Finite Group Databasebeta

← #36525560

Subgroups — #36525560

Order 22784.

592 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26030328C211yesmaximal, characteristic
2#26029257C211yesmaximal, characteristic
3#26029203C211yesmaximal, characteristic, complemented
4#26019158C211yesmaximal, characteristic, complemented
5#26019145C211yesmaximal, characteristic, complemented
6#26019126C211yesmaximal, characteristic, complemented
7#26019106C211yesmaximal, characteristic, complemented
8#26019099C211yesmaximal, characteristic, complemented
9#26019098C211yesmaximal, characteristic, complemented
10#26019061C211yesmaximal, characteristic, complemented
11#26019056C211yesmaximal, characteristic, complemented
12#26019043C211yesmaximal, characteristic, complemented
13#26018648C211yesmaximal, characteristic, complemented
14#26018547C211yesmaximal, characteristic, complemented
15C89×#1495C211yesmaximal, characteristic
16C2×C4×Dic178C2211yescharacteristic
17#19151926C411yescharacteristic
18#19151909C411yescharacteristic
19#19151902C411yescharacteristic
20#19151902C411yescharacteristic
21#19151817C411yescharacteristic
22#19151817C411yescharacteristic
23#19151753C411yescharacteristic
24#19151736C411yescharacteristic
25#19151729C2211yescharacteristic
26#19151727—21no—
27#19151727—21no—
28#19151725C2211yescharacteristic
29#19151725C2211yescharacteristic
30Q8×#10623494C2211yescharacteristic
31#19151722C2211yescharacteristic
32#19151722C2211yescharacteristic
33C4×#13083564C2211yescharacteristic
34C4×#13083564C2211yescharacteristic
35C4×#13083562C2211yescharacteristic
36C4×#13083562C2211yescharacteristic
37#19151676—21no—
38#19151676C2211yescharacteristic
39#19151670—21no—
40D4×#10623494C2211yescharacteristic
41#19151659—21no—
42#19151651C2211yescharacteristic
43#19151651C2211yescharacteristic
44#19151621—21no—
45#19151554—21no—
46C4×#13083558C2211yescharacteristic
47M16×Dic89—21no—
48#19151350C2211yescharacteristic
49#19151350C2211yescharacteristic
50#19151348C2211yescharacteristic
51#19151348C2211yescharacteristic
52#19151330C2211yescharacteristic
53#19151330C2211yescharacteristic
54#19151330C2211yescharacteristic
55#19151330C2211yescharacteristic
56#19151320C2211yescharacteristic
57#19151320C2211yescharacteristic
58#19151319C2211yescharacteristic
59C8×#10623494C2211yescharacteristic
60#529×C89C2211yescharacteristic
61#431×C89C2211yescharacteristic
62#423×C89C2211yescharacteristic
63Q8⋊C8×C89C2211yescharacteristic
64Q8⋊C8×C89C2211yescharacteristic
65D4⋊C8×C89C2211yescharacteristic
66D4⋊C8×C89C2211yescharacteristic
67C22×Dic178C2×C411yescharacteristic
68C2×#13083564C2×C411yescharacteristic
69C2×#13083564C2×C411yescharacteristic
70C2×#13083562C2×C411yescharacteristic
71C2×#13083562C2×C411yescharacteristic
72C2×#13083558—21no—
73C2×#13083558D411yescharacteristic
74C2×#13083558D411yescharacteristic
75C22×#10623494—21no—
76C4×Dic178—21no—
77C4×Dic178—21no—
78C4×Dic178C2×C411yescharacteristic
79C4×Dic178C2×C411yescharacteristic
80C4×Dic178C2311yescharacteristic
81C4×Dic178C2311yescharacteristic
82C2×C4⋊Dic89C2×C411yescharacteristic
83C2×#13083553—21no—
84C2×C4×Dic89—21no—
85#15667945C2×C411yescharacteristic
86#15667945C2×C411yescharacteristic
87#15667943C2×C411yescharacteristic
