Finite Group Databasebeta

← #36526005

Subgroups — #36526005

Order 22784.

640 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26029270C211yesmaximal, characteristic
2C4×#15668043C211yesmaximal, characteristic
3#26019152C211yesmaximal, characteristic, complemented
4#26019142C211yesmaximal, characteristic, complemented
5#26019130C211yesmaximal, characteristic, complemented
6#26019121C211yesmaximal, characteristic, complemented
7#26019100C211yesmaximal, characteristic, complemented
8#26019095C211yesmaximal, characteristic, complemented
9#26019069C211yesmaximal, characteristic, complemented
10C8×#13083564C211yesmaximal, characteristic, complemented
11#26019017C211yesmaximal, characteristic, complemented
12#26018699C211yesmaximal, characteristic, complemented
13#26018689C211yesmaximal, characteristic
14#26018555C211yesmaximal, characteristic, complemented
15Q8×M16×C89C211yesmaximal, characteristic, complemented
16C2×#15668043C411yescharacteristic
17C4×#13083574C2211yescharacteristic
18C8×Dic178C2211yescharacteristic
19C8×Dic178C2211yescharacteristic
20#19151703C411yescharacteristic
21#19151703C411yescharacteristic
22#19151696C411yescharacteristic
23#19151696C411yescharacteristic
24C4×#13083564C2211yescharacteristic
25C4×#13083564C2211yescharacteristic
26C4×#13083561C2211yescharacteristic
27C4×#13083561C2211yescharacteristic
28#19151677C2211yescharacteristic
29#19151677C2211yescharacteristic
30#19151674—21no—
31#19151674—21no—
32#19151655C2211yescharacteristic
33#19151653—21no—
34#19151638C411yescharacteristic
35#19151607C411yescharacteristic
36#19151597C411yescharacteristic
37C4×#13083558C2211yescharacteristic
38C4×#13083558C2211yescharacteristic
39#19151550—21no—
40#19151350C2211yescharacteristic
41#19151350C2211yescharacteristic
42#19151348C2211yescharacteristic
43#19151348C2211yescharacteristic
44Q8⋊#10623494C2211yescharacteristic
45Q8⋊#10623494C2211yescharacteristic
46Q8⋊#10623494C2211yescharacteristic
47Q8⋊#10623494C2211yescharacteristic
48#19151328C2211yescharacteristic
49#19151328C2211yescharacteristic
50#19151325C2211yescharacteristic
51#19151325C2211yescharacteristic
52#19151320C2211yescharacteristic
53#19151320C2211yescharacteristic
54#19151319C2211yescharacteristic
55C8×#10623494C2211yescharacteristic
56C2×C4×Q8×C89C2211yescharacteristic
57#446×C89—21no—
58#446×C89—21no—
59#446×C89C2211yescharacteristic
60#446×C89C2211yescharacteristic
61C8×Q8×C89C2211yescharacteristic
62C8×Q8×C89C2211yescharacteristic
63#423×C89—21no—
64#423×C89C2211yescharacteristic
65C4×M16×C89—21no—
66C4×M16×C89C2211yescharacteristic
67C2×#13083574C2×C411yescharacteristic
68#15668043—24no—
69C2×#13083564C2×C411yescharacteristic
70C2×#13083564C2×C411yescharacteristic
71C2×#13083561C2×C411yescharacteristic
72C2×#13083561C2×C411yescharacteristic
73C2×#13083558—21no—
74C2×#13083558C2×C411yescharacteristic
75C2×#13083558C2×C411yescharacteristic
76C2×#13083557—21no—
77C2×#13083557—21no—
78C4×Dic178C2×C411yescharacteristic
