Finite Group Databasebeta

← #38754194

Subgroups — #38754194

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×#20263350C211yesmaximal, characteristic, complemented
2C2×D8×D197C211yesmaximal, characteristic, complemented
3#27161724C211yesmaximal, characteristic, complemented
4D4×D788C211yesmaximal, characteristic, complemented
5#27161468C211yesmaximal, characteristic, complemented
6#27161333C211yesmaximal, characteristic, complemented
7#27161309C211yesmaximal, characteristic, complemented
8#27161294C211yesmaximal, characteristic, complemented
9#27161274C211yesmaximal, characteristic, complemented
10#27161231C211yesmaximal, characteristic, complemented
11#27161225C211yesmaximal, characteristic, complemented
12#27161214C211yesmaximal, characteristic, complemented
13#27161214C211yesmaximal, characteristic, complemented
14#27161152C211yesmaximal, characteristic, complemented
15#442×C197C211yesmaximal, characteristic, complemented
16C2×#15691357C2211yescharacteristic, complemented
17C22×D788—21no—
18C2×D4×D197C2211yescharacteristic
19C2×D4×D197C2211yescharacteristic, complemented
20C2×D4×D197C2211yescharacteristic, complemented
21#20263350—24no—
22C2×#15691347C2211yescharacteristic, complemented
23C2×#15691347C2211yescharacteristic, complemented
24C2×#15691346C2211yescharacteristic, complemented
25C2×D1576C2211yescharacteristic, complemented
26C2×D1576C2211yescharacteristic, complemented
27C2×#15691343C2211yescharacteristic
28C2×#15691343C2211yescharacteristic, complemented
29C2×#15691343C2211yescharacteristic, complemented
30D8×D197—24no—
31C2×#15691341C2211yescharacteristic, complemented
32C2×C8×D197C2211yescharacteristic, complemented
33#20263323C2211yescharacteristic, complemented
34#20263314C2211yescharacteristic, complemented
35#20263310—21no—
36#20263306C2211yescharacteristic, complemented
37#20263301C2211yescharacteristic
38#20263301C2211yescharacteristic, complemented
39#20263301C2211yescharacteristic, complemented
40#20263300—21no—
41#20263300—21no—
42#20263296—21no—
43#20263296—21no—
44#20263296—21no—
45C4×D788C2211yescharacteristic
46C2×#15691335—21no—
47C4⋊#13096802C2211yescharacteristic, complemented
48#20263249C2211yescharacteristic, complemented
49#20263246C2211yescharacteristic, complemented
50#20263246C2211yescharacteristic, complemented
51#20263246C2211yescharacteristic, complemented
52#20263238C2211yescharacteristic, complemented
53#20263238C2211yescharacteristic, complemented
54C2×D8×C197C2211yescharacteristic
55C4×D4×C197C2211yescharacteristic, complemented
56C4×D4×C197C2211yescharacteristic, complemented
57#106×C197C2211yescharacteristic, complemented
58#102×C197C2211yescharacteristic, complemented
59#102×C197C2211yescharacteristic, complemented
60#97×C197C2211yescharacteristic, complemented
61C23×D197—21no—
62C23×D197—21no—
63C23×D197—21no—
64#15691357—22no—
65#15691357—41no—
66C2×Dic394C2311yescharacteristic
67C2×D788—21no—
68C2×D788—21no—
69C2×D788—21no—
70C2×D788—21no—
71C2×D788—21no—
