Finite Group Databasebeta

← #38751963

Subgroups — #38751963

Order 25216.

806 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#15691359C211yesmaximal, characteristic, complemented
2C2×#20263353C212yesmaximal, complemented
3#121×D197C211yesmaximal, characteristic, complemented
4#27161227C212yesmaximal, complemented
5#27161223C212yesmaximal, complemented
6#27161218C211yesmaximal, characteristic, complemented
7#27161215C211yesmaximal, characteristic, complemented
8#27161212C211yesmaximal, characteristic, complemented
9#27161079C212yesmaximal, complemented
10#27161041C211yesmaximal, characteristic, complemented
11#448×C197C211yesmaximal, characteristic, complemented
12C2×#15691359—28no—
13C2×#15691359—21no—
14C2×#15691359—21no—
15C2×#15691359C2212yes—
16C2×#15691359C2212yescomplemented
17C22×D788—21no—
18C22×D788C2211yescharacteristic, complemented
19C2×D4×D197—21no—
20C2×D4×D197C2211yescharacteristic, complemented
21C22×C4×D197—21no—
22C22×C4×D197C2211yescharacteristic
23#20263353—28no—
24C2×#15691345C2212yescomplemented
25C2×D1576C2212yescomplemented
26C2×#15691343C2212yescomplemented
27C2×#15691341C2212yescomplemented
28#20263301C2211yescharacteristic, complemented
29#20263300C2211yescharacteristic
30#20263299—21no—
31#20263298—21no—
32#20263297C2211yescharacteristic, complemented
33C4⋊C4×D197C2211yescharacteristic, complemented
34C22⋊C4×D197—21no—
35#20263250C2212yescomplemented
36#20263248C2212yescomplemented
37#20263246C2212yescomplemented
38C22⋊#13096802C2211yescharacteristic, complemented
39#20263238C2212yescomplemented
40C22×Q8×C197C2211yescharacteristic, complemented
41C2×SD16×C197C2212yescomplemented
42#121×C197C2211yescharacteristic
43#103×C197C2212yescomplemented
44C22⋊C8×C197C2211yescharacteristic, complemented
45C23×D197—21no—
46C23×D197—21no—
47C23×D197D411yescharacteristic
48#15691359—28no—
49#15691359—28no—
50#15691359—24no—
51#15691359—24no—
52#15691359—24no—
53#15691359D418yescomplemented
54C2×D788—28no—
55C2×D788—22no—
56C2×D788—22no—
57C2×D788—24no—
58C2×D788—21no—
59C2×D788—21no—
60C2×D788C2311yescharacteristic
61C2×D788C2312yes—
62C2×C22⋊D197—21no—
63C2×C22⋊D197—21no—
64C2×C22⋊D197—21no—
65D4×D197—24no—
66D4×D197—44no—
67C22×Dic197D411yescharacteristic
68C2×C4×D197—28no—
69C2×C4×D197—22no—
70C2×C4×D197—22no—
71C2×C4×D197—24no—
72C2×C4×D197—21no—
73C2×C4×D197D411yescharacteristic
74C2×C4×D197D411yescharacteristic
75C2×C4×D197C2311yescharacteristic
76C2×C4×D197C2311yescharacteristic
77#15691345—24no—
78D1576—24no—
79#15691343—24no—
80#15691341—24no—
81C2×#13096802C2312yes—
82C4⋊Dic197C2311yescharacteristic
83#15691337—21no—
84C22⋊Dic197—21no—
85#15691335—21no—
86#15691335—21no—
87C2×Q8×C197—24no—
88C2×Q8×C197—21no—
89C2×Q8×C197C2312yes—
90C2×D4×C197—21no—
91C2×D4×C197C2311yescharacteristic
92C22×C4×C197—21no—
93C22×C4×C197C2311yescharacteristic
94SD16×C197—24no—
95C2×C8×C197C2312yes—
96C4⋊C4×C197C2311yescharacteristic
97C22⋊C4×C197—21no—
98C22×D197—24no—
99C22×D197—22no—
100C22×D197—22no—
101C22×D197—22no—
102C22×D197—21no—
103C22×D197—41no—
104C22×D197—42no—
105C22×D197—42no—
106C22×D197—42no—
107C22×D197—42no—
108C22×D197—42no—
109C22×D197C2×D411yescharacteristic
110C22×D197C2×D411yescharacteristic
111D788—28no—
