Finite Group Databasebeta

← #38745691

Subgroups — #38745691

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×#20263295C211yesmaximal, characteristic
2C2×C4×C22⋊D197C211yesmaximal, characteristic, complemented
3C2×#20263286C211yesmaximal, characteristic, complemented
4C22×C22⋊Dic197C211yesmaximal, characteristic, complemented
5#27160979C211yesmaximal, characteristic, complemented
6#27160973C211yesmaximal, characteristic, complemented
7#27160967C211yesmaximal, characteristic, complemented
8#27160958C211yesmaximal, characteristic, complemented
9#27160956C211yesmaximal, characteristic, complemented
10#27160955C211yesmaximal, characteristic, complemented
11#27160952C211yesmaximal, characteristic, complemented
12#27160951C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic197C211yesmaximal, characteristic, complemented
14C4×C22⋊Dic197C211yesmaximal, characteristic, complemented
15C22×C22⋊C4×C197C211yesmaximal, characteristic, complemented
16C22×C22⋊D197C2211yescharacteristic
17C22×C22⋊D197C2211yescharacteristic
18C22×C22⋊D197C2211yescharacteristic
19C23×Dic197C2211yescharacteristic
20C23×Dic197C2211yescharacteristic
21C22×C4×D197C2211yescharacteristic, complemented
22#20263295C418yescomplemented
23C4×C22⋊D197—24no—
24#20263286—24no—
25C2×#15691337C2211yescharacteristic, complemented
26C2×#15691337C2211yescharacteristic, complemented
27C2×#15691337C2211yescharacteristic, complemented
28C2×C22⋊Dic197—24no—
29C2×C22⋊Dic197C2211yescharacteristic
30C2×C22⋊Dic197C2211yescharacteristic
31C2×C22⋊Dic197C2211yescharacteristic
32C2×C22⋊Dic197C2211yescharacteristic
33C2×C22⋊Dic197C2211yescharacteristic
34C2×C22⋊Dic197C2211yescharacteristic
35C2×C22⋊Dic197C2211yescharacteristic
36C2×#15691335C2211yescharacteristic, complemented
37C2×#15691335C2211yescharacteristic, complemented
38C2×#15691335C2211yescharacteristic, complemented
39C2×C4×Dic197C2211yescharacteristic, complemented
40C2×C4×Dic197C2211yescharacteristic, complemented
41C2×C4×Dic197C2211yescharacteristic, complemented
42#20263233C2211yescharacteristic, complemented
43#20263233C2211yescharacteristic, complemented
44#20263233C2211yescharacteristic, complemented
45#20263233C2211yescharacteristic, complemented
46#20263233C2211yescharacteristic, complemented
47#20263233C2211yescharacteristic, complemented
48C25×C197C2211yescharacteristic
49C23×C4×C197C2211yescharacteristic
50C23×C4×C197C2211yescharacteristic
51C2×C22⋊C4×C197—24no—
52C2×C22⋊C4×C197C2211yescharacteristic
53C2×C22⋊C4×C197C2211yescharacteristic
54C2×C22⋊C4×C197C2211yescharacteristic
55C2×C22⋊C4×C197C2211yescharacteristic
56C23×D197C2311yescharacteristic
57C2×C22⋊D197—22no—
58C2×C22⋊D197—22no—
59C2×C22⋊D197—22no—
60C2×C22⋊D197—22no—
61C2×C22⋊D197—41no—
62C2×C22⋊D197—41no—
63C2×C22⋊D197C2×C414yescomplemented
64C2×C22⋊D197C2×C414yescomplemented
65C2×C22⋊D197C2×C414yescomplemented
66C22×Dic197—22no—
67C22×Dic197—22no—
68C22×Dic197—22no—
69C22×Dic197—22no—
70C22×Dic197—41no—
71C22×Dic197—41no—
72C22×Dic197C2311yescharacteristic
73C22×Dic197C2311yescharacteristic
74C22×Dic197C2311yescharacteristic
75C22×Dic197C2311yescharacteristic
