Finite Group Databasebeta

← #36556684

Subgroups — #36556684

Order 22784.

822 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26028840C211yesmaximal, characteristic
2#26028813C211yesmaximal, characteristic
3#26028668C211yesmaximal, characteristic
4#26020854C211yesmaximal, characteristic, complemented
5#26020853C211yesmaximal, characteristic, complemented
6#26020852C211yesmaximal, characteristic, complemented
7#26020851C211yesmaximal, characteristic, complemented
8#26020850C211yesmaximal, characteristic, complemented
9#26020849C211yesmaximal, characteristic, complemented
10#26020848C211yesmaximal, characteristic, complemented
11#26019977C211yesmaximal, characteristic
12#339×D89C211yesmaximal, characteristic
13#339×D89C211yesmaximal, characteristic
14#26019944C211yesmaximal, characteristic
15C89×#1922C211yesmaximal, characteristic, complemented
16C22×#13083557C2211yescharacteristic
17C22×#13083557C2211yescharacteristic
18C2×M16×D89C2211yescharacteristic
19C2×M16×D89C2211yescharacteristic
20#117×D89C2211yescharacteristic
21#117×D89C2211yescharacteristic
22C2×C42×D89C2211yescharacteristic
23#19151639C411yescharacteristic, complemented
24#19151639C411yescharacteristic, complemented
25#19151639C411yescharacteristic, complemented
26#19151639C411yescharacteristic, complemented
27#19151637C411yescharacteristic, complemented
28#19151637C411yescharacteristic, complemented
29#19151637C411yescharacteristic, complemented
30#19151637C411yescharacteristic, complemented
31#19151636C2211yescharacteristic
32#19151634C2211yescharacteristic
33#19151634C2211yescharacteristic
34#19151626C2211yescharacteristic
35#19151625C2211yescharacteristic
36#19151624C2211yescharacteristic
37#19151623—21no—
38#19151623—21no—
39#19151620—21no—
40#19151620—21no—
41#19151556C2211yescharacteristic
42#19151555—21no—
43#19151554C2211yescharacteristic
44#19151553C2211yescharacteristic
45#19151552—21no—
46#19151550—21no—
47C4×#13083557—21no—
48#19151363C2211yescharacteristic
49#19151362C2211yescharacteristic
50#19151362C2211yescharacteristic
51#19151359C2211yescharacteristic
52#19151359C2211yescharacteristic
53#19151358C2211yescharacteristic
54#19151358C2211yescharacteristic
55#19151357C2211yescharacteristic
56#19151357C2211yescharacteristic
57#19151353C2211yescharacteristic
58#19151352C2211yescharacteristic
59#19151352C2211yescharacteristic
60#427×C89C2211yescharacteristic
61#424×C89C2211yescharacteristic
62#405×C89C2211yescharacteristic
63#341×C89C2211yescharacteristic
64#339×C89C2211yescharacteristic
65#339×C89C2211yescharacteristic
66#336×C89C2211yescharacteristic
67C22×C4×D89—21no—
68C22×C4×D89C2311yescharacteristic
69C2×#13083558C2311yescharacteristic
70C2×#13083558C2311yescharacteristic
71C2×#13083557—21no—
72C2×#13083557—22no—
73C2×#13083557—22no—
74C2×#13083557—22no—
75C2×#13083557—22no—
76C2×#13083557—21no—
77C2×#13083557C2×C411yescharacteristic
78C2×#13083557C2×C411yescharacteristic
