Finite Group Databasebeta

← #38754412

Subgroups — #38754412

Order 25216.

504 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263349C211yesmaximal, characteristic, complemented
2C22×#15691345C211yesmaximal, characteristic, complemented
3#27161740C211yesmaximal, characteristic, complemented
4#27161720C211yesmaximal, characteristic, complemented
5#27161464C211yesmaximal, characteristic, complemented
6#27161311C211yesmaximal, characteristic, complemented
7#27161310C211yesmaximal, characteristic, complemented
8#27161303C211yesmaximal, characteristic, complemented
9#27161297C211yesmaximal, characteristic, complemented
10#27161240C211yesmaximal, characteristic, complemented
11#27161227C211yesmaximal, characteristic, complemented
12#27161215C211yesmaximal, characteristic, complemented
13#27161164C211yesmaximal, characteristic, complemented
14#27161142C211yesmaximal, characteristic, complemented
15#451×C197C211yesmaximal, characteristic, complemented
16C2×#15691359C2211yescharacteristic
17C2×#15691359C2211yescharacteristic, complemented
18C22×D788C2211yescharacteristic
19#20263349—24no—
20C2×#15691345—24no—
21C2×#15691345C2211yescharacteristic
22C2×#15691345C2211yescharacteristic
23C2×#15691345C2211yescharacteristic
24C2×#15691345C2211yescharacteristic
25C2×#15691345C2211yescharacteristic, complemented
26C2×D1576C2211yescharacteristic
27C2×D1576C2211yescharacteristic
28C22×#13096802C2211yescharacteristic
29C2×C8×D197C2211yescharacteristic, complemented
30#20263315—21no—
31#20263315C2211yescharacteristic
32#20263315C2211yescharacteristic, complemented
33#20263311—21no—
34#20263302—21no—
35#20263302C2211yescharacteristic, complemented
36#20263301—21no—
37#20263301C2211yescharacteristic
38#20263300—21no—
39#20263300C2211yescharacteristic
40#20263296—21no—
41Q8×Dic197C2211yescharacteristic, complemented
42#20263288C2211yescharacteristic
43C4×C22⋊D197—21no—
44C4×C22⋊D197C2211yescharacteristic
45C22⋊C4×D197—21no—
46#20263257C2211yescharacteristic, complemented
47#20263255C2211yescharacteristic, complemented
48#20263250C2211yescharacteristic, complemented
49#20263248C2211yescharacteristic
50#20263248C2211yescharacteristic
51#20263246C2211yescharacteristic, complemented
52#20263238C2211yescharacteristic, complemented
53C8×Dic197C2211yescharacteristic, complemented
54C22×Q8×C197C2211yescharacteristic
55C2×Q16×C197C2211yescharacteristic, complemented
56C2×Q16×C197C2211yescharacteristic, complemented
57C22⋊Q8×C197C2211yescharacteristic
58#103×C197C2211yescharacteristic, complemented
59#103×C197C2211yescharacteristic, complemented
60C22⋊C8×C197C2211yescharacteristic
61C23×D197—21no—
62#15691359—22no—
63#15691359—22no—
64#15691359—41no—
65#15691359—41no—
66C2×D788—22no—
67C2×D788—21no—
68C2×D788—21no—
69C2×D788—21no—
70C2×D788—21no—
71C2×D788—41no—