88#15667943C2×C411yescharacteristic
89C4⋊#10623494—21no—
90C4⋊#10623494—21no—
91C4⋊#10623494C2×C411yescharacteristic
92C4⋊#10623494C2×C411yescharacteristic
93C4⋊#10623494C2311yescharacteristic
94C4⋊#10623494C2311yescharacteristic
95#15667941—21no—
96#15667941—21no—
97#15667941—21no—
98#15667941—21no—
99#15667936C2×C411yescharacteristic
100#15667936C2×C411yescharacteristic
101#15667935C2×C411yescharacteristic
102#15667935C2×C411yescharacteristic
103#15667935C2×C411yescharacteristic
104#15667935C2×C411yescharacteristic
105#15667933C2×C411yescharacteristic
106#15667933C2×C411yescharacteristic
107C22⋊#10623494—21no—
108C22⋊#10623494—21no—
109C22⋊#10623494—21no—
110C22⋊#10623494C2×C411yescharacteristic
111C22⋊#10623494C2×C411yescharacteristic
112#15667923—21no—
113#15667923D411yescharacteristic
114#15667923D411yescharacteristic
115#15667922—21no—
116#15667922—21no—
117C4×#10623494—21no—
118C4×#10623494—21no—
119C4×#10623494C2311yescharacteristic
120C4×#10623494C2311yescharacteristic
121C4×#10623494C2311yescharacteristic
122C4×#10623494C2311yescharacteristic
123C8×Dic89—21no—
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×Dic178—22no—
145C2×Dic178—21no—
146C2×Dic178—41no—
147C2×Dic178C22×C411yescharacteristic
148C2×Dic178C22×C411yescharacteristic
149C22×Dic89—21no—
150C22×Dic89—21no—
151#13083564—22no—
152#13083564—41no—
153#13083562—22no—
154#13083562—41no—
155#13083558—22no—
156#13083558—41no—
157#13083558—42no—
158C2×#10623494—21no—
159C2×#10623494—21no—
160C2×#10623494—42no—
161C2×#10623494C22×C411yescharacteristic
162C2×#10623494C22×C411yescharacteristic
163C2×#10623494C22×C411yescharacteristic
164C2×#10623494C22×C411yescharacteristic
165C2×#10623494C2×D411yescharacteristic
166C2×#10623494C2×D411yescharacteristic
167C2×#10623494C2×D411yescharacteristic
168C2×#10623494C2×D411yescharacteristic
169C4⋊Dic89—21no—
170C4⋊Dic89C22×C411yescharacteristic
171C4⋊Dic89C22×C411yescharacteristic
172#13083553—21no—
173#13083553—21no—
174#13083553—21no—
175#13083553—21no—
176C4×Dic89—21no—
177C4×Dic89—21no—
178C4×Dic89—21no—
179C4×Dic89—21no—
180C2×Q8×C89C22×C411yescharacteristic
181C2×D4×C89—21no—
182C2×D4×C89C22×C411yescharacteristic
183C22×C4×C89—21no—
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×C89C22×C411yescharacteristic
197C4⋊C4×C89C22×C411yescharacteristic
198C22⋊C4×C89—21no—
199C22⋊C4×C89—21no—
200C22⋊C4×C89—21no—
201C22⋊C4×C89—21no—
202C22⋊C4×C89—21no—
203C42×C89—21no—
204C42×C89C22×C411yescharacteristic
205C42×C89C2×D411yescharacteristic
206C42×C89#4111yescharacteristic
207C42×C89C2411yescharacteristic
208C23×C89—21no—
209C23×C89C4×D411yescharacteristic
210Q8×C89—21no—
211Q8×C89—21no—
212D4×C89—21no—
213D4×C89—21no—
214D4×C89—81no—
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—41no—
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
231C2×Dic89—21no—
232C2×Dic89—21no—
233C2×Dic89—21no—
234C2×Dic89—21no—
235C2×Dic89—42no—
236C2×Dic89—81no—
237Dic178—21no—
238Dic178—21no—
239Dic178—21no—
240Dic178—42no—
241Dic178#13112yes—
242C8×C89—41no—
243C8×C89—41no—
244C8×C89—41no—
245C8×C89—41no—
246#10623494—21no—
247#10623494—21no—
248#10623494—21no—
249#10623494—21no—
250#10623494—21no—
251#10623494—21no—
252#10623494—21no—