112D788—24no—
113D788—28no—
114D788—22no—
115D788—22no—
116D788—42no—
117D788C2×D414yescomplemented
118C22⋊D197—42no—
119C22⋊D197—42no—
120C22⋊D197—42no—
121C2×Dic197—22no—
122C2×Dic197—21no—
123C2×Dic197C2×D411yescharacteristic
124C2×Dic197C2×D411yescharacteristic
125C4×D197—28no—
126C4×D197—22no—
127C4×D197—22no—
128C4×D197—28no—
129C4×D197—22no—
130C4×D197—42no—
131C4×D197C2×D414yes—
132#13096802—22no—
133C23×C197—21no—
134C23×C197C2×D411yescharacteristic
135Q8×C197—24no—
136Q8×C197—22no—
137Q8×C197C2×D414yescomplemented
138D4×C197—22no—
139D4×C197—42no—
140C2×C4×C197—24no—
141C2×C4×C197—22no—
142C2×C4×C197—22no—
143C2×C4×C197—21no—
144C2×C4×C197C2×D411yescharacteristic
145C2×C4×C197C2411yescharacteristic
146C8×C197—22no—
147C22×C197—21no—
148C22×C197—21no—
149C22×C197—41no—
150C22×C197—41no—
151C22×C197—41no—
152C22×C197AGL(1,ℤ/8)12yescomplemented
153C22×C197C22×D411yescharacteristic
154C2×D197—21no—
155C2×D197—21no—
156C2×D197—21no—
157C2×D197—24no—
158C2×D197—21no—
159C2×D197—21no—
160C2×D197—44no—
161C2×D197—48no—
162C2×D197—42no—
163C2×D197—42no—
164C2×D197—41no—
165C2×D197—41no—
166C2×D197—41no—
167C2×D197—84no—
168C2×D197#12012yes—
169C4×C197—24no—
170C4×C197—21no—
171C4×C197—41no—
172C4×C197C22×D412yes—
173Dic197—21no—
174Dic197—41no—
175Dic197#12012yes—
176C2×C197—22no—
177C2×C197—81no—
178C2×C197C2×#12011yescharacteristic
179C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
180C2×C197#57511yescharacteristic
181D197—22no—
182D197—41no—
183D197—44no—
184D197—81no—
185C197#300611yescharacteristic, complemented, Sylow
186#3006—1971nomaximal, Sylow
187C22×#41—1971no—
188C2×AGL(1,ℤ/8)—1972no—
189C2×#121—1971no—
190#449—1972no—
191#449—1972no—
192#448—1971no—
193#448D19711yescharacteristic, complemented
194#447—1971no—
195#447—1971no—
196#417—1972no—
197#407—1971no—
198C2×#41—3948no—
199C2×#41—3941no—
200C2×#41—3941no—
201C2×#41—1972no—
202C2×#41—1972no—
203C22×Q8C2×D19711yescharacteristic, complemented
204C22×D4—3941no—
205C22×D4—3941no—
206C22×D4—1971no—
207C22×D4—1971no—
208C23×C4—3941no—
209C23×C4—1971no—
210AGL(1,ℤ/8)—3948no—
211C2×SD16—1972no—
212C2×SD16C2×D19712yescomplemented
213C2×D8—1972no—
214C2×D8—1972no—
215C2×M16—1972no—
216#121—3941no—
217#121—3941no—
218#121—1971no—
219#121—1971no—
220#121—1971no—
221#121C2×D19711yescharacteristic
222C2×C4⋊C4—1971no—
223C2×C22⋊C4—3941no—
224#103—1972no—
225#103C2×D19712yescomplemented
226#102—1972no—
227#102—1972no—
228C22⋊C8—1971no—
229C22⋊C8—1972no—
230C22⋊C8C2×D19711yescharacteristic, complemented
231C24—3941no—
232C24—3941no—
233C24—1971no—
234#41—3948no—
235#41—3948no—
236#41—3944no—
237#41—3944no—
238#41—3944no—
239#41—1978no—
240C2×Q8—24no—
241C2×Q8—21no—
242C2×Q8C22×D19712yes—
243C2×D4—21no—
244C2×D4—7884no—
245C2×D4—3941no—
246C2×D4—3948no—
247C2×D4—3942no—
248C2×D4—3942no—
249C2×D4—3944no—
250C2×D4—3941no—
251C2×D4—3941no—
252C2×D4—3944no—
253C2×D4—3941no—
254C2×D4—3941no—
255C2×D4—1971no—
256C2×D4—1972no—
257C2×D4C22×D19711yescharacteristic
258C22×C4—21no—
259C22×C4—3941no—
260C22×C4—3948no—
261C22×C4—3942no—
262C22×C4—3942no—
263C22×C4—3944no—
264C22×C4—1971no—
265C22×C4—1971no—
266C22×C4—1971no—
267C22×C4—1971no—
268C22×C4—1971no—
269C22×C4C22×D19711yescharacteristic
270SD16—24no—
