Finite Group Databasebeta

← #38751932

Subgroups — #38751932

Order 25216.

764 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D197C211yesmaximal, characteristic, complemented
2C2×#20263348C211yesmaximal, characteristic, complemented
3C2×#20263346C211yesmaximal, characteristic, complemented
4#27161591C211yesmaximal, characteristic, complemented
5#27161244C211yesmaximal, characteristic, complemented
6#27161242C211yesmaximal, characteristic, complemented
7#27161236C211yesmaximal, characteristic, complemented
8#27161231C211yesmaximal, characteristic, complemented
9#27161227C211yesmaximal, characteristic, complemented
10#27161225C211yesmaximal, characteristic, complemented
11#27161219C211yesmaximal, characteristic, complemented
12#27161214C211yesmaximal, characteristic, complemented
13#27161073C211yesmaximal, characteristic, complemented
14C2×#20263238C211yesmaximal, characteristic, complemented
15#416×C197C211yesmaximal, characteristic, complemented
16C2×#15691359C2211yescharacteristic
17C2×#15691358—21no—
18C2×#15691358C2211yescharacteristic
19C2×#15691358C2211yescharacteristic, complemented
20C2×#15691357—21no—
21C2×#15691357C2211yescharacteristic
22#41×D197—24no—
23#41×D197—24no—
24C2×Q8×D197C2211yescharacteristic
25C2×D4×D197—21no—
26C2×D4×D197C2211yescharacteristic
27C22×C4×D197—21no—
28C22×C4×D197C2211yescharacteristic
29#20263348—24no—
30#20263346—24no—
31C2×Dic788C2211yescharacteristic, complemented
32C2×#15691348C2211yescharacteristic, complemented
33C2×#15691347C2211yescharacteristic, complemented
34C2×#15691347C2211yescharacteristic, complemented
35C2×#15691346C2211yescharacteristic, complemented
36C2×#15691345C2211yescharacteristic, complemented
37C2×D1576C2211yescharacteristic, complemented
38C2×#15691343C2211yescharacteristic, complemented
39C22×#13096802C2211yescharacteristic, complemented
40#20263310—21no—
41#20263306C2211yescharacteristic
42#20263306C2211yescharacteristic, complemented
43#20263301C2211yescharacteristic
44#20263301C2211yescharacteristic, complemented
45#20263296—21no—
46C4×D788—21no—
47C4×D788—21no—
48#20263249C2211yescharacteristic, complemented
49#20263249C2211yescharacteristic, complemented
50#20263246C2211yescharacteristic, complemented
51#20263246C2211yescharacteristic, complemented
52#20263238C2211yescharacteristic
53#20263238C2211yescharacteristic
54#20263238C2211yescharacteristic, complemented
55#20263238C2211yescharacteristic, complemented
56C2×#41×C197C2211yescharacteristic
57C22×C8×C197C2211yescharacteristic, complemented
58#117×C197C2211yescharacteristic, complemented
59#103×C197C2211yescharacteristic, complemented
60#103×C197C2211yescharacteristic, complemented
61#102×C197C2211yescharacteristic, complemented
62#102×C197C2211yescharacteristic, complemented
63C23×D197—21no—
64C23×D197D411yescharacteristic
65#15691359—22no—
66#15691359—22no—
67#15691358—28no—
68#15691358—22no—
69#15691358—22no—
70#15691358—22no—
71#15691358D414yescomplemented
72#15691357—28no—
73#15691357—22no—
74#15691357—22no—
75C2×Dic394—21no—
76C2×Dic394C2311yescharacteristic
77C2×Dic394C2311yescharacteristic
78Q8×D197—22no—
79Q8×D197—22no—
80C2×D788—21no—
81C2×D788—21no—
82C2×D788—21no—
83C2×D788C2311yescharacteristic