79C2×#13083557C2×C411yescharacteristic
80C2×#13083557C2×C411yescharacteristic
81C2×#13083557D411yescharacteristic
82C2×#13083557D411yescharacteristic
83C2×#13083557D411yescharacteristic
84C2×#13083557D411yescharacteristic
85M16×D89—22no—
86M16×D89—22no—
87M16×D89—22no—
88M16×D89Q814yes—
89C22×#10623494C2311yescharacteristic
90C22×#10623494C2311yescharacteristic
91C2×C8×D89—21no—
92C2×C8×D89—21no—
93#15667971C2311yescharacteristic
94#15667971C2311yescharacteristic
95#15667970C2×C411yescharacteristic
96#15667970C2×C411yescharacteristic
97#15667970C2×C411yescharacteristic
98#15667970C2×C411yescharacteristic
99#15667969—21no—
100#15667969—21no—
101#15667968—21no—
102#15667968—21no—
103C4⋊C4×D89C2×C411yescharacteristic
104C4⋊C4×D89C2×C411yescharacteristic
105C4⋊C4×D89C2×C411yescharacteristic
106C4⋊C4×D89C2×C411yescharacteristic
107C22⋊C4×D89—21no—
108C22⋊C4×D89—21no—
109C2×C4×Dic89—21no—
110C2×C4×Dic89C2311yescharacteristic
111C42×D89—21no—
112C42×D89—21no—
113C42×D89—22no—
114C42×D89—22no—
115#15667946C2×C411yescharacteristic, complemented
116#15667946C2×C411yescharacteristic, complemented
117#15667945C2×C411yescharacteristic, complemented
118#15667945C2×C411yescharacteristic, complemented
119#15667945C2×C411yescharacteristic, complemented
120#15667945C2×C411yescharacteristic, complemented
121#15667944C2×C411yescharacteristic, complemented
122#15667944C2×C411yescharacteristic, complemented
123#15667943C2×C411yescharacteristic, complemented
124#15667943C2×C411yescharacteristic, complemented
125#15667943C2×C411yescharacteristic, complemented
126#15667943C2×C411yescharacteristic, complemented
127C4⋊#10623494—21no—
128C4⋊#10623494—21no—
129#15667941—21no—
130#15667941—21no—
131#15667941—21no—
132#15667941—21no—
133#15667923—21no—
134#15667922—21no—
135#15667922—21no—
136C4×#10623494—21no—
137C8×Dic89—21no—
138C8×Dic89—21no—
139C2×M16×C89C2311yescharacteristic
140C2×M16×C89C2311yescharacteristic
141C22×C8×C89C2311yescharacteristic
142C22×C8×C89C2311yescharacteristic
143#117×C89C2311yescharacteristic
144#117×C89C2311yescharacteristic
145C2×C42×C89C2311yescharacteristic
146#107×C89C2×C411yescharacteristic, complemented
147#107×C89C2×C411yescharacteristic, complemented
148#106×C89C2×C411yescharacteristic, complemented
149#106×C89C2×C411yescharacteristic, complemented
150C4⋊C8×C89—21no—
151C4⋊C8×C89—21no—
152#97×C89—21no—
153C4×C8×C89—21no—
154C23×D89#4111yescharacteristic
155C22×Dic89—21no—
156C22×Dic89#4111yescharacteristic
157C2×C4×D89—21no—
158C2×C4×D89—21no—
159C2×C4×D89—21no—
160C2×C4×D89—21no—
161C2×C4×D89—42no—
162C2×C4×D89—42no—
163C2×C4×D89—42no—
164C2×C4×D89C22×C411yescharacteristic
165C2×C4×D89C22×C411yescharacteristic
166C2×C4×D89C2×D411yescharacteristic
167C2×C4×D89C2×D411yescharacteristic
168C2×C4×D89C2×Q812yes—
169C2×C4×D89#4112yes—
170C2×C4×D89#4112yes—
171C2×C4×D89#4112yes—
172#13083558—21no—
173#13083558—21no—
174#13083558—21no—