79C4×Dic178C2311yescharacteristic
80C4×D356C2×C411yescharacteristic
81C4×D356C2311yescharacteristic
82C4×C22⋊D89—21no—
83C4×C22⋊D89—21no—
84#15667971C2×C411yescharacteristic
85#15667969—21no—
86C42×D89—21no—
87#15667945C2×C411yescharacteristic
88#15667945C2×C411yescharacteristic
89#15667943C2×C411yescharacteristic
90#15667943C2×C411yescharacteristic
91C4⋊#10623494C2311yescharacteristic
92C4⋊#10623494C2311yescharacteristic
93#15667941—21no—
94#15667941—21no—
95#15667936C2×C411yescharacteristic
96#15667936C2×C411yescharacteristic
97#15667936C2×C411yescharacteristic
98#15667936C2×C411yescharacteristic
99#15667935C2×C411yescharacteristic
100#15667935C2×C411yescharacteristic
101#15667932C2×C411yescharacteristic
102#15667932C2×C411yescharacteristic
103C22⋊#10623494—21no—
104#15667924—21no—
105#15667924—21no—
106#15667923—21no—
107#15667923—21no—
108#15667923C2×C411yescharacteristic
109#15667923C2×C411yescharacteristic
110C4×#10623494C2311yescharacteristic
111C4×#10623494C2311yescharacteristic
112C4×#10623494C2311yescharacteristic
113C4×#10623494C2311yescharacteristic
114C8×Dic89—21no—
115C8×Dic89—21no—
116C22×Q8×C89C2×C411yescharacteristic
117C2×M16×C89—21no—
118C2×M16×C89D411yescharacteristic
119C2×M16×C89D411yescharacteristic
120C4×Q8×C89—21no—
121C4×Q8×C89—21no—
122C4×Q8×C89C2×C411yescharacteristic
123C4×Q8×C89C2×C411yescharacteristic
124C4×Q8×C89C2311yescharacteristic
125C4×Q8×C89C2311yescharacteristic
126C2×C4⋊C4×C89—21no—
127C2×C4⋊C4×C89C2×C411yescharacteristic
128C2×C42×C89—21no—
129C2×C42×C89C2311yescharacteristic
130C4⋊C8×C89—21no—
131C4⋊C8×C89—21no—
132C4⋊C8×C89—21no—
133C4⋊C8×C89—21no—
134C4⋊C8×C89—21no—
135C4⋊C8×C89C2311yescharacteristic
136C4⋊C8×C89C2311yescharacteristic
137#97×C89—21no—
138#97×C89—21no—
139#97×C89D411yescharacteristic
140#97×C89D411yescharacteristic
141C4×C8×C89—21no—
142C4×C8×C89—21no—
143C4×C8×C89C2311yescharacteristic
144C4×C8×C89C2311yescharacteristic
145#13083574—22no—
146#13083574—41no—
147C2×Dic178C22×C411yescharacteristic
148C2×D356C22×C411yescharacteristic
149C2×C22⋊D89—21no—
150C2×C4×D89—21no—
151C2×C4×D89—21no—
152C2×C4×D89—21no—
153#13083564—22no—
154#13083564M1614yescomplemented
155#13083561—22no—
156#13083561—22no—
157#13083558—22no—
158#13083558—41no—
159#13083558—42no—
160#13083557—42no—
161#13083557—42no—
162C2×#10623494—21no—
163C2×#10623494—21no—
164C2×#10623494—21no—
165C2×#10623494—21no—
166C2×#10623494C22×C411yescharacteristic
167C2×#10623494C22×C411yescharacteristic
168C2×#10623494C22×C411yescharacteristic
169C2×#10623494C22×C411yescharacteristic
170C4⋊Dic89C22×C411yescharacteristic
171C4⋊Dic89C22×C411yescharacteristic
172#13083553—21no—
173#13083553—21no—
174C22⋊Dic89—21no—
175#13083551—21no—
176#13083551—21no—
177C4×Dic89—21no—
178C4×Dic89—21no—
179C4×Dic89—21no—
180C2×Q8×C89—21no—
181C2×Q8×C89—22no—
182C2×Q8×C89—41no—
183C2×Q8×C89C22×C411yescharacteristic
184C2×Q8×C89C22×C411yescharacteristic
185C22×C4×C89—21no—
186C22×C4×C89—21no—
