Finite Group Databasebeta

← #36526291

Subgroups — #36526291

Order 22784.

588 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26030852C211yesmaximal, characteristic
2#26029250C211yesmaximal, characteristic
3#26029201C211yesmaximal, characteristic, complemented
4#26019157C211yesmaximal, characteristic, complemented
5#26019146C211yesmaximal, characteristic, complemented
6#26019127C211yesmaximal, characteristic, complemented
7#26019126C211yesmaximal, characteristic, complemented
8#26019099C211yesmaximal, characteristic, complemented
9#26019091C211yesmaximal, characteristic, complemented
10#26019068C211yesmaximal, characteristic, complemented
11#26019049C211yesmaximal, characteristic, complemented
12#26019005C211yesmaximal, characteristic, complemented
13#26018666C211yesmaximal, characteristic
14#26018555C211yesmaximal, characteristic, complemented
15C89×#1485C211yesmaximal, characteristic, complemented
16#19152132C2211yescharacteristic
17#19152004C411yescharacteristic
18#19151990C411yescharacteristic
19#19151966C411yescharacteristic
20#19151960C411yescharacteristic
21#19151854C411yescharacteristic
22#19151854C411yescharacteristic
23#19151846C411yescharacteristic
24#19151846C411yescharacteristic
25#19151729—21no—
26#19151729—21no—
27#19151729C2211yescharacteristic
28#19151729C2211yescharacteristic
29#19151725C2211yescharacteristic
30Q8×#10623494C2211yescharacteristic
31Q8×#10623494C2211yescharacteristic
32#19151722C2211yescharacteristic
33C4×#13083564C2211yescharacteristic
34C4×#13083564C2211yescharacteristic
35C4×#13083561C2211yescharacteristic
36C4×#13083561C2211yescharacteristic
37#19151675C2211yescharacteristic
38#19151673—21no—
39#19151672—21no—
40#19151663C2211yescharacteristic
41#19151657—21no—
42#19151625—21no—
43C2×C4⋊#10623494C2211yescharacteristic
44#19151554—21no—
45#19151552—21no—
46C4×#13083558C2211yescharacteristic
47C4×#13083558C2211yescharacteristic
48#19151350C2211yescharacteristic
49#19151350C2211yescharacteristic
50#19151348C2211yescharacteristic
51#19151348C2211yescharacteristic
52#19151330C2211yescharacteristic
53#19151330C2211yescharacteristic
54#19151327C2211yescharacteristic
55#19151327C2211yescharacteristic
56#19151320C2211yescharacteristic
57#19151320C2211yescharacteristic
58#19151319C2211yescharacteristic
59C8×#10623494C2211yescharacteristic
60#533×C89C2211yescharacteristic
61#431×C89C2211yescharacteristic
62C4×M16×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×#13083558C2×C411yescharacteristic
72C2×#13083558C2×C411yescharacteristic
73C2×#13083558D411yescharacteristic
74C2×#13083558D411yescharacteristic
75C2×#13083557—21no—
76C22×#10623494—21no—
77C2×C8×D89—21no—
78#15668006—21no—
79#15668004C2×C411yescharacteristic
80#15668000C2×C411yescharacteristic
81#15667995—21no—
82#15667991C2×C411yescharacteristic
83#15667986C2×C411yescharacteristic
84C4×Dic178C2311yescharacteristic
85C4×D356C2311yescharacteristic
86C4×C22⋊D89—21no—
87#15667969—21no—
88#15667945C2×C411yescharacteristic
89#15667945C2×C411yescharacteristic
90#15667943C2×C411yescharacteristic
91#15667943C2×C411yescharacteristic
92C4⋊#10623494—21no—
93C4⋊#10623494—21no—
94C4⋊#10623494—21no—
95C4⋊#10623494—21no—
96C4⋊#10623494—21no—
97C4⋊#10623494C2311yescharacteristic
98C4⋊#10623494C2311yescharacteristic
99#15667941—21no—
100#15667941—21no—
101#15667936C2×C411yescharacteristic
102#15667936C2×C411yescharacteristic
103#15667936C2×C411yescharacteristic
