Finite Group Databasebeta

← #38754309

Subgroups — #38754309

Order 25216.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263354C211yesmaximal, characteristic, complemented
2C2×#20263353C211yesmaximal, characteristic, complemented
3#27161731C211yesmaximal, characteristic, complemented
4#27161713C211yesmaximal, characteristic, complemented
5#27161450C211yesmaximal, characteristic, complemented
6#27161326C211yesmaximal, characteristic, complemented
7#27161283C211yesmaximal, characteristic, complemented
8#27161273C211yesmaximal, characteristic, complemented
9#27161231C211yesmaximal, characteristic, complemented
10#27161229C211yesmaximal, characteristic, complemented
11#27161227C211yesmaximal, characteristic, complemented
12#27161222C211yesmaximal, characteristic, complemented
13#27161161C211yesmaximal, characteristic, complemented
14#27161155C211yesmaximal, characteristic, complemented
15#468×C197C211yesmaximal, characteristic, complemented
16C2×#15691359C2211yescharacteristic, complemented
17C2×#15691358C2211yescharacteristic, complemented
18C22×D788C2211yescharacteristic
19C2×D4×D197—21no—
20C2×D4×D197C2211yescharacteristic
21#20263354—24no—
22#20263353—24no—
23C2×#15691347C2211yescharacteristic
24C2×#15691347C2211yescharacteristic, complemented
25C2×#15691346C2211yescharacteristic, complemented
26C2×#15691345C2211yescharacteristic
27C2×#15691345C2211yescharacteristic, complemented
28C2×D1576C2211yescharacteristic
29C2×D1576C2211yescharacteristic, complemented
30C2×#15691343C2211yescharacteristic, complemented
31C2×#15691341C2211yescharacteristic, complemented
32#20263324C2211yescharacteristic, complemented
33#20263322C2211yescharacteristic
34#20263311—21no—
35#20263308—21no—
36#20263306C2211yescharacteristic, complemented
37#20263301C2211yescharacteristic
38#20263299—21no—
39#20263299—21no—
40#20263296—21no—
41#20263294—21no—
42#20263288C2211yescharacteristic
43#20263288C2211yescharacteristic, complemented
44C4×C22⋊D197C2211yescharacteristic
45C4⋊C4×D197—21no—
46#20263255C2211yescharacteristic, complemented
47#20263249C2211yescharacteristic, complemented
48#20263246C2211yescharacteristic
49#20263246C2211yescharacteristic, complemented
50#20263246C2211yescharacteristic, complemented
51C22⋊#13096802C2211yescharacteristic, complemented
52#20263238C2211yescharacteristic, complemented
53C8×Dic197C2211yescharacteristic, complemented
54C2×SD16×C197C2211yescharacteristic, complemented
55C2×M16×C197C2211yescharacteristic, complemented
56C22⋊Q8×C197C2211yescharacteristic, complemented
57#121×C197C2211yescharacteristic
58#107×C197C2211yescharacteristic, complemented
59#103×C197C2211yescharacteristic, complemented
60#102×C197C2211yescharacteristic, complemented
61C23×D197—21no—
62C23×D197—21no—
63#15691359—22no—
64#15691359—41no—
65#15691358—22no—
66#15691358—41no—
67C2×Dic394C2311yescharacteristic
68C2×D788—22no—
69C2×D788—21no—
70C2×D788—21no—
71C2×D788—21no—
72C2×D788—41no—
73C2×D788C2311yescharacteristic
74C2×D788C2311yescharacteristic