187C22×C4×C89C22×C411yescharacteristic
188C22×C4×C89C2×D411yescharacteristic
189C22×C4×C89#4111yescharacteristic
190M16×C89—22no—
191M16×C89—41no—
192M16×C89—42no—
193C2×C8×C89—21no—
194C2×C8×C89—21no—
195C2×C8×C89C2×D411yescharacteristic
196C2×C8×C89C2×D411yescharacteristic
197C2×C8×C89C2×D411yescharacteristic
198C2×C8×C89C2×D411yescharacteristic
199C4⋊C4×C89—21no—
200C4⋊C4×C89—21no—
201C4⋊C4×C89—21no—
202C4⋊C4×C89—21no—
203C4⋊C4×C89—21no—
204C4⋊C4×C89C22×C411yescharacteristic
205C4⋊C4×C89C22×C411yescharacteristic
206C42×C89—21no—
207C42×C89—21no—
208C42×C89—21no—
209C42×C89—21no—
210C42×C89C22×C411yescharacteristic
211C42×C89C2×D411yescharacteristic
212C42×C89#4111yescharacteristic
213C42×C89C2411yescharacteristic
214C23×C89C4×D411yescharacteristic
215C22×D89—21no—
216Q8×C89—21no—
217Q8×C89—21no—
218Q8×C89—21no—
219Q8×C89—42no—
220Q8×C89C2×M1612yes—
221C2×C4×C89—21no—
222C2×C4×C89—21no—
223C2×C4×C89—21no—
224C2×C4×C89—21no—
225C2×C4×C89—21no—
226C2×C4×C89—21no—
227C2×C4×C89—21no—
228C2×C4×C89—21no—
229C2×C4×C89—41no—
230C2×C4×C89—42no—
231C2×C4×C89—81no—
232C2×C4×C89C4×D411yescharacteristic
233C2×C4×C89C4×D411yescharacteristic
234C2×C4×C89C4×D411yescharacteristic
235C2×C4×C89C23×C411yescharacteristic
236C2×C4×C89C22×D411yescharacteristic
237C2×C4×C89C2×#4111yescharacteristic
238C22⋊D89—42no—
239C2×Dic89—21no—
240C2×Dic89—21no—
241C2×Dic89—21no—
242D356—21no—
243D356—21no—
244C4×D89—42no—
245C4×D89—41no—
246C4×D89—41no—
247C4×D89—81no—
248Dic178—21no—
249Dic178C2×M1612yes—
250C8×C89—21no—
251C8×C89—21no—
252C8×C89—21no—
253C8×C89—41no—
254C8×C89—41no—
255C8×C89#13712yescomplemented
256#10623494—22no—
257#10623494—21no—
258#10623494—21no—
259#10623494—21no—
260#10623494—41no—
261#10623494—41no—
262#10623494—41no—
263#10623494C2×M1612yes—
264C22×C89—21no—
265C22×C89—21no—
266C22×C89—21no—
267C22×C89C2×C4×D411yescharacteristic
268C2×D89—41no—
269C2×D89—41no—
270C2×D89—41no—
271C4×C89—21no—
272C4×C89—22no—
273C4×C89—21no—
274C4×C89—21no—
275C4×C89—21no—
276C4×C89—41no—
277C4×C89—41no—
278C4×C89—41no—
279C4×C89C22×M1611yescharacteristic
280C4×C89#56811yescharacteristic
281C4×C89C2×#13711yescharacteristic
282C4×C89#57511yescharacteristic
283Dic89—22no—
284Dic89—41no—
285Dic89—41no—
286#18049—891nomaximal, Sylow
287C2×C89—41no—
288C2×C89D4×M1611yescharacteristic
289C2×C89C4×#13711yescharacteristic
290C2×C89#295611yescharacteristic
291D89—81no—
292#2971—891no—
293Q8×M16D8911yescharacteristic, complemented
294C4×#137—891no—
295#1592—891no—
296#1590—891no—
297#1588—891no—
298#1587—891no—
299#1584—891no—
300#1583—891no—
301#1579—891no—
302C8×Q16—891no—
303#1569—891no—
304#1491—891no—
305#1489—891no—
306#1455—891no—
307C89#1804911yescharacteristic, complemented, Sylow
308C2×#137—891no—
309C4×#41—891no—
310C2×C4×Q8C2×D8911yescharacteristic
311#446—21no—
312#446—21no—
313#446C2×D8911yescharacteristic
314#446C2×D8911yescharacteristic
315C8×Q8—891no—
316C8×Q8—891no—
317C8×Q8C2×D8911yescharacteristic
