Finite Group Databasebeta

← #38754667

Subgroups — #38754667

Order 25216.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27161725C211yesmaximal, characteristic, complemented
2#27161441C211yesmaximal, characteristic, complemented
3#27161409C211yesmaximal, characteristic, complemented
4#27161397C211yesmaximal, characteristic, complemented
5#27161366C211yesmaximal, characteristic, complemented
6#27161330C211yesmaximal, characteristic, complemented
7#27161306C211yesmaximal, characteristic, complemented
8#27161294C211yesmaximal, characteristic, complemented
9#27161222C211yesmaximal, characteristic, complemented
10#27161212C211yesmaximal, characteristic, complemented
11#27161189C211yesmaximal, characteristic, complemented
12#27161151C211yesmaximal, characteristic, complemented
13#27161078C211yesmaximal, characteristic, complemented
14C2×#20263246C211yesmaximal, characteristic, complemented
15#547×C197C211yesmaximal, characteristic, complemented
16C2×#15691358C2211yescharacteristic
17C22×D788C2211yescharacteristic
18C2×#15691346C2211yescharacteristic, complemented
19C2×#15691343C2211yescharacteristic
20C2×#15691343C2211yescharacteristic
21C2×#15691342C2211yescharacteristic, complemented
22C22×#13096802C2211yescharacteristic
23#20263314C2211yescharacteristic
24#20263311—21no—
25#20263305C2211yescharacteristic
26#20263300C2211yescharacteristic
27#20263299—21no—
28#20263294—21no—
29C4×D788C2211yescharacteristic
30C22⋊C4×D197—21no—
31#20263257C2211yescharacteristic, complemented
32#20263257C2211yescharacteristic, complemented
33#20263257C2211yescharacteristic, complemented
34C4⋊#13096802C2211yescharacteristic, complemented
35#20263249C2211yescharacteristic, complemented
36#20263249C2211yescharacteristic, complemented
37#20263249C2211yescharacteristic, complemented
38#20263247C2211yescharacteristic, complemented
39#20263246C2211yescharacteristic
40#20263246C2211yescharacteristic
41#20263246C2211yescharacteristic
42#20263246C2211yescharacteristic
43#20263246C2211yescharacteristic
44#20263246C2211yescharacteristic
45C22⋊#13096802C2211yescharacteristic
46C22⋊#13096802C2211yescharacteristic, complemented
47#20263237C2211yescharacteristic, complemented
48C2×#41×C197—21no—
49C4⋊Q8×C197C2211yescharacteristic, complemented
50#123×C197—21no—
51#123×C197—21no—
52C22⋊Q8×C197—21no—
53#121×C197C2211yescharacteristic
54#121×C197C2211yescharacteristic, complemented
55C4×D4×C197C2211yescharacteristic
56C4×D4×C197C2211yescharacteristic, complemented
57C2×C4⋊C4×C197C2211yescharacteristic
58C2×C4⋊C4×C197C2211yescharacteristic, complemented
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#15691346—41no—
73#15691343—22no—
74#15691343—41no—
75#15691342—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—
88#41×C197—44no—
89C2×Q8×C197—21no—
90C2×D4×C197—21no—
91C2×D4×C197—21no—
92C2×D4×C197D411yescharacteristic
93C2×D4×C197C2311yescharacteristic
94C22×C4×C197—21no—
95C22×C4×C197—21no—
96C22×C4×C197—21no—
97C22×C4×C197C2311yescharacteristic
98C22×C4×C197C2311yescharacteristic
99C4⋊C4×C197—21no—
100C4⋊C4×C197—21no—
101C4⋊C4×C197—21no—
102C4⋊C4×C197D411yescharacteristic
103C4⋊C4×C197C2311yescharacteristic
104C4⋊C4×C197C2311yescharacteristic
105C4⋊C4×C197C2311yescharacteristic
106C22⋊C4×C197—21no—
107C22⋊C4×C197—21no—
108C22⋊C4×C197—21no—
109C22⋊C4×C197D411yescharacteristic
110C22⋊C4×C197D411yescharacteristic
111C42×C197C2311yescharacteristic
