Finite Group Databasebeta

← #38754188

Subgroups — #38754188

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×#20263352C211yesmaximal, characteristic, complemented
2C22×#15691343C211yesmaximal, characteristic, complemented
3#27161725C211yesmaximal, characteristic, complemented
4#27161466C211yesmaximal, characteristic, complemented
5#27161330C211yesmaximal, characteristic, complemented
6#27161306C211yesmaximal, characteristic, complemented
7#27161294C211yesmaximal, characteristic, complemented
8#27161271C211yesmaximal, characteristic, complemented
9#27161230C211yesmaximal, characteristic, complemented
10#27161222C211yesmaximal, characteristic, complemented
11#27161213C211yesmaximal, characteristic, complemented
12#27161212C211yesmaximal, characteristic, complemented
13#27161189C211yesmaximal, characteristic, complemented
14#27161151C211yesmaximal, characteristic, complemented
15D42×C197C211yesmaximal, characteristic, complemented
16C2×#15691358C2211yescharacteristic, complemented
17C22×D788C2211yescharacteristic, complemented
18#20263352—24no—
19C2×#15691346C2211yescharacteristic, complemented
20C2×#15691346C2211yescharacteristic, complemented
21C2×#15691346C2211yescharacteristic, complemented
22C2×#15691343—24no—
23C2×#15691343C2211yescharacteristic
24C2×#15691343C2211yescharacteristic
25C2×#15691343C2211yescharacteristic, complemented
26C2×#15691343C2211yescharacteristic, complemented
27C2×#15691343C2211yescharacteristic, complemented
28C2×#15691343C2211yescharacteristic, complemented
29C2×#15691342C2211yescharacteristic, complemented
30C22×#13096802C2211yescharacteristic, complemented
31#20263314C2211yescharacteristic, complemented
32#20263311—21no—
33#20263305C2211yescharacteristic, complemented
34#20263300C2211yescharacteristic, complemented
35#20263299—21no—
36#20263294—21no—
37C4×D788C2211yescharacteristic, complemented
38C22⋊C4×D197—21no—
39#20263257C2211yescharacteristic, complemented
40C4⋊#13096802C2211yescharacteristic, complemented
41#20263249C2211yescharacteristic, complemented
42#20263247C2211yescharacteristic, complemented
43#20263247C2211yescharacteristic, complemented
44#20263247C2211yescharacteristic, complemented
45#20263246C2211yescharacteristic, complemented
46#20263246C2211yescharacteristic, complemented
47C22⋊#13096802C2211yescharacteristic, complemented
48C22⋊#13096802C2211yescharacteristic, complemented
49#20263237C2211yescharacteristic, complemented
50C22×D4×C197—21no—
51C22×D4×C197C2211yescharacteristic
52C22×D4×C197C2211yescharacteristic, complemented
53#127×C197C2211yescharacteristic, complemented
54#121×C197—21no—
55#121×C197C2211yescharacteristic
56#121×C197C2211yescharacteristic, complemented
57#120×C197—21no—
58#120×C197—21no—
59C4×D4×C197C2211yescharacteristic
60C4×D4×C197C2211yescharacteristic, complemented
61C23×D197—21no—
62#15691358—22no—
63#15691358—41no—
64C2×Dic394C2311yescharacteristic
65C2×D788—22no—
66C2×D788—21no—
67C2×D788—41no—
68C2×D788C2311yescharacteristic
69C2×D788C2311yescharacteristic
70C2×C22⋊D197—21no—
71C2×C22⋊D197—21no—
72C2×C4×D197—21no—
73C2×C4×D197—21no—
74#15691346—22no—
75#15691346—22no—
76#15691346—41no—
77#15691343—22no—
78#15691343—22no—
79#15691343—22no—
80#15691343—22no—
81#15691343—41no—
82#15691343D414yescomplemented
83#15691342—22no—
84C2×#13096802—22no—
85C2×#13096802C2311yescharacteristic
86C2×#13096802C2311yescharacteristic
87C2×#13096802C2311yescharacteristic
