Finite Group Databasebeta

← #36526211

Subgroups — #36526211

Order 22784.

600 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26030816C211yesmaximal, characteristic
2#26029256C211yesmaximal, characteristic, complemented
3#26019147C211yesmaximal, characteristic, complemented
4#26019135C211yesmaximal, characteristic, complemented
5#26019095C211yesmaximal, characteristic, complemented
6#26019093C211yesmaximal, characteristic, complemented
7#26019088C211yesmaximal, characteristic, complemented
8#26019086C211yesmaximal, characteristic, complemented
9C8×#13083564C211yesmaximal, characteristic, complemented
10C8×#13083559C211yesmaximal, characteristic, complemented
11#26018998C211yesmaximal, characteristic, complemented
12#26018660C211yesmaximal, characteristic, complemented
13#26018658C211yesmaximal, characteristic, complemented
14M16×#10623494C211yesmaximal, characteristic, complemented
15C89×#2969C211yesmaximal, characteristic, complemented
16#19152132C2211yescharacteristic
17#19151997C411yescharacteristic
18#19151980C411yescharacteristic
19#19151955C411yescharacteristic
20#19151952C411yescharacteristic
21#19151837C411yescharacteristic
22#19151832C411yescharacteristic
23#19151829C411yescharacteristic
24#19151825C411yescharacteristic
25#19151728C2211yescharacteristic
26C8×Dic178C2211yescharacteristic
27C4×#13083564C2211yescharacteristic
28C4×#13083561C2211yescharacteristic
29C4×#13083562C2211yescharacteristic
30C4×#13083559C2211yescharacteristic
31#19151677C2211yescharacteristic
32#19151674—21no—
33#19151667—21no—
34C8×D356C2211yescharacteristic
35#19151660—21no—
36#19151651C2211yescharacteristic
37#19151651C2211yescharacteristic
38#19151555—21no—
39C2×C4×#10623494C2211yescharacteristic
40#19151350C2211yescharacteristic
41#19151350C2211yescharacteristic
42#19151348C2211yescharacteristic
43#19151348C2211yescharacteristic
44Q8⋊#10623494C2211yescharacteristic
45Q8⋊#10623494C2211yescharacteristic
46#19151328C2211yescharacteristic
47#19151328C2211yescharacteristic
48D4⋊#10623494C2211yescharacteristic
49D4⋊#10623494C2211yescharacteristic
50#19151325C2211yescharacteristic
51#19151325C2211yescharacteristic
52#19151319C2211yescharacteristic
53#19151319C2211yescharacteristic
54C8×#10623494C2211yescharacteristic
55C8×#10623494C2211yescharacteristic
56#529×C89C2211yescharacteristic
57#446×C89C2211yescharacteristic
58C8×Q8×C89C2211yescharacteristic
59#436×C89—21no—
60#436×C89C2211yescharacteristic
61#435×C89—21no—
62C8×D4×C89C2211yescharacteristic
63#433×C89—21no—
64#431×C89C2211yescharacteristic
65#405×C89—21no—
66C4×M16×C89C2211yescharacteristic
67C2×#13083564C2×C411yescharacteristic
68C2×#13083562C2×C411yescharacteristic
69C2×#13083561C2×C411yescharacteristic
70C2×#13083559C2×C411yescharacteristic
71C2×#13083557—21no—
72C22×#10623494C2×C411yescharacteristic