72C2×D788C2311yescharacteristic
73C2×D788C2311yescharacteristic
74C2×D788C2311yescharacteristic
75C2×C22⋊D197—21no—
76C2×C22⋊D197D411yescharacteristic
77C2×C4×D197—21no—
78C2×C4×D197—21no—
79C2×C4×D197—21no—
80C2×C4×D197C2311yescharacteristic
81#15691345—22no—
82#15691345—22no—
83#15691345—22no—
84#15691345—22no—
85#15691345D414yescomplemented
86D1576—22no—
87D1576—41no—
88C2×#13096802—22no—
89C2×#13096802C2311yescharacteristic
90C2×#13096802C2311yescharacteristic
91C8×D197—22no—
92C4⋊Dic197—21no—
93C4⋊Dic197C2311yescharacteristic
94#15691337—21no—
95#15691337D411yescharacteristic
96C22⋊Dic197D411yescharacteristic
97#15691335—21no—
98#15691335—21no—
99#15691335—21no—
100#15691335—21no—
101#15691335—21no—
102#15691335—21no—
103#15691335—21no—
104#15691335—21no—
105#15691335D411yescharacteristic
106C4×Dic197—21no—
107C4×Dic197C2311yescharacteristic
108C2×Q8×C197—21no—
109C2×Q8×C197—22no—
110C2×Q8×C197—41no—
111C2×Q8×C197C2311yescharacteristic
112C2×Q8×C197C2311yescharacteristic
113C2×Q8×C197C2311yescharacteristic
114C22×C4×C197—21no—
115C22×C4×C197C2311yescharacteristic
116Q16×C197—22no—
117Q16×C197—41no—
118C2×C8×C197C2311yescharacteristic
119C2×C8×C197C2311yescharacteristic
120C4⋊C4×C197—21no—
121C4⋊C4×C197C2311yescharacteristic
122C22⋊C4×C197—21no—
123C22×D197—21no—
124C22×D197—21no—
125C22×D197—21no—
126C22×D197—42no—
127C22×D197—42no—
128C22×D197—81no—
129C22×D197C2×D411yescharacteristic
130D788—21no—
131D788—21no—
132D788—21no—
133D788—21no—
134D788—21no—
135D788—42no—
136D788—42no—
137D788—42no—
138D788—81no—
139D788C2×D412yes—
140C22⋊D197—22no—
141C22⋊D197—42no—
142C2×Dic197—21no—
143C2×Dic197C2×D411yescharacteristic
144C2×Dic197C2×D411yescharacteristic
145C2×Dic197C2×D411yescharacteristic
146C4×D197—21no—
147C4×D197—21no—
148C4×D197—42no—
149C4×D197—42no—
150C4×D197—81no—
151#13096802—21no—
152#13096802C2×D412yes—
153C23×C197C2×D411yescharacteristic
154Q8×C197—21no—
155Q8×C197—21no—
156Q8×C197—21no—
157Q8×C197—21no—
158Q8×C197—21no—
159Q8×C197—42no—
160Q8×C197C2×D412yes—
161C2×C4×C197—21no—
162C2×C4×C197—21no—
163C2×C4×C197—21no—
164C2×C4×C197—21no—
165C2×C4×C197—21no—
166C2×C4×C197—21no—
167C2×C4×C197—42no—
168C2×C4×C197C2×D411yescharacteristic
169C2×C4×C197C2411yescharacteristic
170C8×C197—21no—
171C8×C197—21no—
172C22×C197—21no—
173C22×C197—21no—
174C22×C197#13512yescomplemented
175C22×C197C22×D411yescharacteristic
176C2×D197—21no—
177C2×D197—22no—
178C2×D197—21no—
179C2×D197—21no—
180C2×D197—41no—
181C2×D197—41no—
182C2×D197—41no—
183C2×D197—41no—
184C2×D197—41no—
185C2×D197—41no—
186C2×D197—41no—
187C2×D197—41no—
188C2×D197—84no—
189C4×C197—21no—
190C4×C197—22no—
191C4×C197—41no—
192C4×C197—41no—
193C4×C197—41no—
194C4×C197C22×D411yescharacteristic
195C4×C197H(5,𝔽2)11yescharacteristic
196Dic197—21no—
197Dic197—21no—
198Dic197—21no—
