Finite Group Databasebeta

← #38754196

Subgroups — #38754196

Order 25216.

568 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263353C211yesmaximal, characteristic, complemented
2C22×#15691343C211yesmaximal, characteristic, complemented
3#27161720C211yesmaximal, characteristic, complemented
4D4×C22⋊D197C211yesmaximal, characteristic, complemented
5#27161465C211yesmaximal, characteristic, complemented
6#27161331C211yesmaximal, characteristic, complemented
7#27161310C211yesmaximal, characteristic, complemented
8#27161296C211yesmaximal, characteristic, complemented
9#27161273C211yesmaximal, characteristic, complemented
10#27161223C211yesmaximal, characteristic, complemented
11#27161215C211yesmaximal, characteristic, complemented
12#27161214C211yesmaximal, characteristic, complemented
13#27161188C211yesmaximal, characteristic, complemented
14#27161147C211yesmaximal, characteristic, complemented
15#450×C197C211yesmaximal, characteristic, complemented
16C2×#15691359C2211yescharacteristic, complemented
17C22×D788C2211yescharacteristic, complemented
18C22×C22⋊D197—21no—
19C2×D4×D197C2211yescharacteristic, complemented
20#20263353—24no—
21C2×#15691345C2211yescharacteristic, complemented
22C2×D1576C2211yescharacteristic, complemented
23C2×D1576C2211yescharacteristic, complemented
24C2×#15691343—24no—
25C2×#15691343C2211yescharacteristic
26C2×#15691343C2211yescharacteristic
27C2×#15691343C2211yescharacteristic, complemented
28C2×#15691343C2211yescharacteristic, complemented
29C2×#15691341C2211yescharacteristic, complemented
30C22×#13096802C2211yescharacteristic, complemented
31#20263322C2211yescharacteristic, complemented
32#20263315C2211yescharacteristic, complemented
33#20263311—21no—
34#20263301—21no—
35#20263301C2211yescharacteristic, complemented
36#20263300C2211yescharacteristic
37#20263298—21no—
38#20263297C2211yescharacteristic, complemented
39#20263296—21no—
40#20263295—21no—
41#20263294—21no—
42D4×Dic197C2211yescharacteristic, complemented
43#20263288C2211yescharacteristic, complemented
44C4×C22⋊D197C2211yescharacteristic
45C22⋊C4×D197—21no—
46#20263257C2211yescharacteristic, complemented
47#20263255C2211yescharacteristic, complemented
48#20263250C2211yescharacteristic, complemented
49#20263248C2211yescharacteristic, complemented
50#20263248C2211yescharacteristic, complemented
51#20263246C2211yescharacteristic, complemented
52#20263238C2211yescharacteristic, complemented
53#20263236C2211yescharacteristic, complemented
54C22×D4×C197C2211yescharacteristic
55C2×SD16×C197C2211yescharacteristic, complemented
56C2×SD16×C197C2211yescharacteristic, complemented
57C22⋊Q8×C197C2211yescharacteristic, complemented
58#102×C197C2211yescharacteristic, complemented
59#102×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⋊D197—21no—
78C2×C22⋊D197—44no—
79C2×C22⋊D197D411yescharacteristic
80D4×D197—22no—
81D4×D197—41no—
82C22×Dic197—21no—
83C2×C4×D197—21no—
84C2×C4×D197—21no—