84C2×D788C2311yescharacteristic
85C2×C22⋊D197—21no—
86C2×C22⋊D197—21no—
87C2×C22⋊D197—21no—
88C2×C22⋊D197—21no—
89D4×D197—28no—
90D4×D197—22no—
91D4×D197—22no—
92C22×Dic197—21no—
93C22×Dic197D411yescharacteristic
94C2×C4×D197—28no—
95C2×C4×D197—21no—
96C2×C4×D197—21no—
97C2×C4×D197—21no—
98C2×C4×D197—22no—
99C2×C4×D197—22no—
100C2×C4×D197—21no—
101C2×C4×D197D411yescharacteristic
102C2×C4×D197D411yescharacteristic
103C2×C4×D197C2311yescharacteristic
104C2×C4×D197C2311yescharacteristic
105Dic788—22no—
106#15691348—22no—
107#15691347—22no—
108#15691347—22no—
109#15691346—22no—
110#15691345—22no—
111D1576—22no—
112#15691343—22no—
113C2×#13096802—22no—
114C2×#13096802C2311yescharacteristic
115C2×#13096802C2311yescharacteristic
116C4⋊Dic197—21no—
117C4⋊Dic197—21no—
118#15691335—21no—
119#15691335—21no—
120#15691335—21no—
121#15691335—21no—
122#41×C197—22no—
123#41×C197D414yescomplemented
124C2×Q8×C197C2311yescharacteristic
125C2×D4×C197—21no—
126C2×D4×C197C2311yescharacteristic
127C22×C4×C197—21no—
128C22×C4×C197C2311yescharacteristic
129C2×C8×C197—22no—
130C2×C8×C197C2311yescharacteristic
131C2×C8×C197C2311yescharacteristic
132C4⋊C4×C197C2311yescharacteristic
133C4⋊C4×C197C2311yescharacteristic
134C22⋊C4×C197—21no—
135C42×C197—21no—
136C22×D197—24no—
137C22×D197—22no—
138C22×D197—21no—
139C22×D197—22no—
140C22×D197—21no—
141C22×D197—41no—
142C22×D197—42no—
143C22×D197—42no—
144C22×D197C2×D411yescharacteristic
145C22×D197C2×D411yescharacteristic
146Dic394—24no—
147Dic394—21no—
148Dic394—21no—
149Dic394—21no—
150Dic394C2×D412yes—
151D788—24no—
152D788—21no—
153D788—21no—
154D788—21no—
155D788—42no—
156D788—42no—
157D788C2×D412yes—
158C22⋊D197—24no—
159C22⋊D197—24no—
160C22⋊D197—24no—
161C22⋊D197—42no—
162C2×Dic197—24no—
163C2×Dic197—22no—
164C2×Dic197—21no—
165C2×Dic197—21no—
166C2×Dic197C2×D411yescharacteristic
167C2×Dic197C2×D411yescharacteristic
168C4×D197—24no—
169C4×D197—24no—
170C4×D197—22no—
171C4×D197—24no—
172C4×D197—22no—
173C4×D197—21no—
174C4×D197—21no—
175C4×D197—21no—
176C4×D197—21no—
177C4×D197—42no—
178#13096802—21no—
179#13096802—21no—
180C23×C197—21no—
181C23×C197C2×D411yescharacteristic
182Q8×C197—21no—
183Q8×C197C2×D412yes—
184D4×C197—24no—
185D4×C197—21no—
186D4×C197C2×D412yes—
187C2×C4×C197—21no—
188C2×C4×C197—24no—
189C2×C4×C197—21no—
190C2×C4×C197—21no—
191C2×C4×C197C2×D412yes—
192C2×C4×C197C2×D411yescharacteristic
193C2×C4×C197C2×D412yes—
194C2×C4×C197C2411yescharacteristic
195C8×C197—21no—
196C8×C197—21no—
197C22×C197—21no—
198C22×C197—21no—
199C22×C197—22no—
200C22×C197—41no—
201C22×C197—41no—
202C22×C197#12012yes—
203C22×C197C22×D411yescharacteristic
204C2×D197—21no—
205C2×D197—21no—
206C2×D197—21no—
207C2×D197—22no—
208C2×D197—21no—
209C2×D197—21no—
210C2×D197—44no—
211C2×D197—48no—
212C2×D197—41no—
213C2×D197—41no—
214C2×D197—41no—
215C2×D197—41no—
216C2×D197—41no—
217C2×D197#13512yescomplemented
218C4×C197—22no—
219C4×C197—41no—
220C4×C197—41no—
221C4×C197#12012yes—
222C4×C197C22×D411yescharacteristic
223C4×C197C22×D411yescharacteristic
224Dic197—21no—
225Dic197—22no—
226Dic197—21no—
227Dic197—41no—
228C2×C197—22no—
229C2×C197—42no—
230C2×C197C2×#12011yescharacteristic