112C22×D197—21no—
113C22×D197—21no—
114C22×D197—42no—
115C22×D197—42no—
116C22×D197—81no—
117Dic394—21no—
118Dic394—21no—
119D788—21no—
120D788—21no—
121D788—21no—
122D788—42no—
123D788#4112yescomplemented
124C22⋊D197—42no—
125C22⋊D197—81no—
126C2×Dic197—21no—
127C2×Dic197—21no—
128C4×D197—42no—
129C4×D197—81no—
130#13096802—21no—
131#13096802—21no—
132#13096802—21no—
133#13096802—21no—
134C23×C197—21no—
135C23×C197C2×D411yescharacteristic
136C23×C197C2×D411yescharacteristic
137Q8×C197—42no—
138D4×C197—22no—
139D4×C197—21no—
140D4×C197—21no—
141D4×C197—42no—
142D4×C197—42no—
143C2×C4×C197—21no—
144C2×C4×C197—21no—
145C2×C4×C197—21no—
146C2×C4×C197—21no—
147C2×C4×C197—21no—
148C2×C4×C197—21no—
149C2×C4×C197—21no—
150C2×C4×C197—42no—
151C2×C4×C197—42no—
152C2×C4×C197—42no—
153C2×C4×C197C2×D411yescharacteristic
154C2×C4×C197C2×D411yescharacteristic
155C2×C4×C197C2×D411yescharacteristic
156C2×C4×C197C2×D411yescharacteristic
157C2×C4×C197C2411yescharacteristic
158C22×C197—21no—
159C22×C197—21no—
160C22×C197—21no—
161C22×C197—21no—
162C22×C197—21no—
163C22×C197—41no—
164C22×C197—41no—
165C22×C197—41no—
166C22×C197AGL(1,ℤ/8)12yes—
167C22×C197C22×D411yescharacteristic
168C2×D197—22no—
169C2×D197—41no—
170C2×D197—41no—
171C2×D197—41no—
172C2×D197—41no—
173C2×D197—41no—
174C2×D197—84no—
175C4×C197—21no—
176C4×C197—21no—
177C4×C197—21no—
178C4×C197—21no—
179C4×C197—41no—
180C4×C197—41no—
181C4×C197—41no—
182C4×C197C2×#4111yescharacteristic
183C4×C197H(5,𝔽2)11yescharacteristic
184Dic197—41no—
185Dic197—41no—
186C2×C197—22no—
187C2×C197—41no—
188C2×C197—81no—
189C2×C197#54611yescharacteristic
190C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
191C2×C197#57711yescharacteristic
192D197—42no—
193D197—81no—
194C197#330611yescharacteristic, complemented, Sylow
195#3306—1971nomaximal, Sylow
196#547D19711yescharacteristic, complemented
197#546—1971no—
198#489—1971no—
199#483—1971no—
200#481—1971no—
201#476—1971no—
202#469—1971no—
203#465—1971no—
204#463—1971no—
205#449—1971no—
206#447—1971no—
207#442—1971no—
208#435—1971no—
209#417—1971no—
210C2×#102—1971no—
211C2×#41—21no—
212C2×#41—1971no—
213C22×D4—1971no—
214C2×SD16—1971no—
215C2×D8—1971no—
216C2×D8—1971no—
217C2×M16—1971no—
218C22×C8—1971no—
219C4⋊Q8C2×D19711yescharacteristic, complemented
220#124—1971no—
221#123—21no—
222#123—21no—
223#123—3941no—
224C22⋊Q8—21no—
225C22⋊Q8—1971no—
226#121—3941no—
227#121—1971no—
228#121C2×D19711yescharacteristic
229#121C2×D19711yescharacteristic, complemented
230#120—3941no—
231C4×D4—1971no—
232C4×D4C2×D19711yescharacteristic
233C4×D4C2×D19711yescharacteristic, complemented
234C2×C4⋊C4C2×D19711yescharacteristic
235C2×C4⋊C4C2×D19711yescharacteristic, complemented
236C2×C22⋊C4—3941no—
237#106—1971no—
238#106—1971no—
239#106—1971no—
240C4⋊C8—1971no—
241#103—1971no—
242#103—1971no—
243#103—1971no—
244#102—1971no—
245#102—1971no—
246#102—1971no—
247#102—1971no—
248#102—1971no—
249#102—1971no—
250#102—1971no—
251C22⋊C8—1971no—
252C22⋊C8—1971no—
253#97—1971no—
254C24—3941no—
255#41—44no—
256#41—7881no—
257#41—3942no—
258C2×Q8—21no—
259C2×Q8—1971no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—7881no—
263C2×D4—3942no—
264C2×D4—3941no—
265C2×D4—3941no—
266C2×D4—3941no—
267C2×D4—1971no—
268C2×D4—1971no—
269C2×D4C22⋊D19711yescharacteristic
270C2×D4C22×D19711yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—3941no—
275C22×C4—3941no—