76C22×Dic197C2311yescharacteristic
77C22×Dic197C2311yescharacteristic
78C22×Dic197C2311yescharacteristic
79C2×C4×D197—22no—
80C2×C4×D197—22no—
81C2×C4×D197—41no—
82#15691337—22no—
83#15691337—22no—
84#15691337—41no—
85C22⋊Dic197—22no—
86C22⋊Dic197—22no—
87C22⋊Dic197—22no—
88C22⋊Dic197C2×C414yescomplemented
89C22⋊Dic197C2×C414yescomplemented
90C22⋊Dic197C2×C414yescomplemented
91C22⋊Dic197D414yescomplemented
92#15691335—22no—
93#15691335—22no—
94#15691335—41no—
95C4×Dic197—22no—
96C4×Dic197—22no—
97C4×Dic197—41no—
98C24×C197—22no—
99C24×C197—22no—
100C24×C197—22no—
101C24×C197—22no—
102C24×C197—41no—
103C24×C197—41no—
104C24×C197C2×C414yescomplemented
105C24×C197C2311yescharacteristic
106C24×C197C2311yescharacteristic
107C24×C197C2311yescharacteristic
108C22×C4×C197—22no—
109C22×C4×C197—22no—
110C22×C4×C197—22no—
111C22×C4×C197—22no—
112C22×C4×C197—41no—
113C22×C4×C197—41no—
114C22×C4×C197C2311yescharacteristic
115C22×C4×C197C2311yescharacteristic
116C22×C4×C197C2311yescharacteristic
117C22×C4×C197C2311yescharacteristic
118C22⋊C4×C197—22no—
119C22⋊C4×C197—22no—
120C22⋊C4×C197—22no—
121C22⋊C4×C197D414yescomplemented
122C22×D197—21no—
123C22×D197—21no—
124C22×D197—21no—
125C22×D197—21no—
126C22×D197—21no—
127C22×D197—21no—
128C22×D197C22×C412yes—
129C22⋊D197—28no—
130C22⋊D197—28no—
131C22⋊D197—44no—
132C2×Dic197—21no—
133C2×Dic197—21no—
134C2×Dic197—21no—
135C2×Dic197—21no—
136C2×Dic197—21no—
137C2×Dic197—21no—
138C2×Dic197—21no—
139C2×Dic197—21no—
140C2×Dic197—21no—
141C2×Dic197—21no—
142C2×Dic197—21no—
143C2×Dic197—21no—
144C2×Dic197—21no—
145C2×Dic197—21no—
146C2×Dic197—21no—
147C2×Dic197—21no—
148C2×Dic197—44no—
149C2×Dic197—44no—
150C2×Dic197C22×C412yes—
151C2×Dic197C22×C412yes—
152C2×Dic197C22×C412yes—
153C2×Dic197C2×D412yes—
154C2×Dic197C2×D412yes—
155C4×D197—44no—
156C23×C197—21no—
157C23×C197—21no—
158C23×C197—21no—
159C23×C197—21no—
160C23×C197—21no—
161C23×C197—21no—
162C23×C197—21no—
163C23×C197—21no—
164C23×C197—28no—
165C23×C197—28no—
166C23×C197—21no—
167C23×C197—21no—
168C23×C197—21no—
169C23×C197—21no—
170C23×C197—21no—
171C23×C197—21no—
172C23×C197—44no—
173C23×C197—44no—
174C23×C197—44no—
175C23×C197—44no—
176C23×C197—44no—
177C23×C197C22×C412yes—
178C23×C197C22×C412yes—
179C23×C197C22×C412yes—
180C23×C197C2×D412yes—
181C23×C197C2×D412yes—
182C23×C197#4112yes—
183C23×C197#4112yes—
184C23×C197C2411yescharacteristic
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×C197—44no—
196C2×C4×C197—44no—
197C2×C4×C197C2×D412yes—
198C2×C4×C197C2×D412yes—
199C22×C197—22no—
200C22×C197—22no—
201C22×C197—22no—
202C22×C197—22no—
203C22×C197—22no—
204C22×C197—22no—
205C22×C197—22no—
206C22×C197—22no—
207C22×C197—22no—
208C22×C197—22no—
209C22×C197—22no—
210C22×C197—22no—
211C22×C197—22no—
212C22×C197—22no—
213C22×C197—22no—
214C22×C197—416no—
215C22×C197—41no—
216C22×C197—41no—
217C22×C197—41no—
218C22×C197—41no—
219C22×C197C4×D414yescomplemented
220C22×C197C4×D414yescomplemented
221C22×C197C23×C411yescharacteristic
222C22×C197C22×D411yescharacteristic
223C22×C197C22×D411yescharacteristic
224C22×C197C2×#4111yescharacteristic
