Finite Group Databasebeta

← C89×#26085 #36403933

Subgroups — C89×#26085 #36403933

Order 22784.

678 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C89×#3163C211yesmaximal, characteristic, complemented
2C89×#3084C211yesmaximal, characteristic
3C89×#3083C211yesmaximal, characteristic
4C89×#2913C211yesmaximal, characteristic
5C89×#2090C211yesmaximal, characteristic, complemented
6C89×#2089C211yesmaximal, characteristic, complemented
7C89×#2061C211yesmaximal, characteristic, complemented
8C89×#2060C211yesmaximal, characteristic, complemented
9C89×#1952C211yesmaximal, characteristic, complemented
10C89×#1946C211yesmaximal, characteristic, complemented
11C89×#1860C211yesmaximal, characteristic
12C89×#1859C211yesmaximal, characteristic
13C89×#1847C211yesmaximal, characteristic, complemented
14C89×#1815C211yesmaximal, characteristic
15C89×#1765C211yesmaximal, characteristic
16C2×#131×C89C2211yescharacteristic
17#544×C89C2211yescharacteristic
18#544×C89C2211yescharacteristic
19#520×C89C2211yescharacteristic
20#520×C89C2211yescharacteristic
21#501×C89—21no—
22#501×C89—21no—
23#499×C89—21no—
24#498×C89—21no—
25#479×C89—21no—
26#479×C89—21no—
27#478×C89—21no—
28#478×C89—21no—
29#477×C89—21no—
30#476×C89—21no—
31#475×C89—21no—
32#474×C89—21no—
33C2×#108×C89C2211yescharacteristic
34#428×C89—21no—
35#428×C89—21no—
36#428×C89C2211yescharacteristic
37#428×C89C2211yescharacteristic
38#428×C89C2211yescharacteristic
39#428×C89C2211yescharacteristic
40#428×C89C2211yescharacteristic
41#428×C89C2211yescharacteristic
42#427×C89—21no—
43#427×C89C2211yescharacteristic
44C2×#107×C89—21no—
45C2×#107×C89C2211yescharacteristic
46C2×#107×C89C2211yescharacteristic
47C2×#106×C89C2211yescharacteristic
48C2×#106×C89C2211yescharacteristic
49#424×C89C2211yescharacteristic
50#424×C89C2211yescharacteristic
51C2×#104×C89C2211yescharacteristic
52#417×C89C2211yescharacteristic
53#417×C89C2211yescharacteristic
54#416×C89C2211yescharacteristic
55#405×C89C2211yescharacteristic
56C2×#97×C89C2211yescharacteristic
57#384×C89C2211yescharacteristic
58#382×C89C2211yescharacteristic
59#343×C89C2211yescharacteristic
60#343×C89C2211yescharacteristic
61#343×C89C2211yescharacteristic
62#343×C89C2211yescharacteristic
63#341×C89C2211yescharacteristic
64#339×C89C2211yescharacteristic
65#339×C89C2211yescharacteristic
66#339×C89C2211yescharacteristic
67C2×#41×C89C2311yescharacteristic
68#131×C89—22no—
69#131×C89Q814yescomplemented
70C2×M16×C89—21no—
71C2×M16×C89C2311yescharacteristic
72C2×M16×C89C2311yescharacteristic
73C2×M16×C89C2311yescharacteristic
74C2×M16×C89C2311yescharacteristic
75C22×C8×C89—21no—
76C22×C8×C89C2311yescharacteristic
77C22×C8×C89C2311yescharacteristic
78C4⋊Q8×C89—21no—
79C4⋊Q8×C89—21no—
80C4⋊Q8×C89—21no—
81C4⋊Q8×C89—21no—
82#125×C89—21no—
83#125×C89—21no—
84#125×C89—21no—
85#125×C89—21no—
86C22⋊Q8×C89—41no—
87C22⋊Q8×C89—41no—
88C4×Q8×C89—21no—
89C4×Q8×C89—21no—
90C4×D4×C89—21no—
91C4×D4×C89—21no—
92C4×D4×C89—41no—
93C4×D4×C89—41no—
94#117×C89—21no—
95#117×C89—21no—
96#117×C89C2311yescharacteristic