318C8×Q8C2×D8911yescharacteristic
319#441—891no—
320#441—891no—
321#440—891no—
322#440—891no—
323C4×Q16—891no—
324C4×Q16—891no—
325C4×SD16—891no—
326C4×SD16—891no—
327#436—891no—
328#436—891no—
329#436—1781no—
330#436—1781no—
331#432—891no—
332#432—1781no—
333#428—891no—
334#423—21no—
335#423C2×D8911yescharacteristic
336#419—891no—
337#417—891no—
338C4×M16—21no—
339C4×M16—891no—
340C4×M16—891no—
341C4×M16—1781no—
342C4×M16C2×D8911yescharacteristic
343#335—891no—
344#335—891no—
345#334—891no—
346#334—891no—
347Q8⋊C8—891no—
348Q8⋊C8—891no—
349Q8⋊C8—891no—
350Q8⋊C8—891no—
351Q8⋊C8—891no—
352Q8⋊C8—891no—
353D4⋊C8—891no—
354D4⋊C8—891no—
355#322—891no—
356#322—891no—
357#322—891no—
358C82—891no—
359C2×#41—891no—
360C22×Q8C4×D8911yescharacteristic
361#137—1784no—
362C2×Q16—891no—
363C2×Q16—891no—
364C2×SD16—891no—
365C2×SD16—891no—
366C2×M16—21no—
367C2×M16—891no—
368C2×M16—891no—
369C2×M16—1781no—
370C2×M16—1781no—
371C2×M16—1781no—
372C2×M16C22⋊D8911yescharacteristic
373C2×M16C22⋊D8911yescharacteristic
374C4×Q8—21no—
375C4×Q8—21no—
376C4×Q8—891no—
377C4×Q8—891no—
378C4×Q8C4×D8911yescharacteristic
379C4×Q8C4×D8911yescharacteristic
380C4×Q8C22×D8911yescharacteristic
381C4×Q8C22×D8911yescharacteristic
382C4×D4—891no—
383C4×D4—891no—
384C4×D4—1781no—
385C4×D4—1781no—
386#117—891no—
387#117—1781no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4C4×D8911yescharacteristic
390C2×C42—21no—
391C2×C42—1781no—
392C2×C42C22×D8911yescharacteristic
393#107—891no—
394#107—891no—
395#106—891no—
396#106—891no—
397C4⋊C8—21no—
398C4⋊C8—21no—
399C4⋊C8—21no—
400C4⋊C8—21no—
401C4⋊C8—21no—
402C4⋊C8—891no—
403C4⋊C8—891no—
404C4⋊C8—1781no—
405C4⋊C8—1781no—
406C4⋊C8C22×D8911yescharacteristic
407C4⋊C8C22×D8911yescharacteristic
408#103—891no—
409#103—891no—
410#103—891no—
411#103—891no—
412#103—891no—
413#103—891no—
414#102—891no—
415#102—891no—
416C22⋊C8—1781no—
417C22⋊C8—1781no—
418C22⋊C8—1781no—
419#97—21no—
420#97—21no—
421#97—891no—
422#97—891no—
423#97—1781no—
424#97—1781no—
425#97C22⋊D8911yescharacteristic
426#97C22⋊D8911yescharacteristic
427C4×C8—21no—
428C4×C8—21no—
429C4×C8—891no—
430C4×C8—891no—
431C4×C8—891no—
432C4×C8—891no—
433C4×C8—1781no—
434C4×C8—1781no—
435C4×C8C22×D8911yescharacteristic
436C4×C8C22×D8911yescharacteristic
437#41—3561no—
438#41—1782no—
439C2×Q8—21no—
440C2×Q8—22no—
441C2×Q8—41no—
442C2×Q8—891no—
443C2×Q8C2×C4×D8911yescharacteristic
444C2×Q8C2×C4×D8911yescharacteristic
445C2×D4—891no—
446C2×D4—1781no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—1781no—
450C22×C4—1781no—
451C22×C4—1781no—
452C22×C4C2×C4×D8911yescharacteristic
453C22×C4C2×C22⋊D8911yescharacteristic
454C22×C4#1308357411yescharacteristic
455Q16—894no—
456Q16—1782no—
457SD16—1782no—
458SD16—1782no—
459M16—22no—
460M16—41no—
461M16—42no—
462M16—3562no—
463M16—3562no—
464M16—3562no—
465M16—3561no—
466M16—1782no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—891no—
470C2×C8—891no—
471C2×C8—891no—
472C2×C8—891no—
473C2×C8—1781no—