72C2×D788—44no—
73C2×D788D411yescharacteristic
74C2×D788C2311yescharacteristic
75C2×D788C2311yescharacteristic
76C2×D788C2311yescharacteristic
77C2×C22⋊D197—21no—
78C2×C22⋊D197—21no—
79C2×C22⋊D197—21no—
80C2×C22⋊D197—21no—
81D4×D197—22no—
82D4×D197—22no—
83D4×D197—22no—
84D4×D197—41no—
85D4×D197—41no—
86D4×D197—41no—
87C22×Dic197—21no—
88C2×C4×D197C2311yescharacteristic
89C2×C4×D197C2311yescharacteristic
90#15691347—22no—
91#15691347—41no—
92#15691346—22no—
93D1576—22no—
94D1576—41no—
95#15691343—22no—
96#15691343—22no—
97#15691343—22no—
98#15691341—22no—
99C2×#13096802C2311yescharacteristic
100C2×#13096802C2311yescharacteristic
101C8×D197—22no—
102C4⋊Dic197—21no—
103C4⋊Dic197D411yescharacteristic
104#15691335—21no—
105#15691335—21no—
106#15691335—21no—
107#15691335—21no—
108#15691335—21no—
109#15691335—21no—
110#15691335D411yescharacteristic
111#15691335D411yescharacteristic
112C2×D4×C197C2311yescharacteristic
113C2×D4×C197C2311yescharacteristic
114C22×C4×C197—21no—
115C22×C4×C197—21no—
116D8×C197D414yescomplemented
117C2×C8×C197C2311yescharacteristic
118C2×C8×C197C2311yescharacteristic
119C4⋊C4×C197C2311yescharacteristic
120C4⋊C4×C197C2311yescharacteristic
121C22⋊C4×C197—21no—
122C22⋊C4×C197—21no—
123C42×C197C2311yescharacteristic
124C22×D197—21no—
125C22×D197—21no—
126C22×D197—21no—
127C22×D197—42no—
128C22×D197—42no—
129C22×D197—42no—
130C22×D197—42no—
131C22×D197—42no—
132C22×D197—42no—
133C22×D197—81no—
134C22×D197—81no—
135C22×D197—81no—
136C22×D197C2×D411yescharacteristic
137C22×D197C2×D411yescharacteristic
138Dic394—21no—
139Dic394—21no—
140D788—22no—
141D788—21no—
142D788—21no—
143D788—21no—
144D788—21no—
145D788—21no—
146D788—21no—
147D788—42no—
148D788—42no—
149D788—42no—
150D788—81no—
151D788—81no—
152C22⋊D197—42no—
153C22⋊D197—42no—
154C22⋊D197—42no—
155C22⋊D197—42no—
156C2×Dic197—21no—
157C2×Dic197—42no—
158C2×Dic197C2×D411yescharacteristic
159C2×Dic197C2×D411yescharacteristic
160C4×D197—21no—
161C4×D197—21no—
162C4×D197—21no—
163C4×D197—21no—
164#13096802—21no—
165#13096802—21no—
166C23×C197—21no—
167C23×C197—21no—
168D4×C197—21no—
169D4×C197—21no—
170D4×C197C2×D412yes—
171D4×C197C2×D412yescomplemented
172C2×C4×C197—21no—
173C2×C4×C197—21no—
174C2×C4×C197—42no—
175C2×C4×C197—81no—
176C2×C4×C197C2×D411yescharacteristic
177C2×C4×C197C2×D411yescharacteristic
178C2×C4×C197C2411yescharacteristic
179C8×C197—21no—
180C8×C197C2×D412yescomplemented
181C22×C197—22no—
182C22×C197—22no—
183C22×C197—41no—
184C22×C197—41no—
185C22×C197—41no—
186C22×C197—41no—
187C22×C197C22×D411yescharacteristic
188C2×D197—21no—
189C2×D197—21no—
190C2×D197—21no—
191C2×D197—21no—
192C2×D197—21no—
193C2×D197—41no—
194C2×D197—41no—
195C2×D197—41no—
196C2×D197—41no—
197C2×D197—41no—
198C2×D197—41no—
199C2×D197—41no—
200C2×D197—41no—
201C2×D197—41no—
202C2×D197—84no—
203C2×D197—84no—
204C2×D197—84no—
205C2×D197AGL(1,ℤ/8)12yescomplemented
206C4×C197—21no—
207C4×C197—21no—
208C4×C197—41no—
209C4×C197—41no—
210C4×C197C22×D411yescharacteristic
