Finite Group Databasebeta

← #38745690

Subgroups — #38745690

Order 25216.

928 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#20263296C211yesmaximal, characteristic
2C2×C4×D788C211yesmaximal, characteristic, complemented
3C2×#20263288C211yesmaximal, characteristic, complemented
4C2×C22⋊C4×D197C211yesmaximal, characteristic, complemented
5C2×C22⋊C4×D197C211yesmaximal, characteristic, complemented
6#27160979C211yesmaximal, characteristic, complemented
7#27160972C211yesmaximal, characteristic, complemented
8#27160957C211yesmaximal, characteristic, complemented
9#27160957C211yesmaximal, characteristic, complemented
10#27160955C211yesmaximal, characteristic, complemented
11#27160953C211yesmaximal, characteristic, complemented
12#27160950C211yesmaximal, characteristic, complemented
13C4×#15691335C211yesmaximal, characteristic, complemented
14#27160926C211yesmaximal, characteristic, complemented
15#388×C197C211yesmaximal, characteristic, complemented
16C24×D197C2211yescharacteristic
17C22×D788C2211yescharacteristic
18C22×D788C2211yescharacteristic
19C22×C22⋊D197C2211yescharacteristic
20C22×C4×D197C2211yescharacteristic
21C22×C4×D197C2211yescharacteristic
22C22×C4×D197C2211yescharacteristic
23C22×C4×D197C2211yescharacteristic
24C22×C4×D197C2211yescharacteristic, complemented
25#20263296C418yescomplemented
26C4×D788—24no—
27#20263288—24no—
28C2×C4⋊Dic197C2211yescharacteristic, complemented
29C2×#15691337C2211yescharacteristic, complemented
30C2×C22⋊Dic197C2211yescharacteristic
31C2×C22⋊Dic197C2211yescharacteristic
32C2×#15691335C2211yescharacteristic
33C2×#15691335C2211yescharacteristic
34C2×#15691335C2211yescharacteristic
35C2×#15691335C2211yescharacteristic
36C2×#15691335C2211yescharacteristic
37C2×#15691335C2211yescharacteristic
38C2×#15691335C2211yescharacteristic, complemented
39C2×#15691335C2211yescharacteristic, complemented
40C2×#15691335C2211yescharacteristic, complemented
41C22⋊C4×D197—24no—
42C22⋊C4×D197—24no—
43C2×C4×Dic197C2211yescharacteristic, complemented
44C2×C4×Dic197C2211yescharacteristic, complemented
45#20263233C2211yescharacteristic, complemented
46#20263233C2211yescharacteristic, complemented
47#20263233C2211yescharacteristic, complemented
48#20263233C2211yescharacteristic, complemented
49C2×C4⋊C4×C197C2211yescharacteristic, complemented
50C2×C22⋊C4×C197C2211yescharacteristic
51C2×C22⋊C4×C197C2211yescharacteristic
52C2×C22⋊C4×C197C2211yescharacteristic
53C2×C42×C197C2211yescharacteristic, complemented
54#95×C197C2211yescharacteristic, complemented
55#95×C197C2211yescharacteristic, complemented
56C23×D197—22no—
57C23×D197—22no—
58C23×D197—22no—
59C23×D197—22no—
60C23×D197—41no—
61C23×D197—41no—
62C23×D197C2×C414yescomplemented