88C2×#13096802C2311yescharacteristic
89C4⋊Dic197C2311yescharacteristic
90#15691337—21no—
91C22⋊Dic197—21no—
92#15691335—21no—
93#15691335—21no—
94#15691335—21no—
95#15691335—21no—
96C24×C197—21no—
97C24×C197—21no—
98C2×D4×C197—22no—
99C2×D4×C197—22no—
100C2×D4×C197—21no—
101C2×D4×C197—21no—
102C2×D4×C197—21no—
103C2×D4×C197—21no—
104C2×D4×C197—21no—
105C2×D4×C197—21no—
106C2×D4×C197—41no—
107C2×D4×C197—41no—
108C2×D4×C197—44no—
109C2×D4×C197D411yescharacteristic
110C2×D4×C197C2311yescharacteristic
111C2×D4×C197C2311yescharacteristic
112C2×D4×C197C2311yescharacteristic
113C22×C4×C197—21no—
114C22×C4×C197C2311yescharacteristic
115C22×C4×C197C2311yescharacteristic
116C4⋊C4×C197D411yescharacteristic
117C4⋊C4×C197C2311yescharacteristic
118C22⋊C4×C197—21no—
119C22⋊C4×C197—21no—
120C22⋊C4×C197—21no—
121C22⋊C4×C197D411yescharacteristic
122C22⋊C4×C197D411yescharacteristic
123C42×C197C2311yescharacteristic
124C22×D197—21no—
125C22×D197—21no—
126C22×D197—42no—
127C22×D197—42no—
128C22×D197—81no—
129Dic394—21no—
130Dic394—21no—
131D788—21no—
132D788—21no—
133D788—21no—
134D788—42no—
135D788C2×D412yescomplemented
136C22⋊D197—42no—
137C22⋊D197—81no—
138C2×Dic197—21no—
139C2×Dic197—21no—
140C4×D197—42no—
141C4×D197—81no—
142#13096802—21no—
143#13096802—21no—
144#13096802—21no—
145#13096802C2×D412yescomplemented
146C23×C197—21no—
147C23×C197—21no—
148C23×C197—21no—
149C23×C197—42no—
150C23×C197—42no—
151C23×C197—42no—
152C23×C197—42no—
153C23×C197—81no—
154C23×C197—81no—
155C23×C197C2×D411yescharacteristic
156C23×C197C2×D411yescharacteristic
157D4×C197—22no—
158D4×C197—21no—
159D4×C197—21no—
160D4×C197—21no—
161D4×C197—21no—
162D4×C197—21no—
163D4×C197—42no—
164D4×C197—42no—
165D4×C197—42no—
166D4×C197—42no—
167D4×C197—42no—
168D4×C197—81no—
169D4×C197C2×D412yes—
170C2×C4×C197—21no—
171C2×C4×C197—21no—
172C2×C4×C197—21no—
173C2×C4×C197—21no—
174C2×C4×C197—21no—
175C2×C4×C197—42no—
176C2×C4×C197C2×D411yescharacteristic
177C2×C4×C197C2×D411yescharacteristic
178C2×C4×C197C2×D411yescharacteristic
179C2×C4×C197C2×D411yescharacteristic
180C2×C4×C197C2411yescharacteristic
181C22×C197—21no—
182C22×C197—21no—
183C22×C197—21no—
184C22×C197—22no—
185C22×C197—21no—
186C22×C197—21no—
187C22×C197—41no—
188C22×C197—41no—
189C22×C197—41no—
190C22×C197—41no—
191C22×C197—41no—
192C22×C197—41no—
193C22×C197—41no—
194C22×C197—41no—
195C22×C197—84no—
196C22×C197—84no—
197C22×C197AGL(1,ℤ/8)12yescomplemented
198C22×C197C22×D411yescharacteristic
199C2×D197—22no—
200C2×D197—41no—
201C2×D197—41no—
202C2×D197—41no—
203C2×D197—41no—
204C2×D197—41no—
205C2×D197—84no—
206C4×C197—21no—
207C4×C197—21no—
208C4×C197—21no—
209C4×C197—21no—
210C4×C197—41no—
211C4×C197C22×D411yescharacteristic
212C4×C197H(5,𝔽2)11yescharacteristic
213Dic197—41no—
214Dic197—41no—
215C2×C197—22no—
216C2×C197—41no—
217C2×C197—42no—
218C2×C197—81no—
219C2×C197—81no—
220C2×C197D4211yescharacteristic
221C2×C197C2×AGL(1,ℤ/8)11yescharacteristic
222C2×C197#57611yescharacteristic
223D197—42no—
224D197—81no—
225C197#326411yescharacteristic, complemented, Sylow
226#3264—1971nomaximal, Sylow
227C2×AGL(1,ℤ/8)—1971no—
228C22×D8—1971no—
229#546—1971no—
230D42D19711yescharacteristic, complemented
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#442—1971no—
241#435—1971no—
242C2×#41—1971no—
243C22×D4—21no—
244C22×D4—1971no—
245C22×D4C2×D19711yescharacteristic
246C22×D4C2×D19711yescharacteristic, complemented