104#15667936C2×C411yescharacteristic
105#15667935C2×C411yescharacteristic
106#15667935C2×C411yescharacteristic
107#15667932C2×C411yescharacteristic
108#15667932C2×C411yescharacteristic
109#15667924—21no—
110#15667924—21no—
111#15667923—21no—
112#15667923—21no—
113#15667923C2×C411yescharacteristic
114#15667923C2×C411yescharacteristic
115#15667923D411yescharacteristic
116#15667923D411yescharacteristic
117#15667922—21no—
118C4×#10623494—21no—
119C4×#10623494—21no—
120C4×#10623494C2311yescharacteristic
121C4×#10623494C2311yescharacteristic
122C4×#10623494C2311yescharacteristic
123C4×#10623494C2311yescharacteristic
124C8×Dic89—21no—
125C2×M16×C89—21no—
126C4⋊Q8×C89C2×C411yescharacteristic
127C4⋊Q8×C89C2×C411yescharacteristic
128#125×C89—21no—
129C22⋊Q8×C89—21no—
130C22⋊Q8×C89C2×C411yescharacteristic
131C22⋊Q8×C89C2×C411yescharacteristic
132C4×Q8×C89—21no—
133C4×Q8×C89C2311yescharacteristic
134C4×Q8×C89C2311yescharacteristic
135#117×C89—21no—
136C2×C42×C89C2311yescharacteristic
137C4⋊C8×C89C2311yescharacteristic
138C4⋊C8×C89C2311yescharacteristic
139C22⋊C8×C89—21no—
140#97×C89—21no—
141C4×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—41no—
150#13083561—22no—
151#13083561—41no—
152#13083558—22no—
153#13083558—22no—
154#13083558—41no—
155#13083558—41no—
156#13083557—42no—
157C2×#10623494—21no—
158C2×#10623494—21no—
159C2×#10623494—42no—
160C2×#10623494C22×C411yescharacteristic
161C2×#10623494C22×C411yescharacteristic
162C2×#10623494C22×C411yescharacteristic
163C2×#10623494C22×C411yescharacteristic
164C2×#10623494C2×D411yescharacteristic
165C2×#10623494C2×D411yescharacteristic
166C2×#10623494C2×D411yescharacteristic
167C2×#10623494C2×D411yescharacteristic
168C8×D89—42no—
169C4⋊Dic89C22×C411yescharacteristic
170C4⋊Dic89C22×C411yescharacteristic
171#13083553—21no—
172#13083553—21no—
173#13083553—21no—
174#13083553—21no—
175C22⋊Dic89—21no—
176#13083551—21no—
177#13083551—21no—
178#13083551—21no—
179#13083551—21no—
180C4×Dic89—21no—
181C2×Q8×C89—21no—
182C2×Q8×C89C22×C411yescharacteristic
183C2×Q8×C89C22×C411yescharacteristic
184C22×C4×C89C22×C411yescharacteristic
185C22×C4×C89C2×D411yescharacteristic
186C22×C4×C89#4111yescharacteristic
187M16×C89—42no—
188C2×C8×C89—21no—
189C2×C8×C89—21no—
190C2×C8×C89—21no—
191C2×C8×C89—21no—
192C4⋊C4×C89—21no—
193C4⋊C4×C89—21no—
194C4⋊C4×C89—21no—
195C4⋊C4×C89—21no—
196C4⋊C4×C89—21no—
197C4⋊C4×C89—21no—
198C4⋊C4×C89—21no—
199C4⋊C4×C89C22×C411yescharacteristic
200C4⋊C4×C89C22×C411yescharacteristic
201C22⋊C4×C89—21no—
202C22⋊C4×C89—21no—
203C42×C89—21no—
204C42×C89—21no—
205C42×C89C22×C411yescharacteristic
206C42×C89C2×D411yescharacteristic
207C42×C89#4111yescharacteristic
208C42×C89C2411yescharacteristic
209C23×C89C4×D411yescharacteristic
210C22×D89—21no—
211Q8×C89—21no—
212Q8×C89—21no—
213Q8×C89—21no—
214Q8×C89—21no—
215Q8×C89—81no—
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—21no—
223C2×C4×C89—21no—
224C2×C4×C89—41no—
225C2×C4×C89C4×D411yescharacteristic
226C2×C4×C89C4×D411yescharacteristic
227C2×C4×C89C4×D411yescharacteristic
228C2×C4×C89C23×C411yescharacteristic
229C2×C4×C89C22×D411yescharacteristic
230C2×C4×C89C2×#4111yescharacteristic
231C22⋊D89—81no—
232C2×Dic89—21no—
233C2×Dic89—21no—
234C2×Dic89—21no—
235D356—21no—
236D356—21no—
237C4×D89—41no—
238C4×D89—41no—
239Dic178—21no—
240Dic178—21no—
241C8×C89—22no—
242C8×C89—41no—
243C8×C89—41no—
244C8×C89—41no—
245#10623494—21no—