63C23×D197C2311yescharacteristic
64C23×D197C2311yescharacteristic
65C23×D197C2311yescharacteristic
66C2×D788—22no—
67C2×D788—22no—
68C2×D788—22no—
69C2×D788—22no—
70C2×D788C2×C414yescomplemented
71C2×D788C2×C414yescomplemented
72C2×C22⋊D197—41no—
73C2×C22⋊D197—41no—
74C2×C22⋊D197C2×C414yescomplemented
75C22×Dic197C2311yescharacteristic
76C22×Dic197C2311yescharacteristic
77C22×Dic197C2311yescharacteristic
78C22×Dic197C2311yescharacteristic
79C22×Dic197C2311yescharacteristic
80C2×C4×D197—22no—
81C2×C4×D197—22no—
82C2×C4×D197—22no—
83C2×C4×D197—22no—
84C2×C4×D197—22no—
85C2×C4×D197—22no—
86C2×C4×D197—22no—
87C2×C4×D197—22no—
88C2×C4×D197—22no—
89C2×C4×D197—22no—
90C2×C4×D197—41no—
91C2×C4×D197—41no—
92C2×C4×D197—41no—
93C2×C4×D197—41no—
94C2×C4×D197—41no—
95C4⋊Dic197—22no—
96#15691337—41no—
97C22⋊Dic197—22no—
98C22⋊Dic197—22no—
99#15691335—22no—
100#15691335—22no—
101#15691335—22no—
102#15691335—22no—
103#15691335—41no—
104#15691335C2×C414yescomplemented
105#15691335C2×C414yescomplemented
106#15691335D414yescomplemented
107#15691335D414yescomplemented
108C4×Dic197—22no—
109C4×Dic197—41no—
110C24×C197C2311yescharacteristic
111C22×C4×C197C2311yescharacteristic
112C22×C4×C197C2311yescharacteristic
113C22×C4×C197C2311yescharacteristic
114C22×C4×C197C2311yescharacteristic
115C22×C4×C197C2311yescharacteristic
116C22×C4×C197C2311yescharacteristic
117C4⋊C4×C197—22no—
118C22⋊C4×C197—22no—
119C22⋊C4×C197—22no—
120C22⋊C4×C197C2×C414yescomplemented
121C42×C197—22no—
122C22×D197—21no—
123C22×D197—21no—
124C22×D197—21no—
125C22×D197—21no—
126C22×D197—21no—
127C22×D197—21no—
128C22×D197—21no—
129C22×D197—21no—
130C22×D197—21no—
131C22×D197—21no—
132C22×D197—21no—
133C22×D197—21no—
134C22×D197—21no—
135C22×D197—21no—
136C22×D197—28no—
137C22×D197—28no—
138C22×D197—44no—
139C22×D197—44no—
140C22×D197—44no—
141C22×D197—44no—
142C22×D197—44no—
143C22×D197C22×C412yes—
144C22×D197C22×C412yes—
145C22×D197C22×C412yes—
146C22×D197C2×D412yes—
147C22×D197C2×D412yes—
148C22×D197#4112yes—
149C22×D197#4112yes—
150D788—28no—
151D788—28no—
152C22⋊D197—44no—
153C2×Dic197—21no—
154C2×Dic197—21no—
155C2×Dic197—21no—
156C2×Dic197—21no—
157C2×Dic197—21no—
158C2×Dic197—21no—
159C2×Dic197—21no—
160C2×Dic197—21no—
161C2×Dic197—21no—
162C2×Dic197—21no—
163C2×Dic197—21no—
164C2×Dic197—21no—
165C2×Dic197C22×C412yes—
166C2×Dic197C2×D412yes—
167C2×Dic197C2×D412yes—
168C4×D197—44no—
169C4×D197—44no—
170C4×D197—44no—
171C4×D197—44no—
172C4×D197—44no—
173C23×C197—21no—
174C23×C197—21no—
175C23×C197—21no—
176C23×C197—21no—
177C23×C197—21no—
178C23×C197—21no—
179C23×C197C22×C412yes—
180C23×C197C2411yescharacteristic
181C2×C4×C197—21no—
182C2×C4×C197—21no—
183C2×C4×C197—21no—
184C2×C4×C197—21no—