73C22×#10623494C2×C411yescharacteristic
74C2×C8×D89—21no—
75#15668006—21no—
76#15668004C2×C411yescharacteristic
77#15668000C2×C411yescharacteristic
78#15667995—21no—
79#15667991C2×C411yescharacteristic
80#15667986C2×C411yescharacteristic
81C4×Dic178C2311yescharacteristic
82C4×D356C2311yescharacteristic
83C4×C22⋊D89—21no—
84#15667969—21no—
85#15667945C2×C411yescharacteristic
86#15667945C2×C411yescharacteristic
87#15667943C2×C411yescharacteristic
88#15667943C2×C411yescharacteristic
89C4⋊#10623494—21no—
90C4⋊#10623494C2311yescharacteristic
91C4⋊#10623494C2311yescharacteristic
92#15667941—21no—
93#15667941—21no—
94#15667936C2×C411yescharacteristic
95#15667936C2×C411yescharacteristic
96#15667935C2×C411yescharacteristic
97#15667935C2×C411yescharacteristic
98#15667933C2×C411yescharacteristic
99#15667933C2×C411yescharacteristic
100#15667932C2×C411yescharacteristic
101#15667932C2×C411yescharacteristic
102C22⋊#10623494—21no—
103C22⋊#10623494—21no—
104#15667924—21no—
105#15667924—21no—
106#15667922—21no—
107C4×#10623494—21no—
108C4×#10623494C2×C411yescharacteristic
109C4×#10623494C2×C411yescharacteristic
110C4×#10623494C2311yescharacteristic
111C4×#10623494C2311yescharacteristic
112C4×#10623494C2311yescharacteristic
113C4×#10623494C2311yescharacteristic
114C8×Dic89—21no—
115C2×M16×C89—21no—
116C2×M16×C89D411yescharacteristic
117C2×M16×C89D411yescharacteristic
118C22×C8×C89—21no—
119#126×C89—21no—
120#125×C89C2×C411yescharacteristic
121#124×C89C2×C411yescharacteristic
122#123×C89—21no—
123C22⋊Q8×C89C2×C411yescharacteristic
124#121×C89C2×C411yescharacteristic
125C4×Q8×C89C2311yescharacteristic
126C4×D4×C89—21no—
127C4×D4×C89C2311yescharacteristic
128#117×C89—21no—
129C2×C42×C89C2311yescharacteristic
130C4⋊C8×C89—21no—
131C4⋊C8×C89—21no—
132C4⋊C8×C89C2311yescharacteristic
133C4⋊C8×C89C2311yescharacteristic
134C22⋊C8×C89—21no—
135C22⋊C8×C89—21no—
136C22⋊C8×C89—21no—
137#97×C89—21no—
138#97×C89D411yescharacteristic
139#97×C89D411yescharacteristic
140C4×C8×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#13083562—41no—
150#13083561—41no—
151#13083559—22no—
152#13083557—42no—
153C2×#10623494—22no—
154C2×#10623494—21no—
155C2×#10623494—21no—
156C2×#10623494—21no—
157C2×#10623494—21no—
158C2×#10623494—41no—
159C2×#10623494C22×C411yescharacteristic
160C2×#10623494C22×C411yescharacteristic
161C2×#10623494C22×C411yescharacteristic
162C2×#10623494C22×C411yescharacteristic
163C8×D89—42no—
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×C89C22×C411yescharacteristic
177C2×D4×C89—21no—
178C2×D4×C89C22×C411yescharacteristic
179C22×C4×C89—21no—
180C22×C4×C89C22×C411yescharacteristic
181C22×C4×C89C2×D411yescharacteristic
182C22×C4×C89#4111yescharacteristic
183M16×C89—22no—
184M16×C89—22no—
185M16×C89—42no—
186C2×C8×C89—21no—
187C2×C8×C89—21no—
188C2×C8×C89—42no—
189C2×C8×C89C2×D411yescharacteristic