199Dic197—41no—
200C2×C197—22no—
201C2×C197D4211yescharacteristic
202C2×C197C2×#13511yescharacteristic
203C2×C197#57711yescharacteristic
204D197—41no—
205D197—42no—
206D197—81no—
207D197—81no—
208C197#328211yescharacteristic, complemented, Sylow
209#3282—1971nomaximal, Sylow
210C2×#135—1971no—
211C22×SD16—1971no—
212#548—1971no—
213#546—1971no—
214#495—1971no—
215#465—1971no—
216#465—1971no—
217#464—1971no—
218#463—1971no—
219#452—1971no—
220#451D19711yescharacteristic, complemented
221#449—1971no—
222#447—1971no—
223C4×SD16—1971no—
224C8×D4—1971no—
225C2×#41—1971no—
226C2×#41—1971no—
227C22×Q8C2×D19711yescharacteristic
228C22×D4—1971no—
229#135—3944no—
230C2×Q16C2×D19711yescharacteristic, complemented
231C2×Q16C2×D19711yescharacteristic, complemented
232C2×SD16—3944no—
233C2×SD16—1971no—
234C2×SD16—1971no—
235C2×SD16—1971no—
236C2×SD16—1971no—
237C2×SD16—1971no—
238C2×D8—1971no—
239C2×D8—1971no—
240C22×C8—1971no—
241C22×C8—1971no—
242#124—3941no—
243#124—1971no—
244#124—1971no—
245#123—3941no—
246C22⋊Q8—3941no—
247C22⋊Q8—1971no—
248C22⋊Q8C2×D19711yescharacteristic
249#121—3941no—
250#121—3941no—
251#121—1971no—
252#121—1971no—
253#120—3941no—
254C4×Q8—1971no—
255C4×D4—3941no—
256C4×D4—1971no—
257C4×D4—1971no—
258C2×C22⋊C4—3941no—
259#106—1971no—
260C4⋊C8—1971no—
261#103—1971no—
262#103C2×D19711yescharacteristic, complemented
263#103C2×D19711yescharacteristic, complemented
264#102—1971no—
265#102—1971no—
266#102—1971no—
267C22⋊C8—1971no—
268C22⋊C8C2×D19711yescharacteristic
269C4×C8—1971no—
270C24—3941no—
271#41—7881no—
272#41—7881no—
273#41—3942no—
274#41—3942no—
275C2×Q8—21no—
276C2×Q8—22no—
277C2×Q8—41no—
278C2×Q8C22×D19711yescharacteristic
279C2×Q8C22×D19711yescharacteristic
280C2×Q8C22×D19711yescharacteristic
281C2×D4—7881no—
282C2×D4—3941no—
283C2×D4—3941no—
284C2×D4—3942no—
285C2×D4—3941no—
286C2×D4—3941no—
287C2×D4—3941no—
288C2×D4—1971no—
289C2×D4—1971no—
290C2×D4—1971no—
291C2×D4—1971no—
292C22×C4—21no—
293C22×C4—3941no—
294C22×C4—3941no—
295C22×C4—3941no—
296C22×C4—1971no—
297C22×C4C22×D19711yescharacteristic
298Q16—22no—
299Q16—41no—
300SD16—3942no—
301SD16—3942no—
302SD16—3942no—
303SD16—3942no—
304SD16—1974no—
305D8—7881no—
306D8—3942no—
307C2×C8—3942no—
308C2×C8—3942no—
309C2×C8—1971no—
310C2×C8—1971no—
311C2×C8C22×D19711yescharacteristic
312C2×C8C22×D19711yescharacteristic
313C4⋊C4—21no—
314C4⋊C4—3941no—
315C4⋊C4—3941no—
316C4⋊C4—1971no—
317C4⋊C4—1971no—
318C4⋊C4C22×D19711yescharacteristic
319C22⋊C4—21no—
320C22⋊C4—3941no—
321C22⋊C4—3941no—
322C22⋊C4—3941no—
323C22⋊C4—3941no—
324C22⋊C4—3941no—
325C22⋊C4—3941no—
326C22⋊C4—3941no—
327C22⋊C4—3941no—
328C22⋊C4—1971no—
329C22⋊C4—1971no—
330C42—3941no—
331C42—1971no—
332C23—7882no—
333C23—7882no—
334C23—15761no—
335C23—3941no—
336C23—3941no—
337C23—3941no—
338C23—1971no—