185C2×C4×C197—21no—
186C2×C4×C197—21no—
187C2×C4×C197—21no—
188C2×C4×C197—21no—
189C2×C4×C197—21no—
190C2×C4×C197—21no—
191C2×C4×C197—21no—
192C2×C4×C197—21no—
193C2×C4×C197—21no—
194C2×C4×C197—21no—
195C2×C4×C197C22×C412yes—
196C2×C4×C197C22×C412yes—
197C2×C4×C197C2×D412yes—
198C2×C4×C197C2×D412yes—
199C22×C197—22no—
200C22×C197—22no—
201C22×C197—22no—
202C22×C197—41no—
203C22×C197—41no—
204C22×C197—41no—
205C22×C197—41no—
206C22×C197C23×C411yescharacteristic
207C22×C197C22×D411yescharacteristic
208C22×C197C22×D411yescharacteristic
209C22×C197C2×#4111yescharacteristic
210C22×C197C2×#4111yescharacteristic
211C22×C197H(5,𝔽2)11yescharacteristic
212C22×C197H(5,𝔽2)11yescharacteristic
213C2×D197—22no—
214C2×D197—22no—
215C2×D197—22no—
216C2×D197—22no—
217C2×D197—22no—
218C2×D197—22no—
219C2×D197—22no—
220C2×D197—22no—
221C2×D197—22no—
222C2×D197—22no—
223C2×D197—22no—
224C2×D197—22no—
225C2×D197—22no—
226C2×D197—22no—
227C2×D197—22no—
228C2×D197—416no—
229C2×D197—41no—
230C2×D197—41no—
231C2×D197—41no—
232C2×D197—41no—
233C2×D197C4×D414yescomplemented
234C2×D197C4×D414yescomplemented
235C4×C197—22no—
236C4×C197—22no—
237C4×C197—22no—
238C4×C197—22no—
239C4×C197—41no—
240C4×C197—41no—
241Dic197—22no—
242Dic197—22no—
243Dic197—22no—
244Dic197—41no—
245Dic197—41no—
246C2×C197—42no—
247C2×C197C2×C4×D411yescharacteristic
248C2×C197C2×C4×D411yescharacteristic
249C2×C197#51811yescharacteristic
250C2×C197D4211yescharacteristic
251C2×C197#54611yescharacteristic
252C2×C197#54611yescharacteristic
253C2×C197#55111yescharacteristic
254D197—24no—
255D197—24no—
256D197—42no—
257C197#235811yescharacteristic, complemented, Sylow
258#2358—1971nomaximal, Sylow
259C2×#120—1971no—
260C2×C4×D4—1971no—
261C2×C4×D4—1971no—
262C22×C22⋊C4—1971no—
263C22×C22⋊C4—1971no—
264#390—1971no—
265#388—1971no—
266#388D19711yescharacteristic, complemented
267#386—1971no—
268#386—1971no—
269#386—1971no—
270#385—1971no—
271#385—1971no—
272C4×C22⋊C4—1971no—
273C4×C22⋊C4—1971no—
274C25—1971no—
275C22×D4—1971no—
276C22×D4—1971no—
277C22×D4—1971no—
278C23×C4—1971no—
279C23×C4—1971no—
280C23×C4—1971no—
281C23×C4—1971no—
282C23×C4—1971no—
283#120—1978no—
284C4×D4—3944no—
285C4×D4—3944no—
286C2×C4⋊C4—1971no—
287C2×C4⋊C4—1971no—
288C2×C4⋊C4C2×D19711yescharacteristic, complemented
289C2×C22⋊C4—3944no—
290C2×C22⋊C4—3944no—
291C2×C22⋊C4—1971no—
292C2×C22⋊C4—1971no—
293C2×C22⋊C4—1971no—
294C2×C22⋊C4—1971no—
295C2×C22⋊C4—1971no—
296C2×C22⋊C4—1971no—
297C2×C22⋊C4—1971no—
298C2×C22⋊C4—1971no—
299C2×C22⋊C4—1971no—
300C2×C22⋊C4—1971no—
301C2×C22⋊C4—1971no—
302C2×C22⋊C4C2×D19711yescharacteristic
303C2×C22⋊C4C2×D19711yescharacteristic
304C2×C22⋊C4C2×D19711yescharacteristic
305C2×C42—1971no—
306C2×C42—1971no—
