Finite Group Databasebeta

← #36525960

Subgroups — #36525960

Order 22784.

720 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26029267C211yesmaximal, characteristic, complemented
2C4×#15668038C211yesmaximal, characteristic
3#26019154C211yesmaximal, characteristic, complemented
4#26019140C211yesmaximal, characteristic, complemented
5#26019114C211yesmaximal, characteristic, complemented
6#26019109C211yesmaximal, characteristic, complemented
7#26019088C211yesmaximal, characteristic, complemented
8#26019079C211yesmaximal, characteristic, complemented
9#26019074C211yesmaximal, characteristic, complemented
10C8×#13083559C211yesmaximal, characteristic, complemented
11#26019017C211yesmaximal, characteristic, complemented
12#26018695C211yesmaximal, characteristic, complemented
13#26018689C211yesmaximal, characteristic, complemented
14#26018555C211yesmaximal, characteristic, complemented
15D4×M16×C89C211yesmaximal, characteristic, complemented
16C2×#15668038C411yescharacteristic
17C4×#13083574C2211yescharacteristic
18#19151728C2211yescharacteristic, complemented
19#19151728C2211yescharacteristic, complemented
20#19151708C411yescharacteristic
21#19151708C411yescharacteristic
22#19151698C411yescharacteristic
23#19151698C411yescharacteristic
24C4×#13083562C2211yescharacteristic
25C4×#13083562C2211yescharacteristic
26C4×#13083559C2211yescharacteristic
27C4×#13083559C2211yescharacteristic
28#19151667—21no—
29#19151667—21no—
30C8×D356C2211yescharacteristic, complemented
31C8×D356C2211yescharacteristic, complemented
32#19151655C2211yescharacteristic
33#19151653—21no—
34#19151638C411yescharacteristic
35#19151603C411yescharacteristic
36#19151597C411yescharacteristic
37C4×#13083558C2211yescharacteristic
38C4×#13083558C2211yescharacteristic
39#97×D89—21no—
40#19151350C2211yescharacteristic, complemented
41#19151350C2211yescharacteristic, complemented
42#19151348C2211yescharacteristic, complemented
43#19151348C2211yescharacteristic, complemented
44#19151328C2211yescharacteristic, complemented
45#19151328C2211yescharacteristic, complemented
46D4⋊#10623494C2211yescharacteristic
47D4⋊#10623494C2211yescharacteristic
48D4⋊#10623494C2211yescharacteristic
49D4⋊#10623494C2211yescharacteristic
50#19151325C2211yescharacteristic, complemented
51#19151325C2211yescharacteristic, complemented
52#19151320C2211yescharacteristic, complemented
53#19151320C2211yescharacteristic, complemented
54#19151319C2211yescharacteristic, complemented
55C8×#10623494C2211yescharacteristic, complemented
56C22×M16×C89—21no—
57C2×C4×D4×C89C2211yescharacteristic
58#436×C89C2211yescharacteristic
59#436×C89C2211yescharacteristic
60#435×C89—21no—
61#435×C89—21no—
62C8×D4×C89C2211yescharacteristic
63C8×D4×C89C2211yescharacteristic
64#423×C89C2211yescharacteristic
65#407×C89—21no—
66C4×M16×C89C2211yescharacteristic
67C2×#13083574C2×C411yescharacteristic
68#15668038—24no—
69C2×#13083562C2×C411yescharacteristic