75C2×D788C2311yescharacteristic
76C2×C22⋊D197—21no—
77C2×C22⋊D197—21no—
78C2×C22⋊D197—21no—
79C2×C22⋊D197—21no—
80C2×C22⋊D197D411yescharacteristic
81D4×D197—22no—
82D4×D197—41no—
83D4×D197—44no—
84C2×C4×D197—21no—
85C2×C4×D197—21no—
86C2×C4×D197—21no—
87C2×C4×D197—21no—
88C2×C4×D197C2311yescharacteristic
89#15691347—22no—
90#15691347—22no—
91#15691346—41no—
92#15691345—22no—
93#15691345—41no—
94D1576—22no—
95D1576D414yescomplemented
96#15691343—22no—
97#15691341—22no—
98C2×#13096802C2311yescharacteristic
99C2×#13096802C2311yescharacteristic
100C4⋊Dic197—21no—
101#15691337—21no—
102#15691337—21no—
103#15691337D411yescharacteristic
104C22⋊Dic197D411yescharacteristic
105#15691335—21no—
106#15691335—21no—
107#15691335—21no—
108#15691335—21no—
109#15691335D411yescharacteristic
110C4×Dic197C2311yescharacteristic
111C2×Q8×C197C2311yescharacteristic
112C2×D4×C197—21no—
113C2×D4×C197C2311yescharacteristic
114C22×C4×C197C2311yescharacteristic
115SD16×C197—22no—
116M16×C197—22no—
117C2×C8×C197C2311yescharacteristic
118C2×C8×C197C2311yescharacteristic
119C4⋊C4×C197—21no—
120C4⋊C4×C197C2311yescharacteristic
121C4⋊C4×C197C2311yescharacteristic
122C22⋊C4×C197—21no—
123C22⋊C4×C197—21no—
124C22×D197—21no—
125C22×D197—21no—
126C22×D197—21no—
127C22×D197—42no—
128C22×D197—42no—
129C22×D197—42no—
130C22×D197—42no—
131C22×D197—81no—
132C22×D197—81no—
133C22×D197C2×D411yescharacteristic
134Dic394—21no—
135Dic394—21no—
136D788—21no—
137D788—21no—
138D788—21no—
139D788—21no—
140D788—42no—
141D788—42no—
142D788—42no—
143D788C2×D412yes—
144D788C2×D412yescomplemented
145C22⋊D197—22no—
146C22⋊D197—42no—
147C22⋊D197—42no—
148C22⋊D197—42no—
149C22⋊D197—81no—
150C2×Dic197—21no—
151C2×Dic197C2×D411yescharacteristic
152C2×Dic197C2×D411yescharacteristic
153C2×Dic197C2×D411yescharacteristic
154C4×D197—21no—
155C4×D197—21no—
156C4×D197—42no—
157C4×D197—42no—
158C4×D197—42no—
159C4×D197—42no—
160#13096802—21no—
161#13096802—21no—
162C23×C197—21no—
163C23×C197C2×D411yescharacteristic
164Q8×C197—21no—
165Q8×C197—21no—
166D4×C197—21no—
167D4×C197—21no—
168D4×C197—42no—
169C2×C4×C197—21no—
170C2×C4×C197—21no—
171C2×C4×C197—21no—
172C2×C4×C197—21no—
173C2×C4×C197—21no—
174C2×C4×C197C2×D411yescharacteristic
175C2×C4×C197C2411yescharacteristic
176C8×C197—21no—
177C8×C197C2×D412yescomplemented
178C22×C197—21no—
179C22×C197—21no—
180C22×C197—21no—
181C22×C197—41no—
182C22×C197—41no—
183C22×C197—41no—
184C22×C197C22×D411yescharacteristic
185C2×D197—21no—
186C2×D197—22no—
187C2×D197—22no—
188C2×D197—21no—
189C2×D197—21no—
190C2×D197—41no—
191C2×D197—41no—
192C2×D197—41no—
193C2×D197—41no—
194C2×D197—41no—
195C2×D197—41no—
196C2×D197—41no—
197C2×D197—84no—
198C2×D197—84no—
199C4×C197—21no—
200C4×C197—41no—
201C4×C197—41no—
202C4×C197—41no—
203C4×C197C22×D411yescharacteristic
204C4×C197H(5,𝔽2)11yescharacteristic
205Dic197—21no—
206Dic197—21no—
207Dic197—41no—
208Dic197AGL(1,ℤ/8)12yescomplemented
209C2×C197—41no—
210C2×C197—81no—
211C2×C197D4211yescharacteristic
