Finite Group Databasebeta

← #36526044

Subgroups — #36526044

Order 22784.

674 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26029384C211yesmaximal, characteristic, complemented
2#26029289C211yesmaximal, characteristic, complemented
3C4×#15668037C211yesmaximal, characteristic
4#26019163C211yesmaximal, characteristic, complemented
5#26019134C211yesmaximal, characteristic, complemented
6#26019127C211yesmaximal, characteristic, complemented
7#26019119C211yesmaximal, characteristic, complemented
8#26019092C211yesmaximal, characteristic, complemented
9#26019084C211yesmaximal, characteristic, complemented
10#26019067C211yesmaximal, characteristic, complemented
11#26019049C211yesmaximal, characteristic, complemented
12#26019020C211yesmaximal, characteristic, complemented
13#26018678C211yesmaximal, characteristic, complemented
14#26018547C211yesmaximal, characteristic, complemented
15C89×#1488C211yesmaximal, characteristic, complemented
16C2×#15668037C411yescharacteristic
17C2×C4×D356C2211yescharacteristic
18Q8×#10623494C2211yescharacteristic, complemented
19Q8×#10623494C2211yescharacteristic, complemented
20#19151708C411yescharacteristic
21#19151708C411yescharacteristic
22#19151696C411yescharacteristic
23#19151696C411yescharacteristic
24C4×#13083561C2211yescharacteristic
25C4×#13083561C2211yescharacteristic
26C4×#13083559C2211yescharacteristic
27C4×#13083559C2211yescharacteristic
28#19151676—21no—
29#19151676—21no—
30#19151676C2211yescharacteristic, complemented
31#19151676C2211yescharacteristic, complemented
32#19151675C2211yescharacteristic
33#19151675C2211yescharacteristic
34#19151671—21no—
35#19151671—21no—
36#19151663C2211yescharacteristic
37#19151663C2211yescharacteristic
38#19151651—21no—
39#19151651C2211yescharacteristic
40#19151641C411yescharacteristic
41#19151622—21no—
42#19151622C2211yescharacteristic
43#19151602C411yescharacteristic
44#19151599C411yescharacteristic
45#19151560—21no—
46C22⋊C8×D89—21no—
47C4×#13083558C2211yescharacteristic
48#19151350C2211yescharacteristic, complemented
49#19151350C2211yescharacteristic, complemented
50#19151348C2211yescharacteristic, complemented
51#19151348C2211yescharacteristic, complemented
52#19151327C2211yescharacteristic
53#19151327C2211yescharacteristic
54#19151327C2211yescharacteristic
55#19151327C2211yescharacteristic
56#19151320C2211yescharacteristic, complemented
57#19151320C2211yescharacteristic, complemented
58#19151319C2211yescharacteristic, complemented
59C8×#10623494C2211yescharacteristic, complemented
60C4×#41×C89C2211yescharacteristic
61#432×C89C2211yescharacteristic
62#431×C89C2211yescharacteristic
63Q8⋊C8×C89C2211yescharacteristic, complemented
64Q8⋊C8×C89C2211yescharacteristic, complemented
65D4⋊C8×C89C2211yescharacteristic, complemented
66D4⋊C8×C89C2211yescharacteristic, complemented
67C22×D356C2×C411yescharacteristic
68C22×C4×D89—21no—
69#15668037—24no—
70C2×#13083561C2×C411yescharacteristic
71C2×#13083561C2×C411yescharacteristic
72C2×#13083559C2×C411yescharacteristic
73C2×#13083559C2×C411yescharacteristic
74C2×#13083558—21no—
75C2×#13083558—21no—
76C2×#13083558—21no—
77C2×#13083558C2×C411yescharacteristic
78C2×#13083558C2×C411yescharacteristic
79C2×#13083557—21no—
80C2×#13083557—21no—
81C2×C8×D89—21no—
82C2×C8×D89—21no—
83C4×D356—21no—
84C4×D356—21no—
85C4×D356C2×C411yescharacteristic
86C4×D356C2×C411yescharacteristic
87C4×D356C2311yescharacteristic
88C4×D356C2311yescharacteristic
89C2×C4⋊Dic89C2×C411yescharacteristic
90C2×#13083551—21no—
91#15667945C2×C411yescharacteristic
92#15667945C2×C411yescharacteristic
93#15667943C2×C411yescharacteristic
94#15667943C2×C411yescharacteristic
95C4⋊#10623494—21no—
96C4⋊#10623494—21no—
97C4⋊#10623494—21no—