70C2×#13083562C2×C411yescharacteristic
71C2×#13083559C2×C411yescharacteristic
72C2×#13083559C2×C411yescharacteristic
73C2×#13083558—21no—
74C2×#13083558C2×C411yescharacteristic
75C2×#13083558C2×C411yescharacteristic
76C2×C8×D89—21no—
77C2×C8×D89—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#15667935C2×C411yescharacteristic
96#15667935C2×C411yescharacteristic
97#15667933C2×C411yescharacteristic
98#15667933C2×C411yescharacteristic
99#15667933C2×C411yescharacteristic
100#15667933C2×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
110#15667922—21no—
111#15667922—21no—
112C4×#10623494C2311yescharacteristic
113C4×#10623494C2311yescharacteristic
114C4×#10623494C2311yescharacteristic
115C4×#10623494C2311yescharacteristic
116C22×D4×C89C2×C411yescharacteristic
117C23×C4×C89—21no—
118C2×M16×C89—21no—
119C2×M16×C89—21no—
120C2×M16×C89—21no—
121C2×M16×C89—44no—
122C2×M16×C89D411yescharacteristic
123C2×M16×C89D411yescharacteristic
124C22×C8×C89—21no—
125C22×C8×C89—21no—
126C4×D4×C89—21no—
127C4×D4×C89—21no—
128C4×D4×C89C2×C411yescharacteristic
129C4×D4×C89C2×C411yescharacteristic
130C4×D4×C89C2311yescharacteristic
131C4×D4×C89C2311yescharacteristic
132C2×C4⋊C4×C89C2×C411yescharacteristic
133C2×C22⋊C4×C89—21no—
134C2×C42×C89C2311yescharacteristic
135C4⋊C8×C89—21no—
136C4⋊C8×C89C2311yescharacteristic
137C4⋊C8×C89C2311yescharacteristic
138C22⋊C8×C89—21no—
139C22⋊C8×C89—21no—
140C22⋊C8×C89—21no—
141C22⋊C8×C89—21no—
142#97×C89D411yescharacteristic
143#97×C89D411yescharacteristic
144C4×C8×C89C2311yescharacteristic
145C4×C8×C89C2311yescharacteristic
146#13083574—22no—
147#13083574—41no—
148C2×Dic178C22×C411yescharacteristic
149C2×D356C22×C411yescharacteristic
150C2×C22⋊D89—21no—
151C2×C4×D89—21no—
152C2×C4×D89—21no—
153C2×C4×D89—21no—
154#13083562—22no—
155#13083562—22no—
156#13083559—22no—
157#13083559M1614yescomplemented
158#13083558—22no—
159#13083558—41no—
160#13083558—42no—
161C2×#10623494—21no—
162C2×#10623494—21no—
163C2×#10623494—21no—
164C2×#10623494—21no—
165C2×#10623494C22×C411yescharacteristic
166C2×#10623494C22×C411yescharacteristic
167C2×#10623494C22×C411yescharacteristic
168C2×#10623494C22×C411yescharacteristic
169C8×D89—42no—
170C8×D89—42no—
171C4⋊Dic89C22×C411yescharacteristic
172C4⋊Dic89C22×C411yescharacteristic
173#13083553—21no—
174#13083553—21no—
175C22⋊Dic89—21no—
176#13083551—21no—
177#13083551—21no—
178C4×Dic89—21no—
179C4×Dic89—21no—
180C4×Dic89—21no—
181C24×C89—21no—
182C2×D4×C89—22no—
183C2×D4×C89—21no—
184C2×D4×C89—41no—
185C2×D4×C89C22×C411yescharacteristic
186C2×D4×C89C22×C411yescharacteristic
187C22×C4×C89—21no—
188C22×C4×C89—21no—
189C22×C4×C89—21no—
190C22×C4×C89—21no—
191C22×C4×C89—21no—
192C22×C4×C89—42no—
193C22×C4×C89—42no—
194C22×C4×C89C22×C411yescharacteristic
195C22×C4×C89C2×D411yescharacteristic
196C22×C4×C89#4111yescharacteristic
197M16×C89—22no—
198M16×C89—41no—
199M16×C89—42no—
200M16×C89—42no—