253#10623494—41no—
254#10623494—41no—
255#10623494#13712yes—
256C22×C89—21no—
257C22×C89—21no—
258C22×C89—21no—
259C22×C89—41no—
260C22×C89—41no—
261C22×C89—41no—
262C22×C89C2×C4×D411yescharacteristic
263C4×C89—21no—
264C4×C89—21no—
265C4×C89—21no—
266C4×C89—21no—
267C4×C89—41no—
268C4×C89—41no—
269C4×C89—41no—
270C4×C89C2×#13111yescharacteristic
271C4×C89#56811yescharacteristic
272C4×C89C2×#13711yescharacteristic
273C4×C89#57511yescharacteristic
274Dic89—22no—
275Dic89—41no—
276Dic89—41no—
277Dic89—41no—
278#18012—891nomaximal, Sylow
279C2×C89—41no—
280C2×C89—81no—
281C2×C89#293311yescharacteristic
282C2×C89C4×#13711yescharacteristic
283C2×C89#295711yescharacteristic
284#3108—891no—
285#2969—891no—
286Q8×M16—891no—
287#1593—891no—
288#1590—891no—
289#1587—891no—
290#1585—891no—
291#1584—891no—
292#1583—891no—
293SD16⋊C8—891no—
294C8×Q16—891no—
295#1574—891no—
296#1495D8911yescharacteristic
297#1479—891no—
298#1453—891no—
299C89#1801211yescharacteristic, complemented, Sylow
300#529C2×D8911yescharacteristic
301C2×C4×Q8—891no—
302#483—891no—
303#480—891no—
304#479—891no—
305#479—891no—
306#464—891no—
307#464—891no—
308#451—891no—
309#448—891no—
310#446—891no—
311#446—891no—
312#446—891no—
313#446—1781no—
314#446—1781no—
315C8×Q8—891no—
316C8×Q8—891no—
317C8×Q8—891no—
318C4×Q16—891no—
319C4×Q16—891no—
320C4×SD16—891no—
321C4×SD16—891no—
322#436—891no—
323#436—1781no—
324#435—1781no—
325C8×D4—891no—
326#433—1781no—
327#431—891no—
328#431—891no—
329#431C2×D8911yescharacteristic
330#423—1781no—
331#423C2×D8911yescharacteristic
332#405—1781no—
333C4×M16—891no—
334C4×M16—1781no—
335#335—891no—
336#335—891no—
337#334—891no—
338#334—891no—
339Q8⋊C8—891no—
340Q8⋊C8—891no—
341Q8⋊C8—891no—
342Q8⋊C8—891no—
343Q8⋊C8C2×D8911yescharacteristic
344Q8⋊C8C2×D8911yescharacteristic
345D4⋊C8C2×D8911yescharacteristic
346D4⋊C8C2×D8911yescharacteristic
347#322—891no—
348#322—891no—
349#322—891no—
350C82—891no—
351C22×Q8—891no—
352C2×Q16—891no—
353C2×Q16—891no—
354C2×SD16—891no—
355C2×SD16—891no—
356C2×M16—21no—
357C2×M16—891no—
358C2×M16—891no—
359C2×M16—1781no—
360C22×C8—1781no—
361#126—21no—
362#125C4×D8911yescharacteristic
363#124C4×D8911yescharacteristic
364#123—21no—
365C22⋊Q8C4×D8911yescharacteristic
366#121C4×D8911yescharacteristic
367C4×Q8—891no—
368C4×Q8—891no—
369C4×Q8—891no—
370C4×Q8—891no—
371C4×Q8—1781no—
372C4×Q8—1781no—
373C4×Q8C22×D8911yescharacteristic
374C4×D4—21no—
375C4×D4C22×D8911yescharacteristic
376#117—21no—
377C2×C4⋊C4—891no—
378C2×C4⋊C4—1781no—
379C2×C42—1781no—
380C2×C42C22×D8911yescharacteristic
381#107—891no—
382#107—891no—
383#106—891no—
384#106—891no—
385C4⋊C8—21no—
386C4⋊C8—21no—
387C4⋊C8—891no—
388C4⋊C8—891no—
389C4⋊C8—891no—
390C4⋊C8—891no—
391C4⋊C8—1781no—
392C4⋊C8—1781no—
393C4⋊C8—1781no—
394C4⋊C8—1781no—
395C4⋊C8—1781no—
396C4⋊C8—1781no—
397C4⋊C8C22×D8911yescharacteristic
398C4⋊C8C22×D8911yescharacteristic
399#103—891no—
400#103—891no—
401#103—891no—
402#103—891no—
403#103—891no—
404#103—891no—
405#102—891no—
406#102—891no—
407C22⋊C8—21no—
408C22⋊C8—891no—
409C22⋊C8—891no—
410C22⋊C8—1781no—
411C22⋊C8—1781no—
412C22⋊C8—1781no—
413#97—891no—
414#97—891no—
415#97—1781no—
416#97—1781no—
417#97—1781no—
418C4×C8—891no—
419C4×C8—891no—
420C4×C8—891no—
