Finite Group Databasebeta

← #36525548

Subgroups — #36525548

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#26030332C211yesmaximal, characteristic
2#26029250C211yesmaximal, characteristic
3Q8×#13083558C211yesmaximal, characteristic, complemented
4#26019158C211yesmaximal, characteristic, complemented
5#26019139C211yesmaximal, characteristic, complemented
6#26019120C211yesmaximal, characteristic, complemented
7#26019112C211yesmaximal, characteristic, complemented
8#26019105C211yesmaximal, characteristic, complemented
9#26019092C211yesmaximal, characteristic, complemented
10#26019067C211yesmaximal, characteristic, complemented
11#26019056C211yesmaximal, characteristic, complemented
12#26019040C211yesmaximal, characteristic, complemented
13#26018718C211yesmaximal, characteristic
14#26018547C211yesmaximal, characteristic, complemented
15C89×#1479C211yesmaximal, characteristic, complemented
16#19152132C2211yescharacteristic
17#19151928C411yescharacteristic
18#19151911C411yescharacteristic
19#19151905C411yescharacteristic
20#19151905C411yescharacteristic
21#19151820C411yescharacteristic
22#19151820C411yescharacteristic
23#19151754C411yescharacteristic
24#19151737C411yescharacteristic
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#19151651C2211yescharacteristic
43#19151622—21no—
44#19151622C2211yescharacteristic
45#19151552—21no—
46C4×#13083558—21no—
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
60C2×C4×Q8×C89C2211yescharacteristic
61#431×C89C2211yescharacteristic
62#431×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×#13083558—21no—
72C2×#13083558D411yescharacteristic
73C2×#13083558D411yescharacteristic
74C2×#13083557—21no—
75C2×C8×D89—21no—
76#15668006—21no—
77#15668004C2×C411yescharacteristic
78#15668000C2×C411yescharacteristic
79#15667995—21no—
80#15667991C2×C411yescharacteristic
81#15667986C2×C411yescharacteristic
82C4×Dic178C2311yescharacteristic
83C4×D356C2311yescharacteristic
84C4×C22⋊D89—21no—
85#15667969—21no—
86#15667945C2×C411yescharacteristic
87#15667945C2×C411yescharacteristic
88#15667943C2×C411yescharacteristic
89#15667943C2×C411yescharacteristic
90C4⋊#10623494—21no—
91C4⋊#10623494—21no—
92C4⋊#10623494—21no—
93C4⋊#10623494—21no—
94C4⋊#10623494—21no—
95C4⋊#10623494C2×C411yescharacteristic
96C4⋊#10623494C2×C411yescharacteristic
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
109C22⋊#10623494C2×C411yescharacteristic
110C22⋊#10623494C2×C411yescharacteristic
111#15667924—21no—
112#15667924—21no—
113#15667923—21no—
114#15667923—21no—
115#15667923D411yescharacteristic
116#15667923D411yescharacteristic
117#15667922—21no—
118C4×#10623494—21no—
119C4×#10623494—21no—
120C4×#10623494C2311yescharacteristic
121C4×#10623494C2311yescharacteristic
122C4×#10623494C2311yescharacteristic
123C4×#10623494C2311yescharacteristic
124C8×Dic89—21no—
125C22×Q8×C89C2×C411yescharacteristic
126C4×Q8×C89—21no—
127C4×Q8×C89—21no—
128C4×Q8×C89C2×C411yescharacteristic
129C4×Q8×C89C2×C411yescharacteristic
130C4×Q8×C89C2311yescharacteristic
131C4×Q8×C89C2311yescharacteristic
132C2×C4⋊C4×C89—21no—
133C2×C4⋊C4×C89C2×C411yescharacteristic
134C2×C42×C89—21no—
135C2×C42×C89C2311yescharacteristic
136C4⋊C8×C89—21no—
137C4⋊C8×C89C2311yescharacteristic
138C4⋊C8×C89C2311yescharacteristic
139C22⋊C8×C89—21no—
140C22⋊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#13083564—41no—
150#13083561—22no—
151#13083561—41no—
152#13083558—22no—