201M16×C89—42no—
202C2×C8×C89—21no—
203C2×C8×C89—21no—
204C2×C8×C89—42no—
205C2×C8×C89—42no—
206C2×C8×C89C2×D411yescharacteristic
207C2×C8×C89C2×D411yescharacteristic
208C2×C8×C89C2×D411yescharacteristic
209C2×C8×C89C2×D411yescharacteristic
210C4⋊C4×C89—21no—
211C4⋊C4×C89C22×C411yescharacteristic
212C4⋊C4×C89C22×C411yescharacteristic
213C22⋊C4×C89—21no—
214C22⋊C4×C89—21no—
215C22⋊C4×C89—21no—
216C22⋊C4×C89—21no—
217C42×C89C22×C411yescharacteristic
218C42×C89C2×D411yescharacteristic
219C42×C89#4111yescharacteristic
220C42×C89C2411yescharacteristic
221C23×C89—21no—
222C23×C89—21no—
223C23×C89—42no—
224C23×C89—42no—
225C23×C89—81no—
226C23×C89C4×D411yescharacteristic
227C22×D89—21no—
228D4×C89—21no—
229D4×C89—21no—
230D4×C89—21no—
231D4×C89—42no—
232D4×C89C2×M1612yes—
233C2×C4×C89—21no—
234C2×C4×C89—21no—
235C2×C4×C89—21no—
236C2×C4×C89—21no—
237C2×C4×C89—21no—
238C2×C4×C89—21no—
239C2×C4×C89—41no—
240C2×C4×C89—42no—
241C2×C4×C89—42no—
242C2×C4×C89—42no—
243C2×C4×C89—41no—
244C2×C4×C89—41no—
245C2×C4×C89—41no—
246C2×C4×C89—41no—
247C2×C4×C89C4×D411yescharacteristic
248C2×C4×C89C4×D411yescharacteristic
249C2×C4×C89C4×D411yescharacteristic
250C2×C4×C89C23×C411yescharacteristic
251C2×C4×C89C22×D411yescharacteristic
252C2×C4×C89C2×#4111yescharacteristic
253C22⋊D89—42no—
254C2×Dic89—21no—
255C2×Dic89—21no—
256C2×Dic89—21no—
257D356—21no—
258D356C2×M1612yescomplemented
259C4×D89—42no—
260C4×D89—41no—
261C4×D89—41no—
262C4×D89—81no—
263Dic178—21no—
264Dic178—21no—
265C8×C89—21no—
266C8×C89—21no—
267C8×C89—21no—
268C8×C89—41no—
269C8×C89—41no—
270C8×C89AGL(1,ℤ/8)12yescomplemented
271#10623494—22no—
272#10623494—21no—
273#10623494—21no—
274#10623494—21no—
275#10623494—41no—
276#10623494—41no—
277#10623494—41no—
278#10623494C2×M1612yescomplemented
279C22×C89—21no—
280C22×C89—22no—
281C22×C89—21no—
282C22×C89—21no—
283C22×C89—41no—
284C22×C89—41no—
285C22×C89—41no—
286C22×C89—41no—
287C22×C89—41no—
288C22×C89—84no—
289C22×C89C2×C4×D411yescharacteristic
290C2×D89—22no—
291C2×D89—41no—
292C2×D89—41no—
293C4×C89—21no—
294C4×C89—21no—
295C4×C89—21no—
296C4×C89—21no—
297C4×C89—41no—
298C4×C89—41no—
299C4×C89C22×M1611yescharacteristic
300C4×C89#56811yescharacteristic
301C4×C89C2×AGL(1,ℤ/8)11yescharacteristic
302C4×C89#57511yescharacteristic
303Dic89—41no—
304Dic89—41no—
305Dic89—41no—
306#18046—891nomaximal, Sylow
307C2×C89—41no—
308C2×C89—42no—
309C2×C89—81no—
310C2×C89D4×M1611yescharacteristic
311C2×C89C4×AGL(1,ℤ/8)11yescharacteristic
312C2×C89#295611yescharacteristic
313D89—42no—
314#2970—891no—
315C4×AGL(1,ℤ/8)—891no—
316D4×M16D8911yescharacteristic, complemented
317#1592—891no—
318#1590—891no—
319#1586—891no—
320#1585—891no—
321#1582—891no—
322#1581—891no—
323#1580—891no—
324C8×D8—891no—
325#1569—891no—
326#1490—891no—
327#1489—891no—
328#1455—891no—
329C89#1804611yescharacteristic, complemented, Sylow
330C2×AGL(1,ℤ/8)—891no—
331C22×M16—21no—
332C4×#41—891no—