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×D8—1971no—
258C2×M16—1971no—
259C22×C8—1971no—
260#127C2×D19711yescharacteristic, complemented
261#124—1971no—
262#123—3941no—
263C22⋊Q8—1971no—
264#121—21no—
265#121—3941no—
266#121—1971no—
267#121C2×D19711yescharacteristic
268#121C2×D19711yescharacteristic, complemented
269#120—21no—
270#120—21no—
271#120—3941no—
272C4×D4—1971no—
273C4×D4C2×D19711yescharacteristic
274C4×D4C2×D19711yescharacteristic, complemented
275C2×C22⋊C4—3941no—
276#106—1971no—
277C4⋊C8—1971no—
278#103—1971no—
279#102—1971no—
280#102—1971no—
281#102—1971no—
282#102—1971no—
283#102—1971no—
284C22⋊C8—1971no—
285C22⋊C8—1971no—
286#97—1971no—
287C24—21no—
288C24—21no—
289C24—3941no—
290#41—7881no—
291#41—3942no—
292C2×Q8—1971no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—44no—
304C2×D4—7881no—
305C2×D4—3942no—
306C2×D4—3941no—
307C2×D4—3941no—
308C2×D4—3941no—
309C2×D4—1971no—
310C2×D4—1971no—
311C2×D4C22⋊D19711yescharacteristic
312C2×D4C22×D19711yescharacteristic
313C2×D4C22×D19711yescharacteristic
314C2×D4C22×D19711yescharacteristic
315C22×C4—21no—
316C22×C4—3941no—
317C22×C4—3941no—
318C22×C4C22×D19711yescharacteristic
319C22×C4C22×D19711yescharacteristic
320SD16—7881no—
321SD16—3942no—
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×C8—1971no—
334C2×C8—1971no—
335C4⋊C4—3941no—
336C4⋊C4—1971no—
337C4⋊C4C22⋊D19711yescharacteristic
338C4⋊C4C22×D19711yescharacteristic
339C22⋊C4—21no—
340C22⋊C4—21no—
341C22⋊C4—21no—
342C22⋊C4—3941no—
343C22⋊C4—3941no—
344C22⋊C4—3941no—
345C22⋊C4—3941no—
346C22⋊C4—3941no—
347C22⋊C4C22⋊D19711yescharacteristic
348C22⋊C4C22⋊D19711yescharacteristic
349C42C22×D19711yescharacteristic
350C23—21no—
351C23—21no—
352C23—21no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—81no—
358C23—81no—
359C23—7882no—
360C23—7882no—
361C23—15761no—
362C23—3941no—
363C23—3941no—
364C23C2×C22⋊D19711yescharacteristic
365C23C2×C22⋊D19711yescharacteristic
366Q8—3941no—
367Q8—3941no—
368D4—22no—
369D4—21no—
370D4—21no—
371D4—21no—
372D4—21no—
373D4—21no—
374D4—42no—
375D4—42no—
376D4—42no—
377D4—42no—
378D4—42no—
379D4—81no—
380D4—7882no—
381D4—7882no—
382D4—15761no—
383D4—3941no—
384D4—3941no—
385D4—3941no—
386D4—1972no—
387D4D4×D19712yes—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—42no—
394C2×C4—7882no—
395C2×C4—15761no—
396C2×C4—3941no—
397C2×C4—3941no—
398C2×C4C2×C22⋊D19711yescharacteristic
399C2×C4C2×C22⋊D19711yescharacteristic
400C2×C4C2×C22⋊D19711yescharacteristic
401C2×C4C2×C22⋊D19711yescharacteristic
402C2×C4C23×D19711yescharacteristic
403C8—3941no—
404C8—3941no—
405C8—3941no—
406C8—1972no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—84no—
422C22—84no—
423C22—7881no—
424C22—7881no—
425C22—7881no—
426C22—7881no—
427C22—7881no—
428C22—15764no—
429C22—3942no—
430C22#2026335212yescomplemented
431C22C22×C22⋊D19711yescharacteristic
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—41no—
437C4—7881no—
438C4—7881no—
439C4C2×D4×D19711yescharacteristic
440C4#2026337111yescharacteristic
441C2—22no—
442C2—41no—
443C2—42no—
444C2—81no—
445C2—81no—
446C2—7882no—
447C2—15761no—
448C2D4×C22⋊D19711yescharacteristic
449C2C2×#2026335211yescharacteristic
450C2#2716188211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
FittingD42×C197C2
socleC22×C197C22×D4
hypercenterD42D197
hypocenterC197#3264
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C4×D4 ◁ D42