276C22×C4C22×D19711yescharacteristic
277C22×C4C22×D19711yescharacteristic
278SD16—7881no—
279D8—7881no—
280D8—3942no—
281M16—7881no—
282C2×C8—7881no—
283C2×C8—1971no—
284C2×C8—1971no—
285C2×C8—1971no—
286C2×C8—1971no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—3941no—
291C4⋊C4—1971no—
292C4⋊C4C22⋊D19711yescharacteristic
293C4⋊C4C22×D19711yescharacteristic
294C4⋊C4C22×D19711yescharacteristic
295C4⋊C4C22×D19711yescharacteristic
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—3941no—
300C22⋊C4—3941no—
301C22⋊C4—3941no—
302C22⋊C4—3941no—
303C22⋊C4—3941no—
304C22⋊C4C22⋊D19711yescharacteristic
305C22⋊C4C22⋊D19711yescharacteristic
306C42C22×D19711yescharacteristic
307C23—21no—
308C23—7882no—
309C23—7882no—
310C23—15761no—
311C23—3941no—
312C23—3941no—
313C23C2×C22⋊D19711yescharacteristic
314C23C2×C22⋊D19711yescharacteristic
315Q8—42no—
316Q8—3941no—
317Q8—3941no—
318D4—22no—
319D4—21no—
320D4—21no—
321D4—42no—
322D4—42no—
323D4—7882no—
324D4—7882no—
325D4—15761no—
326D4—3941no—
327D4—3941no—
328D4—3941no—
329D4—1972no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—7882no—
341C2×C4—15761no—
342C2×C4—3941no—
343C2×C4—3941no—
344C2×C4C2×C22⋊D19711yescharacteristic
345C2×C4C2×C22⋊D19711yescharacteristic
346C2×C4C2×C22⋊D19711yescharacteristic
347C2×C4C2×C22⋊D19711yescharacteristic
348C2×C4C23×D19711yescharacteristic
349C8—3941no—
350C8—3941no—
351C8—3941no—
352C8—3941no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—7881no—
362C22—7881no—
363C22—7881no—
364C22—7881no—
365C22—7881no—
366C22—15764no—
367C22—3942no—
368C22#2026335212yes—
369C22C22×C22⋊D19711yescharacteristic
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—7881no—
378C4—7881no—
379C4#41×D19711yescharacteristic
380C4#2026337211yescharacteristic
381C2—22no—
382C2—41no—
383C2—81no—
384C2—7882no—
385C2—15761no—
386C2#2716171711yescharacteristic
387C2C2×#2026335211yescharacteristic
388C2#2716188711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#547×C197C2
socleC22×C197C22×D4
hypercenter#547D197
hypocenterC197#3306
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #27161725C2semidirect
2. #27161441C2semidirect
3. #27161409C2semidirect
4. #27161397C2semidirect
5. #27161366C2semidirect
6. #27161330C2semidirect
7. #27161306C2semidirect
8. #27161294C2semidirect
9. #27161222C2semidirect
10. #27161212C2semidirect
11. #27161189C2semidirect
12. #27161151C2semidirect
13. #27161078C2semidirect
14. C2×#20263246C2semidirect
15. #547×C197C2semidirect
18. C2×#15691346C22semidirect
21. C2×#15691342C22semidirect
31. #20263257C22semidirect
32. #20263257C22semidirect
33. #20263257C22semidirect
34. C4⋊#13096802C22semidirect
35. #20263249C22semidirect
36. #20263249C22semidirect
37. #20263249C22semidirect
38. #20263247C22semidirect
46. C22⋊#13096802C22semidirect
47. #20263237C22semidirect
49. C4⋊Q8×C197C22semidirect
54. #121×C197C22semidirect
56. C4×D4×C197C22semidirect
58. C2×C4⋊C4×C197C22semidirect
123. D788#41semidirect
194. C197#3306semidirect
196. #547D197semidirect
219. C4⋊Q8C2×D197semidirect
229. #121C2×D197semidirect
233. C4×D4C2×D197semidirect
235. C2×C4⋊C4C2×D197semidirect

Maximal subgroups

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

subgroupindex
#271617252
#271614412
#271614092
#271613972
#271613662
#271613302
#271613062
#271612942
#271612222
#271612122
#271611892
#271611512
#271610782
C2×#202632462
#547×C1972
#3306197