Finite Group Databasebeta

← #36524497

Subgroups — #36524497

Order 22784.

596 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26030332C211yesmaximal, characteristic
2#26029397C211yesmaximal, characteristic
3#26029333C211yesmaximal, characteristic, complemented
4#26019127C211yesmaximal, characteristic, complemented
5#26019120C211yesmaximal, characteristic, complemented
6#26019099C211yesmaximal, characteristic, complemented
7#26019092C211yesmaximal, characteristic, complemented
8#26019068C211yesmaximal, characteristic, complemented
9#26019067C211yesmaximal, characteristic, complemented
10#26019056C211yesmaximal, characteristic, complemented
11#26019049C211yesmaximal, characteristic, complemented
12#26019043C211yesmaximal, characteristic, complemented
13#26018661C211yesmaximal, characteristic
14#26018552C211yesmaximal, characteristic, complemented
15C89×#1487C211yesmaximal, characteristic, complemented
16#19152132C2211yescharacteristic
17#19151928C411yescharacteristic
18#19151911C411yescharacteristic
19#19151905C411yescharacteristic
20#19151905C411yescharacteristic
21#19151820C411yescharacteristic
22#19151820C411yescharacteristic
23#19151754C411yescharacteristic
24#19151737C411yescharacteristic
25#19151725C2211yescharacteristic
26Q8×#10623494—21no—
27Q8×#10623494—21no—
28Q8×#10623494C2211yescharacteristic
29Q8×#10623494C2211yescharacteristic
30Q8×#10623494C2211yescharacteristic
31Q8×#10623494C2211yescharacteristic
32#19151722C2211yescharacteristic
33C4×#13083564C2211yescharacteristic
34C4×#13083564C2211yescharacteristic
35C4×#13083561C2211yescharacteristic
36C4×#13083561C2211yescharacteristic
37#19151675C2211yescharacteristic
38#19151674—21no—
39#19151667—21no—
40#19151663C2211yescharacteristic
41#19151658—21no—
42#19151651—21no—
43#19151651—21no—
44#19151651C2211yescharacteristic
45#19151651C2211yescharacteristic
46#19151651C2211yescharacteristic
47#19151623—21no—
48#19151350C2211yescharacteristic
49#19151350C2211yescharacteristic
50#19151348C2211yescharacteristic
51#19151348C2211yescharacteristic
52#19151330C2211yescharacteristic
53#19151330C2211yescharacteristic
54#19151327C2211yescharacteristic
55#19151327C2211yescharacteristic
56#19151320C2211yescharacteristic
57#19151320C2211yescharacteristic
58#19151319C2211yescharacteristic
59C8×#10623494C2211yescharacteristic
60C2×C4×Q8×C89C2211yescharacteristic
61#431×C89C2211yescharacteristic
62#423×C89C2211yescharacteristic
63Q8⋊C8×C89C2211yescharacteristic
64Q8⋊C8×C89C2211yescharacteristic
65Q8⋊C8×C89C2211yescharacteristic
66Q8⋊C8×C89C2211yescharacteristic
67C2×#13083564C2×C411yescharacteristic
68C2×#13083564C2×C411yescharacteristic
69C2×#13083561C2×C411yescharacteristic
70C2×#13083561C2×C411yescharacteristic
71C2×#13083557—21no—
72C2×C8×D89—21no—
73#15668006—21no—
74#15668004C2×C411yescharacteristic
75#15668000C2×C411yescharacteristic