271SD16—3944no—
272D8—3944no—
273D8—3944no—
274M16—3944no—
275C2×C8—1972no—
276C2×C8C22×D19712yes—
277C4⋊C4—3941no—
278C4⋊C4—1971no—
279C4⋊C4C22×D19711yescharacteristic
280C22⋊C4—21no—
281C22⋊C4—3941no—
282C22⋊C4—3941no—
283C22⋊C4—3941no—
284C23—21no—
285C23—7882no—
286C23—7881no—
287C23—7882no—
288C23—7882no—
289C23—7882no—
290C23—7882no—
291C23—3944no—
292C23—3942no—
293C23—3941no—
294C23—3942no—
295C23—3942no—
296C23—1971no—
297C23—1971no—
298C23D4×D19711yescharacteristic
299Q8—24no—
300Q8—22no—
301Q8D4×D19714yescomplemented
302D4—22no—
303D4—42no—
304D4—7882no—
305D4—7882no—
306D4—7882no—
307D4—7882no—
308D4—3948no—
309D4—3944no—
310D4—3948no—
311D4—3942no—
312D4—3942no—
313D4—1974no—
314C2×C4—24no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—7882no—
319C2×C4—3948no—
320C2×C4—3942no—
321C2×C4—3948no—
322C2×C4—3942no—
323C2×C4—3941no—
324C2×C4—3942no—
325C2×C4—3942no—
326C2×C4—1971no—
327C2×C4—1971no—
328C2×C4—1974no—
329C2×C4D4×D19711yescharacteristic
330C2×C4C23×D19711yescharacteristic
331C8—22no—
332C8—3942no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—7888no—
339C22—7884no—
340C22—7882no—
341C22—7882no—
342C22—7881no—
343C22—7881no—
344C22—7881no—
345C22—15764no—
346C22—3941no—
347C22—3941no—
348C22—3941no—
349C22—3944no—
350C22—3941no—
351C22—3941no—
352C22—1972no—
353C22#2026335312yescomplemented
354C22C2×D4×D19711yescharacteristic
355C4—24no—
356C4—21no—
357C4—41no—
358C4—7881no—
359C4—3941no—
360C4—1972no—
361C4C2×D4×D19712yes—
362C2—22no—
363C2—81no—
364C2—7884no—
365C2—7881no—
366C2—15761no—
367C2—3942no—
368C2#120×D19711yescharacteristic
369C2C2×#2026335311yescharacteristic
370C2#2716187811yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#448×C197C2
socleC22×C197C22×D4
hypercenter#448D197
hypocenterC197#3006
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. C22×#15691359C2semidirect
2. C2×#20263353C2semidirect
3. #121×D197C2semidirect
4. #27161227C2semidirect
5. #27161223C2semidirect
6. #27161218C2semidirect
7. #27161215C2semidirect
8. #27161212C2semidirect
9. #27161079C2semidirect
10. #27161041C2semidirect
11. #448×C197C2semidirect
16. C2×#15691359C22semidirect
18. C22×D788C22semidirect
20. C2×D4×D197C22semidirect
24. C2×#15691345C22semidirect
25. C2×D1576C22semidirect
26. C2×#15691343C22semidirect
27. C2×#15691341C22semidirect
28. #20263301C22semidirect
32. #20263297C22semidirect
33. C4⋊C4×D197C22semidirect
35. #20263250C22semidirect
36. #20263248C22semidirect
37. #20263246C22semidirect
38. C22⋊#13096802C22semidirect
39. #20263238C22semidirect
40. C22×Q8×C197C22semidirect
41. C2×SD16×C197C22semidirect
43. #103×C197C22semidirect
44. C22⋊C8×C197C22semidirect
53. #15691359D4semidirect
117. D788C2×D4semidirect
137. Q8×C197C2×D4semidirect
152. C22×C197AGL(1,ℤ/8)semidirect
185. C197#3006semidirect
193. #448D197semidirect
203. C22×Q8C2×D197semidirect
212. C2×SD16C2×D197semidirect
225. #103C2×D197semidirect
230. C22⋊C8C2×D197semidirect
301. Q8D4×D197semidirect
353. C22#20263353semidirect

Maximal subgroups

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

subgroupindex
C22×#156913592
C2×#202633532
#121×D1972
#271612272
#271612232
#271612182
#271612152
#271612122
#271610792
#271610412
#448×C1972
#3006197