Complements

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

NH ≅ G/Ntype
1. C2×#20263352C2semidirect
2. C22×#15691343C2semidirect
3. #27161725C2semidirect
4. #27161466C2semidirect
5. #27161330C2semidirect
6. #27161306C2semidirect
7. #27161294C2semidirect
8. #27161271C2semidirect
9. #27161230C2semidirect
10. #27161222C2semidirect
11. #27161213C2semidirect
12. #27161212C2semidirect
13. #27161189C2semidirect
14. #27161151C2semidirect
15. D42×C197C2semidirect
16. C2×#15691358C22semidirect
17. C22×D788C22semidirect
19. C2×#15691346C22semidirect
20. C2×#15691346C22semidirect
21. C2×#15691346C22semidirect
25. C2×#15691343C22semidirect
26. C2×#15691343C22semidirect
27. C2×#15691343C22semidirect
28. C2×#15691343C22semidirect
29. C2×#15691342C22semidirect
30. C22×#13096802C22semidirect
31. #20263314C22semidirect
33. #20263305C22semidirect
34. #20263300C22semidirect
37. C4×D788C22semidirect
39. #20263257C22semidirect
40. C4⋊#13096802C22semidirect
41. #20263249C22semidirect
42. #20263247C22semidirect
43. #20263247C22semidirect
44. #20263247C22semidirect
45. #20263246C22semidirect
46. #20263246C22semidirect
47. C22⋊#13096802C22semidirect
48. C22⋊#13096802C22semidirect
49. #20263237C22semidirect
52. C22×D4×C197C22semidirect
53. #127×C197C22semidirect
56. #121×C197C22semidirect
60. C4×D4×C197C22semidirect
82. #15691343D4semidirect
135. D788C2×D4semidirect
145. #13096802C2×D4semidirect
197. C22×C197AGL(1,ℤ/8)semidirect
225. C197#3264semidirect
230. D42D197semidirect
246. C22×D4C2×D197semidirect
260. #127C2×D197semidirect
268. #121C2×D197semidirect
274. C4×D4C2×D197semidirect
430. C22#20263352semidirect

Maximal subgroups

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

subgroupindex
C2×#202633522
C22×#156913432
#271617252
#271614662
#271613302
#271613062
#271612942
#271612712
#271612302
#271612222
#271612132
#271612122
#271611892
#271611512
D42×C1972
#3264197