76#15667995—21no—
77#15667991C2×C411yescharacteristic
78#15667986C2×C411yescharacteristic
79C4×Dic178C2311yescharacteristic
80C4×D356C2311yescharacteristic
81C4×C22⋊D89—21no—
82#15667969—21no—
83#15667945C2×C411yescharacteristic
84#15667945C2×C411yescharacteristic
85#15667943C2×C411yescharacteristic
86#15667943C2×C411yescharacteristic
87C4⋊#10623494—21no—
88C4⋊#10623494—21no—
89C4⋊#10623494—21no—
90C4⋊#10623494—21no—
91C4⋊#10623494C2×C411yescharacteristic
92C4⋊#10623494C2×C411yescharacteristic
93C4⋊#10623494D411yescharacteristic
94C4⋊#10623494D411yescharacteristic
95C4⋊#10623494C2311yescharacteristic
96C4⋊#10623494C2311yescharacteristic
97#15667941—21no—
98#15667941—21no—
99#15667936C2×C411yescharacteristic
100#15667936C2×C411yescharacteristic
101#15667936C2×C411yescharacteristic
102#15667936C2×C411yescharacteristic
103#15667935C2×C411yescharacteristic
104#15667935C2×C411yescharacteristic
105#15667932C2×C411yescharacteristic
106#15667932C2×C411yescharacteristic
107C22⋊#10623494—21no—
108C22⋊#10623494C2×C411yescharacteristic
109C22⋊#10623494C2×C411yescharacteristic
110C22⋊#10623494D411yescharacteristic
111C22⋊#10623494D411yescharacteristic
112#15667924—21no—
113#15667924—21no—
114#15667922—21no—
115C4×#10623494—21no—
116C4×#10623494—21no—
117C4×#10623494—21no—
118C4×#10623494—21no—
119C4×#10623494—21no—
120C4×#10623494C2311yescharacteristic
121C4×#10623494C2311yescharacteristic
122C4×#10623494C2311yescharacteristic
123C4×#10623494C2311yescharacteristic
124C8×Dic89—21no—
125C22×Q8×C89C2×C411yescharacteristic
126C2×M16×C89—21no—
127C4×Q8×C89—21no—
128C4×Q8×C89—21no—
129C4×Q8×C89C2×C411yescharacteristic
130C4×Q8×C89C2×C411yescharacteristic
131C4×Q8×C89C2311yescharacteristic
132C4×Q8×C89C2311yescharacteristic
133C2×C4⋊C4×C89—21no—
134C2×C4⋊C4×C89C2×C411yescharacteristic
135C2×C42×C89—21no—
136C2×C42×C89C2311yescharacteristic
137C4⋊C8×C89—21no—
138C4⋊C8×C89—21no—
139C4⋊C8×C89C2311yescharacteristic
140C4⋊C8×C89C2311yescharacteristic
141C22⋊C8×C89—21no—
142C4×C8×C89C2311yescharacteristic
143C4×C8×C89C2311yescharacteristic
144C2×Dic178C22×C411yescharacteristic
145C2×D356C22×C411yescharacteristic
146C2×C22⋊D89—21no—
147C2×C4×D89—21no—
148#13083564—22no—
149#13083564—22no—
150#13083561—22no—
151#13083561—22no—
152#13083557—81no—
153C2×#10623494—21no—
154C2×#10623494—21no—
155C2×#10623494C22×C411yescharacteristic
156C2×#10623494C22×C411yescharacteristic
157C2×#10623494C22×C411yescharacteristic
158C2×#10623494C22×C411yescharacteristic
159C2×#10623494C2×D411yescharacteristic
160C2×#10623494C2×D411yescharacteristic
161C2×#10623494C2×D411yescharacteristic
162C2×#10623494C2×D411yescharacteristic
163C8×D89—81no—
164C4⋊Dic89C22×C411yescharacteristic
165C4⋊Dic89C22×C411yescharacteristic
166#13083553—21no—
167#13083553—21no—
168#13083553—21no—
169#13083553—21no—
170C22⋊Dic89—21no—
171#13083551—21no—
172#13083551—21no—