231C2×C197C2×#13511yescharacteristic
232C2×C197#57511yescharacteristic
233D197—22no—
234D197—41no—
235D197—42no—
236D197—81no—
237C197#300211yescharacteristic, complemented, Sylow
238#3002—1971nomaximal, Sylow
239C22×#41—1971no—
240C2×#135—1971no—
241C2×#135—1971no—
242#525—1971no—
243#452—1971no—
244#452—1971no—
245#451—1971no—
246#450—1971no—
247#449—1971no—
248#449—1971no—
249#448—1971no—
250#447—1971no—
251#416—1971no—
252#416D19711yescharacteristic, complemented
253C2×C22⋊C8—1971no—
254C2×#41—3941no—
255C2×#41—3944no—
256C2×#41—3944no—
257C2×#41—3941no—
258C2×#41—1971no—
259C2×#41—1971no—
260C2×#41—1971no—
261C2×#41—1971no—
262C2×#41C2×D19711yescharacteristic
263C22×Q8—1971no—
264C22×D4—3941no—
265C22×D4—1971no—
266C23×C4—3941no—
267C23×C4—1971no—
268#135—3944no—
269#135—3944no—
270C2×Q16—1971no—
271C2×Q16—1971no—
272C2×SD16—1971no—
273C2×SD16—1971no—
274C2×SD16—1971no—
275C2×SD16—1971no—
276C2×D8—1971no—
277C2×D8—1971no—
278C22×C8—1971no—
279C22×C8C2×D19711yescharacteristic, complemented
280#123—3941no—
281C22⋊Q8—1971no—
282C22⋊Q8—1971no—
283#121—1971no—
284#121—1971no—
285#120—3941no—
286C4×D4—3941no—
287C4×D4—3941no—
288#117C2×D19711yescharacteristic, complemented
289#103—1971no—
290#103—1971no—
291#103C2×D19711yescharacteristic, complemented
292#103C2×D19711yescharacteristic, complemented
293#102—1971no—
294#102—1971no—
295#102C2×D19711yescharacteristic, complemented
296#102C2×D19711yescharacteristic, complemented
297C22⋊C8—1971no—
298C22⋊C8—1971no—
299C22⋊C8—1971no—
300C22⋊C8—1971no—
301C24—3941no—
302C24—1971no—
303#41—22no—
304#41—3948no—
305#41—3942no—
306#41—3942no—
307#41—3948no—
308#41—3942no—
309#41—3942no—
310#41—3942no—
311#41—3942no—
312#41—3942no—
313#41—1974no—
314#41D78814yescomplemented
315C2×Q8—3941no—
316C2×Q8—3942no—
317C2×Q8—3942no—
318C2×Q8—1971no—
319C2×Q8—1971no—
320C2×Q8C22×D19711yescharacteristic
321C2×D4—21no—
322C2×D4—3941no—
323C2×D4—3941no—
324C2×D4—3941no—
325C2×D4—3948no—
326C2×D4—3942no—
327C2×D4—3941no—
328C2×D4—3941no—
329C2×D4—3941no—
330C2×D4—3942no—
331C2×D4—3941no—
332C2×D4—1971no—
333C2×D4—1971no—
334C2×D4C22×D19711yescharacteristic
335C22×C4—21no—
336C22×C4—3948no—
337C22×C4—3941no—
338C22×C4—3941no—
339C22×C4—3941no—
340C22×C4—3941no—
341C22×C4—3942no—
342C22×C4—3941no—
343C22×C4—3942no—
344C22×C4—1971no—
345C22×C4—1971no—
346C22×C4—1971no—
347C22×C4—1971no—
348C22×C4—1971no—
349C22×C4C22×D19711yescharacteristic
350Q16—3942no—
351Q16—3942no—
352SD16—3942no—
353SD16—3942no—
354SD16—3942no—
355SD16—3942no—
356D8—3942no—
357D8—3942no—
358C2×C8—22no—
359C2×C8—3942no—
360C2×C8—1971no—
361C2×C8—1971no—
362C2×C8C22×D19711yescharacteristic
363C2×C8C22×D19711yescharacteristic
364C4⋊C4—3941no—
365C4⋊C4—3941no—
366C4⋊C4C22×D19711yescharacteristic
367C4⋊C4C22×D19711yescharacteristic
368C22⋊C4—21no—
369C22⋊C4—3941no—
370C22⋊C4—3941no—
371C22⋊C4—3941no—
372C22⋊C4—3941no—
373C42—21no—
374C23—21no—
375C23—7881no—
376C23—7882no—
377C23—7882no—
378C23—3944no—
379C23—3942no—
380C23—3941no—
381C23—3942no—
382C23—3941no—
383C23—1971no—
384C23—1971no—
385C23D4×D19711yescharacteristic
386Q8—21no—
387Q8—3944no—
388Q8—3941no—
389Q8—3941no—
390Q8—3941no—
391Q8—1972no—
392Q8C2×D78812yes—
393D4—24no—
394D4—21no—
395D4—7882no—
