Finite Group Databasebeta

← #38754200

Subgroups — #38754200

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×#20263353C211yesmaximal, characteristic, complemented
2C22×#15691345C211yesmaximal, characteristic, complemented
3#27161740C211yesmaximal, characteristic, complemented
4#27161713C211yesmaximal, characteristic, complemented
5#27161465C211yesmaximal, characteristic, complemented
6#27161328C211yesmaximal, characteristic, complemented
7#27161313C211yesmaximal, characteristic, complemented
8#27161297C211yesmaximal, characteristic, complemented
9#27161277C211yesmaximal, characteristic, complemented
10#27161228C211yesmaximal, characteristic, complemented
11#27161227C211yesmaximal, characteristic, complemented
12#27161215C211yesmaximal, characteristic, complemented
13#27161182C211yesmaximal, characteristic, complemented
14#27161147C211yesmaximal, characteristic, complemented
15#448×C197C211yesmaximal, characteristic, complemented
16C2×#15691359C2211yescharacteristic, complemented
17C22×D788C2211yescharacteristic
18C2×D4×D197—21no—
19C2×D4×D197C2211yescharacteristic
20#20263353—24no—
21C2×#15691345—24no—
22C2×#15691345C2211yescharacteristic
23C2×#15691345C2211yescharacteristic
24C2×#15691345C2211yescharacteristic, complemented
25C2×#15691345C2211yescharacteristic, complemented
26C2×D1576C2211yescharacteristic
27C2×D1576C2211yescharacteristic, complemented
28C2×#15691343C2211yescharacteristic, complemented
29C2×#15691341C2211yescharacteristic, complemented
30C22×#13096802C2211yescharacteristic, complemented
31#20263322C2211yescharacteristic
32#20263315—21no—
33#20263315C2211yescharacteristic, complemented
34#20263302—21no—
35#20263302C2211yescharacteristic, complemented
36#20263301C2211yescharacteristic
37#20263300—21no—
38#20263300C2211yescharacteristic, complemented
39#20263299—21no—
40#20263296—21no—
41#20263294—21no—
42Q8×Dic197C2211yescharacteristic, complemented
43#20263288C2211yescharacteristic
44C4×C22⋊D197—21no—
45C4×C22⋊D197C2211yescharacteristic
46#20263259C2211yescharacteristic, complemented
47#20263255C2211yescharacteristic, complemented
48#20263248C2211yescharacteristic
49#20263248C2211yescharacteristic, complemented
50#20263247C2211yescharacteristic, complemented
51#20263246C2211yescharacteristic, complemented
52#20263238C2211yescharacteristic, complemented
53#20263236C2211yescharacteristic, complemented
54C22×Q8×C197C2211yescharacteristic, complemented
55C2×SD16×C197C2211yescharacteristic, complemented
56C2×SD16×C197C2211yescharacteristic, complemented
57#121×C197C2211yescharacteristic
58#103×C197C2211yescharacteristic, complemented
59#103×C197C2211yescharacteristic, complemented
60C22⋊C8×C197C2211yescharacteristic, complemented
61C23×D197—21no—
62C23×D197—21no—
63#15691359—22no—
64#15691359—41no—
65C2×D788—22no—
66C2×D788—21no—
67C2×D788—21no—
68C2×D788—21no—
69C2×D788—41no—
70C2×D788C2311yescharacteristic
71C2×D788C2311yescharacteristic
72C2×D788C2311yescharacteristic
73C2×C22⋊D197—21no—
74C2×C22⋊D197—21no—
75C2×C22⋊D197—21no—
76C2×C22⋊D197—21no—
77C2×C22⋊D197D411yescharacteristic
78D4×D197—22no—
79D4×D197—41no—
80D4×D197—44no—
81C2×C4×D197—21no—
82C2×C4×D197—21no—
83C2×C4×D197C2311yescharacteristic
84#15691345—22no—
85#15691345—22no—
86#15691345—22no—
87#15691345D414yescomplemented
88D1576—22no—
89D1576—41no—
90#15691343—22no—
91#15691341—22no—
92C2×#13096802—22no—
93C2×#13096802C2311yescharacteristic
94C2×#13096802C2311yescharacteristic
95C4⋊Dic197—21no—
96C4⋊Dic197C2311yescharacteristic
97#15691337—21no—
98#15691337D411yescharacteristic
99C22⋊Dic197D411yescharacteristic
100#15691335—21no—
101#15691335—21no—
102#15691335—21no—
103#15691335—21no—
104#15691335—21no—
105#15691335—21no—
106#15691335D411yescharacteristic
107C4×Dic197—21no—
108C4×Dic197C2311yescharacteristic
109C2×Q8×C197—21no—
110C2×Q8×C197—22no—