212C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C197#57711yescharacteristic
214D197—41no—
215D197—42no—
216D197—42no—
217D197—81no—
218C197#327211yescharacteristic, complemented, Sylow
219#3272—1971nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1971no—
221C2×AGL(1,ℤ/8)—1971no—
222#547—1971no—
223D42—1971no—
224#492—1971no—
225#468—1971no—
226#468D19711yescharacteristic, complemented
227#461—1971no—
228#459—1971no—
229#450—1971no—
230#450—1971no—
231#449—1971no—
232#449—1971no—
233C4×D8—1971no—
234#436—1971no—
235C2×#41—1971no—
236C2×#41—1971no—
237C22×D4—3941no—
238C22×D4—1971no—
239C22×D4—1971no—
240AGL(1,ℤ/8)—3944no—
241AGL(1,ℤ/8)—3944no—
242C2×SD16—1971no—
243C2×SD16—1971no—
244C2×SD16—1971no—
245C2×SD16—1971no—
246C2×SD16—1971no—
247C2×SD16C2×D19711yescharacteristic, complemented
248C2×D8—1971no—
249C2×D8—1971no—
250C2×D8—1971no—
251C2×M16—1971no—
252C2×M16C2×D19711yescharacteristic, complemented
253C4⋊Q8—1971no—
254#127—1971no—
255#123—3941no—
256#123—3941no—
257C22⋊Q8—1971no—
258C22⋊Q8C2×D19711yescharacteristic, complemented
259#121—3941no—
260#121—3941no—
261#121—1971no—
262#121C2×D19711yescharacteristic
263#120—3941no—
264#120—3941no—
265C4×D4—1971no—
266C4×D4—1971no—
267C4×D4—1971no—
268C2×C4⋊C4—3941no—
269#107C2×D19711yescharacteristic, complemented
270C4⋊C8—1971no—
271#103—1971no—
272#103C2×D19711yescharacteristic, complemented
273#102—1971no—
274#102—1971no—
275#102—1971no—
276#102C2×D19711yescharacteristic, complemented
277C22⋊C8—1971no—
278C22⋊C8—1971no—
279C4×C8—1971no—
280C24—3941no—
281C24—3941no—
282#41—7881no—
283#41—7881no—
284#41—3942no—
285#41—3942no—
286C2×Q8—1971no—
287C2×Q8C22×D19711yescharacteristic
288C2×D4—21no—
289C2×D4—7881no—
290C2×D4—7884no—
291C2×D4—7881no—
292C2×D4—3941no—
293C2×D4—3941no—
294C2×D4—3942no—
295C2×D4—3942no—
296C2×D4—3941no—
297C2×D4—3941no—
298C2×D4—3941no—
299C2×D4—3941no—
300C2×D4—3941no—
301C2×D4—1971no—
302C2×D4—1971no—
303C2×D4—1971no—
304C2×D4—1971no—
305C2×D4C22×D19711yescharacteristic
306C22×C4—3941no—
307C22×C4—3941no—
308C22×C4—3941no—
309C22×C4—3941no—
310C22×C4—1971no—
311C22×C4C22×D19711yescharacteristic
312SD16—22no—
313SD16—7881no—
314SD16—7881no—
315SD16—3942no—
316SD16—3942no—
317SD16—3942no—
318D8—3942no—
319D8—3942no—
320D8—1974no—
321M16—22no—
322M16—3942no—
323C2×C8—1971no—
324C2×C8—1971no—
325C2×C8C22×D19711yescharacteristic
326C2×C8C22×D19711yescharacteristic
327C4⋊C4—21no—
328C4⋊C4—3941no—
329C4⋊C4—3941no—
330C4⋊C4—3941no—
331C4⋊C4—1971no—
332C4⋊C4C22×D19711yescharacteristic
333C4⋊C4C22×D19711yescharacteristic
334C22⋊C4—21no—
335C22⋊C4—21no—
336C22⋊C4—3941no—
337C22⋊C4—3941no—
338C22⋊C4—3941no—
339C22⋊C4—3941no—
340C22⋊C4—1971no—
341C22⋊C4—1971no—
342C42—1971no—
343C23—21no—
344C23—7882no—
345C23—7882no—
346C23—7882no—
347C23—7882no—
348C23—15761no—
349C23—15761no—
350C23—3941no—
351C23—3941no—
352C23—3941no—
353C23—1971no—
354C23D4×D19711yescharacteristic
355Q8—21no—
356Q8—21no—
357Q8—3941no—
358Q8—3941no—
359D4—21no—
360D4—21no—
361D4—42no—
362D4—7882no—
363D4—7882no—
364D4—7882no—