421C4×C8—891no—
422C4×C8—1781no—
423C4×C8—1781no—
424C4×C8—1781no—
425C4×C8—1781no—
426C4×C8C22×D8911yescharacteristic
427C4×C8C22×D8911yescharacteristic
428C2×Q8—891no—
429C2×Q8—891no—
430C2×Q8—3561no—
431C2×Q8—1782no—
432C2×Q8—1781no—
433C2×Q8C2×C4×D8911yescharacteristic
434C2×D4—21no—
435C2×D4C2×C4×D8911yescharacteristic
436C22×C4—21no—
437C22×C4—1781no—
438C22×C4—1781no—
439C22×C4C2×C4×D8911yescharacteristic
440C22×C4D4×D8911yescharacteristic
441C22×C4#1308357311yescharacteristic
442Q16—3561no—
443Q16—1782no—
444SD16—3561no—
445SD16—1782no—
446M16—42no—
447M16—3562no—
448M16—3561no—
449M16—1782no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—891no—
455C2×C8—891no—
456C2×C8—891no—
457C2×C8—891no—
458C2×C8—891no—
459C2×C8—891no—
460C2×C8—891no—
461C2×C8—891no—
462C2×C8—3562no—
463C2×C8—1781no—
464C2×C8—1781no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—891no—
470C4⋊C4—891no—
471C4⋊C4—1781no—
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—21no—
481C22⋊C4—21no—
482C22⋊C4—21no—
483C42—21no—
484C42—1781no—
485C42—1781no—
486C42—1781no—
487C42—1781no—
488C42C2×C4×D8911yescharacteristic
489C42D4×D8911yescharacteristic
490C42#1308357311yescharacteristic
491C42C23×D8911yescharacteristic
492C23—21no—
493C23#1566797211yescharacteristic
494Q8—21no—
495Q8—21no—
496Q8—892no—
497Q8—3562no—
498Q8—1781no—
499Q8—1781no—
500Q8—1781no—
501D4—21no—
502D4—21no—
503D4—81no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—3562no—
515C2×C4—1781no—
516C2×C4—1781no—
517C2×C4—1781no—
518C2×C4—1781no—
519C2×C4—7121no—
520C2×C4#1566797211yescharacteristic
521C2×C4#1566797211yescharacteristic
522C2×C4#1566797211yescharacteristic
523C2×C4C22×C4×D8911yescharacteristic
524C2×C4C2×D4×D8911yescharacteristic
525C2×C4C2×#1308357311yescharacteristic
526C8—41no—
527C8—41no—
528C8—41no—
529C8—41no—
530C8—892no—
531C8—3561no—
532C8—3561no—
533C8—1781no—
534C8—1781no—
535C8—1781no—
536C8—1781no—
537C8—1781no—
538C8—1781no—
539C8—1781no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22C2×#1566797211yescharacteristic
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—3561no—
555C4—3561no—
556C4—3561no—
557C4—1782no—
558C4#131×D8911yescharacteristic
559C4#1915236311yescharacteristic
560C4#137×D8911yescharacteristic
561C4#1915239411yescharacteristic
562C2—41no—
563C2—81no—
564C2#2602898211yescharacteristic
565C2#2602907611yescharacteristic
566C2#2602916511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#1495C2
socleC22×C89C2×C4×D4
prime radical#121×C89C2×C4
cosocleC42×C89C24
hypercenter#1495D89
hypocenterC89#18012
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
3. #26029203C2semidirect
4. #26019158C2semidirect
5. #26019145C2semidirect
6. #26019126C2semidirect
7. #26019106C2semidirect
8. #26019099C2semidirect
9. #26019098C2semidirect
10. #26019061C2semidirect
11. #26019056C2semidirect
12. #26019043C2semidirect
13. #26018648C2semidirect
14. #26018547C2semidirect
299. C89#18012semidirect

Maximal subgroups

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

subgroupindex
#260303282
#260292572
#260292032
#260191582
#260191452
#260191262
#260191062
#260190992
#260190982
#260190612
#260190562
#260190432
#260186482
#260185472
C89×#14952
#1801289