98C4⋊#10623494D411yescharacteristic
99C4⋊#10623494D411yescharacteristic
100C4⋊#10623494C2311yescharacteristic
101C4⋊#10623494C2311yescharacteristic
102#15667936C2×C411yescharacteristic
103#15667936C2×C411yescharacteristic
104#15667933C2×C411yescharacteristic
105#15667933C2×C411yescharacteristic
106#15667932C2×C411yescharacteristic
107#15667932C2×C411yescharacteristic
108#15667932C2×C411yescharacteristic
109#15667932C2×C411yescharacteristic
110C22⋊#10623494—21no—
111C22⋊#10623494—21no—
112C22⋊#10623494D411yescharacteristic
113C22⋊#10623494D411yescharacteristic
114#15667924—21no—
115#15667924—21no—
116#15667924—21no—
117#15667924—21no—
118#15667923C2×C411yescharacteristic
119#15667923C2×C411yescharacteristic
120C4×#10623494—21no—
121C4×#10623494—21no—
122C4×#10623494C2311yescharacteristic
123C4×#10623494C2311yescharacteristic
124C4×#10623494C2311yescharacteristic
125C4×#10623494C2311yescharacteristic
126C2×#41×C89C2×C411yescharacteristic
127C4×Q8×C89C2×C411yescharacteristic
128C4×Q8×C89C2311yescharacteristic
129C4×D4×C89—21no—
130C4×D4×C89—21no—
131C4×D4×C89C2×C411yescharacteristic
132C4×D4×C89C2311yescharacteristic
133#117×C89—21no—
134#117×C89C2×C411yescharacteristic
135C2×C42×C89—21no—
136C2×C42×C89C2311yescharacteristic
137C4⋊C8×C89—21no—
138C4⋊C8×C89C2311yescharacteristic
139C4⋊C8×C89C2311yescharacteristic
140C22⋊C8×C89—21no—
141C22⋊C8×C89—21no—
142#97×C89—21no—
143C4×C8×C89C2311yescharacteristic
144C4×C8×C89C2311yescharacteristic
145C23×D89—21no—
146C2×D356—22no—
147C2×D356—21no—
148C2×D356—41no—
149C2×D356C22×C411yescharacteristic
150C2×D356C22×C411yescharacteristic
151C22×Dic89—21no—
152C2×C4×D89—21no—
153C2×C4×D89—21no—
154C2×C4×D89—21no—
155C2×C4×D89—21no—
156C2×C4×D89—81no—
157C2×C4×D89—81no—
158#13083561—22no—
159#13083561M1614yescomplemented
160#13083559—22no—
161#13083559—22no—
162#13083558—22no—
163#13083558—41no—
164#13083558—81no—
165#13083558—81no—
166#13083558—81no—
167#13083557—81no—
168#13083557—81no—
169C2×#10623494—21no—
170C2×#10623494—21no—
171C2×#10623494C22×C411yescharacteristic
172C2×#10623494C22×C411yescharacteristic
173C2×#10623494C22×C411yescharacteristic
174C2×#10623494C22×C411yescharacteristic
175C2×#10623494C2×D411yescharacteristic
176C2×#10623494C2×D411yescharacteristic
177C2×#10623494C2×D411yescharacteristic
178C2×#10623494C2×D411yescharacteristic
179C8×D89—81no—
180C8×D89—81no—
181C4⋊Dic89—21no—
182C4⋊Dic89C22×C411yescharacteristic
183C4⋊Dic89C22×C411yescharacteristic
184#13083551—21no—
185#13083551—21no—
186#13083551—21no—
187#13083551—21no—
188#41×C89—22no—
189#41×C89—41no—
190C2×Q8×C89C22×C411yescharacteristic
191C2×D4×C89—21no—
192C2×D4×C89C22×C411yescharacteristic
193C22×C4×C89—21no—
194C22×C4×C89—21no—
195C22×C4×C89—21no—
196C22×C4×C89C22×C411yescharacteristic
197C22×C4×C89C2×D411yescharacteristic
198C22×C4×C89#4111yescharacteristic
199C2×C8×C89—21no—
200C2×C8×C89—21no—
201C2×C8×C89—21no—
202C2×C8×C89—21no—
203C4⋊C4×C89—21no—
204C4⋊C4×C89—21no—
205C4⋊C4×C89C22×C411yescharacteristic
206C4⋊C4×C89C22×C411yescharacteristic
207C22⋊C4×C89—21no—
208C22⋊C4×C89—21no—
209C22⋊C4×C89—21no—
210C42×C89—21no—
211C42×C89—21no—
212C42×C89—21no—
213C42×C89C22×C411yescharacteristic
214C42×C89C2×D411yescharacteristic
215C42×C89#4111yescharacteristic
216C42×C89C2411yescharacteristic
217C23×C89—21no—
218C23×C89C4×D411yescharacteristic
219C22×D89—21no—
220C22×D89—21no—
221C22×D89—42no—
222C22×D89—42no—
223C22×D89—81no—
224Q8×C89—21no—
225Q8×C89C2×M1612yes—
226D4×C89—21no—
227D4×C89—21no—
228D4×C89—42no—
229C2×C4×C89—21no—
230C2×C4×C89—21no—
231C2×C4×C89—21no—
232C2×C4×C89—21no—
233C2×C4×C89—21no—
234C2×C4×C89—21no—
235C2×C4×C89—21no—
236C2×C4×C89—41no—
237C2×C4×C89—42no—
