Finite Group Databasebeta

← #36525301

Subgroups — #36525301

Order 22784.

576 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#26029401C211yesmaximal, characteristic, complemented
3#26029397C211yesmaximal, characteristic, complemented
4#26019126C211yesmaximal, characteristic, complemented
5#26019120C211yesmaximal, characteristic, complemented
6#26019119C211yesmaximal, characteristic, complemented
7#26019113C211yesmaximal, characteristic, complemented
8#26019098C211yesmaximal, characteristic, complemented
9#26019092C211yesmaximal, characteristic, complemented
10#26019091C211yesmaximal, characteristic, complemented
11#26019085C211yesmaximal, characteristic, complemented
12#26019007C211yesmaximal, characteristic, complemented
13#26018718C211yesmaximal, characteristic, complemented
14#26018545C211yesmaximal, characteristic, complemented
15C89×#1495C211yesmaximal, characteristic, complemented
16#19152132C2211yescharacteristic
17#19151997C411yescharacteristic
18#19151980C411yescharacteristic
19#19151955C411yescharacteristic
20#19151952C411yescharacteristic
21#19151837C411yescharacteristic
22#19151832C411yescharacteristic
23#19151829C411yescharacteristic
24#19151825C411yescharacteristic
25#19151729C2211yescharacteristic
26#19151725C2211yescharacteristic
27Q8×#10623494C2211yescharacteristic
28#19151722C2211yescharacteristic
29C4×#13083564C2211yescharacteristic
30C4×#13083561C2211yescharacteristic
31C4×#13083562C2211yescharacteristic
32C4×#13083559C2211yescharacteristic
33#19151676—21no—
34#19151676C2211yescharacteristic
35#19151675C2211yescharacteristic
36#19151674—21no—
37#19151670—21no—
38#19151667—21no—
39D4×#10623494C2211yescharacteristic
40#19151663C2211yescharacteristic
41#19151659—21no—
42#19151658—21no—
43#19151651C2211yescharacteristic
44#19151625—21no—
45#19151623—21no—
46#19151622C2211yescharacteristic
47C2×C4×#10623494C2211yescharacteristic
48#19151348C2211yescharacteristic
49#19151348C2211yescharacteristic
50#19151348C2211yescharacteristic
51#19151348C2211yescharacteristic
52#19151330C2211yescharacteristic
53#19151330C2211yescharacteristic
54#19151327C2211yescharacteristic
55#19151327C2211yescharacteristic
56#19151320C2211yescharacteristic
57#19151320C2211yescharacteristic
58#19151319C2211yescharacteristic
59#19151319C2211yescharacteristic
60#529×C89C2211yescharacteristic
61#431×C89C2211yescharacteristic
62#423×C89C2211yescharacteristic
63Q8⋊C8×C89C2211yescharacteristic
64Q8⋊C8×C89C2211yescharacteristic
65D4⋊C8×C89C2211yescharacteristic
66D4⋊C8×C89C2211yescharacteristic
67C2×#13083564C2×C411yescharacteristic
68C2×#13083562C2×C411yescharacteristic
69C2×#13083561C2×C411yescharacteristic
70C2×#13083559C2×C411yescharacteristic
71C2×#13083558—21no—
72C2×#13083558—21no—
73C2×#13083557—21no—
74C22×#10623494—21no—
75C22×#10623494C2×C411yescharacteristic
76C22×#10623494C2×C411yescharacteristic
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⋊#10623494D411yescharacteristic
96C4⋊#10623494D411yescharacteristic
97C4⋊#10623494C2311yescharacteristic
98C4⋊#10623494C2311yescharacteristic
99#15667941—21no—
100#15667941—21no—
101#15667936C2×C411yescharacteristic
102#15667936C2×C411yescharacteristic
103#15667935C2×C411yescharacteristic
104#15667935C2×C411yescharacteristic
105#15667933C2×C411yescharacteristic
106#15667933C2×C411yescharacteristic
107#15667932C2×C411yescharacteristic