111C2×Q8×C197—41no—
112C2×Q8×C197C2311yescharacteristic
113C2×Q8×C197C2311yescharacteristic
114C2×D4×C197—21no—
115C2×D4×C197C2311yescharacteristic
116C22×C4×C197—21no—
117C22×C4×C197C2311yescharacteristic
118SD16×C197—22no—
119SD16×C197—41no—
120C2×C8×C197C2311yescharacteristic
121C2×C8×C197C2311yescharacteristic
122C4⋊C4×C197C2311yescharacteristic
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
134D788—21no—
135D788—21no—
136D788—21no—
137D788—21no—
138D788—21no—
139D788—42no—
140D788—42no—
141D788—42no—
142D788C2×D412yes—
143C22⋊D197—22no—
144C22⋊D197—42no—
145C22⋊D197—42no—
146C22⋊D197—42no—
147C22⋊D197—81no—
148C2×Dic197—21no—
149C2×Dic197C2×D411yescharacteristic
150C2×Dic197C2×D411yescharacteristic
151C2×Dic197C2×D411yescharacteristic
152C4×D197—21no—
153C4×D197—21no—
154C4×D197—42no—
155C4×D197—42no—
156#13096802—21no—
157#13096802C2×D412yescomplemented
158C23×C197—21no—
159C23×C197C2×D411yescharacteristic
160Q8×C197—21no—
161Q8×C197—21no—
162Q8×C197—21no—
163Q8×C197—42no—
164Q8×C197C2×D412yescomplemented
165D4×C197—21no—
166D4×C197—21no—
167D4×C197—42no—
168C2×C4×C197—21no—
169C2×C4×C197—21no—
170C2×C4×C197—21no—
171C2×C4×C197—21no—
172C2×C4×C197—21no—
173C2×C4×C197—42no—
174C2×C4×C197C2×D411yescharacteristic
175C2×C4×C197C2411yescharacteristic
176C8×C197—21no—
177C8×C197—21no—
178C22×C197—21no—
179C22×C197—21no—
180C22×C197—41no—
181C22×C197—41no—
182C22×C197—41no—
183C22×C197AGL(1,ℤ/8)12yescomplemented
184C22×C197C22×D411yescharacteristic
185C2×D197—21no—
186C2×D197—22no—
187C2×D197—21no—
188C2×D197—21no—
189C2×D197—41no—
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—22no—
201C4×C197—41no—
202C4×C197—41no—
203C4×C197C22×D411yescharacteristic
204C4×C197H(5,𝔽2)11yescharacteristic
205Dic197—21no—
206Dic197—21no—
207Dic197—21no—
208Dic197—41no—
209C2×C197—22no—
210C2×C197—81no—
211C2×C197D4211yescharacteristic
212C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
213C2×C197#57711yescharacteristic
214D197—41no—
215D197—42no—
216D197—81no—
217D197—81no—
218C197#326511yescharacteristic, complemented, Sylow
219#3265—1971nomaximal, Sylow
220C2×AGL(1,ℤ/8)—1971no—
221C22×SD16—1971no—
222#548—1971no—
223D42—1971no—
224#496—1971no—
225#468—1971no—
226#466—1971no—
227#463—1971no—
228#460—1971no—
229#450—1971no—
230#449—1971no—
231#448D19711yescharacteristic, complemented
232#447—1971no—
233#440—1971no—
234#435—1971no—
235C2×#41—1971no—
236C22×Q8C2×D19711yescharacteristic, complemented
237C22×D4—3941no—
238C22×D4—1971no—
239C22×D4—1971no—
240AGL(1,ℤ/8)—3944no—
241C2×SD16—3944no—
242C2×SD16—1971no—
243C2×SD16—1971no—
244C2×SD16—1971no—
245C2×SD16—1971no—
246C2×SD16C2×D19711yescharacteristic, complemented
247C2×SD16C2×D19711yescharacteristic, complemented
248C2×D8—1971no—
249C2×D8—1971no—
250C2×D8—1971no—
251C2×M16—1971no—
252C22×C8—1971no—
253#127—1971no—
254#124—3941no—
255#124—1971no—
256C22⋊Q8—3941no—
257C22⋊Q8—1971no—
258#121—3941no—
259#121—3941no—
260#121—1971no—
261#121—1971no—
262#121C2×D19711yescharacteristic
263#120—3941no—
264#120—3941no—
265C4×Q8—1971no—
266C4×D4—3941no—
267C4×D4—1971no—
268C4×D4—1971no—
269#107—1971no—
270C4⋊C8—1971no—
271#103C2×D19711yescharacteristic, complemented
272#103C2×D19711yescharacteristic, complemented
273#102—1971no—
274#102—1971no—
275#102—1971no—
276#102—1971no—
277C22⋊C8—1971no—
278C22⋊C8C2×D19711yescharacteristic, complemented
279#97—1971no—
280C24—3941no—
281C24—3941no—
282#41—7881no—
283#41—3942no—
284C2×Q8—21no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8C22×D19711yescharacteristic
288C2×Q8C22×D19711yescharacteristic
289C2×D4—21no—
290C2×D4—7881no—
291C2×D4—7884no—
292C2×D4—7881no—
293C2×D4—3941no—
294C2×D4—3941no—
295C2×D4—3942no—
296C2×D4—3942no—
297C2×D4—3941no—
298C2×D4—3941no—
299C2×D4—3941no—
300C2×D4—3941no—
301C2×D4—3941no—
302C2×D4—1971no—
303C2×D4—1971no—