238C2×C4×C89—42no—
239C2×C4×C89—41no—
240C2×C4×C89—41no—
241C2×C4×C89C4×D411yescharacteristic
242C2×C4×C89C4×D411yescharacteristic
243C2×C4×C89C4×D411yescharacteristic
244C2×C4×C89C23×C411yescharacteristic
245C2×C4×C89C22×D411yescharacteristic
246C2×C4×C89C2×#4111yescharacteristic
247C2×Dic89—21no—
248C2×Dic89—21no—
249C2×Dic89—81no—
250D356—21no—
251D356—21no—
252D356—21no—
253D356—42no—
254D356C2×M1612yes—
255C4×D89—41no—
256C4×D89—41no—
257C4×D89—41no—
258C4×D89—41no—
259C4×D89—81no—
260C4×D89—81no—
261C8×C89—41no—
262C8×C89—41no—
263C8×C89—41no—
264C8×C89—41no—
265#10623494—21no—
266#10623494—21no—
267#10623494—21no—
268#10623494—21no—
269#10623494—21no—
270#10623494—21no—
271#10623494—21no—
272#10623494—41no—
273#10623494—41no—
274#10623494C2×M1612yes—
275C22×C89—21no—
276C22×C89—21no—
277C22×C89—21no—
278C22×C89—41no—
279C22×C89—41no—
280C22×C89—41no—
281C22×C89C2×C4×D411yescharacteristic
282C2×D89—22no—
283C2×D89—41no—
284C2×D89—41no—
285C2×D89—41no—
286C2×D89—41no—
287C2×D89—41no—
288C2×D89—84no—
289C4×C89—21no—
290C4×C89—22no—
291C4×C89—21no—
292C4×C89—21no—
293C4×C89—21no—
294C4×C89—41no—
295C4×C89—41no—
296C4×C89C22×M1611yescharacteristic
297C4×C89#56811yescharacteristic
298C4×C89#57611yescharacteristic
299C4×C89#57711yescharacteristic
300Dic89—41no—
301Dic89—41no—
302#18052—891nomaximal, Sylow
303C2×C89—41no—
304C2×C89—81no—
305C2×C89D4×M1611yescharacteristic
306C2×C89#294611yescharacteristic
307C2×C89#295511yescharacteristic
308D89—42no—
309D89—81no—
310#2991—891no—
311#2974—891no—
312C4×AGL(1,ℤ/8)—891no—
313#1594—891no—
314#1589—891no—
315#1588—891no—
316#1586—891no—
317#1583—891no—
318#1581—891no—
319SD16⋊C8—891no—
320C8×SD16—891no—
321#1569—891no—
322#1488D8911yescharacteristic, complemented
323#1486—891no—
324#1453—891no—
325C89#1805211yescharacteristic, complemented, Sylow
326C2×AGL(1,ℤ/8)—891no—
327C4×#41C2×D8911yescharacteristic
328C2×C4×D4—891no—
329C8×Q8—891no—
330C8×Q8—891no—
331#442—891no—
332#442—891no—
333#440—891no—
334#440—891no—
335C4×SD16—891no—
336C4×SD16—891no—
337C4×D8—891no—
338C4×D8—891no—
339#436—891no—
340#436—891no—
341#436—891no—
342#436—891no—
343#436—1781no—
344#436—1781no—
345#435—1781no—
346#435—1781no—
347C8×D4—891no—
348C8×D4—891no—
349#432C2×D8911yescharacteristic
350#431—891no—
351#431—1781no—
352#431C2×D8911yescharacteristic
353#428—891no—
354#423—891no—
355#423—1781no—
356#418—891no—
357#417—891no—
358#407—1781no—
359C2×C22⋊C8—1781no—
360C4×M16—891no—
361#335—891no—
362#335—891no—
363#334—891no—
364#334—891no—
365Q8⋊C8C2×D8911yescharacteristic, complemented
366Q8⋊C8C2×D8911yescharacteristic, complemented
367D4⋊C8—891no—
368D4⋊C8—891no—
369D4⋊C8—891no—
370D4⋊C8—891no—
371D4⋊C8C2×D8911yescharacteristic, complemented
372D4⋊C8C2×D8911yescharacteristic, complemented
373#322—891no—
374#322—891no—
375#322—891no—
376C82—891no—
377C2×#41C4×D8911yescharacteristic
378C22×D4—891no—
379C23×C4—1781no—
380AGL(1,ℤ/8)—1784no—
381C2×SD16—891no—
382C2×SD16—891no—
383C2×D8—891no—
384C2×D8—891no—
385C2×M16—891no—
386C2×M16—891no—
387C2×M16—1781no—
388C2×M16—1781no—
389C2×M16—1781no—
390C2×M16—1781no—
391C2×M16—1781no—
392C22×C8—1781no—
393C22×C8—1781no—
394C4×Q8C4×D8911yescharacteristic
395C4×Q8C22×D8911yescharacteristic
396C4×D4—21no—
397C4×D4—21no—
398C4×D4—891no—
399C4×D4—891no—
400C4×D4—891no—
401C4×D4—891no—
402C4×D4—1781no—
403C4×D4—1781no—
404C4×D4C4×D8911yescharacteristic
405C4×D4C22×D8911yescharacteristic
406#117—21no—
407#117C4×D8911yescharacteristic
408C2×C4⋊C4—891no—
409C2×C22⋊C4—1781no—
410C2×C42—21no—
411C2×C42C22×D8911yescharacteristic
412#107—891no—
413#107—891no—
414#106—891no—
415#106—891no—
416C4⋊C8—21no—
417C4⋊C8—891no—
418C4⋊C8—891no—
419C4⋊C8—891no—