97#117×C89C2311yescharacteristic
98#117×C89C2311yescharacteristic
99C2×C4⋊C4×C89—21no—
100C2×C4⋊C4×C89—21no—
101C2×C4⋊C4×C89—41no—
102C2×C4⋊C4×C89C2311yescharacteristic
103C2×C4⋊C4×C89C2311yescharacteristic
104C2×C4⋊C4×C89C2311yescharacteristic
105C2×C4⋊C4×C89C2311yescharacteristic
106C2×C42×C89C2311yescharacteristic
107#108×C89—22no—
108#107×C89—21no—
109#107×C89—21no—
110#107×C89—21no—
111#107×C89—21no—
112#107×C89—21no—
113#107×C89—21no—
114#107×C89—21no—
115#107×C89—21no—
116#107×C89—21no—
117#107×C89—21no—
118#107×C89D411yescharacteristic
119#107×C89D411yescharacteristic
120#106×C89—21no—
121#106×C89—21no—
122#106×C89—21no—
123#106×C89—21no—
124#106×C89—21no—
125#106×C89—21no—
126#106×C89—21no—
127#106×C89—21no—
128#106×C89—21no—
129#106×C89—21no—
130#106×C89D411yescharacteristic
131#106×C89D411yescharacteristic
132C4⋊C8×C89—21no—
133C4⋊C8×C89—21no—
134C4⋊C8×C89—21no—
135C4⋊C8×C89—21no—
136#104×C89Q814yescomplemented
137#103×C89—21no—
138#103×C89—21no—
139#103×C89D411yescharacteristic
140#103×C89D411yescharacteristic
141#102×C89—21no—
142#102×C89—21no—
143#102×C89D411yescharacteristic
144#102×C89D411yescharacteristic
145#97×C89—21no—
146#97×C89—22no—
147C4×C8×C89—21no—
148#95×C89—41no—
149#41×C89—21no—
150#41×C89—21no—
151#41×C89C2×Q812yes—
152#41×C89C2×Q812yes—
153C2×Q8×C89—41no—
154C2×Q8×C89—41no—
155C2×Q8×C89C2×D411yescharacteristic
156C2×D4×C89—21no—
157C2×D4×C89C2×D411yescharacteristic
158C22×C4×C89—21no—
159C22×C4×C89—21no—
160C22×C4×C89—21no—
161C22×C4×C89—21no—
162C22×C4×C89—21no—
163C22×C4×C89—21no—
164C22×C4×C89—41no—
165C22×C4×C89—41no—
166C22×C4×C89C2411yescharacteristic
167M16×C89—21no—
168M16×C89—21no—
169M16×C89—24no—
170M16×C89—21no—
171M16×C89—21no—
172M16×C89—21no—
173M16×C89—21no—
174M16×C89C2×Q812yes—
175M16×C89C2×Q812yes—
176C2×C8×C89—21no—
177C2×C8×C89—21no—
178C2×C8×C89—21no—
179C2×C8×C89—21no—
180C2×C8×C89—24no—
181C2×C8×C89—21no—
182C2×C8×C89—21no—
183C2×C8×C89C2×D411yescharacteristic
184C2×C8×C89C2×D411yescharacteristic
185C2×C8×C89C2×D411yescharacteristic
186C2×C8×C89C2×D411yescharacteristic
187C2×C8×C89C2×Q812yes—
188C2×C8×C89#4112yes—
189C4⋊C4×C89—21no—
190C4⋊C4×C89—21no—
191C4⋊C4×C89—21no—
192C4⋊C4×C89—21no—
193C4⋊C4×C89—21no—
194C4⋊C4×C89—21no—
195C4⋊C4×C89—21no—
196C4⋊C4×C89—21no—
197C4⋊C4×C89—41no—
198C4⋊C4×C89—41no—
199C4⋊C4×C89—41no—
200C4⋊C4×C89—41no—
201C4⋊C4×C89—41no—
202C4⋊C4×C89—41no—
203C4⋊C4×C89—41no—
204C4⋊C4×C89—41no—
205C4⋊C4×C89—82no—
206C4⋊C4×C89C2×D411yescharacteristic
207C4⋊C4×C89C2×D411yescharacteristic
208C4⋊C4×C89C2×D411yescharacteristic
209C4⋊C4×C89C2×D411yescharacteristic
210C4⋊C4×C89C2×D411yescharacteristic
211C4⋊C4×C89C2×D411yescharacteristic
212C22⋊C4×C89—21no—
213C22⋊C4×C89—21no—
214C22⋊C4×C89—21no—
215C22⋊C4×C89—41no—
216C22⋊C4×C89—41no—
217C42×C89—21no—
218C42×C89—21no—
219C42×C89—21no—
220C42×C89—21no—
221C42×C89—41no—
222C42×C89—41no—
223C42×C89C2×Q812yes—
224C23×C89—21no—
225C23×C89H(5,𝔽2)11yescharacteristic
226Q8×C89—21no—
227Q8×C89—161no—
228Q8×C89—161no—
229Q8×C89C4⋊Q812yes—
230D4×C89—22no—