190C2×C8×C89C2×D411yescharacteristic
191C2×C8×C89C2×D411yescharacteristic
192C2×C8×C89C2×D411yescharacteristic
193C4⋊C4×C89—21no—
194C4⋊C4×C89—21no—
195C4⋊C4×C89—21no—
196C4⋊C4×C89—21no—
197C4⋊C4×C89C22×C411yescharacteristic
198C4⋊C4×C89C22×C411yescharacteristic
199C22⋊C4×C89—21no—
200C22⋊C4×C89—21no—
201C22⋊C4×C89—21no—
202C22⋊C4×C89—21no—
203C22⋊C4×C89—21no—
204C42×C89—21no—
205C42×C89C22×C411yescharacteristic
206C42×C89C2×D411yescharacteristic
207C42×C89#4111yescharacteristic
208C42×C89C2411yescharacteristic
209C23×C89—21no—
210C23×C89C4×D411yescharacteristic
211C22×D89—21no—
212Q8×C89—21no—
213Q8×C89—21no—
214D4×C89—21no—
215D4×C89—21no—
216D4×C89—81no—
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×C89—41no—
226C2×C4×C89—41no—
227C2×C4×C89C4×D411yescharacteristic
228C2×C4×C89C4×D411yescharacteristic
229C2×C4×C89C4×D411yescharacteristic
230C2×C4×C89C23×C411yescharacteristic
231C2×C4×C89C22×D411yescharacteristic
232C2×C4×C89C2×#4111yescharacteristic
233C22⋊D89—81no—
234C2×Dic89—21no—
235C2×Dic89—21no—
236C2×Dic89—21no—
237D356—21no—
238D356—21no—
239C4×D89—41no—
240C4×D89—41no—
241Dic178—21no—
242Dic178—21no—
243C8×C89—21no—
244C8×C89—21no—
245C8×C89—41no—
246C8×C89—41no—
247C8×C89#13512yes—
248C8×C89#13512yes—
249#10623494—21no—
250#10623494—21no—
251#10623494—21no—
252#10623494—41no—
253#10623494—41no—
254#10623494—41no—
255#10623494—41no—
256#10623494#13112yes—
257C22×C89—21no—
258C22×C89—21no—
259C22×C89—21no—
260C22×C89—41no—
261C22×C89—41no—
262C22×C89—41no—
263C22×C89C2×C4×D411yescharacteristic
264C2×D89—41no—
265C2×D89—41no—
266C2×D89—41no—
267C4×C89—21no—
268C4×C89—21no—
269C4×C89—21no—
270C4×C89—21no—
271C4×C89—41no—
272C4×C89—41no—
273C4×C89—41no—
274C4×C89C2×#13111yescharacteristic
275C4×C89#56811yescharacteristic
276C4×C89C2×#13511yescharacteristic
277C4×C89C2×#13511yescharacteristic
278Dic89—41no—
279Dic89—41no—
280Dic89—41no—
281#18069—891nomaximal, Sylow
282C2×C89—41no—
283C2×C89—81no—
284C2×C89#293511yescharacteristic
285C2×C89C4×#13511yescharacteristic
286C2×C89#295711yescharacteristic
287D89—81no—
288#3168—891no—
289#2969—891no—
290#2969D8911yescharacteristic, complemented
291#1591—891no—
292#1589—891no—
293#1583—891no—
294#1583—891no—
295#1582—891no—
296#1582—891no—
297C8×Q16—891no—
298C8×D8—891no—
299#1564—891no—
300#1482—891no—
301#1482—891no—
302C8×M16—891no—
303C89#1806911yescharacteristic, complemented, Sylow
304#529—891no—
305#529C2×D8911yescharacteristic
306#499—891no—
307#495—891no—
308#488—891no—
309#488—891no—
310#467—891no—
311#466—891no—
312#465—891no—
313#465—891no—
314#446—891no—
315#446C2×D8911yescharacteristic
316C8×Q8—891no—
317C8×Q8C2×D8911yescharacteristic
318C4×Q16—891no—
319C4×SD16—891no—
320C4×SD16—891no—
321C4×D8—891no—
322#436—21no—
323#436—891no—
324#436—1781no—
325#436C2×D8911yescharacteristic
326#435—21no—