173#13083551—21no—
174#13083551—21no—
175C4×Dic89—21no—
176C2×Q8×C89—22no—
177C2×Q8×C89—22no—
178C2×Q8×C89—21no—
179C2×Q8×C89C22×C411yescharacteristic
180C2×Q8×C89C22×C411yescharacteristic
181C22×C4×C89—21no—
182C22×C4×C89—21no—
183C22×C4×C89C22×C411yescharacteristic
184C22×C4×C89C2×D411yescharacteristic
185C22×C4×C89#4111yescharacteristic
186M16×C89—81no—
187C2×C8×C89—21no—
188C2×C8×C89—21no—
189C2×C8×C89—21no—
190C2×C8×C89—21no—
191C4⋊C4×C89—21no—
192C4⋊C4×C89—21no—
193C4⋊C4×C89—21no—
194C4⋊C4×C89—21no—
195C4⋊C4×C89—21no—
196C4⋊C4×C89C22×C411yescharacteristic
197C4⋊C4×C89C22×C411yescharacteristic
198C42×C89—21no—
199C42×C89—21no—
200C42×C89—21no—
201C42×C89—21no—
202C42×C89C22×C411yescharacteristic
203C42×C89C2×D411yescharacteristic
204C42×C89#4111yescharacteristic
205C42×C89C2411yescharacteristic
206C23×C89C4×D411yescharacteristic
207C22×D89—21no—
208Q8×C89—24no—
209Q8×C89—21no—
210Q8×C89—21no—
211Q8×C89#13112yes—
212Q8×C89#13112yes—
213C2×C4×C89—24no—
214C2×C4×C89—21no—
215C2×C4×C89—21no—
216C2×C4×C89—21no—
217C2×C4×C89—21no—
218C2×C4×C89—21no—
219C2×C4×C89—21no—
220C2×C4×C89—21no—
221C2×C4×C89—21no—
222C2×C4×C89—41no—
223C2×C4×C89—81no—
224C2×C4×C89C4×D411yescharacteristic
225C2×C4×C89C4×D411yescharacteristic
226C2×C4×C89C4×D411yescharacteristic
227C2×C4×C89C23×C411yescharacteristic
228C2×C4×C89C22×D411yescharacteristic
229C2×C4×C89C2×#4111yescharacteristic
230C22⋊D89—81no—
231C2×Dic89—21no—
232C2×Dic89—21no—
233C2×Dic89—21no—
234D356—21no—
235D356—21no—
236C4×D89—41no—
237C4×D89—41no—
238Dic178—21no—
239Dic178—21no—
240C8×C89—41no—
241C8×C89—41no—
242C8×C89—41no—
243C8×C89—41no—
244#10623494—21no—
245#10623494—21no—
246#10623494—21no—
247#10623494—21no—
248#10623494—21no—
249#10623494—21no—
250#10623494—21no—
251#10623494—21no—
252#10623494—41no—
253#10623494—41no—
254C22×C89—21no—
255C22×C89—21no—
256C22×C89—21no—
257C22×C89C2×C4×D411yescharacteristic
258C2×D89—41no—
259C2×D89—41no—
260C2×D89—41no—
261C4×C89—22no—
262C4×C89—22no—
263C4×C89—21no—
264C4×C89—21no—
265C4×C89—21no—
266C4×C89—21no—
267C4×C89—41no—
268C4×C89—41no—
269C4×C89C2×#13111yescharacteristic
270C4×C89C2×#13111yescharacteristic
271C4×C89#57711yescharacteristic
272C4×C89#57811yescharacteristic
273Dic89—41no—
274Dic89—41no—
275Dic89—41no—
276#17919—891nomaximal, Sylow
277C2×C89—41no—
278C2×C89#292911yescharacteristic
279C2×C89#294711yescharacteristic
280C2×C89#295711yescharacteristic
281D89—81no—
282#3108—891no—
283#2993—891no—
284#2982—891no—
285#1588—891no—
286#1587—891no—
287#1584—891no—
288#1583—891no—
289#1579—891no—
290SD16⋊C8—891no—
291C8×Q16—891no—
292C8×SD16—891no—
293#1574—891no—
294#1487D8911yescharacteristic, complemented
295#1482—891no—
296#1454—891no—
297C89#1791911yescharacteristic, complemented, Sylow
298#529—891no—
299C2×C4×Q8C2×D8911yescharacteristic
300#483—891no—
301#480—891no—
302#479—891no—
303#479—891no—