211C4×C197H(5,𝔽2)11yescharacteristic
212Dic197—21no—
213Dic197—21no—
214Dic197—41no—
215C2×C197—42no—
216C2×C197—42no—
217C2×C197D4211yescharacteristic
218C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
219C2×C197#57611yescharacteristic
220D197—22no—
221D197—41no—
222D197—81no—
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#450—1971no—
237#449—1971no—
238#447—1971no—
239#447—1971no—
240#442D19711yescharacteristic, complemented
241#435—1971no—
242C2×#41—1971no—
243C22×D4—3941no—
244C22×D4—1971no—
245C22×D4—1971no—
246C22×D4—1971no—
247AGL(1,ℤ/8)—3944no—
248C2×SD16—1971no—
249C2×SD16—1971no—
250C2×SD16—1971no—
251C2×D8—3944no—
252C2×D8—1971no—
253C2×D8—1971no—
254C2×D8—1971no—
255C2×D8—1971no—
256C2×D8—1971no—
257C2×D8C2×D19711yescharacteristic
258C2×M16—1971no—
259C22×C8—1971no—
260#127—1971no—
261#124—1971no—
262#123—3941no—
263C22⋊Q8—1971no—
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×D4C2×D19711yescharacteristic, complemented
274C4×D4C2×D19711yescharacteristic, complemented
275C2×C22⋊C4—3941no—
276#106C2×D19711yescharacteristic, complemented
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⋊C8—1971no—
286#97C2×D19711yescharacteristic, complemented
287C24—3941no—
288C24—3941no—
289C24—3941no—
290#41—7881no—
291#41—3942no—
292C2×Q8—1971no—
293C2×D4—7884no—
294C2×D4—7881no—
295C2×D4—7881no—
296C2×D4—7881no—
297C2×D4—3941no—
298C2×D4—3941no—
299C2×D4—3941no—
300C2×D4—3941no—
301C2×D4—3942no—
302C2×D4—3942no—
303C2×D4—3942no—
304C2×D4—3941no—
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—21no—
316C22×C4—21no—
317C22×C4—3941no—
318C22×C4—1971no—
319C22×C4—1971no—
320SD16—7881no—
321SD16—3942no—
322SD16—3942no—
323D8—7881no—
324D8—3942no—
325D8—3942no—
326D8—3942no—
327D8—3942no—
328D8D78814yescomplemented
329M16—3942no—
330C2×C8—3942no—
331C2×C8—1971no—
332C2×C8—1971no—
333C2×C8C22×D19711yescharacteristic
334C2×C8C22×D19711yescharacteristic
335C4⋊C4—3941no—
336C4⋊C4—1971no—
337C4⋊C4C22×D19711yescharacteristic
338C4⋊C4C22×D19711yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—3941no—
342C22⋊C4—3941no—
343C22⋊C4—3941no—
344C22⋊C4—3941no—
345C22⋊C4—3941no—
346C22⋊C4—3941no—
347C22⋊C4—1971no—
348C22⋊C4—1971no—
349C42C22×D19711yescharacteristic
350C23—21no—
351C23—21no—
352C23—7882no—
353C23—7882no—
354C23—7882no—
355C23—7882no—
356C23—7882no—
357C23—7882no—
358C23—15761no—
359C23—15761no—
360C23—15761no—
361C23—3941no—
362C23—3941no—
363C23—3941no—
364C23—1971no—
365C23—1971no—
366Q8—3941no—
367Q8—3941no—
368D4—21no—
369D4—21no—
370D4—7882no—
371D4—7882no—
372D4—7882no—
373D4—7882no—
374D4—7882no—
375D4—7882no—
376D4—7882no—
377D4—15761no—
378D4—15761no—
379D4—3942no—
380D4—3941no—
381D4—3941no—
382D4—3941no—
383D4—3941no—
384D4—3941no—
385D4—3941no—
386D4C2×D78812yes—
387D4C2×D78812yescomplemented
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—42no—
391C2×C4—81no—
392C2×C4—7882no—
393C2×C4—3941no—
394C2×C4—3941no—
395C2×C4—3941no—
396C2×C4—3941no—
397C2×C4—3941no—
398C2×C4—1971no—
