197C2211yescharacteristic
56C4×Q8×C197C2211yescharacteristic
57C4×D4×C197—21no—
58C4×D4×C197C2211yescharacteristic
59#15691357—44no—
60C2×Dic394—21no—
61C2×D788C2311yescharacteristic
62C2×C22⋊D197—21no—
63C2×C22⋊D197—21no—
64C22×Dic197—21no—
65C22×Dic197—21no—
66C2×C4×D197—21no—
67#15691345—22no—
68#15691345—41no—
69#15691343—41no—
70#15691342—41no—
71C2×#13096802—41no—
72C2×#13096802C2311yescharacteristic
73C2×#13096802C2311yescharacteristic
74C2×#13096802C2311yescharacteristic
75C2×#13096802C2311yescharacteristic
76C4⋊Dic197—21no—
77C4⋊Dic197—21no—
78C4⋊Dic197—21no—
79C4⋊Dic197C2311yescharacteristic
80C4⋊Dic197C2311yescharacteristic
81C4⋊Dic197C2311yescharacteristic
82#15691335—21no—
83#15691335—21no—
84#15691335—21no—
85#41×C197—22no—
86#41×C197—41no—
87C2×Q8×C197—21no—
88C2×Q8×C197—22no—
89C2×Q8×C197—41no—
90C2×Q8×C197C2311yescharacteristic
91C2×Q8×C197C2311yescharacteristic
92C2×D4×C197—21no—
93C2×D4×C197—21no—
94C2×D4×C197D411yescharacteristic
95C2×D4×C197C2311yescharacteristic
96C22×C4×C197—21no—
97C22×C4×C197—21no—
98C22×C4×C197C2311yescharacteristic
99C22×C4×C197C2311yescharacteristic
100C4⋊C4×C197—21no—
101C4⋊C4×C197—21no—
102C4⋊C4×C197D411yescharacteristic
103C4⋊C4×C197C2311yescharacteristic
104C22⋊C4×C197—21no—
105C22⋊C4×C197—21no—
106C22⋊C4×C197—21no—
107C22⋊C4×C197—21no—
108C22⋊C4×C197D411yescharacteristic
109C22⋊C4×C197D411yescharacteristic
110C42×C197—21no—
111C42×C197C2311yescharacteristic
112C22×D197—21no—
113Dic394—42no—
114D788—21no—
115D788—21no—
116C22⋊D197—42no—
117C22⋊D197—42no—
118C2×Dic197—21no—
119C2×Dic197—21no—
120C2×Dic197—21no—
121C2×Dic197—42no—
122C2×Dic197—42no—
123C4×D197—42no—
124#13096802—21no—
125#13096802—21no—
126#13096802—21no—
127#13096802—21no—
128C23×C197—21no—
129C23×C197C2×D411yescharacteristic
130C23×C197C2×D411yescharacteristic
131Q8×C197—21no—
132Q8×C197—21no—
133Q8×C197—21no—
134Q8×C197—42no—
135Q8×C197#4112yescomplemented
136D4×C197—22no—
137D4×C197—21no—
138D4×C197—21no—
139D4×C197—42no—
140D4×C197—42no—
141C2×C4×C197—21no—
142C2×C4×C197—21no—
143C2×C4×C197—21no—
144C2×C4×C197—21no—
145C2×C4×C197—21no—
146C2×C4×C197—21no—
147C2×C4×C197—21no—
148C2×C4×C197—42no—
149C2×C4×C197—42no—
150C2×C4×C197C2×D411yescharacteristic
151C2×C4×C197C2×D411yescharacteristic
152C2×C4×C197C2×D411yescharacteristic
153C2×C4×C197C2×D411yescharacteristic
154C2×C4×C197C2411yescharacteristic
155C22×C197—21no—
156C22×C197—21no—
157C22×C197—21no—
158C22×C197—21no—
159C22×C197—21no—
160C22×C197—41no—
161C22×C197—41no—
162C22×C197—41no—
163C22×C197AGL(1,ℤ/8)12yes—
164C22×C197C22×D411yescharacteristic
165C2×D197—41no—
166C2×D197—41no—
167C2×D197—41no—
168C4×C197—21no—
169C4×C197—21no—
170C4×C197—22no—
171C4×C197—21no—
172C4×C197—21no—
173C4×C197—41no—
174C4×C197—41no—
175C4×C197C2×#4111yescharacteristic
176C4×C197H(5,𝔽2)11yescharacteristic
177Dic197—41no—
178Dic197—41no—
179Dic197—41no—
180C2×C197—22no—
181C2×C197—41no—
182C2×C197—81no—
183C2×C197#54611yescharacteristic
184C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C197#57811yescharacteristic
186D197—81no—
187C197#330811yescharacteristic, complemented, Sylow
188#3308—1971nomaximal, Sylow
189#548D19711yescharacteristic, complemented
190#547—1971no—
191#489—1971no—
192#484—1971no—
193#480—1971no—
194#477—1971no—
195#468—1971no—
196#466—1971no—
197#463—1971no—
198#449—1971no—
199#448—1971no—
200#440—1971no—
201#435—1971no—
202#417—1971no—
203C2×#103—1971no—
204C2×#41—3941no—
205C2×#41C2×D19711yescharacteristic
206C22×Q8C2×D19711yescharacteristic
207C2×SD16—1971no—
208C2×SD16—1971no—
209C2×D8—1971no—
210C2×M16—1971no—
211C22×C8—1971no—
212C4⋊Q8—1971no—
213#124—21no—
214#124C2×D19711yescharacteristic
215#123—3941no—
216#123—3941no—
217C22⋊Q8—21no—
218C22⋊Q8—3941no—