153#13083558—41no—
154#13083558—42no—
155#13083557—42no—
156C2×#10623494—21no—
157C2×#10623494—21no—
158C2×#10623494C22×C411yescharacteristic
159C2×#10623494C22×C411yescharacteristic
160C2×#10623494C22×C411yescharacteristic
161C2×#10623494C22×C411yescharacteristic
162C2×#10623494C2×D411yescharacteristic
163C2×#10623494C2×D411yescharacteristic
164C2×#10623494C2×D411yescharacteristic
165C2×#10623494C2×D411yescharacteristic
166C8×D89—42no—
167C4⋊Dic89C22×C411yescharacteristic
168C4⋊Dic89C22×C411yescharacteristic
169#13083553—21no—
170#13083553—21no—
171#13083553—21no—
172#13083553—21no—
173C22⋊Dic89—21no—
174#13083551—21no—
175#13083551—21no—
176#13083551—21no—
177#13083551—21no—
178C4×Dic89—21no—
179C2×Q8×C89—21no—
180C2×Q8×C89—22no—
181C2×Q8×C89—41no—
182C2×Q8×C89C22×C411yescharacteristic
183C2×Q8×C89C22×C411yescharacteristic
184C22×C4×C89—21no—
185C22×C4×C89—21no—
186C22×C4×C89C22×C411yescharacteristic
187C22×C4×C89C2×D411yescharacteristic
188C22×C4×C89#4111yescharacteristic
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×C89—21no—
198C4⋊C4×C89C22×C411yescharacteristic
199C4⋊C4×C89C22×C411yescharacteristic
200C42×C89—21no—
201C42×C89—21no—
202C42×C89—21no—
203C42×C89—21no—
204C42×C89C22×C411yescharacteristic
205C42×C89C2×D411yescharacteristic
206C42×C89#4111yescharacteristic
207C42×C89C2411yescharacteristic
208C23×C89C4×D411yescharacteristic
209C22×D89—21no—
210Q8×C89—21no—
211Q8×C89—21no—
212Q8×C89—21no—
213Q8×C89—42no—
214Q8×C89#13112yes—
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—21no—
223C2×C4×C89—41no—
224C2×C4×C89—42no—
225C2×C4×C89—81no—
226C2×C4×C89C4×D411yescharacteristic
227C2×C4×C89C4×D411yescharacteristic
228C2×C4×C89C4×D411yescharacteristic
229C2×C4×C89C23×C411yescharacteristic
230C2×C4×C89C22×D411yescharacteristic
231C2×C4×C89C2×#4111yescharacteristic
232C22⋊D89—81no—
233C2×Dic89—21no—
234C2×Dic89—21no—
235C2×Dic89—21no—
236D356—21no—
237D356—21no—
238C4×D89—41no—
239C4×D89—41no—
240Dic178—21no—
241Dic178—21no—
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×C89C2×C4×D411yescharacteristic
260C2×D89—41no—
261C2×D89—41no—
262C2×D89—41no—
263C4×C89—21no—
264C4×C89—22no—
265C4×C89—21no—
266C4×C89—21no—
267C4×C89—21no—
268C4×C89—41no—
269C4×C89—41no—
270C4×C89—41no—
271C4×C89C2×#13111yescharacteristic
272C4×C89#56811yescharacteristic
273C4×C89C2×#13711yescharacteristic
274C4×C89#57511yescharacteristic
275Dic89—41no—
276Dic89—41no—
277Dic89—41no—
278#18012—891nomaximal, Sylow
279C2×C89—41no—
280C2×C89#293311yescharacteristic
281C2×C89C4×#13711yescharacteristic
282C2×C89#295711yescharacteristic
283D89—81no—
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#1495—891no—
297#1479D8911yescharacteristic, complemented
298#1453—891no—
299C89#1801211yescharacteristic, complemented, Sylow
300#529—891no—
301C2×C4×Q8C2×D8911yescharacteristic
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#431C2×D8911yescharacteristic
329#431C2×D8911yescharacteristic
330#423—891no—
331#423—1781no—
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⋊C8C2×D8911yescharacteristic
342Q8⋊C8C2×D8911yescharacteristic
343Q8⋊C8C2×D8911yescharacteristic
344Q8⋊C8C2×D8911yescharacteristic
345D4⋊C8—891no—
346D4⋊C8—891no—
347#322—891no—
348#322—891no—
349#322—891no—
350C82—891no—
351C22×Q8C4×D8911yescharacteristic
352C2×Q16—891no—
353C2×Q16—891no—
354C2×SD16—891no—
355C2×SD16—891no—
356C2×M16—891no—
357C2×M16—891no—
358C2×M16—1781no—