304C2×D4—1971no—
305C2×D4—1971no—
306C2×D4C22×D19711yescharacteristic
307C22×C4—21no—
308C22×C4—3941no—
309C22×C4—3941no—
310C22×C4—1971no—
311C22×C4C22×D19711yescharacteristic
312SD16—22no—
313SD16—41no—
314SD16—3942no—
315SD16—3942no—
316SD16—3942no—
317SD16—1974no—
318D8—7881no—
319D8—3942no—
320D8—3942no—
321M16—3942no—
322C2×C8—3942no—
323C2×C8—1971no—
324C2×C8—1971no—
325C2×C8C22×D19711yescharacteristic
326C2×C8C22×D19711yescharacteristic
327C4⋊C4—3941no—
328C4⋊C4—3941no—
329C4⋊C4—1971no—
330C4⋊C4—1971no—
331C4⋊C4C22×D19711yescharacteristic
332C22⋊C4—21no—
333C22⋊C4—3941no—
334C22⋊C4—3941no—
335C22⋊C4—3941no—
336C22⋊C4—3941no—
337C22⋊C4—3941no—
338C22⋊C4—3941no—
339C22⋊C4—1971no—
340C22⋊C4—1971no—
341C42—3941no—
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—21no—
358Q8—42no—
359Q8D4×D19712yescomplemented
360D4—21no—
361D4—21no—
362D4—42no—
363D4—7882no—
364D4—7882no—
365D4—7882no—
366D4—7882no—
367D4—7882no—
368D4—7882no—
369D4—15761no—
370D4—3942no—
371D4—3941no—
372D4—3941no—
373D4—3941no—
374D4—3941no—
375D4—3941no—
376D4—1972no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—42no—
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—21no—
395C8—3941no—
396C8—1972no—
397C22—21no—
398C22—21no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—7881no—
403C22—7881no—
404C22—7881no—
405C22—7881no—
406C22—7881no—
407C22—7881no—
408C22—7881no—
409C22—7881no—
410C22—15764no—
411C22—15764no—
412C22—3941no—
413C22—3941no—
414C22—3941no—
415C22—3942no—
416C22#2026335312yescomplemented
417C22C2×D4×D19711yescharacteristic
418C4—21no—
419C4—22no—
420C4—41no—
421C4—41no—
422C4—7881no—
423C4—3941no—
424C4—3941no—
425C4—3941no—
426C4C2×D4×D19711yescharacteristic
427C4#2026337111yescharacteristic
428C2—22no—
429C2—81no—
430C2—7882no—
431C2—7881no—
432C2—15761no—
433C2—15761no—
434C2#2716171111yescharacteristic
435C2C2×#2026335311yescharacteristic
436C2#2716188811yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#448×C197C2
socleC22×C197C22×D4
hypercenter#448D197
hypocenterC197#3265
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. C2×#20263353C2semidirect
2. C22×#15691345C2semidirect
3. #27161740C2semidirect
4. #27161713C2semidirect
5. #27161465C2semidirect
6. #27161328C2semidirect
7. #27161313C2semidirect
8. #27161297C2semidirect
9. #27161277C2semidirect
10. #27161228C2semidirect
11. #27161227C2semidirect
12. #27161215C2semidirect
13. #27161182C2semidirect
14. #27161147C2semidirect
15. #448×C197C2semidirect
16. C2×#15691359C22semidirect
24. C2×#15691345C22semidirect
25. C2×#15691345C22semidirect
27. C2×D1576C22semidirect
28. C2×#15691343C22semidirect
29. C2×#15691341C22semidirect
30. C22×#13096802C22semidirect
33. #20263315C22semidirect
35. #20263302C22semidirect
38. #20263300C22semidirect
42. Q8×Dic197C22semidirect
46. #20263259C22semidirect
47. #20263255C22semidirect
49. #20263248C22semidirect
50. #20263247C22semidirect
51. #20263246C22semidirect
52. #20263238C22semidirect
53. #20263236C22semidirect
54. C22×Q8×C197C22semidirect
55. C2×SD16×C197C22semidirect
56. C2×SD16×C197C22semidirect
58. #103×C197C22semidirect
59. #103×C197C22semidirect
60. C22⋊C8×C197C22semidirect
87. #15691345D4semidirect
157. #13096802C2×D4semidirect
164. Q8×C197C2×D4semidirect
183. C22×C197AGL(1,ℤ/8)semidirect
218. C197#3265semidirect
231. #448D197semidirect
236. C22×Q8C2×D197semidirect
246. C2×SD16C2×D197semidirect
247. C2×SD16C2×D197semidirect
271. #103C2×D197semidirect
272. #103C2×D197semidirect
278. C22⋊C8C2×D197semidirect
359. Q8D4×D197semidirect
416. C22#20263353semidirect

Maximal subgroups

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

subgroupindex
C2×#202633532
C22×#156913452
#271617402
#271617132
#271614652
#271613282
#271613132
#271612972
#271612772
#271612282
#271612272
#271612152
#271611822
#271611472
#448×C1972
#3265197