225C22×C197C2×#4111yescharacteristic
226C22×C197H(5,𝔽2)11yescharacteristic
227C22×C197H(5,𝔽2)11yescharacteristic
228C2×D197—22no—
229C2×D197—22no—
230C2×D197—22no—
231C2×D197—41no—
232C2×D197—41no—
233C2×D197—41no—
234C2×D197—41no—
235C4×C197—22no—
236C4×C197—22no—
237C4×C197—41no—
238C4×C197—41no—
239Dic197—22no—
240Dic197—22no—
241Dic197—22no—
242Dic197—22no—
243Dic197—22no—
244Dic197—41no—
245Dic197—41no—
246C2×C197—24no—
247C2×C197—24no—
248C2×C197—42no—
249C2×C197C2×C4×D411yescharacteristic
250C2×C197C2×C4×D411yescharacteristic
251C2×C197#51811yescharacteristic
252C2×C197D4211yescharacteristic
253C2×C197#54611yescharacteristic
254C2×C197#54611yescharacteristic
255C2×C197#55111yescharacteristic
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⋊C4D19711yescharacteristic, complemented
264#390—1971no—
265#388—1971no—
266#388—1971no—
267#386—1971no—
268#386—1971no—
269#386—1971no—
270#385—1971no—
271#385—1971no—
272C4×C22⋊C4—1971no—
273C4×C22⋊C4—1971no—
274C25C2×D19711yescharacteristic
275C22×D4—1971no—
276C22×D4—1971no—
277C22×D4—1971no—
278C23×C4—1971no—
279C23×C4—1971no—
280C23×C4—1971no—
281C23×C4C2×D19711yescharacteristic
282C23×C4C2×D19711yescharacteristic
283#120—1978no—
284C4×D4—3944no—
285C4×D4—3944no—
286C2×C4⋊C4—1971no—
287C2×C4⋊C4—1971no—
288C2×C4⋊C4—1971no—
289C2×C22⋊C4—24no—
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⋊C4C2×D19711yescharacteristic
302C2×C22⋊C4C2×D19711yescharacteristic
303C2×C22⋊C4C2×D19711yescharacteristic
304C2×C22⋊C4C2×D19711yescharacteristic
305C2×C42—1971no—
306C2×C42—1971no—
307C2×C42—1971no—
308#95—1971no—
309#95—1971no—
310#95—1971no—
311#95—1971no—
312#95—1971no—
313#95—1971no—
314C24—22no—
315C24—22no—
316C24—22no—
317C24—22no—
318C24—41no—
319C24—41no—
320C24—1971no—
321C24C4×D19714yescomplemented
322C24C22×D19711yescharacteristic
323C24C22×D19711yescharacteristic
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—22no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—7881no—
341C22×C4—7881no—
342C22×C4—7881no—
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×C4—1971no—
355C22×C4—1971no—
356C22×C4C22×D19711yescharacteristic
357C22×C4C22×D19711yescharacteristic
358C22×C4C22×D19711yescharacteristic
359C22×C4C22×D19711yescharacteristic
360C4⋊C4—7881no—
361C4⋊C4—3942no—
362C4⋊C4—3942no—
363C22⋊C4—22no—
364C22⋊C4—22no—
365C22⋊C4—22no—
366C22⋊C4—7881no—
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⋊C4C22⋊D19714yescomplemented
377C42—7881no—
378C42—3942no—
379C42—3942no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—28no—
389C23—28no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—44no—
397C23—44no—
398C23—44no—
399C23—44no—
400C23—44no—
401C23—3941no—
402C23—3941no—
403C23—3941no—
404C23—3941no—
405C23—3941no—
406C23—3941no—
407C23—1972no—
408C23C2×C4×D19712yes—
409C23C2×C4×D19712yes—
410C23C2×C4×D19712yes—
411C23D4×D19712yes—
412C23C2×C22⋊D19712yes—
413C23#1569135712yes—
414C23#1569135812yes—
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—44no—
430C2×C4—44no—
431C2×C4—7884no—
432C2×C4—7884no—
433C2×C4—7884no—
434C2×C4—3941no—