231D4×C89—22no—
232D4×C89—21no—
233D4×C89C4⋊Q812yes—
234C2×C4×C89—21no—
235C2×C4×C89—22no—
236C2×C4×C89—22no—
237C2×C4×C89—21no—
238C2×C4×C89—21no—
239C2×C4×C89—21no—
240C2×C4×C89—21no—
241C2×C4×C89—21no—
242C2×C4×C89—21no—
243C2×C4×C89—22no—
244C2×C4×C89—41no—
245C2×C4×C89—41no—
246C2×C4×C89—41no—
247C2×C4×C89—41no—
248C2×C4×C89—41no—
249C2×C4×C89—41no—
250C2×C4×C89—81no—
251C2×C4×C89—81no—
252C2×C4×C89—81no—
253C2×C4×C89—81no—
254C2×C4×C89—161no—
255C2×C4×C89—161no—
256C2×C4×C89C22×D411yescharacteristic
257C2×C4×C89C22×D411yescharacteristic
258C2×C4×C89C22×Q811yescharacteristic
259C2×C4×C89C22×Q811yescharacteristic
260C2×C4×C89C2×#4111yescharacteristic
261C2×C4×C892−1+411yescharacteristic
262C8×C89—22no—
263C8×C89—21no—
264C8×C89—21no—
265C8×C89—22no—
266C8×C89—41no—
267C8×C89—41no—
268C8×C89C22⋊Q812yes—
269C8×C89C22⋊Q812yes—
270C22×C89—21no—
271C22×C89—21no—
272C22×C89—22no—
273C22×C89—41no—
274C22×C89—41no—
275C22×C89#54411yescharacteristic
276C22×C89D4211yescharacteristic
277C22×C89#55411yescharacteristic
278C4×C89—22no—
279C4×C89—42no—
280C4×C89—41no—
281C4×C89—41no—
282C4×C89—41no—
283C4×C89—41no—
284C4×C89—41no—
285C4×C89—41no—
286C4×C89—81no—
287C4×C89—81no—
288C4×C89—81no—
289C4×C89—81no—
290C4×C89C2×C22⋊Q811yescharacteristic
291C4×C89C2×C4⋊Q811yescharacteristic
292C4×C89#54711yescharacteristic
293C4×C89D4×Q811yescharacteristic
294#26085C8911yesmaximal, characteristic, direct product factor, Sylow
295C2×C89—21no—
296C2×C89—21no—
297C2×C89—42no—
298C2×C89#244011yescharacteristic
299C2×C89#328811yescharacteristic
300C2×C89#328911yescharacteristic
301#3163C2×C8911yescharacteristic, complemented
302#3084C2×C8911yescharacteristic
303#3083C2×C8911yescharacteristic
304#2913C2×C8911yescharacteristic
305#2090C2×C8911yescharacteristic, complemented
306#2089C2×C8911yescharacteristic, complemented
307#2061C2×C8911yescharacteristic, complemented
308#2060C2×C8911yescharacteristic, complemented
309#1952C2×C8911yescharacteristic, complemented
310#1946C2×C8911yescharacteristic, complemented
311#1860C2×C8911yescharacteristic
312#1859C2×C8911yescharacteristic
313#1847C2×C8911yescharacteristic, complemented
314#1815C2×C8911yescharacteristic
315#1765C2×C8911yescharacteristic
316C89#2608511yescharacteristic, direct product factor, Sylow
317C2×#131C22×C8911yescharacteristic
318#544C22×C8911yescharacteristic
319#544C22×C8911yescharacteristic
320#520C22×C8911yescharacteristic
321#520C22×C8911yescharacteristic
322#501—21no—
323#501—21no—
324#499—21no—
325#498—21no—
326#479—21no—
327#479—21no—
328#478—21no—
329#478—21no—
330#477—21no—
331#476—21no—
332#475—21no—
333#474—21no—
334C2×#108C22×C8911yescharacteristic
335#428—21no—
336#428—21no—
337#428C22×C8911yescharacteristic
338#428C22×C8911yescharacteristic
339#428C22×C8911yescharacteristic
340#428C22×C8911yescharacteristic
341#428C22×C8911yescharacteristic
342#428C22×C8911yescharacteristic
343#427—21no—
344#427C22×C8911yescharacteristic
345C2×#107—21no—
346C2×#107C22×C8911yescharacteristic
347C2×#107C22×C8911yescharacteristic