175#13083558C2×Q812yes—
176#13083557—22no—
177#13083557—22no—
178#13083557—22no—
179#13083557—22no—
180#13083557—24no—
181#13083557—22no—
182#13083557—22no—
183#13083557—22no—
184#13083557—22no—
185#13083557—42no—
186C2×#10623494—21no—
187C2×#10623494—21no—
188C2×#10623494—21no—
189C2×#10623494—21no—
190C2×#10623494—21no—
191C2×#10623494—21no—
192C2×#10623494C22×C411yescharacteristic
193C2×#10623494C22×C411yescharacteristic
194C2×#10623494C2×D411yescharacteristic
195C2×#10623494C2×D411yescharacteristic
196C8×D89—24no—
197C8×D89—42no—
198C4⋊Dic89C22×C411yescharacteristic
199C4⋊Dic89C22×C411yescharacteristic
200C4⋊Dic89C22×C411yescharacteristic
201C4⋊Dic89C22×C411yescharacteristic
202#13083553—21no—
203#13083553—21no—
204#13083553—21no—
205#13083553—21no—
206C22⋊Dic89—21no—
207C22⋊Dic89—21no—
208#13083551—21no—
209#13083551—21no—
210#13083551—21no—
211#13083551—21no—
212C4×Dic89—21no—
213C4×Dic89—21no—
214C4×Dic89—21no—
215C4×Dic89—21no—
216C4×Dic89—21no—
217C4×Dic89—21no—
218C4×Dic89—21no—
219C4×Dic89—21no—
220C4×Dic89—42no—
221C22×C4×C89—21no—
222C22×C4×C89C2411yescharacteristic
223M16×C89—21no—
224M16×C89—21no—
225M16×C89—21no—
226M16×C89C2×Q812yes—
227C2×C8×C89—21no—
228C2×C8×C89—21no—
229C2×C8×C89—21no—
230C2×C8×C89—21no—
231C2×C8×C89—21no—
232C2×C8×C89—21no—
233C2×C8×C89C22×C411yescharacteristic
234C2×C8×C89C22×C411yescharacteristic
235C2×C8×C89C2×D411yescharacteristic
236C2×C8×C89C2×D411yescharacteristic
237C4⋊C4×C89C22×C411yescharacteristic
238C4⋊C4×C89C22×C411yescharacteristic
239C4⋊C4×C89C22×C411yescharacteristic
240C4⋊C4×C89C22×C411yescharacteristic
241C22⋊C4×C89—21no—
242C22⋊C4×C89—21no—
243C42×C89—21no—
244C42×C89—21no—
245C42×C89—21no—
246C42×C89—21no—
247C23×C89C2×#4111yescharacteristic
248C22×D89—41no—
249C22×D89—41no—
250C22×D89C4×D411yescharacteristic
251C22×D89C4×D411yescharacteristic
252C22×D89#12512yes—
253C22×D89#12612yes—
254C2×C4×C89—22no—
255C2×C4×C89—21no—
256C2×C4×C89—21no—
257C2×C4×C89—21no—
258C2×C4×C89—21no—
259C2×C4×C89—41no—
260C2×C4×C89—41no—
261C2×C4×C89C23×C411yescharacteristic
262C2×C4×C89C22×D411yescharacteristic
263C2×C4×C89C22×Q811yescharacteristic
264C2×C4×C89C2×#4111yescharacteristic
265C2×C4×C89C2×#4111yescharacteristic
266C2×C4×C89C2×#4111yescharacteristic
267C2×Dic89—22no—
268C2×Dic89—21no—
269C2×Dic89—21no—
270C2×Dic89—21no—
271C2×Dic89—21no—
272C2×Dic89—41no—
273C2×Dic89—41no—
274C2×Dic89C4×D411yescharacteristic
275C2×Dic89C4×D411yescharacteristic
276C2×Dic89#12512yes—
277C2×Dic89#12612yes—
278C4×D89—48no—
279C4×D89—81no—
280C4×D89—81no—
281C4×D89—81no—
282C4×D89—81no—
283C4×D89#11712yes—
284C4×D89#11712yes—
285C4×D89C4×Q812yes—
286C4×D89C4×Q812yes—
287C4×D89C22⋊Q812yes—
288C4×D89C22⋊Q812yes—
289C4×D89#12312yes—
290C4×D89#12312yes—
291C8×C89—21no—
292C8×C89—21no—
293C8×C89—22no—
294C8×C89—21no—
295C8×C89—21no—
296C8×C89—41no—
297#10623494—21no—
298#10623494—21no—
299#10623494—22no—
300#10623494—21no—