435C2×C4—3941no—
436C2×C4—3941no—
437C2×C4—3941no—
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×C4—1972no—
454C2×C4—1972no—
455C2×C4C2×C22⋊D19712yes—
456C2×C4C2×C22⋊D19712yes—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—416no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—7881no—
478C22—7881no—
479C22—7881no—
480C22—7881no—
481C22—3942no—
482C22—3942no—
483C22—3942no—
484C22#2026328614yescomplemented
485C22C4×C22⋊D19714yescomplemented
486C22C22×C4×D19711yescharacteristic
487C22C2×D4×D19711yescharacteristic
488C22C22×C22⋊D19711yescharacteristic
489C22C2×#1569135711yescharacteristic
490C22C2×#1569135811yescharacteristic
491C22#2026337111yescharacteristic
492C22#2026337111yescharacteristic
493C4—22no—
494C4—22no—
495C4—41no—
496C4—41no—
497C4—7881no—
498C4—7881no—
499C4—3942no—
500C4—3942no—
501C4—3942no—
502C4—3942no—
503C4—3942no—
504C2—24no—
505C2—24no—
506C2—42no—
507C2—7882no—
508C2C2×#2026328611yescharacteristic
509C2C2×C4×C22⋊D19711yescharacteristic
510C2#2716155211yescharacteristic
511C2D4×C22⋊D19711yescharacteristic
512C2#2716171911yescharacteristic
513C2#2716172311yescharacteristic
514C2#2716175511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D197
commutatorC22×C197C23×C4
FittingC22×C22⋊C4×C197C2
socle, cosocleC23×C197C24
prime radicalC2×#20263295C2
hypercenterC22×C22⋊C4D197
hypocenterC197#2358
LCSG ▷ C22×C197 ▷ C197
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D197C2semidirect
3. C2×#20263286C2semidirect
4. C22×C22⋊Dic197C2semidirect
5. #27160979C2semidirect
6. #27160973C2semidirect
7. #27160967C2semidirect
8. #27160958C2semidirect
9. #27160956C2semidirect
10. #27160955C2semidirect
11. #27160952C2semidirect
12. #27160951C2semidirect
13. C22⋊C4×Dic197C2semidirect
14. C4×C22⋊Dic197C2semidirect
15. C22×C22⋊C4×C197C2semidirect
21. C22×C4×D197C22semidirect
22. #20263295C4semidirect
25. C2×#15691337C22semidirect
26. C2×#15691337C22semidirect
27. C2×#15691337C22semidirect
36. C2×#15691335C22semidirect
37. C2×#15691335C22semidirect
38. C2×#15691335C22semidirect
39. C2×C4×Dic197C22semidirect
40. C2×C4×Dic197C22semidirect
41. C2×C4×Dic197C22semidirect
42. #20263233C22semidirect
43. #20263233C22semidirect
44. #20263233C22semidirect
45. #20263233C22semidirect
46. #20263233C22semidirect
47. #20263233C22semidirect
63. C2×C22⋊D197C2×C4semidirect
64. C2×C22⋊D197C2×C4semidirect
65. C2×C22⋊D197C2×C4semidirect
88. C22⋊Dic197C2×C4semidirect
89. C22⋊Dic197C2×C4semidirect
90. C22⋊Dic197C2×C4semidirect
91. C22⋊Dic197D4semidirect
104. C24×C197C2×C4semidirect
121. C22⋊C4×C197D4semidirect
219. C22×C197C4×D4semidirect
220. C22×C197C4×D4semidirect
257. C197#2358semidirect
263. C22×C22⋊C4D197semidirect
321. C24C4×D197semidirect
376. C22⋊C4C22⋊D197semidirect
484. C22#20263286semidirect
485. C22C4×C22⋊D197semidirect

Maximal subgroups

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

subgroupindex
C2×#202632952
C2×C4×C22⋊D1972
C2×#202632862
C22×C22⋊Dic1972
#271609792
#271609732
#271609672
#271609582
#271609562
#271609552
#271609522
#271609512
C22⋊C4×Dic1972
C4×C22⋊Dic1972
C22×C22⋊C4×C1972
#2358197