304#464—891no—
305#464—891no—
306#451—891no—
307#448—891no—
308#446—891no—
309C8×Q8—891no—
310C8×Q8—891no—
311C8×Q8—891no—
312C8×Q8—891no—
313C8×Q8—891no—
314C8×Q8—1781no—
315C8×Q8—1781no—
316C4×Q16—891no—
317C4×Q16—891no—
318C4×SD16—891no—
319C4×SD16—891no—
320#436—891no—
321#436—1781no—
322#435—1781no—
323C8×D4—891no—
324#433—1781no—
325#431—891no—
326#431—891no—
327#431—891no—
328#431—1781no—
329#431—1781no—
330#431C2×D8911yescharacteristic
331#424—1781no—
332#423C2×D8911yescharacteristic
333#335—891no—
334#335—891no—
335#334—891no—
336#334—891no—
337Q8⋊C8—891no—
338Q8⋊C8—891no—
339Q8⋊C8C2×D8911yescharacteristic
340Q8⋊C8C2×D8911yescharacteristic
341Q8⋊C8C2×D8911yescharacteristic
342Q8⋊C8C2×D8911yescharacteristic
343D4⋊C8—891no—
344D4⋊C8—891no—
345#322—891no—
346#322—891no—
347#322—891no—
348C82—891no—
349C22×Q8C4×D8911yescharacteristic
350C2×Q16—891no—
351C2×Q16—891no—
352C2×SD16—891no—
353C2×SD16—891no—
354C2×M16—21no—
355C2×M16—1781no—
356C22×C8—1781no—
357#126—1781no—
358#125—891no—
359#124—891no—
360#123—1781no—
361C22⋊Q8—891no—
362#121—891no—
363C4×Q8—21no—
364C4×Q8—21no—
365C4×Q8—891no—
366C4×Q8C4×D8911yescharacteristic
367C4×Q8C4×D8911yescharacteristic
368C4×Q8C22×D8911yescharacteristic
369C4×Q8C22×D8911yescharacteristic
370C4×D4—891no—
371C4×D4—1781no—
372#117—1781no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4C4×D8911yescharacteristic
375C2×C42—21no—
376C2×C42C22×D8911yescharacteristic
377#107—891no—
378#107—891no—
379#106—891no—
380#106—891no—
381C4⋊C8—21no—
382C4⋊C8—21no—
383C4⋊C8—891no—
384C4⋊C8—891no—
385C4⋊C8—891no—
386C4⋊C8—891no—
387C4⋊C8—891no—
388C4⋊C8—891no—
389C4⋊C8—1781no—
390C4⋊C8—1781no—
391C4⋊C8—1781no—
392C4⋊C8—1781no—
393C4⋊C8—1781no—
394C4⋊C8—1781no—
395C4⋊C8C22×D8911yescharacteristic
396C4⋊C8C22×D8911yescharacteristic
397#103—891no—
398#103—891no—
399#103—891no—
400#103—891no—
401#103—891no—
402#103—891no—
403#102—891no—
404#102—891no—
405C22⋊C8—21no—
406C22⋊C8—891no—
407C22⋊C8—891no—
408C22⋊C8—891no—
409C22⋊C8—891no—
410C22⋊C8—1781no—
411C22⋊C8—1781no—
412C22⋊C8—1781no—
413#97—1781no—
414C4×C8—891no—
415C4×C8—891no—
416C4×C8—891no—
417C4×C8—891no—
418C4×C8—1781no—
419C4×C8—1781no—
420C4×C8—1781no—
421C4×C8—1781no—
422C4×C8—1781no—
423C4×C8—1781no—
424C4×C8C22×D8911yescharacteristic
425C4×C8C22×D8911yescharacteristic
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—21no—
429C2×Q8—891no—
430C2×Q8C2×C4×D8911yescharacteristic
431C2×Q8C2×C4×D8911yescharacteristic
432C2×D4—891no—
433C2×D4—1781no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—1781no—
437C22×C4C2×C4×D8911yescharacteristic
438C22×C4D4×D8911yescharacteristic
439C22×C4#1308357311yescharacteristic
440Q16—1782no—
441Q16—1782no—
442SD16—1782no—
443SD16—1782no—
444M16—81no—
445M16—7121no—
446C2×C8—21no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—891no—
451C2×C8—891no—
452C2×C8—891no—