301#10623494—21no—
302#10623494—41no—
303C22×C89—21no—
304C22×C89—21no—
305C22×C89C2×C4×D411yescharacteristic
306C22×C89C2×#12511yescharacteristic
307C22×C89C2×#12611yescharacteristic
308C2×D89—21no—
309C2×D89—21no—
310C2×D89—21no—
311C2×D89—21no—
312C2×D89—42no—
313C2×D89—42no—
314C2×D89#38712yes—
315C2×D89#38712yes—
316C4×C89—42no—
317C4×C89—41no—
318C4×C89—41no—
319C4×C89—41no—
320C4×C89—41no—
321C4×C89C2×#11711yescharacteristic
322C4×C89C2×C4×Q811yescharacteristic
323C4×C89C2×C22⋊Q811yescharacteristic
324C4×C89C2×#12311yescharacteristic
325Dic89—42no—
326Dic89—41no—
327Dic89—41no—
328Dic89—41no—
329Dic89—41no—
330Dic89#38712yes—
331Dic89#38712yes—
332#21042—891nomaximal, Sylow
333C2×C89—21no—
334C2×C89—21no—
335C2×C89C2×#38711yescharacteristic
336C2×C89#295611yescharacteristic
337C2×C89#295611yescharacteristic
338D89—22no—
339D89—22no—
340#2911—891no—
341#2905—891no—
342#2873—891no—
343#1922—891no—
344#1922—891no—
345#1922—891no—
346#1922—891no—
347#1922—891no—
348#1922—891no—
349#1922—891no—
350#1922D8911yescharacteristic, complemented
351#1738—891no—
352C2×#339—891no—
353C2×#339—891no—
354#1728—891no—
355C89#2104211yescharacteristic, complemented, Sylow
356C22×M16—891no—
357C22×M16—891no—
358C22×M16—891no—
359C22×M16—891no—
360C2×#117—891no—
361C2×#117—891no—
362C22×C42—891no—
363#428—891no—
364#428—891no—
365#428—891no—
366#428—891no—
367#428—891no—
368#428—891no—
369#428—891no—
370#428—891no—
371#427—891no—
372#427—891no—
373#427—891no—
374#427C2×D8911yescharacteristic
375#424—891no—
376#424—891no—
377#424—891no—
378#424—1781no—
379#424—1781no—
380#424C2×D8911yescharacteristic
381#423—1781no—
382#423—1781no—
383#405—891no—
384#405—891no—
385#405—891no—
386#405—1781no—
387#405—1781no—
388#405C2×D8911yescharacteristic
389C4×M16—1781no—
390C4×M16—1781no—
391#341—891no—
392#341—891no—
393#341—891no—
394#341C2×D8911yescharacteristic
395#339—891no—
396#339—891no—
397#339—891no—
398#339—891no—
399#339—891no—
400#339—891no—
401#339C2×D8911yescharacteristic
402#339C2×D8911yescharacteristic
403#336—891no—
404#336—891no—
405#336—891no—
406#336C2×D8911yescharacteristic
407C23×C4—891no—
408C23×C4—1781no—
409C2×M16—891no—
410C2×M16—891no—
411C2×M16—891no—
412C2×M16—891no—
413C2×M16—891no—
414C2×M16—891no—
415C2×M16—891no—
416C2×M16—891no—
417C2×M16—891no—
418C2×M16—894no—
419C2×M16—891no—
420C2×M16—1781no—
421C2×M16—1782no—
422C2×M16—1782no—
423C2×M16—1782no—
424C2×M16—1782no—
425C2×M16—1781no—
426C2×M16—1782no—
427C2×M16—1782no—
428C2×M16—1782no—
429C2×M16C22×D8911yescharacteristic
430C2×M16C22×D8911yescharacteristic
431C22×C8—891no—
432C22×C8—891no—
433C22×C8—1781no—
434C22×C8—1781no—
435C22×C8C22×D8911yescharacteristic
436C22×C8C22×D8911yescharacteristic
437#117—891no—
438#117—891no—
439#117—891no—
440#117—891no—
441#117—891no—
442#117—891no—
443#117—1781no—
444#117—1781no—
445#117—1781no—
446#117—1781no—
447#117C22×D8911yescharacteristic