339C23D4×D19711yescharacteristic
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—21no—
344Q8—21no—
345Q8—42no—
346Q8D4×D19712yes—
347D4—7882no—
348D4—7882no—
349D4—7882no—
350D4—7882no—
351D4—15761no—
352D4—3942no—
353D4—3941no—
354D4—3941no—
355D4—3941no—
356D4—3941no—
357D4—3941no—
358D4—1972no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—42no—
366C2×C4—7882no—
367C2×C4—7882no—
368C2×C4—15761no—
369C2×C4—3941no—
370C2×C4—3941no—
371C2×C4—3941no—
372C2×C4—1971no—
373C2×C4—1971no—
374C2×C4—1971no—
375C2×C4D4×D19711yescharacteristic
376C2×C4C23×D19711yescharacteristic
377C8—21no—
378C8—21no—
379C8—3941no—
380C8—1972no—
381C22—21no—
382C22—21no—
383C22—7881no—
384C22—7881no—
385C22—7881no—
386C22—7881no—
387C22—7881no—
388C22—7881no—
389C22—7881no—
390C22—7881no—
391C22—15764no—
392C22—3941no—
393C22—3941no—
394C22—3942no—
395C22—3941no—
396C22#2026334912yescomplemented
397C22C2×D4×D19711yescharacteristic
398C4—21no—
399C4—22no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—7881no—
404C4—3941no—
405C4—3941no—
406C4—3941no—
407C4C2×D4×D19711yescharacteristic
408C4#2026337111yescharacteristic
409C2—22no—
410C2—7882no—
411C2—7881no—
412C2—15761no—
413C2—15761no—
414C2#2716171111yescharacteristic
415C2C2×#2026334911yescharacteristic
416C2#2716188811yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#451×C197C2
socleC22×C197C22×D4
hypercenter#451D197
hypocenterC197#3282
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C2×#20263349C2semidirect
2. C22×#15691345C2semidirect
3. #27161740C2semidirect
4. #27161720C2semidirect
5. #27161464C2semidirect
6. #27161311C2semidirect
7. #27161310C2semidirect
8. #27161303C2semidirect
9. #27161297C2semidirect
10. #27161240C2semidirect
11. #27161227C2semidirect
12. #27161215C2semidirect
13. #27161164C2semidirect
14. #27161142C2semidirect
15. #451×C197C2semidirect
17. C2×#15691359C22semidirect
25. C2×#15691345C22semidirect
29. C2×C8×D197C22semidirect
32. #20263315C22semidirect
35. #20263302C22semidirect
41. Q8×Dic197C22semidirect
46. #20263257C22semidirect
47. #20263255C22semidirect
48. #20263250C22semidirect
51. #20263246C22semidirect
52. #20263238C22semidirect
53. C8×Dic197C22semidirect
55. C2×Q16×C197C22semidirect
56. C2×Q16×C197C22semidirect
58. #103×C197C22semidirect
59. #103×C197C22semidirect
85. #15691345D4semidirect
174. C22×C197#135semidirect
208. C197#3282semidirect
220. #451D197semidirect
230. C2×Q16C2×D197semidirect
231. C2×Q16C2×D197semidirect
262. #103C2×D197semidirect
263. #103C2×D197semidirect
396. C22#20263349semidirect

Maximal subgroups

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

subgroupindex
C2×#202633492
C22×#156913452
#271617402
#271617202
#271614642
#271613112
#271613102
#271613032
#271612972
#271612402
#271612272
#271612152
#271611642
#271611422
#451×C1972
#3282197