85C2×C4×D197C2311yescharacteristic
86#15691345—22no—
87D1576—22no—
88D1576—41no—
89#15691343—22no—
90#15691343—22no—
91#15691343—22no—
92#15691343D414yescomplemented
93#15691341—22no—
94C2×#13096802—22no—
95C2×#13096802C2311yescharacteristic
96C2×#13096802C2311yescharacteristic
97C4⋊Dic197C2311yescharacteristic
98#15691337D411yescharacteristic
99C22⋊Dic197—21no—
100C22⋊Dic197—21no—
101C22⋊Dic197D411yescharacteristic
102#15691335—21no—
103#15691335—21no—
104#15691335—21no—
105#15691335—21no—
106#15691335—21no—
107#15691335D411yescharacteristic
108C4×Dic197C2311yescharacteristic
109C24×C197—21no—
110C2×Q8×C197C2311yescharacteristic
111C2×D4×C197—22no—
112C2×D4×C197—21no—
113C2×D4×C197—41no—
114C2×D4×C197C2311yescharacteristic
115C2×D4×C197C2311yescharacteristic
116C22×C4×C197C2311yescharacteristic
117SD16×C197—22no—
118SD16×C197—41no—
119C2×C8×C197C2311yescharacteristic
120C2×C8×C197C2311yescharacteristic
121C4⋊C4×C197—21no—
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—81no—
142D788C2×D412yescomplemented
143C22⋊D197—22no—
144C22⋊D197—42no—
145C22⋊D197—42no—
146C22⋊D197—42no—
147C22⋊D197—42no—
148C22⋊D197—81no—
149C2×Dic197—21no—
150C2×Dic197—42no—
151C2×Dic197C2×D411yescharacteristic
152C2×Dic197C2×D411yescharacteristic
153C2×Dic197C2×D411yescharacteristic
154C4×D197—21no—
155C4×D197—21no—
156C4×D197—42no—
157C4×D197—81no—
158#13096802—21no—
159#13096802C2×D412yescomplemented
160C23×C197—21no—
161C23×C197—21no—
162C23×C197—42no—
163C23×C197—42no—
164C23×C197—81no—
165C23×C197C2×D411yescharacteristic
166Q8×C197—21no—
167Q8×C197—21no—
168D4×C197—21no—
169D4×C197—21no—
170D4×C197—21no—
171D4×C197—42no—
172D4×C197C2×D412yes—
173C2×C4×C197—21no—
174C2×C4×C197—21no—
175C2×C4×C197—21no—
176C2×C4×C197—21no—
177C2×C4×C197C2×D411yescharacteristic
178C2×C4×C197C2411yescharacteristic
179C8×C197—21no—
180C8×C197—21no—
181C22×C197—21no—
182C22×C197—21no—
183C22×C197—22no—
184C22×C197—41no—
185C22×C197—41no—
186C22×C197—41no—
187C22×C197—41no—
188C22×C197—41no—
189C22×C197—84no—
190C22×C197AGL(1,ℤ/8)12yescomplemented
191C22×C197C22×D411yescharacteristic
192C2×D197—21no—
193C2×D197—22no—
194C2×D197—21no—
195C2×D197—21no—
196C2×D197—41no—
197C2×D197—41no—
198C2×D197—41no—
199C2×D197—41no—
200C2×D197—41no—
201C2×D197—41no—
202C2×D197—41no—
203C2×D197—41no—
204C2×D197—84no—
205C2×D197—84no—
206C4×C197—21no—
207C4×C197—41no—
208C4×C197—41no—
209C4×C197C22×D411yescharacteristic
210C4×C197H(5,𝔽2)11yescharacteristic
211Dic197—21no—
212Dic197—21no—
213Dic197—21no—
214Dic197—41no—
215C2×C197—22no—
216C2×C197—42no—
217C2×C197—81no—
218C2×C197D4211yescharacteristic
219C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
220C2×C197#57611yescharacteristic
221D197—41no—
222D197—42no—
223D197—81no—
224D197—81no—
225C197#326411yescharacteristic, complemented, Sylow
226#3264—1971nomaximal, Sylow
227C2×AGL(1,ℤ/8)—1971no—
228C22×D8—1971no—
229#546—1971no—
230D42—1971no—
231#496—1971no—
232#469—1971no—
233#465—1971no—