359C2×M16—1781no—
360C22×C8—1781no—
361#126—1781no—
362#125—891no—
363#124—891no—
364#123—1781no—
365C22⋊Q8—891no—
366#121—891no—
367C4×Q8—21no—
368C4×Q8—21no—
369C4×Q8—891no—
370C4×Q8C4×D8911yescharacteristic
371C4×Q8C4×D8911yescharacteristic
372C4×Q8C22×D8911yescharacteristic
373C4×Q8C22×D8911yescharacteristic
374C4×D4—891no—
375C4×D4—1781no—
376#117—1781no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4C4×D8911yescharacteristic
379C2×C42—21no—
380C2×C42C22×D8911yescharacteristic
381#107—891no—
382#107—891no—
383#106—891no—
384#106—891no—
385C4⋊C8—21no—
386C4⋊C8—891no—
387C4⋊C8—891no—
388C4⋊C8—891no—
389C4⋊C8—891no—
390C4⋊C8—1781no—
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—21no—
409C22⋊C8—891no—
410C22⋊C8—891no—
411C22⋊C8—1781no—
412C22⋊C8—1781no—
413#97—891no—
414#97—891no—
415#97—1781no—
416#97—1781no—
417#97—1781no—
418C4×C8—21no—
419C4×C8—891no—
420C4×C8—891no—
421C4×C8—891no—
422C4×C8—891no—
423C4×C8—1781no—
424C4×C8—1781no—
425C4×C8—1781no—
426C4×C8C22×D8911yescharacteristic
427C4×C8C22×D8911yescharacteristic
428C2×Q8—21no—
429C2×Q8—22no—
430C2×Q8—41no—
431C2×Q8—891no—
432C2×Q8C2×C4×D8911yescharacteristic
433C2×Q8C2×C4×D8911yescharacteristic
434C2×D4—891no—
435C2×D4—1781no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—1781no—
439C22×C4C2×C4×D8911yescharacteristic
440C22×C4D4×D8911yescharacteristic
441C22×C4#1308357311yescharacteristic
442Q16—3561no—
443Q16—1782no—
444SD16—3561no—
445SD16—1782no—
446M16—3562no—
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—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—1781no—
479C22⋊C4—1781no—
480C22⋊C4—1781no—
481C22⋊C4—1781no—
482C22⋊C4—1781no—
483C42—21no—
484C42—21no—
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—42no—
498Q8—1781no—
499Q8—1781no—
500Q8#1566801812yes—
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—42no—
514C2×C4—81no—
515C2×C4—3561no—
516C2×C4—3561no—
517C2×C4—1781no—
518C2×C4—1781no—
519C2×C4—1781no—
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—3561no—
544C22—3561no—
545C22—3561no—
546C22C2×#1566797211yescharacteristic
547C4—21no—
548C4—21no—
549C4—21no—
550C4—21no—
551C4—22no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—3561no—
556C4—3561no—
557C4—3561no—
558C4C2×#1566801811yescharacteristic
559C4#1915236411yescharacteristic
560C4#137×D8911yescharacteristic
561C4#1915239711yescharacteristic
562C2—41no—
563C2—7121no—
564C2#2602897311yescharacteristic
565C2#2602906711yescharacteristic
566C2#2602916511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#1479C2
socleC22×C89C2×C4×D4
prime radical#15667986C2×C4
cosocleC42×C89C24
hypercenter#1479D89
hypocenterC89#18012
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1479

Complements

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

NH ≅ G/Ntype
3. Q8×#13083558C2semidirect
4. #26019158C2semidirect
5. #26019139C2semidirect
6. #26019120C2semidirect
7. #26019112C2semidirect
8. #26019105C2semidirect
9. #26019092C2semidirect
10. #26019067C2semidirect
11. #26019056C2semidirect
12. #26019040C2semidirect
14. #26018547C2semidirect
15. C89×#1479C2semidirect
297. #1479D89semidirect
299. C89#18012semidirect

Maximal subgroups

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

subgroupindex
#260303322
#260292502
Q8×#130835582
#260191582
#260191392
#260191202
#260191122
#260191052
#260190922
#260190672
#260190562
#260190402
#260187182
#260185472
C89×#14792
#1801289