327#435—1781no—
328C8×D4—891no—
329C8×D4C2×D8911yescharacteristic
330#433—21no—
331#433—1781no—
332#431—891no—
333#431—891no—
334#431C2×D8911yescharacteristic
335#405—21no—
336#405—1781no—
337C4×M16C2×D8911yescharacteristic
338C2×C4×C8—891no—
339#335—891no—
340#335—891no—
341#334—891no—
342#334—891no—
343Q8⋊C8—891no—
344Q8⋊C8—891no—
345Q8⋊C8—891no—
346Q8⋊C8—891no—
347D4⋊C8—891no—
348D4⋊C8—891no—
349D4⋊C8—891no—
350D4⋊C8—891no—
351#322—891no—
352#322—891no—
353C82—891no—
354C82—891no—
355C2×Q16—891no—
356C2×SD16—891no—
357C2×SD16—891no—
358C2×D8—891no—
359C2×M16—21no—
360C2×M16—1781no—
361C2×M16C22⋊D8911yescharacteristic
362C2×M16C22⋊D8911yescharacteristic
363C22×C8—21no—
364C22×C8—891no—
365C22×C8—891no—
366C22×C8—1781no—
367#126—21no—
368#126—1781no—
369#125—891no—
370#125C4×D8911yescharacteristic
371#124—891no—
372#124C4×D8911yescharacteristic
373#123—21no—
374#123—1781no—
375C22⋊Q8—891no—
376C22⋊Q8C4×D8911yescharacteristic
377#121—891no—
378#121C4×D8911yescharacteristic
379C4×Q8—891no—
380C4×Q8C22×D8911yescharacteristic
381C4×D4—21no—
382C4×D4—891no—
383C4×D4—1781no—
384C4×D4C22×D8911yescharacteristic
385#117—21no—
386#117—1781no—
387C2×C42C22×D8911yescharacteristic
388#107—891no—
389#107—891no—
390#106—891no—
391#106—891no—
392C4⋊C8—21no—
393C4⋊C8—21no—
394C4⋊C8—891no—
395C4⋊C8—891no—
396C4⋊C8—1781no—
397C4⋊C8—1781no—
398C4⋊C8—1781no—
399C4⋊C8C22×D8911yescharacteristic
400C4⋊C8C22×D8911yescharacteristic
401#103—891no—
402#103—891no—
403#103—891no—
404#103—891no—
405#102—891no—
406#102—891no—
407#102—891no—
408#102—891no—
409C22⋊C8—21no—
410C22⋊C8—21no—
411C22⋊C8—21no—
412C22⋊C8—1781no—
413C22⋊C8—1781no—
414C22⋊C8—1781no—
415C22⋊C8—1781no—
416#97—21no—
417#97—1781no—
418#97C22⋊D8911yescharacteristic
419#97C22⋊D8911yescharacteristic
420C4×C8—21no—
421C4×C8—21no—
422C4×C8—891no—
423C4×C8—891no—
424C4×C8—891no—
425C4×C8—891no—
426C4×C8—891no—
427C4×C8—891no—
428C4×C8—1781no—
429C4×C8—1781no—
430C4×C8C22×D8911yescharacteristic
431C4×C8C22×D8911yescharacteristic
432C2×Q8—891no—
433C2×Q8C2×C4×D8911yescharacteristic
434C2×D4—21no—
435C2×D4—891no—
436C2×D4—1781no—
437C2×D4C2×C4×D8911yescharacteristic
438C22×C4—21no—
439C22×C4—1781no—
440C22×C4C2×C4×D8911yescharacteristic
441C22×C4C2×C22⋊D8911yescharacteristic
442C22×C4#1308357411yescharacteristic
443Q16—1782no—
444SD16—3561no—
445SD16—3561no—
446D8—1782no—
447M16—22no—
448M16—22no—
449M16—42no—
450M16—3562no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—42no—
454C2×C8—891no—
455C2×C8—891no—
456C2×C8—891no—
457C2×C8—891no—
458C2×C8—3562no—
459C2×C8—3561no—
460C2×C8—1782no—
461C2×C8—1781no—
462C2×C8—1781no—
463C2×C8—1781no—
464C2×C8—1781no—
465C2×C8C2×C22⋊D8911yescharacteristic
466C2×C8C2×C22⋊D8911yescharacteristic
467C2×C8C2×C22⋊D8911yescharacteristic
468C2×C8C2×C22⋊D8911yescharacteristic