333C2×C4×D4C2×D8911yescharacteristic
334#446—891no—
335#446—891no—
336#442—891no—
337#442—891no—
338#440—891no—
339#440—891no—
340C4×SD16—891no—
341C4×SD16—891no—
342C4×D8—891no—
343C4×D8—891no—
344#436C2×D8911yescharacteristic
345#436C2×D8911yescharacteristic
346#435—21no—
347#435—21no—
348#435—1781no—
349#435—1781no—
350C8×D4—891no—
351C8×D4—891no—
352C8×D4C2×D8911yescharacteristic
353C8×D4C2×D8911yescharacteristic
354#432—891no—
355#432—1781no—
356#428—891no—
357#423C2×D8911yescharacteristic
358#418—891no—
359#417—891no—
360#407—21no—
361C4×M16—891no—
362C4×M16—891no—
363C4×M16C2×D8911yescharacteristic
364C2×#97—1781no—
365#335—891no—
366#335—891no—
367#334—891no—
368#334—891no—
369Q8⋊C8—891no—
370Q8⋊C8—891no—
371D4⋊C8—891no—
372D4⋊C8—891no—
373D4⋊C8—891no—
374D4⋊C8—891no—
375D4⋊C8—891no—
376D4⋊C8—891no—
377#322—891no—
378#322—891no—
379#322—891no—
380C82—891no—
381C2×#41—891no—
382C22×D4C4×D8911yescharacteristic
383C23×C4—21no—
384AGL(1,ℤ/8)—1784no—
385C2×SD16—891no—
386C2×SD16—891no—
387C2×D8—891no—
388C2×D8—891no—
389C2×M16—21no—
390C2×M16—21no—
391C2×M16—21no—
392C2×M16—44no—
393C2×M16—891no—
394C2×M16—891no—
395C2×M16—1781no—
396C2×M16C22⋊D8911yescharacteristic
397C2×M16C22⋊D8911yescharacteristic
398C22×C8—21no—
399C22×C8—21no—
400C22×C8—1781no—
401C22×C8—1781no—
402C4×Q8—891no—
403C4×Q8—891no—
404C4×D4—21no—
405C4×D4—21no—
406C4×D4—891no—
407C4×D4—891no—
408C4×D4—1781no—
409C4×D4—1781no—
410C4×D4C4×D8911yescharacteristic
411C4×D4C4×D8911yescharacteristic
412C4×D4C22×D8911yescharacteristic
413C4×D4C22×D8911yescharacteristic
414#117—891no—
415#117—1781no—
416C2×C4⋊C4C4×D8911yescharacteristic
417C2×C22⋊C4—21no—
418C2×C42—1781no—
419C2×C42C22×D8911yescharacteristic
420#107—891no—
421#107—891no—
422#106—891no—
423#106—891no—
424C4⋊C8—21no—
425C4⋊C8—891no—
426C4⋊C8—891no—
427C4⋊C8—1781no—
428C4⋊C8—1781no—
429C4⋊C8C22×D8911yescharacteristic
430C4⋊C8C22×D8911yescharacteristic
431#103—891no—
432#103—891no—
433#102—891no—
434#102—891no—
435#102—891no—
436#102—891no—
437#102—891no—
438#102—891no—
439C22⋊C8—21no—
440C22⋊C8—21no—
441C22⋊C8—21no—
442C22⋊C8—21no—
443C22⋊C8—1781no—
444C22⋊C8—1781no—
445C22⋊C8—1781no—
446#97—891no—
447#97—891no—
448#97—1781no—
449#97—1781no—
450#97—1781no—
451#97—1781no—
452#97C22⋊D8911yescharacteristic
453#97C22⋊D8911yescharacteristic
454C4×C8—891no—
455C4×C8—891no—
456C4×C8—891no—
457C4×C8—891no—
458C4×C8C22×D8911yescharacteristic
459C4×C8C22×D8911yescharacteristic
460C24—21no—
461#41—3561no—
462#41—1782no—
463C2×Q8—891no—
464C2×D4—22no—
465C2×D4—21no—
466C2×D4—41no—
467C2×D4—891no—
468C2×D4—1781no—
469C2×D4C2×C4×D8911yescharacteristic
470C2×D4C2×C4×D8911yescharacteristic
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—42no—
477C22×C4—42no—
478C22×C4—1781no—
479C22×C4—1781no—
480C22×C4—1781no—
481C22×C4C2×C4×D8911yescharacteristic
482C22×C4C2×C22⋊D8911yescharacteristic
483C22×C4#1308357411yescharacteristic