348C2×#106C22×C8911yescharacteristic
349C2×#106C22×C8911yescharacteristic
350#424C22×C8911yescharacteristic
351#424C22×C8911yescharacteristic
352C2×#104C22×C8911yescharacteristic
353#417C22×C8911yescharacteristic
354#417C22×C8911yescharacteristic
355#416C22×C8911yescharacteristic
356#405C22×C8911yescharacteristic
357C2×#97C22×C8911yescharacteristic
358#384C22×C8911yescharacteristic
359#382C22×C8911yescharacteristic
360#343C22×C8911yescharacteristic
361#343C22×C8911yescharacteristic
362#343C22×C8911yescharacteristic
363#343C22×C8911yescharacteristic
364#341C22×C8911yescharacteristic
365#339C22×C8911yescharacteristic
366#339C22×C8911yescharacteristic
367#339C22×C8911yescharacteristic
368C2×#41C23×C8911yescharacteristic
369#131—22no—
370#131Q8×C8914yescomplemented
371C2×M16—21no—
372C2×M16C23×C8911yescharacteristic
373C2×M16C23×C8911yescharacteristic
374C2×M16C23×C8911yescharacteristic
375C2×M16C23×C8911yescharacteristic
376C22×C8—21no—
377C22×C8C23×C8911yescharacteristic
378C22×C8C23×C8911yescharacteristic
379C4⋊Q8—21no—
380C4⋊Q8—21no—
381C4⋊Q8—21no—
382C4⋊Q8—21no—
383#125—21no—
384#125—21no—
385#125—21no—
386#125—21no—
387C22⋊Q8—41no—
388C22⋊Q8—41no—
389C4×Q8—21no—
390C4×Q8—21no—
391C4×D4—21no—
392C4×D4—21no—
393C4×D4—41no—
394C4×D4—41no—
395#117—21no—
396#117—21no—
397#117C23×C8911yescharacteristic
398#117C23×C8911yescharacteristic
399#117C23×C8911yescharacteristic
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—41no—
403C2×C4⋊C4C23×C8911yescharacteristic
404C2×C4⋊C4C23×C8911yescharacteristic
405C2×C4⋊C4C23×C8911yescharacteristic
406C2×C4⋊C4C23×C8911yescharacteristic
407C2×C42C23×C8911yescharacteristic
408#108—22no—
409#107—21no—
410#107—21no—
411#107—21no—
412#107—21no—
413#107—21no—
414#107—21no—
415#107—21no—
416#107—21no—
417#107—21no—
418#107—21no—
419#107D4×C8911yescharacteristic
420#107D4×C8911yescharacteristic
421#106—21no—
422#106—21no—
423#106—21no—
424#106—21no—
425#106—21no—
426#106—21no—
427#106—21no—
428#106—21no—
429#106—21no—
430#106—21no—
431#106D4×C8911yescharacteristic
432#106D4×C8911yescharacteristic
433C4⋊C8—21no—
434C4⋊C8—21no—
435C4⋊C8—21no—
436C4⋊C8—21no—
437#104Q8×C8914yescomplemented
438#103—21no—
439#103—21no—
440#103D4×C8911yescharacteristic
441#103D4×C8911yescharacteristic
442#102—21no—
443#102—21no—
444#102D4×C8911yescharacteristic
445#102D4×C8911yescharacteristic
446#97—21no—
447#97—22no—
448C4×C8—21no—
449#95—41no—
450#41—21no—
451#41—21no—
452#41C2×Q8×C8912yes—
453#41C2×Q8×C8912yes—
454C2×Q8—41no—
455C2×Q8—41no—
456C2×Q8C2×D4×C8911yescharacteristic
457C2×D4—21no—
458C2×D4C2×D4×C8911yescharacteristic
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—41no—
466C22×C4—41no—
467C22×C4C24×C8911yescharacteristic
468M16—21no—
469M16—21no—
470M16—24no—
471M16—21no—
472M16—21no—
473M16—21no—
474M16—21no—
475M16C2×Q8×C8912yes—
476M16C2×Q8×C8912yes—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—24no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8C2×D4×C8911yescharacteristic
485C2×C8C2×D4×C8911yescharacteristic
486C2×C8C2×D4×C8911yescharacteristic