234#463—1971no—
235#459—1971no—
236#450D19711yescharacteristic, complemented
237#449—1971no—
238#447—1971no—
239#447—1971no—
240#442—1971no—
241#435—1971no—
242C2×#41—1971no—
243C22×D4—3941no—
244C22×D4—1971no—
245C22×D4—1971no—
246C22×D4C2×D19711yescharacteristic
247AGL(1,ℤ/8)—3944no—
248C2×SD16—1971no—
249C2×SD16C2×D19711yescharacteristic, complemented
250C2×SD16C2×D19711yescharacteristic, complemented
251C2×D8—3944no—
252C2×D8—1971no—
253C2×D8—1971no—
254C2×D8—1971no—
255C2×D8—1971no—
256C2×D8—1971no—
257C2×D8—1971no—
258C2×M16—1971no—
259C22×C8—1971no—
260#127—1971no—
261#124—1971no—
262#123—3941no—
263C22⋊Q8C2×D19711yescharacteristic, complemented
264#121—3941no—
265#121—3941no—
266#121—1971no—
267#121—1971no—
268#121—1971no—
269#120—3941no—
270#120—3941no—
271#120—3941no—
272C4×D4—1971no—
273C4×D4—1971no—
274C4×D4—1971no—
275C2×C22⋊C4—3941no—
276#106—1971no—
277C4⋊C8—1971no—
278#103—1971no—
279#102—1971no—
280#102—1971no—
281#102—1971no—
282#102C2×D19711yescharacteristic, complemented
283#102C2×D19711yescharacteristic, complemented
284C22⋊C8—1971no—
285C22⋊C8C2×D19711yescharacteristic, complemented
286#97—1971no—
287C24—21no—
288C24—3941no—
289C24—3941no—
290#41—7881no—
291#41—3942no—
292C2×Q8C22×D19711yescharacteristic
293C2×D4—22no—
294C2×D4—21no—
295C2×D4—41no—
296C2×D4—7881no—
297C2×D4—7884no—
298C2×D4—7881no—
299C2×D4—3941no—
300C2×D4—3942no—
301C2×D4—3941no—
302C2×D4—3941no—
303C2×D4—3941no—
304C2×D4—3942no—
305C2×D4—3941no—
306C2×D4—3941no—
307C2×D4—3941no—
308C2×D4—3941no—
309C2×D4—1971no—
310C2×D4—1971no—
311C2×D4—1971no—
312C2×D4—1971no—
313C2×D4C22×D19711yescharacteristic
314C2×D4C22×D19711yescharacteristic
315C22×C4—3941no—
316C22×C4—3941no—
317C22×C4—3941no—
318C22×C4—1971no—
319C22×C4C22×D19711yescharacteristic
320SD16—22no—
321SD16—41no—
322SD16—3942no—
323D8—7881no—
324D8—3942no—
325D8—3942no—
326D8—3942no—
327D8—3942no—
328D8—1974no—
329M16—3942no—
330C2×C8—3942no—
331C2×C8—1971no—
332C2×C8—1971no—
333C2×C8C22×D19711yescharacteristic
334C2×C8C22×D19711yescharacteristic
335C4⋊C4—21no—
336C4⋊C4—1971no—
337C4⋊C4—1971no—
338C4⋊C4C22×D19711yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—3941no—
341C22⋊C4—3941no—
342C22⋊C4—3941no—
343C22⋊C4—3941no—
344C22⋊C4—3941no—
345C22⋊C4—3941no—
346C22⋊C4—3941no—
347C22⋊C4—1971no—
348C22⋊C4—1971no—
349C42—1971no—
350C23—21no—
351C23—21no—
352C23—42no—
353C23—42no—
354C23—81no—
355C23—7882no—
356C23—7882no—
357C23—7882no—
358C23—7882no—
359C23—15761no—
360C23—15761no—
361C23—3941no—
362C23—3941no—
363C23—3941no—
364C23—1971no—
365C23D4×D19711yescharacteristic
366Q8—21no—
367Q8—21no—
368D4—21no—
369D4—21no—
370D4—21no—
371D4—42no—
372D4—7882no—
373D4—7882no—
374D4—7882no—
375D4—7882no—
376D4—7882no—
377D4—7882no—
378D4—15761no—
379D4—15761no—
380D4—3942no—
381D4—3941no—
382D4—3941no—
383D4—3941no—
384D4—3941no—
385D4—3941no—
386D4—1972no—
387D4D4×D19712yes—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—7882no—