448#117C22×D8911yescharacteristic
449C2×C4⋊C4—891no—
450C2×C4⋊C4—891no—
451C2×C4⋊C4—891no—
452C2×C4⋊C4—891no—
453C2×C22⋊C4—1781no—
454C2×C22⋊C4—1781no—
455C2×C42—891no—
456C2×C42—1782no—
457C2×C42—1782no—
458C2×C42—1781no—
459C2×C42—1781no—
460C2×C42—1781no—
461C2×C42C22×D8911yescharacteristic
462#107—891no—
463#107—891no—
464#107—891no—
465#107—891no—
466#107—891no—
467#107—891no—
468#107C4×D8911yescharacteristic, complemented
469#107C4×D8911yescharacteristic, complemented
470#106—891no—
471#106—891no—
472#106—891no—
473#106—891no—
474#106—891no—
475#106—891no—
476#106C4×D8911yescharacteristic, complemented
477#106C4×D8911yescharacteristic, complemented
478C4⋊C8—21no—
479C4⋊C8—21no—
480C4⋊C8—1781no—
481C4⋊C8—1781no—
482C4⋊C8—1781no—
483C4⋊C8—1781no—
484C4⋊C8—1781no—
485C4⋊C8—1781no—
486#97—21no—
487#97—1781no—
488#97—1781no—
489#97—1781no—
490C4×C8—21no—
491C4×C8—1781no—
492C4×C8—1781no—
493C4×C8—1781no—
494C24—891no—
495C22×C4—21no—
496C22×C4—891no—
497C22×C4—891no—
498C22×C4—891no—
499C22×C4—891no—
500C22×C4—891no—
501C22×C4—892no—
502C22×C4—892no—
503C22×C4—892no—
504C22×C4—892no—
505C22×C4—3562no—
506C22×C4—3562no—
507C22×C4—3562no—
508C22×C4—1781no—
509C22×C4—1781no—
510C22×C4—1781no—
511C22×C4—1781no—
512C22×C4—1781no—
513C22×C4C23×D8911yescharacteristic
514M16—21no—
515M16—21no—
516M16—21no—
517M16—892no—
518M16—3562no—
519M16—1782no—
520M16—1782no—
521M16—1782no—
522M16—1782no—
523M16—1784no—
524M16—1782no—
525M16—1782no—
526M16—1782no—
527M16—1782no—
528M16—1781no—
529M16—1781no—
530M16—1781no—
531M16Q8×D8912yes—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—891no—
539C2×C8—891no—
540C2×C8—891no—
541C2×C8—891no—
542C2×C8—3562no—
543C2×C8—1784no—
544C2×C8—1781no—
545C2×C8—1781no—
546C2×C8—1781no—
547C2×C8—1781no—
548C2×C8—1781no—
549C2×C8—1781no—
550C2×C8C2×C4×D8911yescharacteristic
551C2×C8C2×C4×D8911yescharacteristic
552C2×C8D4×D8911yescharacteristic
553C2×C8D4×D8911yescharacteristic
554C4⋊C4—891no—
555C4⋊C4—891no—
556C4⋊C4—891no—
557C4⋊C4—891no—
558C4⋊C4—1781no—
559C4⋊C4—1781no—
560C4⋊C4—1781no—
561C4⋊C4—1781no—
562C4⋊C4C2×C4×D8911yescharacteristic
563C4⋊C4C2×C4×D8911yescharacteristic
564C4⋊C4C2×C4×D8911yescharacteristic
565C4⋊C4C2×C4×D8911yescharacteristic
566C22⋊C4—21no—
567C22⋊C4—21no—
568C22⋊C4—1781no—
569C22⋊C4—1781no—
570C22⋊C4—1781no—
571C22⋊C4—1781no—
572C22⋊C4—1781no—
573C22⋊C4—1781no—
574C42—21no—
575C42—21no—
576C42—21no—
577C42—21no—
578C42—3562no—
579C42—1781no—
580C42—1781no—
581C42—1781no—
582C42—1781no—
583C42—1781no—
584C42—1781no—
585C42—1781no—
586C42—1781no—
587C23—892no—
588C23—892no—
589C23—891no—
590C23—891no—
591C23—3561no—
592C23—3561no—
593C23#41×D8911yescharacteristic
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—891no—
602C2×C4—891no—
603C2×C4—892no—
604C2×C4—892no—
605C2×C4—892no—
606C2×C4—892no—
607C2×C4—892no—
608C2×C4—892no—
609C2×C4—892no—
610C2×C4—892no—
611C2×C4—892no—
612C2×C4—892no—
613C2×C4—3568no—