420C4⋊C8—891no—
421C4⋊C8—1781no—
422C4⋊C8—1781no—
423C4⋊C8—1781no—
424C4⋊C8C22×D8911yescharacteristic
425C4⋊C8C22×D8911yescharacteristic
426#103—891no—
427#103—891no—
428#102—891no—
429#102—891no—
430#102—891no—
431#102—891no—
432#102—891no—
433#102—891no—
434C22⋊C8—21no—
435C22⋊C8—21no—
436C22⋊C8—891no—
437C22⋊C8—891no—
438C22⋊C8—1781no—
439C22⋊C8—1781no—
440C22⋊C8—1781no—
441C22⋊C8—1781no—
442C22⋊C8—1781no—
443C22⋊C8—1781no—
444#97—21no—
445#97—891no—
446#97—891no—
447C4×C8—891no—
448C4×C8—891no—
449C4×C8—891no—
450C4×C8—891no—
451C4×C8—1781no—
452C4×C8—1781no—
453C4×C8C22×D8911yescharacteristic
454C4×C8C22×D8911yescharacteristic
455C24—1781no—
456#41—22no—
457#41—41no—
458C2×Q8C2×C4×D8911yescharacteristic
459C2×D4—21no—
460C2×D4—891no—
461C2×D4—891no—
462C2×D4—3561no—
463C2×D4—1782no—
464C2×D4—1781no—
465C2×D4C2×C4×D8911yescharacteristic
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—1781no—
470C22×C4—1781no—
471C22×C4—1781no—
472C22×C4—1781no—
473C22×C4—1781no—
474C22×C4—7121no—
475C22×C4—7121no—
476C22×C4C2×C4×D8911yescharacteristic
477C22×C4D4×D8911yescharacteristic
478C22×C4#1308357311yescharacteristic
479SD16—894no—
480SD16—1782no—
481D8—1782no—
482D8—1782no—
483M16—3561no—
484M16—1782no—
485M16—7121no—
486M16—7121no—
487M16—7121no—
488M16—7121no—
489M16—7121no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—891no—
495C2×C8—891no—
496C2×C8—891no—
497C2×C8—891no—
498C2×C8—891no—
499C2×C8—891no—
500C2×C8—891no—
501C2×C8—891no—
502C2×C8—1781no—
503C2×C8—1781no—
504C2×C8—7121no—
505C2×C8—7121no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—891no—
509C4⋊C4—891no—
510C4⋊C4—1781no—
511C4⋊C4C2×C4×D8911yescharacteristic
512C4⋊C4C2×C4×D8911yescharacteristic
513C22⋊C4—21no—
514C22⋊C4—21no—
515C22⋊C4—21no—
516C22⋊C4—1781no—
517C22⋊C4—1781no—
518C22⋊C4—1781no—
519C22⋊C4—1781no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42C2×C4×D8911yescharacteristic
524C42D4×D8911yescharacteristic
525C42#1308357311yescharacteristic
526C42C23×D8911yescharacteristic
527C23—21no—
528C23—3562no—
529C23—3562no—
530C23—1781no—
531C23—1781no—
532C23—7121no—
533C23#1566797211yescharacteristic
534Q8—21no—
535Q8M16×D8912yes—
536D4—21no—
537D4—21no—
538D4—42no—
539D4—892no—
540D4—3562no—
541D4—1781no—
542D4—1781no—
543D4—1781no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—3561no—
557C2×C4—3561no—
558C2×C4—3561no—
559C2×C4—3561no—
560C2×C4—1781no—
561C2×C4—1781no—
562C2×C4—7121no—
563C2×C4—7121no—
564C2×C4—7121no—
565C2×C4#1566797211yescharacteristic
566C2×C4#1566797211yescharacteristic
567C2×C4#1566797211yescharacteristic
568C2×C4C22×C4×D8911yescharacteristic
569C2×C4C2×D4×D8911yescharacteristic
570C2×C4C2×#1308357311yescharacteristic
571C8—41no—
572C8—41no—
573C8—41no—
574C8—41no—
575C8—892no—
576C8—3561no—
577C8—3561no—
578C8—1781no—
579C8—1781no—
580C8—1781no—
581C8—1781no—
582C8—1781no—
583C8—1781no—
584C8—1781no—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—3561no—
592C22—3561no—
593C22—3561no—
594C22—3561no—
595C22—3561no—
596C22—1782no—
597C22—7124no—
598C22C2×#1566797211yescharacteristic
599C4—21no—
600C4—21no—
601C4—21no—
602C4—21no—
603C4—22no—
604C4—41no—
605C4—41no—
606C4—3561no—
607C4—3561no—
608C4C2×M16×D8911yescharacteristic
609C4#1915236411yescharacteristic
610C4#1915240011yescharacteristic
611C4#1915240511yescharacteristic
612C2—41no—
613C2—81no—
614C2—3562no—
615C2—7121no—
616C2#2602898411yescharacteristic
617C2#2602908711yescharacteristic
618C2#2602915111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#1488C2
socleC22×C89C2×C4×D4
prime radicalC2×#15668037C4
cosocleC42×C89C24
hypercenter#1488D89
hypocenterC89#18052
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1488