474C2×C8—1781no—
475C2×C8—1781no—
476C2×C8—1781no—
477C2×C8C2×C22⋊D8911yescharacteristic
478C2×C8C2×C22⋊D8911yescharacteristic
479C2×C8C2×C22⋊D8911yescharacteristic
480C2×C8C2×C22⋊D8911yescharacteristic
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—891no—
487C4⋊C4—891no—
488C4⋊C4—1781no—
489C4⋊C4—1781no—
490C4⋊C4C2×C4×D8911yescharacteristic
491C4⋊C4C2×C4×D8911yescharacteristic
492C22⋊C4—1781no—
493C22⋊C4—1781no—
494C22⋊C4—1781no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—1781no—
500C42—1781no—
501C42—1781no—
502C42C2×C4×D8911yescharacteristic
503C42C2×C22⋊D8911yescharacteristic
504C42#1308357411yescharacteristic
505C42C23×D8911yescharacteristic
506C23—1781no—
507C23C4×C22⋊D8911yescharacteristic
508Q8—21no—
509Q8—21no—
510Q8—21no—
511Q8—42no—
512Q8—892no—
513Q8—1781no—
514Q8M16×D8912yes—
515D4—3562no—
516D4—1781no—
517D4—1781no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—81no—
529C2×C4—3562no—
530C2×C4—3561no—
531C2×C4—3561no—
532C2×C4—1781no—
533C2×C4—1781no—
534C2×C4—1781no—
535C2×C4—7121no—
536C2×C4C4×C22⋊D8911yescharacteristic
537C2×C4C4×C22⋊D8911yescharacteristic
538C2×C4C4×C22⋊D8911yescharacteristic
539C2×C4C22×C4×D8911yescharacteristic
540C2×C4C22×C22⋊D8911yescharacteristic
541C2×C4C2×#1308357411yescharacteristic
542C8—21no—
543C8—21no—
544C8—21no—
545C8—41no—
546C8—41no—
547C8—892no—
548C8—3561no—
549C8—3561no—
550C8—3561no—
551C8—1782no—
552C8—1781no—
553C8—1781no—
554C8—1781no—
555C8#1566804312yescomplemented
556C22—21no—
557C22—21no—
558C22—21no—
559C22—3561no—
560C22—3561no—
561C22—3561no—
562C22C2×C4×C22⋊D8911yescharacteristic
563C4—21no—
564C4—21no—
565C4—21no—
566C4—21no—
567C4—22no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—3561no—
572C4—3561no—
573C4—1782no—
574C4C2×M16×D8911yescharacteristic
575C4#1915236411yescharacteristic
576C4C2×#1566804311yescharacteristic
577C4#1915239611yescharacteristic
578C2—41no—
579C2—7121no—
580C2M16×C22⋊D8911yescharacteristic
581C2C4×#1566804311yescharacteristic
582C2#2602915911yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingQ8×M16×C89C2
socleC22×C89C2×C4×D4
prime radicalC2×#13083574C2×C4
cosocleC42×C89C24
hypercenterQ8×M16D89
hypocenterC89#18049
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
3. #26019152C2semidirect
4. #26019142C2semidirect
5. #26019130C2semidirect
6. #26019121C2semidirect
7. #26019100C2semidirect
8. #26019095C2semidirect
9. #26019069C2semidirect
10. C8×#13083564C2semidirect
11. #26019017C2semidirect
12. #26018699C2semidirect
14. #26018555C2semidirect
15. Q8×M16×C89C2semidirect
154. #13083564M16semidirect
255. C8×C89#137semidirect
293. Q8×M16D89semidirect
307. C89#18049semidirect
555. C8#15668043semidirect

Maximal subgroups

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

subgroupindex
#260292702
C4×#156680432
#260191522
#260191422
#260191302
#260191212
#260191002
#260190952
#260190692
C8×#130835642
#260190172
#260186992
#260186892
#260185552
Q8×M16×C892
#1804989