246#10623494—21no—
247#10623494—21no—
248#10623494—21no—
249#10623494—21no—
250#10623494—21no—
251#10623494—41no—
252#10623494—41no—
253#10623494#13112yes—
254#10623494#13712yes—
255C22×C89—21no—
256C22×C89—21no—
257C22×C89—21no—
258C22×C89C2×C4×D411yescharacteristic
259C2×D89—41no—
260C2×D89—41no—
261C2×D89—41no—
262C4×C89—21no—
263C4×C89—21no—
264C4×C89—21no—
265C4×C89—21no—
266C4×C89—41no—
267C4×C89—41no—
268C4×C89—41no—
269C4×C89—41no—
270C4×C89C2×#13111yescharacteristic
271C4×C89#56811yescharacteristic
272C4×C89C2×#13711yescharacteristic
273C4×C89#57511yescharacteristic
274Dic89—41no—
275Dic89—41no—
276Dic89—41no—
277#18077—891nomaximal, Sylow
278C2×C89—41no—
279C2×C89#293511yescharacteristic
280C2×C89C4×#13711yescharacteristic
281C2×C89#295611yescharacteristic
282D89—81no—
283#3172—891no—
284#2969—891no—
285#2962—891no—
286#1592—891no—
287#1591—891no—
288#1588—891no—
289#1587—891no—
290#1584—891no—
291#1582—891no—
292#1579—891no—
293C8×SD16—891no—
294#1565—891no—
295#1485D8911yescharacteristic, complemented
296#1483—891no—
297#1455—891no—
298C89#1807711yescharacteristic, complemented, Sylow
299#533C2×D8911yescharacteristic
300#529—891no—
301#501—891no—
302#497—891no—
303#491—891no—
304#489—891no—
305#470—891no—
306#470—891no—
307#468—891no—
308#468—891no—
309#446—891no—
310#446—891no—
311#446—891no—
312#446—1781no—
313#446—1781no—
314C8×Q8—891no—
315C8×Q8—891no—
316C8×Q8—891no—
317C4×Q16—891no—
318C4×Q16—891no—
319C4×SD16—891no—
320C4×SD16—891no—
321#436—891no—
322#436—1781no—
323#435—1781no—
324C8×D4—891no—
325#433—1781no—
326#431C2×D8911yescharacteristic
327#424—1781no—
328C2×C4⋊C8—891no—
329#405—1781no—
330#405—1781no—
331C4×M16—891no—
332C4×M16—891no—
333C4×M16C2×D8911yescharacteristic
334#335—891no—
335#335—891no—
336#334—891no—
337#334—891no—
338Q8⋊C8—891no—
339Q8⋊C8—891no—
340Q8⋊C8C2×D8911yescharacteristic
341Q8⋊C8C2×D8911yescharacteristic
342Q8⋊C8C2×D8911yescharacteristic
343Q8⋊C8C2×D8911yescharacteristic
344D4⋊C8—891no—
345D4⋊C8—891no—
346#322—891no—
347#322—891no—
348#322—891no—
349C82—891no—
350C2×Q16—891no—
351C2×Q16—891no—
352C2×SD16—891no—
353C2×SD16—891no—
354C2×M16—21no—
355C2×M16—891no—
356C2×M16—891no—
357C2×M16—891no—
358C2×M16—891no—
359C2×M16—1781no—
360C22×C8—1781no—
361C22×C8—1781no—
362C4⋊Q8C4×D8911yescharacteristic
363C4⋊Q8C4×D8911yescharacteristic
364#126—1781no—
365#125—21no—
366#125—891no—
367#124—891no—
368#123—1781no—
369C22⋊Q8—21no—
370C22⋊Q8—891no—
371C22⋊Q8C4×D8911yescharacteristic
372C22⋊Q8C4×D8911yescharacteristic
373#121—891no—
374C4×Q8—21no—
375C4×Q8—891no—
376C4×Q8C22×D8911yescharacteristic
377C4×Q8C22×D8911yescharacteristic
378C4×D4—891no—
379C4×D4—1781no—
380#117—21no—
381#117—1781no—
382C2×C42C22×D8911yescharacteristic
383#107—891no—
384#107—891no—
385#106—891no—
386#106—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⋊C8—1781no—
396C4⋊C8C22×D8911yescharacteristic
397C4⋊C8C22×D8911yescharacteristic
398#103—891no—
399#103—891no—
400#103—891no—
401#103—891no—
402#103—891no—
403#103—891no—
404#102—891no—
405#102—891no—
406C22⋊C8—21no—
407C22⋊C8—1781no—
408C22⋊C8—1781no—
409#97—21no—
410#97—891no—
411#97—891no—
412#97—891no—
413#97—891no—
414#97—1781no—
415#97—1781no—
416#97—1781no—
417C4×C8—21no—
418C4×C8—891no—
419C4×C8—891no—
420C4×C8—891no—
421C4×C8—891no—
422C4×C8—1781no—
423C4×C8—1781no—
424C4×C8—1781no—
425C4×C8C22×D8911yescharacteristic
426C4×C8C22×D8911yescharacteristic