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—891no—
474C4⋊C4—891no—
475C4⋊C4—1781no—
476C4⋊C4—1781no—
477C4⋊C4—1781no—
478C4⋊C4—1781no—
479C4⋊C4C2×C4×D8911yescharacteristic
480C4⋊C4C2×C4×D8911yescharacteristic
481C22⋊C4—21no—
482C22⋊C4—21no—
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—21no—
486C22⋊C4—1781no—
487C22⋊C4—1781no—
488C22⋊C4—1781no—
489C22⋊C4—1781no—
490C22⋊C4—1781no—
491C42—21no—
492C42—1781no—
493C42C2×C4×D8911yescharacteristic
494C42C2×C22⋊D8911yescharacteristic
495C42#1308357411yescharacteristic
496C42C23×D8911yescharacteristic
497C23—21no—
498C23—1781no—
499C23C4×C22⋊D8911yescharacteristic
500Q8—21no—
501Q8—21no—
502Q8—1781no—
503Q8—1781no—
504D4—21no—
505D4—21no—
506D4—81no—
507D4—1781no—
508D4—1781no—
509D4—7121no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—3561no—
521C2×C4—3561no—
522C2×C4—1781no—
523C2×C4—1781no—
524C2×C4—1781no—
525C2×C4C4×C22⋊D8911yescharacteristic
526C2×C4C4×C22⋊D8911yescharacteristic
527C2×C4C4×C22⋊D8911yescharacteristic
528C2×C4C22×C4×D8911yescharacteristic
529C2×C4C22×C22⋊D8911yescharacteristic
530C2×C4C2×#1308357411yescharacteristic
531C8—21no—
532C8—21no—
533C8—41no—
534C8—41no—
535C8—892no—
536C8—3561no—
537C8—3561no—
538C8—3561no—
539C8—3561no—
540C8—1781no—
541C8—1781no—
542C8—1781no—
543C8#1566803212yes—
544C8#1566803212yes—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—3561no—
552C22—3561no—
553C22—3561no—
554C22C2×C4×C22⋊D8911yescharacteristic
555C4—21no—
556C4—21no—
557C4—21no—
558C4—21no—
559C4—41no—
560C4—41no—
561C4—41no—
562C4—3561no—
563C4—3561no—
564C4—3561no—
565C4#131×D8911yescharacteristic
566C4#1915236311yescharacteristic
567C4C2×#1566803211yescharacteristic
568C4C2×#1566803211yescharacteristic
569C2—41no—
570C2—81no—
571C2—7121no—
572C2#2602899411yescharacteristic
573C2C4×#1566803211yescharacteristic
574C2#2602916711yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#2969C2
socleC22×C89C2×C4×D4
prime radical#19151832C4
cosocleC42×C89C24
hypercenter#2969D89
hypocenterC89#18069
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #2969

Complements

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

NH ≅ G/Ntype
2. #26029256C2semidirect
3. #26019147C2semidirect
4. #26019135C2semidirect
5. #26019095C2semidirect
6. #26019093C2semidirect
7. #26019088C2semidirect
8. #26019086C2semidirect
9. C8×#13083564C2semidirect
10. C8×#13083559C2semidirect
11. #26018998C2semidirect
12. #26018660C2semidirect
13. #26018658C2semidirect
14. M16×#10623494C2semidirect
15. C89×#2969C2semidirect
290. #2969D89semidirect
303. C89#18069semidirect

Maximal subgroups

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

subgroupindex
#260308162
#260292562
#260191472
#260191352
#260190952
#260190932
#260190882
#260190862
C8×#130835642
C8×#130835592
#260189982
#260186602
#260186582
M16×#106234942
C89×#29692
#1806989