307C2×C42C2×D19711yescharacteristic, complemented
308#95—1971no—
309#95—1971no—
310#95—1971no—
311#95—1971no—
312#95C2×D19711yescharacteristic, complemented
313#95C2×D19711yescharacteristic, complemented
314C24—7881no—
315C24—7881no—
316C24—3942no—
317C24—3942no—
318C24—3942no—
319C24—3942no—
320C24—1974no—
321C24—1971no—
322C24—1971no—
323C24—1971no—
324C24C22×D19711yescharacteristic
325C2×D4—7881no—
326C2×D4—7881no—
327C2×D4—3942no—
328C2×D4—3942no—
329C2×D4—3942no—
330C2×D4—3942no—
331C2×D4—1974no—
332C2×D4—1974no—
333C2×D4—1974no—
334C22×C4—7881no—
335C22×C4—7881no—
336C22×C4—7881no—
337C22×C4—7881no—
338C22×C4—7881no—
339C22×C4—3942no—
340C22×C4—3942no—
341C22×C4—3942no—
342C22×C4—3942no—
343C22×C4—3942no—
344C22×C4—3942no—
345C22×C4—3942no—
346C22×C4—3942no—
347C22×C4—3942no—
348C22×C4—3942no—
349C22×C4—1971no—
350C22×C4—1971no—
351C22×C4—1971no—
352C22×C4—1971no—
353C22×C4—1971no—
354C22×C4C22×D19711yescharacteristic
355C22×C4C22×D19711yescharacteristic
356C22×C4C22×D19711yescharacteristic
357C22×C4C22×D19711yescharacteristic
358C22×C4C22×D19711yescharacteristic
359C22×C4C22×D19711yescharacteristic
360C4⋊C4—22no—
361C4⋊C4—7881no—
362C4⋊C4—3942no—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—7881no—
366C22⋊C4—3942no—
367C22⋊C4—3942no—
368C22⋊C4—3942no—
369C22⋊C4—3942no—
370C22⋊C4—3942no—
371C22⋊C4—3942no—
372C22⋊C4—1974no—
373C22⋊C4—1974no—
374C22⋊C4—1974no—
375C22⋊C4—1974no—
376C22⋊C4C4×D19714yescomplemented
377C42—22no—
378C42—7881no—
379C42—3942no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—7884no—
387C23—7884no—
388C23—7884no—
389C23—7884no—
390C23—7884no—
391C23—3948no—
392C23—3948no—
393C23—3941no—
394C23—3941no—
395C23—3941no—
396C23—3941no—
397C23—3941no—
398C23—3941no—
399C23—3941no—
400C23—3941no—
401C23—3941no—
402C23—3941no—
403C23—3941no—
404C23—3941no—
405C23—3941no—
406C23—3941no—
407C23—1972no—
408C23—1972no—
409C23—1972no—
410C23—1972no—
411C23—1972no—
412C23—1972no—
413C23—1972no—
414C23C2×C4×D19712yes—
415C23C23×D19711yescharacteristic
416D4—7884no—
417D4—3948no—
418D4—3948no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—7884no—
434C2×C4—7884no—
435C2×C4—7884no—
436C2×C4—7884no—
437C2×C4—7884no—
438C2×C4—3941no—
439C2×C4—3941no—
440C2×C4—3941no—
441C2×C4—3941no—
442C2×C4—3941no—
443C2×C4—3941no—
444C2×C4—3941no—
445C2×C4—3941no—
446C2×C4—3941no—
447C2×C4—3941no—
448C2×C4—3941no—
449C2×C4—3941no—
450C2×C4—1972no—
451C2×C4—1972no—
452C2×C4—1972no—
453C2×C4C2×C4×D19712yes—
454C2×C4C2×C4×D19712yes—
455C2×C4D4×D19712yes—
456C2×C4D4×D19712yes—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—78816no—
465C22—7881no—
466C22—7881no—
467C22—7881no—
468C22—7881no—
469C22—3942no—
470C22—3942no—