399C2×C4—1971no—
400C2×C4D4×D19711yescharacteristic
401C2×C4D4×D19711yescharacteristic
402C2×C4C23×D19711yescharacteristic
403C8—21no—
404C8—3941no—
405C8—3941no—
406C8C2×D78812yescomplemented
407C22—22no—
408C22—22no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—7881no—
414C22—7881no—
415C22—7881no—
416C22—7881no—
417C22—7881no—
418C22—7881no—
419C22—7881no—
420C22—7881no—
421C22—7881no—
422C22—15764no—
423C22—15764no—
424C22—15764no—
425C22—3941no—
426C22—3941no—
427C22—3941no—
428C22—3941no—
429C22—3941no—
430C22—1972no—
431C22C2×D4×D19711yescharacteristic
432C4—21no—
433C4—21no—
434C4—41no—
435C4—41no—
436C4—7881no—
437C4—3941no—
438C4—3941no—
439C4C22×D78811yescharacteristic
440C4#2026337211yescharacteristic
441C2—42no—
442C2—42no—
443C2—7881no—
444C2—15761no—
445C2—15761no—
446C2—15761no—
447C2—3942no—
448C2D4×D78811yescharacteristic
449C2AGL(1,ℤ/8)×D19711yescharacteristic
450C2#2716188311yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#442×C197C2
socleC22×C197C22×D4
hypercenter#442D197
hypocenterC197#3264
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. C2×#20263350C2semidirect
2. C2×D8×D197C2semidirect
3. #27161724C2semidirect
4. D4×D788C2semidirect
5. #27161468C2semidirect
6. #27161333C2semidirect
7. #27161309C2semidirect
8. #27161294C2semidirect
9. #27161274C2semidirect
10. #27161231C2semidirect
11. #27161225C2semidirect
12. #27161214C2semidirect
13. #27161214C2semidirect
14. #27161152C2semidirect
15. #442×C197C2semidirect
16. C2×#15691357C22semidirect
19. C2×D4×D197C22semidirect
20. C2×D4×D197C22semidirect
22. C2×#15691347C22semidirect
23. C2×#15691347C22semidirect
24. C2×#15691346C22semidirect
25. C2×D1576C22semidirect
26. C2×D1576C22semidirect
28. C2×#15691343C22semidirect
29. C2×#15691343C22semidirect
31. C2×#15691341C22semidirect
32. C2×C8×D197C22semidirect
33. #20263323C22semidirect
34. #20263314C22semidirect
36. #20263306C22semidirect
38. #20263301C22semidirect
39. #20263301C22semidirect
47. C4⋊#13096802C22semidirect
48. #20263249C22semidirect
49. #20263246C22semidirect
50. #20263246C22semidirect
51. #20263246C22semidirect
52. #20263238C22semidirect
53. #20263238C22semidirect
55. C4×D4×C197C22semidirect
56. C4×D4×C197C22semidirect
57. #106×C197C22semidirect
58. #102×C197C22semidirect
59. #102×C197C22semidirect
60. #97×C197C22semidirect
116. D8×C197D4semidirect
171. D4×C197C2×D4semidirect
180. C8×C197C2×D4semidirect
205. C2×D197AGL(1,ℤ/8)semidirect
225. C197#3264semidirect
240. #442D197semidirect
273. C4×D4C2×D197semidirect
274. C4×D4C2×D197semidirect
276. #106C2×D197semidirect
282. #102C2×D197semidirect
283. #102C2×D197semidirect
286. #97C2×D197semidirect
328. D8D788semidirect
387. D4C2×D788semidirect
406. C8C2×D788semidirect

Maximal subgroups

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

subgroupindex
C2×#202633502
C2×D8×D1972
#271617242
D4×D7882
#271614682
#271613332
#271613092
#271612942
#271612742
#271612312
#271612252
#271612142
#271611522
#442×C1972
#3264197