427C2×Q8—21no—
428C2×Q8—891no—
429C2×Q8C2×C4×D8911yescharacteristic
430C2×Q8C2×C4×D8911yescharacteristic
431C2×D4—891no—
432C2×D4—1781no—
433C22×C4—1781no—
434C22×C4C2×C4×D8911yescharacteristic
435C22×C4D4×D8911yescharacteristic
436C22×C4#1308357311yescharacteristic
437Q16—3561no—
438Q16—1782no—
439SD16—3561no—
440SD16—1782no—
441M16—42no—
442M16—3562no—
443M16—3561no—
444M16—3561no—
445M16—1782no—
446M16—1782no—
447C2×C8—21no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—891no—
452C2×C8—891no—
453C2×C8—891no—
454C2×C8—891no—
455C2×C8—891no—
456C2×C8—891no—
457C2×C8—891no—
458C2×C8—891no—
459C2×C8—3562no—
460C2×C8—3562no—
461C2×C8—1781no—
462C2×C8—1781no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—891no—
471C4⋊C4—891no—
472C4⋊C4—1781no—
473C4⋊C4—1781no—
474C4⋊C4—1781no—
475C4⋊C4—1781no—
476C4⋊C4C2×C4×D8911yescharacteristic
477C4⋊C4C2×C4×D8911yescharacteristic
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—1781no—
481C22⋊C4—1781no—
482C22⋊C4—1781no—
483C22⋊C4—1781no—
484C22⋊C4—1781no—
485C42—21no—
486C42—21no—
487C42—1781no—
488C42C2×C4×D8911yescharacteristic
489C42D4×D8911yescharacteristic
490C42#1308357311yescharacteristic
491C42C23×D8911yescharacteristic
492C23—1781no—
493C23#1566797211yescharacteristic
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—21no—
498Q8—81no—
499Q8—1781no—
500Q8—1781no—
501D4—1781no—
502D4—1781no—
503D4—7121no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—41no—
513C2×C4—3561no—
514C2×C4—3561no—
515C2×C4—1781no—
516C2×C4—1781no—
517C2×C4—1781no—
518C2×C4#1566797211yescharacteristic
519C2×C4#1566797211yescharacteristic
520C2×C4#1566797211yescharacteristic
521C2×C4C22×C4×D8911yescharacteristic
522C2×C4C2×D4×D8911yescharacteristic
523C2×C4C2×#1308357311yescharacteristic
524C8—22no—
525C8—41no—
526C8—41no—
527C8—41no—
528C8—892no—
529C8—892no—
530C8—3561no—
531C8—3561no—
532C8—1781no—
533C8—1781no—
534C8—1781no—
535C8—1781no—
536C8—1781no—
537C8—1781no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—3561no—
542C22—3561no—
543C22—3561no—
544C22C2×#1566797211yescharacteristic
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—41no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—3561no—
554C4—3561no—
555C4—3561no—
556C4#131×D8911yescharacteristic
557C4#1915236311yescharacteristic
558C4#137×D8911yescharacteristic
559C4#1915239711yescharacteristic
560C2—41no—
561C2—7121no—
562C2#2602899211yescharacteristic
563C2#2602906711yescharacteristic
564C2#2602915811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#1485C2
socleC22×C89C2×C4×D4
prime radical#15667986C2×C4
cosocleC42×C89C24
hypercenter#1485D89
hypocenterC89#18077
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1485

Complements

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

NH ≅ G/Ntype
3. #26029201C2semidirect
4. #26019157C2semidirect
5. #26019146C2semidirect
6. #26019127C2semidirect
7. #26019126C2semidirect
8. #26019099C2semidirect
9. #26019091C2semidirect
10. #26019068C2semidirect
11. #26019049C2semidirect
12. #26019005C2semidirect
14. #26018555C2semidirect
15. C89×#1485C2semidirect
295. #1485D89semidirect
298. C89#18077semidirect

Maximal subgroups

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

subgroupindex
#260308522
#260292502
#260292012
#260191572
#260191462
#260191272
#260191262
#260190992
#260190912
#260190682
#260190492
#260190052
#260186662
#260185552
C89×#14852
#1807789