471C22—3942no—
472C22—3942no—
473C22—3942no—
474C22—3942no—
475C22—3942no—
476C22—3942no—
477C22—3942no—
478C22—3942no—
479C22—3942no—
480C22—3942no—
481C22—3942no—
482C22—3942no—
483C22—3942no—
484C22—1974no—
485C22—1974no—
486C22C22×C4×D19711yescharacteristic
487C22C2×D4×D19711yescharacteristic
488C22C2×D4×D19711yescharacteristic
489C22#41×D19711yescharacteristic
490C22#41×D19711yescharacteristic
491C22#2026337211yescharacteristic
492C22#2026337211yescharacteristic
493C4—22no—
494C4—22no—
495C4—22no—
496C4—22no—
497C4—41no—
498C4—41no—
499C4—7881no—
500C4—7881no—
501C4—3942no—
502C4—3942no—
503C4—3942no—
504C2—42no—
505C2—7882no—
506C2—3944no—
507C2—3944no—
508C2C4×D4×D19711yescharacteristic
509C2C4×D4×D19711yescharacteristic
510C2#2716155311yescharacteristic
511C2#2716171311yescharacteristic
512C2#2716172111yescharacteristic
513C2#2716172211yescharacteristic
514C2#2716175411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D197
commutatorC22×C197C23×C4
Fitting#388×C197C2
socle, cosocleC23×C197C24
prime radicalC2×#20263296C2
hypercenter#388D197
hypocenterC197#2358
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×C4×D788C2semidirect
3. C2×#20263288C2semidirect
4. C2×C22⋊C4×D197C2semidirect
5. C2×C22⋊C4×D197C2semidirect
6. #27160979C2semidirect
7. #27160972C2semidirect
8. #27160957C2semidirect
9. #27160957C2semidirect
10. #27160955C2semidirect
11. #27160953C2semidirect
12. #27160950C2semidirect
13. C4×#15691335C2semidirect
14. #27160926C2semidirect
15. #388×C197C2semidirect
24. C22×C4×D197C22semidirect
25. #20263296C4semidirect
28. C2×C4⋊Dic197C22semidirect
29. C2×#15691337C22semidirect
38. C2×#15691335C22semidirect
39. C2×#15691335C22semidirect
40. C2×#15691335C22semidirect
43. C2×C4×Dic197C22semidirect
44. C2×C4×Dic197C22semidirect
45. #20263233C22semidirect
46. #20263233C22semidirect
47. #20263233C22semidirect
48. #20263233C22semidirect
49. C2×C4⋊C4×C197C22semidirect
53. C2×C42×C197C22semidirect
54. #95×C197C22semidirect
55. #95×C197C22semidirect
62. C23×D197C2×C4semidirect
70. C2×D788C2×C4semidirect
71. C2×D788C2×C4semidirect
74. C2×C22⋊D197C2×C4semidirect
104. #15691335C2×C4semidirect
105. #15691335C2×C4semidirect
106. #15691335D4semidirect
107. #15691335D4semidirect
120. C22⋊C4×C197C2×C4semidirect
233. C2×D197C4×D4semidirect
234. C2×D197C4×D4semidirect
257. C197#2358semidirect
266. #388D197semidirect
288. C2×C4⋊C4C2×D197semidirect
307. C2×C42C2×D197semidirect
312. #95C2×D197semidirect
313. #95C2×D197semidirect
376. C22⋊C4C4×D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202632962
C2×C4×D7882
C2×#202632882
C2×C22⋊C4×D1972
#271609792
#271609722
#271609572
#271609552
#271609532
#271609502
C4×#156913352
#271609262
#388×C1972
#2358197