614C2×C4—3561no—
615C2×C4—3561no—
616C2×C4—1782no—
617C2×C4—1781no—
618C2×C4—1781no—
619C2×C4—1781no—
620C2×C4—1781no—
621C2×C4—7121no—
622C2×C4—7121no—
623C2×C4—7121no—
624C2×C4—7121no—
625C2×C4C22×C4×D8911yescharacteristic
626C2×C4C2×D4×D8911yescharacteristic
627C2×C4C2×Q8×D8911yescharacteristic
628C2×C4#41×D8911yescharacteristic
629C2×C4#41×D8911yescharacteristic
630C2×C4#41×D8911yescharacteristic
631C8—21no—
632C8—21no—
633C8—22no—
634C8—21no—
635C8—21no—
636C8—41no—
637C8—3561no—
638C8—1781no—
639C8—1781no—
640C8—1782no—
641C8—1781no—
642C8—1781no—
643C22—21no—
644C22—21no—
645C22—892no—
646C22—892no—
647C22—3562no—
648C22—3562no—
649C22—1781no—
650C22—1781no—
651C22—1781no—
652C22—1781no—
653C22C4×D4×D8911yescharacteristic
654C22#125×D8911yescharacteristic
655C22#126×D8911yescharacteristic
656C4—42no—
657C4—41no—
658C4—41no—
659C4—41no—
660C4—41no—
661C4—892no—
662C4—892no—
663C4—3562no—
664C4—3561no—
665C4—3561no—
666C4—3561no—
667C4—3561no—
668C4#117×D8911yescharacteristic
669C4C4×Q8×D8911yescharacteristic
670C4C22⋊Q8×D8911yescharacteristic
671C4#123×D8911yescharacteristic
672C2—21no—
673C2—21no—
674C2—1782no—
675C2—1782no—
676C2#387×D8911yescharacteristic
677C2#2602915711yescharacteristic
678C2#2602916011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D89
commutatorC2×C4×C89C23×C4
FrattiniC22×C4C23×D89
FittingC89×#1922C2
socleC22×C89C2×C4×D4
prime radicalC23×D89#41
cosocleC22×C4×C89C24
hypercenter#1922D89
hypocenterC89#21042
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1922

Complements

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

NH ≅ G/Ntype
4. #26020854C2semidirect
5. #26020853C2semidirect
6. #26020852C2semidirect
7. #26020851C2semidirect
8. #26020850C2semidirect
9. #26020849C2semidirect
10. #26020848C2semidirect
15. C89×#1922C2semidirect
23. #19151639C4semidirect
24. #19151639C4semidirect
25. #19151639C4semidirect
26. #19151639C4semidirect
27. #19151637C4semidirect
28. #19151637C4semidirect
29. #19151637C4semidirect
30. #19151637C4semidirect
115. #15667946C2×C4semidirect
116. #15667946C2×C4semidirect
117. #15667945C2×C4semidirect
118. #15667945C2×C4semidirect
119. #15667945C2×C4semidirect
120. #15667945C2×C4semidirect
121. #15667944C2×C4semidirect
122. #15667944C2×C4semidirect
123. #15667943C2×C4semidirect
124. #15667943C2×C4semidirect
125. #15667943C2×C4semidirect
126. #15667943C2×C4semidirect
146. #107×C89C2×C4semidirect
147. #107×C89C2×C4semidirect
148. #106×C89C2×C4semidirect
149. #106×C89C2×C4semidirect
350. #1922D89semidirect
355. C89#21042semidirect
468. #107C4×D89semidirect
469. #107C4×D89semidirect
476. #106C4×D89semidirect
477. #106C4×D89semidirect

Maximal subgroups

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

subgroupindex
#260288402
#260288132
#260286682
#260208542
#260208532
#260208522
#260208512
#260208502
#260208492
#260208482
#260199772
#339×D892
#260199442
C89×#19222
#2104289