108#15667932C2×C411yescharacteristic
109C22⋊#10623494—21no—
110C22⋊#10623494—21no—
111C22⋊#10623494D411yescharacteristic
112C22⋊#10623494D411yescharacteristic
113#15667924—21no—
114#15667924—21no—
115#15667923—21no—
116#15667922—21no—
117C4×#10623494—21no—
118C4×#10623494C2×C411yescharacteristic
119C4×#10623494C2×C411yescharacteristic
120C4×#10623494C2311yescharacteristic
121C4×#10623494C2311yescharacteristic
122C4×#10623494C2311yescharacteristic
123C4×#10623494C2311yescharacteristic
124C8×Dic89—21no—
125C2×M16×C89—21no—
126#126×C89—21no—
127#125×C89C2×C411yescharacteristic
128#124×C89C2×C411yescharacteristic
129#123×C89—21no—
130C22⋊Q8×C89C2×C411yescharacteristic
131#121×C89C2×C411yescharacteristic
132C4×Q8×C89C2311yescharacteristic
133C4×D4×C89—21no—
134C4×D4×C89C2311yescharacteristic
135#117×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—41no—
149#13083562—41no—
150#13083561—41no—
151#13083559—41no—
152#13083558—81no—
153#13083558—81no—
154#13083557—81no—
155C2×#10623494—21no—
156C2×#10623494—21no—
157C2×#10623494—41no—
158C2×#10623494—41no—
159C2×#10623494—81no—
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—81no—
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×C89C22×C411yescharacteristic
182C2×D4×C89—21no—
183C2×D4×C89C22×C411yescharacteristic
184C22×C4×C89—21no—
185C22×C4×C89C22×C411yescharacteristic
186C22×C4×C89C2×D411yescharacteristic
187C22×C4×C89#4111yescharacteristic
188M16×C89—81no—
189C2×C8×C89—21no—
190C2×C8×C89—21no—
191C2×C8×C89—21no—
192C2×C8×C89—21no—
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—41no—
244C8×C89—41no—
245C8×C89—41no—
246C8×C89—41no—
247#10623494—21no—
248#10623494—21no—
249#10623494—21no—
250#10623494—21no—
251#10623494—21no—
252#10623494—21no—
253#10623494—21no—
254#10623494—21no—
255#10623494—41no—
256#10623494—41no—
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×C89#56811yescharacteristic
275C4×C89#56811yescharacteristic
276C4×C89#57711yescharacteristic
277C4×C89#57711yescharacteristic
278Dic89—41no—
279Dic89—41no—
280Dic89—41no—
281#17993—891nomaximal, Sylow
282C2×C89—41no—
283C2×C89—81no—
284C2×C89#293211yescharacteristic
285C2×C89#295211yescharacteristic
286C2×C89#295711yescharacteristic
287D89—81no—
288#3168—891no—
289#2993—891no—
290#2993—891no—
291#1587—891no—
292#1587—891no—
293#1586—891no—
294#1586—891no—
295#1583—891no—
296#1583—891no—
297#1582—891no—
298#1582—891no—
299#1566—891no—
300#1495—891no—
301#1495D8911yescharacteristic, complemented
302#1452—891no—
303C89#1799311yescharacteristic, 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#446—891no—
316C8×Q8—891no—
317C8×Q8—891no—
318C4×Q16—891no—
319C4×SD16—891no—
320C4×SD16—891no—
321C4×D8—891no—
322#436—891no—
323#436—891no—
324#436—1781no—
325#436—1781no—
326#435—1781no—
327#435—1781no—
328C8×D4—891no—
329C8×D4—891no—
330#433—1781no—
331#433—1781no—
332#431—891no—
333#431C2×D8911yescharacteristic
334#424—1781no—
335#424—1781no—
336#423—891no—
337#423C2×D8911yescharacteristic
338C2×C4×C8—891no—
339#334—891no—
340#334—891no—
341#334—891no—
342#334—891no—
343Q8⋊C8—891no—
344Q8⋊C8—891no—
345Q8⋊C8C2×D8911yescharacteristic
346Q8⋊C8C2×D8911yescharacteristic
347D4⋊C8—891no—
348D4⋊C8—891no—
349D4⋊C8C2×D8911yescharacteristic
350D4⋊C8C2×D8911yescharacteristic
351#322—891no—
352#322—891no—
353#322—891no—
354#322—891no—
355C2×Q16—891no—
356C2×SD16—891no—