393C2×C4—7882no—
394C2×C4—15761no—
395C2×C4—3941no—
396C2×C4—3941no—
397C2×C4—3941no—
398C2×C4—1971no—
399C2×C4—1971no—
400C2×C4—1971no—
401C2×C4D4×D19711yescharacteristic
402C2×C4C23×D19711yescharacteristic
403C8—21no—
404C8—21no—
405C8—3941no—
406C8—1972no—
407C22—21no—
408C22—21no—
409C22—22no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—84no—
416C22—7881no—
417C22—7881no—
418C22—7881no—
419C22—7881no—
420C22—7881no—
421C22—7881no—
422C22—7881no—
423C22—7881no—
424C22—15764no—
425C22—15764no—
426C22—3941no—
427C22—3941no—
428C22—3942no—
429C22—3941no—
430C22#2026335312yescomplemented
431C22C2×D4×D19711yescharacteristic
432C4—21no—
433C4—41no—
434C4—41no—
435C4—7881no—
436C4—3941no—
437C4—3941no—
438C4—3941no—
439C4C2×D4×D19711yescharacteristic
440C4#2026337111yescharacteristic
441C2—22no—
442C2—42no—
443C2—81no—
444C2—7882no—
445C2—7881no—
446C2—15761no—
447C2—15761no—
448C2#2716171111yescharacteristic
449C2C2×#2026335311yescharacteristic
450C2#2716188311yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#450×C197C2
socleC22×C197C22×D4
hypercenter#450D197
hypocenterC197#3264
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C2×#20263353C2semidirect
2. C22×#15691343C2semidirect
3. #27161720C2semidirect
4. D4×C22⋊D197C2semidirect
5. #27161465C2semidirect
6. #27161331C2semidirect
7. #27161310C2semidirect
8. #27161296C2semidirect
9. #27161273C2semidirect
10. #27161223C2semidirect
11. #27161215C2semidirect
12. #27161214C2semidirect
13. #27161188C2semidirect
14. #27161147C2semidirect
15. #450×C197C2semidirect
16. C2×#15691359C22semidirect
17. C22×D788C22semidirect
19. C2×D4×D197C22semidirect
21. C2×#15691345C22semidirect
22. C2×D1576C22semidirect
23. C2×D1576C22semidirect
27. C2×#15691343C22semidirect
28. C2×#15691343C22semidirect
29. C2×#15691341C22semidirect
30. C22×#13096802C22semidirect
31. #20263322C22semidirect
32. #20263315C22semidirect
35. #20263301C22semidirect
38. #20263297C22semidirect
42. D4×Dic197C22semidirect
43. #20263288C22semidirect
46. #20263257C22semidirect
47. #20263255C22semidirect
48. #20263250C22semidirect
49. #20263248C22semidirect
50. #20263248C22semidirect
51. #20263246C22semidirect
52. #20263238C22semidirect
53. #20263236C22semidirect
55. C2×SD16×C197C22semidirect
56. C2×SD16×C197C22semidirect
57. C22⋊Q8×C197C22semidirect
58. #102×C197C22semidirect
59. #102×C197C22semidirect
60. C22⋊C8×C197C22semidirect
92. #15691343D4semidirect
142. D788C2×D4semidirect
159. #13096802C2×D4semidirect
190. C22×C197AGL(1,ℤ/8)semidirect
225. C197#3264semidirect
236. #450D197semidirect
249. C2×SD16C2×D197semidirect
250. C2×SD16C2×D197semidirect
263. C22⋊Q8C2×D197semidirect
282. #102C2×D197semidirect
283. #102C2×D197semidirect
285. C22⋊C8C2×D197semidirect
430. 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×#156913432
#271617202
D4×C22⋊D1972
#271614652
#271613312
#271613102
#271612962
#271612732
#271612232
#271612152
#271612142
#271611882
#271611472
#450×C1972
#3264197