219C22⋊Q8C2×D19711yescharacteristic
220#121—21no—
221#121—1971no—
222#121—1971no—
223#121C2×D19711yescharacteristic
224C4×Q8C2×D19711yescharacteristic
225C4×D4—21no—
226C4×D4—1971no—
227C4×D4C2×D19711yescharacteristic
228C2×C4⋊C4—1971no—
229C2×C4⋊C4—1971no—
230#107—1971no—
231#107—1971no—
232#107—1971no—
233C4⋊C8—1971no—
234#103—1971no—
235#103—1971no—
236#103—1971no—
237#103—1971no—
238#103—1971no—
239#103—1971no—
240#102—1971no—
241#102—1971no—
242#102—1971no—
243#102—1971no—
244C22⋊C8—1971no—
245C22⋊C8—1971no—
246#97—1971no—
247#41—22no—
248#41—41no—
249#41—7884no—
250C2×Q8—21no—
251C2×Q8—22no—
252C2×Q8—41no—
253C2×Q8—3941no—
254C2×Q8C22×D19711yescharacteristic
255C2×Q8C22×D19711yescharacteristic
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—3941no—
259C2×D4—3941no—
260C2×D4—1971no—
261C2×D4C22⋊D19711yescharacteristic
262C2×D4C22×D19711yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—3941no—
266C22×C4—3941no—
267C22×C4—3941no—
268C22×C4C22×D19711yescharacteristic
269C22×C4C22×D19711yescharacteristic
270SD16—7881no—
271SD16—3942no—
272D8—7881no—
273M16—7881no—
274C2×C8—7881no—
275C2×C8—1971no—
276C2×C8—1971no—
277C2×C8—1971no—
278C2×C8—1971no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—3941no—
282C4⋊C4—3941no—
283C4⋊C4—3941no—
284C4⋊C4—1971no—
285C4⋊C4—1971no—
286C4⋊C4—1971no—
287C4⋊C4C22⋊D19711yescharacteristic
288C4⋊C4C22×D19711yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—3941no—
294C22⋊C4—3941no—
295C22⋊C4—3941no—
296C22⋊C4C22⋊D19711yescharacteristic
297C22⋊C4C22⋊D19711yescharacteristic
298C42—21no—
299C42C22×D19711yescharacteristic
300C23—21no—
301C23—3941no—
302C23C2×C22⋊D19711yescharacteristic
303C23C2×C22⋊D19711yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—21no—
307Q8—42no—
308Q8—7882no—
309Q8#1569135712yescomplemented
310D4—22no—
311D4—21no—
312D4—21no—
313D4—42no—
314D4—42no—
315D4—7882no—
316D4—7882no—
317D4—3941no—
318D4—3941no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—7882no—
329C2×C4—7882no—
330C2×C4—7882no—
331C2×C4—3941no—
332C2×C4—3941no—
333C2×C4—3941no—
334C2×C4C2×C22⋊D19711yescharacteristic
335C2×C4C2×C22⋊D19711yescharacteristic
336C2×C4C2×C22⋊D19711yescharacteristic
337C2×C4C2×C22⋊D19711yescharacteristic
338C2×C4C23×D19711yescharacteristic
339C8—3941no—
340C8—3941no—
341C8—3941no—
342C8—3941no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—7881no—
352C22—7881no—
353C22—7881no—
354C22#2026335112yes—
355C22C22×C22⋊D19711yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—22no—
361C4—41no—
362C4—41no—
363C4—7881no—
364C4—7881no—
365C4—7881no—
366C4C2×#1569135711yescharacteristic
367C4#2026337111yescharacteristic
368C2—22no—
369C2—41no—
370C2—81no—
371C2—15761no—
372C2#2716172311yescharacteristic
373C2C2×#2026335111yescharacteristic
374C2#2716189411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#548×C197C2
socleC22×C197C22×D4
hypercenter#548D197
hypocenterC197#3308
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #27161733C2semidirect
2. #27161438C2semidirect
3. #27161412C2semidirect
4. #27161390C2semidirect
5. #27161374C2semidirect
6. #27161327C2semidirect
7. #27161313C2semidirect
8. #27161291C2semidirect
9. #27161224C2semidirect
10. #27161218C2semidirect
11. #27161177C2semidirect
12. #27161151C2semidirect
13. #27161080C2semidirect
14. C2×#20263250C2semidirect
15. #548×C197C2semidirect
135. Q8×C197#41semidirect
187. C197#3308semidirect
189. #548D197semidirect
309. Q8#15691357semidirect

Maximal subgroups

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

subgroupindex
#271617332
#271614382
#271614122
#271613902
#271613742
#271613272
#271613132
#271612912
#271612242
#271612182
#271611772
#271611512
#271610802
C2×#202632502
#548×C1972
#3308197