357C2×SD16—891no—
358C2×D8—891no—
359C2×M16—21no—
360C2×M16—1781no—
361C2×M16—1781no—
362C2×M16—1781no—
363C22×C8—891no—
364C22×C8—891no—
365C22×C8—1781no—
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—891no—
397C4⋊C8—891no—
398C4⋊C8—1781no—
399C4⋊C8—1781no—
400C4⋊C8—1781no—
401C4⋊C8—1781no—
402C4⋊C8—1781no—
403C4⋊C8C22×D8911yescharacteristic
404C4⋊C8C22×D8911yescharacteristic
405#103—891no—
406#103—891no—
407#103—891no—
408#103—891no—
409#102—891no—
410#102—891no—
411#102—891no—
412#102—891no—
413C22⋊C8—21no—
414C22⋊C8—891no—
415C22⋊C8—891no—
416C22⋊C8—1781no—
417C22⋊C8—1781no—
418C22⋊C8—1781no—
419C22⋊C8—1781no—
420#97—1781no—
421#97—1781no—
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×C4D4×D8911yescharacteristic
442C22×C4#1308357311yescharacteristic
443Q16—3561no—
444SD16—3561no—
445SD16—3561no—
446D8—3561no—
447M16—81no—
448M16—7121no—
449M16—7121no—
450M16—7121no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—891no—
456C2×C8—891no—
457C2×C8—891no—
458C2×C8—891no—
459C2×C8—891no—
460C2×C8—891no—
461C2×C8—891no—
462C2×C8—891no—
463C2×C8—3561no—
464C2×C8—3561no—
465C2×C8—1781no—
466C2×C8—1781no—
467C2×C8—7121no—
468C2×C8—7121no—
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
494C42D4×D8911yescharacteristic
495C42#1308357311yescharacteristic
496C42C23×D8911yescharacteristic
497C23—21no—
498C23—1781no—
499C23#1566797211yescharacteristic
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×C4#1566797211yescharacteristic
526C2×C4#1566797211yescharacteristic
527C2×C4#1566797211yescharacteristic
528C2×C4C22×C4×D8911yescharacteristic
529C2×C4C2×D4×D8911yescharacteristic
530C2×C4C2×#1308357311yescharacteristic
531C8—41no—
532C8—41no—
533C8—41no—
534C8—41no—
535C8—3561no—
536C8—3561no—
537C8—1781no—
538C8—1781no—
539C8—1781no—
540C8—1781no—
541C8—1781no—
542C8—1781no—
543C8—1781no—
544C8—1781no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—3561no—
552C22—3561no—
553C22—3561no—
554C22C2×#1566797211yescharacteristic
555C4—21no—
556C4—21no—
557C4—21no—
558C4—21no—
559C4—41no—
560C4—41no—
561C4—41no—
562C4—3561no—
563C4—3561no—
564C4—3561no—
565C4#1915236311yescharacteristic
566C4#1915236311yescharacteristic
567C4#1915240711yescharacteristic
568C4#1915240811yescharacteristic
569C2—41no—
570C2—81no—
571C2—7121no—
572C2#2602896711yescharacteristic
573C2#2602913111yescharacteristic
574C2#2602916511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#1495C2
socleC22×C89C2×C4×D4
prime radical#19151832C4
cosocleC42×C89C24
hypercenter#1495D89
hypocenterC89#17993
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1495

Complements

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

NH ≅ G/Ntype
2. #26029401C2semidirect
3. #26029397C2semidirect
4. #26019126C2semidirect
5. #26019120C2semidirect
6. #26019119C2semidirect
7. #26019113C2semidirect
8. #26019098C2semidirect
9. #26019092C2semidirect
10. #26019091C2semidirect
11. #26019085C2semidirect
12. #26019007C2semidirect
13. #26018718C2semidirect
14. #26018545C2semidirect
15. C89×#1495C2semidirect
301. #1495D89semidirect
303. C89#17993semidirect

Maximal subgroups

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

subgroupindex
#260308162
#260294012
#260293972
#260191262
#260191202
#260191192
#260191132
#260190982
#260190922
#260190912
#260190852
#260190072
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