453C2×C8—891no—
454C2×C8—891no—
455C2×C8—891no—
456C2×C8—891no—
457C2×C8—891no—
458C2×C8—1781no—
459C2×C8—1781no—
460C2×C8—7121no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—891no—
467C4⋊C4—891no—
468C4⋊C4—1781no—
469C4⋊C4—1781no—
470C4⋊C4—1781no—
471C4⋊C4—1781no—
472C4⋊C4C2×C4×D8911yescharacteristic
473C4⋊C4C2×C4×D8911yescharacteristic
474C22⋊C4—1781no—
475C22⋊C4—1781no—
476C22⋊C4—1781no—
477C22⋊C4—1781no—
478C22⋊C4—1781no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—1781no—
484C42C2×C4×D8911yescharacteristic
485C42D4×D8911yescharacteristic
486C42#1308357311yescharacteristic
487C42C23×D8911yescharacteristic
488C23—1781no—
489C23#1566797211yescharacteristic
490Q8—24no—
491Q8—21no—
492Q8—21no—
493Q8—1781no—
494Q8—1781no—
495Q8#1566802012yes—
496Q8#1566802012yes—
497D4—1781no—
498D4—1781no—
499D4—7121no—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—41no—
510C2×C4—81no—
511C2×C4—3561no—
512C2×C4—3561no—
513C2×C4—1781no—
514C2×C4—1781no—
515C2×C4—1781no—
516C2×C4#1566797211yescharacteristic
517C2×C4#1566797211yescharacteristic
518C2×C4#1566797211yescharacteristic
519C2×C4C22×C4×D8911yescharacteristic
520C2×C4C2×D4×D8911yescharacteristic
521C2×C4C2×#1308357311yescharacteristic
522C8—41no—
523C8—41no—
524C8—41no—
525C8—41no—
526C8—3561no—
527C8—3561no—
528C8—1781no—
529C8—1781no—
530C8—1781no—
531C8—1781no—
532C8—1781no—
533C8—1781no—
534C8—1781no—
535C8—1781no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—3561no—
540C22—3561no—
541C22—3561no—
542C22C2×#1566797211yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—22no—
547C4—22no—
548C4—21no—
549C4—41no—
550C4—41no—
551C4—3561no—
552C4—3561no—
553C4—3561no—
554C4C2×#1566802011yescharacteristic
555C4C2×#1566802011yescharacteristic
556C4#1915240411yescharacteristic
557C4#1915241111yescharacteristic
558C2—41no—
559C2—7121no—
560C2#2602894211yescharacteristic
561C2#2602908911yescharacteristic
562C2#2602916511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#1487C2
socleC22×C89C2×C4×D4
prime radical#15667986C2×C4
cosocleC42×C89C24
hypercenter#1487D89
hypocenterC89#17919
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1487

Complements

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

NH ≅ G/Ntype
3. #26029333C2semidirect
4. #26019127C2semidirect
5. #26019120C2semidirect
6. #26019099C2semidirect
7. #26019092C2semidirect
8. #26019068C2semidirect
9. #26019067C2semidirect
10. #26019056C2semidirect
11. #26019049C2semidirect
12. #26019043C2semidirect
14. #26018552C2semidirect
15. C89×#1487C2semidirect
294. #1487D89semidirect
297. C89#17919semidirect

Maximal subgroups

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

subgroupindex
#260303322
#260293972
#260293332
#260191272
#260191202
#260190992
#260190922
#260190682
#260190672
#260190562
#260190492
#260190432
#260186612
#260185522
C89×#14872
#1791989