396D4—7882no—
397D4—7882no—
398D4—3944no—
399D4—3944no—
400D4—3944no—
401D4—3944no—
402D4—3941no—
403D4—3941no—
404D4—3941no—
405D4—1972no—
406D4C2×D78812yes—
407C2×C4—21no—
408C2×C4—24no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—7882no—
412C2×C4—3944no—
413C2×C4—3944no—
414C2×C4—3941no—
415C2×C4—3944no—
416C2×C4—3942no—
417C2×C4—3941no—
418C2×C4—3942no—
419C2×C4—3944no—
420C2×C4—3942no—
421C2×C4—3941no—
422C2×C4—3941no—
423C2×C4—3941no—
424C2×C4—3941no—
425C2×C4—1971no—
426C2×C4—1971no—
427C2×C4D4×D19712yes—
428C2×C4D4×D19711yescharacteristic
429C2×C4C2×D78812yes—
430C2×C4C23×D19711yescharacteristic
431C8—21no—
432C8—21no—
433C8—3941no—
434C8—3941no—
435C22—21no—
436C22—21no—
437C22—22no—
438C22—41no—
439C22—41no—
440C22—7884no—
441C22—7888no—
442C22—7881no—
443C22—7881no—
444C22—7881no—
445C22—7881no—
446C22—7881no—
447C22—3941no—
448C22—3941no—
449C22—3941no—
450C22—3942no—
451C22—3941no—
452C22—3941no—
453C22—1972no—
454C22#2026329612yes—
455C22C2×D4×D19711yescharacteristic
456C4—22no—
457C4—41no—
458C4—41no—
459C4—7881no—
460C4—3941no—
461C4—3942no—
462C4—3941no—
463C4#2026329612yes—
464C4C2×D4×D19711yescharacteristic
465C4C22×D78811yescharacteristic
466C2—22no—
467C2—42no—
468C2—7882no—
469C2—7881no—
470C2—15761no—
471C2—3942no—
472C2C2×#2026329611yescharacteristic
473C2#135×D19711yescharacteristic
474C2#2716187611yescharacteristic
subgroupG/H
centerC2×C4D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#416×C197C2
socleC22×C197C22×D4
hypercenter#416D197
hypocenterC197#3002
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. C2×#41×D197C2semidirect
2. C2×#20263348C2semidirect
3. C2×#20263346C2semidirect
4. #27161591C2semidirect
5. #27161244C2semidirect
6. #27161242C2semidirect
7. #27161236C2semidirect
8. #27161231C2semidirect
9. #27161227C2semidirect
10. #27161225C2semidirect
11. #27161219C2semidirect
12. #27161214C2semidirect
13. #27161073C2semidirect
14. C2×#20263238C2semidirect
15. #416×C197C2semidirect
19. C2×#15691358C22semidirect
31. C2×Dic788C22semidirect
32. C2×#15691348C22semidirect
33. C2×#15691347C22semidirect
34. C2×#15691347C22semidirect
35. C2×#15691346C22semidirect
36. C2×#15691345C22semidirect
37. C2×D1576C22semidirect
38. C2×#15691343C22semidirect
39. C22×#13096802C22semidirect
42. #20263306C22semidirect
44. #20263301C22semidirect
48. #20263249C22semidirect
49. #20263249C22semidirect
50. #20263246C22semidirect
51. #20263246C22semidirect
54. #20263238C22semidirect
55. #20263238C22semidirect
57. C22×C8×C197C22semidirect
58. #117×C197C22semidirect
59. #103×C197C22semidirect
60. #103×C197C22semidirect
61. #102×C197C22semidirect
62. #102×C197C22semidirect
71. #15691358D4semidirect
123. #41×C197D4semidirect
217. C2×D197#135semidirect
237. C197#3002semidirect
252. #416D197semidirect
279. C22×C8C2×D197semidirect
288. #117C2×D197semidirect
291. #103C2×D197semidirect
292. #103C2×D197semidirect
295. #102C2×D197semidirect
296. #102C2×D197semidirect
314. #41D788semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1972
C2×#202633482
C2×#202633462
#271615912
#271612442
#271612422
#271612362
#271612312
#271612272
#271612252
#271612192
#271612142
#271610732
C2×#202632382
#416×C1972
#3002197