Complements

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

NH ≅ G/Ntype
1. #26029384C2semidirect
2. #26029289C2semidirect
4. #26019163C2semidirect
5. #26019134C2semidirect
6. #26019127C2semidirect
7. #26019119C2semidirect
8. #26019092C2semidirect
9. #26019084C2semidirect
10. #26019067C2semidirect
11. #26019049C2semidirect
12. #26019020C2semidirect
13. #26018678C2semidirect
14. #26018547C2semidirect
15. C89×#1488C2semidirect
18. Q8×#10623494C22semidirect
19. Q8×#10623494C22semidirect
30. #19151676C22semidirect
31. #19151676C22semidirect
48. #19151350C22semidirect
49. #19151350C22semidirect
50. #19151348C22semidirect
51. #19151348C22semidirect
56. #19151320C22semidirect
57. #19151320C22semidirect
58. #19151319C22semidirect
59. C8×#10623494C22semidirect
63. Q8⋊C8×C89C22semidirect
64. Q8⋊C8×C89C22semidirect
65. D4⋊C8×C89C22semidirect
66. D4⋊C8×C89C22semidirect
159. #13083561M16semidirect
322. #1488D89semidirect
325. C89#18052semidirect
365. Q8⋊C8C2×D89semidirect
366. Q8⋊C8C2×D89semidirect
371. D4⋊C8C2×D89semidirect
372. D4⋊C8C2×D89semidirect

Maximal subgroups

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

subgroupindex
#260293842
#260292892
C4×#156680372
#260191632
#260191342
#260191272
#260191192
#260190922
#260190842
#260190672
#260190492
#260190202
#260186782
#260185472
C89×#14882
#1805289