487C2×C8C2×D4×C8911yescharacteristic
488C2×C8C2×Q8×C8912yes—
489C2×C8#41×C8912yes—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—41no—
499C4⋊C4—41no—
500C4⋊C4—41no—
501C4⋊C4—41no—
502C4⋊C4—41no—
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—41no—
506C4⋊C4—82no—
507C4⋊C4C2×D4×C8911yescharacteristic
508C4⋊C4C2×D4×C8911yescharacteristic
509C4⋊C4C2×D4×C8911yescharacteristic
510C4⋊C4C2×D4×C8911yescharacteristic
511C4⋊C4C2×D4×C8911yescharacteristic
512C4⋊C4C2×D4×C8911yescharacteristic
513C22⋊C4—21no—
514C22⋊C4—21no—
515C22⋊C4—21no—
516C22⋊C4—41no—
517C22⋊C4—41no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—41no—
523C42—41no—
524C42C2×Q8×C8912yes—
525C23—21no—
526C23H(5,𝔽2)×C8911yescharacteristic
527Q8—21no—
528Q8—161no—
529Q8—161no—
530Q8C4⋊Q8×C8912yes—
531D4—22no—
532D4—22no—
533D4—21no—
534D4C4⋊Q8×C8912yes—
535C2×C4—21no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4—81no—
554C2×C4—81no—
555C2×C4—161no—
556C2×C4—161no—
557C2×C4C22×D4×C8911yescharacteristic
558C2×C4C22×D4×C8911yescharacteristic
559C2×C4C22×Q8×C8911yescharacteristic
560C2×C4C22×Q8×C8911yescharacteristic
561C2×C4C2×#41×C8911yescharacteristic
562C2×C42−1+4×C8911yescharacteristic
563C8—22no—
564C8—21no—
565C8—21no—
566C8—22no—
567C8—41no—
568C8—41no—
569C8C22⋊Q8×C8912yes—
570C8C22⋊Q8×C8912yes—
571C22—21no—
572C22—21no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22#544×C8911yescharacteristic
577C22D42×C8911yescharacteristic
578C22#554×C8911yescharacteristic
579C4—22no—
580C4—42no—
581C4—41no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—41no—
586C4—41no—
587C4—81no—
588C4—81no—
589C4—81no—
590C4—81no—
591C4C2×C22⋊Q8×C8911yescharacteristic
592C4C2×C4⋊Q8×C8911yescharacteristic
593C4#547×C8911yescharacteristic
594C4D4×Q8×C8911yescharacteristic
595C2—21no—
596C2—21no—
597C2—42no—
598C2C89×#244011yescharacteristic
599C2C89×#328811yescharacteristic
600C2C89×#328911yescharacteristic
subgroupG/H
center, socleC22×C89D42
commutator, Frattini, cosocleC22×C4C24×C89
prime radicalC2×#41×C89C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C89 ◁ C22×C4×C89 ◁ G

Complements

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

NH ≅ G/Ntype
1. C89×#3163C2semidirect
5. C89×#2090C2semidirect
6. C89×#2089C2semidirect
7. C89×#2061C2semidirect
8. C89×#2060C2semidirect
9. C89×#1952C2semidirect
10. C89×#1946C2semidirect
13. C89×#1847C2semidirect
69. #131×C89Q8semidirect
136. #104×C89Q8semidirect
294. #26085C89direct (N × H)
301. #3163C2×C89semidirect
305. #2090C2×C89semidirect
306. #2089C2×C89semidirect
307. #2061C2×C89semidirect
308. #2060C2×C89semidirect
309. #1952C2×C89semidirect
310. #1946C2×C89semidirect
313. #1847C2×C89semidirect
316. C89#26085direct (N × H)
370. #131Q8×C89semidirect
437. #104Q8×C89semidirect

Maximal subgroups

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

subgroupindex
C89×#31632
C89×#30842
C89×#30832
C89×#29132
C89×#20902
C89×#20892
C89×#20612
C89×#20602
C89×#19522
C89×#19462
C89×#18602
C89×#18592
C89×#18472
C89×#18152
C89×#17652
#2608589