365D4—7882no—
366D4—7882no—
367D4—7882no—
368D4—15761no—
369D4—3942no—
370D4—3941no—
371D4—3941no—
372D4—3941no—
373D4—3941no—
374D4—1972no—
375D4—1972no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—7882no—
382C2×C4—7882no—
383C2×C4—7882no—
384C2×C4—7882no—
385C2×C4—3941no—
386C2×C4—3941no—
387C2×C4—3941no—
388C2×C4—1971no—
389C2×C4—1971no—
390C2×C4—1971no—
391C2×C4D4×D19711yescharacteristic
392C2×C4C23×D19711yescharacteristic
393C8—21no—
394C8—3941no—
395C8—3941no—
396C8D4×D19712yescomplemented
397C22—21no—
398C22—21no—
399C22—21no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—7881no—
404C22—7881no—
405C22—7881no—
406C22—7881no—
407C22—7881no—
408C22—7881no—
409C22—7881no—
410C22—15764no—
411C22—15764no—
412C22—3941no—
413C22—3941no—
414C22—3942no—
415C22—3941no—
416C22—3942no—
417C22C2×D4×D19711yescharacteristic
418C4—21no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—7881no—
423C4—3941no—
424C4—3941no—
425C4—1972no—
426C4C2×D4×D19711yescharacteristic
427C4#2026337211yescharacteristic
428C2—41no—
429C2—81no—
430C2—7882no—
431C2—7882no—
432C2—7881no—
433C2—15761no—
434C2#2716171311yescharacteristic
435C2AGL(1,ℤ/8)×D19711yescharacteristic
436C2#2716188811yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#468×C197C2
socleC22×C197C22×D4
hypercenter#468D197
hypocenterC197#3272
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. C2×#20263354C2semidirect
2. C2×#20263353C2semidirect
3. #27161731C2semidirect
4. #27161713C2semidirect
5. #27161450C2semidirect
6. #27161326C2semidirect
7. #27161283C2semidirect
8. #27161273C2semidirect
9. #27161231C2semidirect
10. #27161229C2semidirect
11. #27161227C2semidirect
12. #27161222C2semidirect
13. #27161161C2semidirect
14. #27161155C2semidirect
15. #468×C197C2semidirect
16. C2×#15691359C22semidirect
17. C2×#15691358C22semidirect
24. C2×#15691347C22semidirect
25. C2×#15691346C22semidirect
27. C2×#15691345C22semidirect
29. C2×D1576C22semidirect
30. C2×#15691343C22semidirect
31. C2×#15691341C22semidirect
32. #20263324C22semidirect
36. #20263306C22semidirect
43. #20263288C22semidirect
46. #20263255C22semidirect
47. #20263249C22semidirect
49. #20263246C22semidirect
50. #20263246C22semidirect
51. C22⋊#13096802C22semidirect
52. #20263238C22semidirect
53. C8×Dic197C22semidirect
54. C2×SD16×C197C22semidirect
55. C2×M16×C197C22semidirect
56. C22⋊Q8×C197C22semidirect
58. #107×C197C22semidirect
59. #103×C197C22semidirect
60. #102×C197C22semidirect
95. D1576D4semidirect
144. D788C2×D4semidirect
177. C8×C197C2×D4semidirect
208. Dic197AGL(1,ℤ/8)semidirect
218. C197#3272semidirect
226. #468D197semidirect
247. C2×SD16C2×D197semidirect
252. C2×M16C2×D197semidirect
258. C22⋊Q8C2×D197semidirect
269. #107C2×D197semidirect
272. #103C2×D197semidirect
276. #102C2×D197semidirect
396. C8D4×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202633542
C2×#202633532
#271617312
#271617132
#271614502
#271613262
#271612832
#271612732
#271612312
#271612292
#271612272
#271612222
#271611612
#271611552
#468×C1972
#3272197