484SD16—1782no—
485SD16—1782no—
486D8—894no—
487D8—1782no—
488M16—22no—
489M16—41no—
490M16—42no—
491M16—42no—
492M16—42no—
493M16—3562no—
494M16—3561no—
495M16—1782no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—42no—
499C2×C8—42no—
500C2×C8—891no—
501C2×C8—891no—
502C2×C8—891no—
503C2×C8—891no—
504C2×C8—3562no—
505C2×C8—3562no—
506C2×C8—1781no—
507C2×C8—1781no—
508C2×C8—1781no—
509C2×C8—1781no—
510C2×C8C2×C22⋊D8911yescharacteristic
511C2×C8C2×C22⋊D8911yescharacteristic
512C2×C8C2×C22⋊D8911yescharacteristic
513C2×C8C2×C22⋊D8911yescharacteristic
514C4⋊C4—21no—
515C4⋊C4—891no—
516C4⋊C4—891no—
517C4⋊C4—1781no—
518C4⋊C4—1781no—
519C4⋊C4C2×C4×D8911yescharacteristic
520C4⋊C4C2×C4×D8911yescharacteristic
521C22⋊C4—21no—
522C22⋊C4—21no—
523C22⋊C4—21no—
524C22⋊C4—21no—
525C22⋊C4—1781no—
526C22⋊C4—1781no—
527C22⋊C4—1781no—
528C42—1781no—
529C42—1781no—
530C42—1781no—
531C42C2×C4×D8911yescharacteristic
532C42C2×C22⋊D8911yescharacteristic
533C42#1308357411yescharacteristic
534C42C23×D8911yescharacteristic
535C23—21no—
536C23—21no—
537C23—42no—
538C23—42no—
539C23—81no—
540C23—1781no—
541C23C4×C22⋊D8911yescharacteristic
542Q8—1781no—
543Q8—1781no—
544D4—21no—
545D4—21no—
546D4—21no—
547D4—42no—
548D4—892no—
549D4—3562no—
550D4—1781no—
551D4M16×D8912yes—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—3562no—
567C2×C4—3561no—
568C2×C4—3561no—
569C2×C4—1781no—
570C2×C4—1781no—
571C2×C4—1781no—
572C2×C4—7121no—
573C2×C4C4×C22⋊D8911yescharacteristic
574C2×C4C4×C22⋊D8911yescharacteristic
575C2×C4C4×C22⋊D8911yescharacteristic
576C2×C4C22×C4×D8911yescharacteristic
577C2×C4C22×C22⋊D8911yescharacteristic
578C2×C4C2×#1308357411yescharacteristic
579C8—21no—
580C8—21no—
581C8—21no—
582C8—41no—
583C8—41no—
584C8—892no—
585C8—3561no—
586C8—3561no—
587C8—3561no—
588C8—1782no—
589C8—1781no—
590C8—1781no—
591C8—1781no—
592C8#1566803812yescomplemented
593C22—21no—
594C22—21no—
595C22—21no—
596C22—22no—
597C22—41no—
598C22—41no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—84no—
603C22—3561no—
604C22—3561no—
605C22—1782no—
606C22C2×C4×C22⋊D8911yescharacteristic
607C4—21no—
608C4—21no—
609C4—21no—
610C4—21no—
611C4—41no—
612C4—41no—
613C4—3561no—
614C4—3561no—
615C4—3561no—
616C4C2×M16×D8911yescharacteristic
617C4#1915236411yescharacteristic
618C4C2×#1566803811yescharacteristic
619C4#1915239611yescharacteristic
620C2—41no—
621C2—42no—
622C2—81no—
623C2—3562no—
624C2M16×C22⋊D8911yescharacteristic
625C2C4×#1566803811yescharacteristic
626C2#2602915911yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingD4×M16×C89C2
socleC22×C89C2×C4×D4
prime radicalC2×#15668038C4
cosocleC42×C89C24
hypercenterD4×M16D89
hypocenterC89#18046
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ D4×M16

Complements

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

NH ≅ G/Ntype
1. #26029267C2semidirect
3. #26019154C2semidirect
4. #26019140C2semidirect
5. #26019114C2semidirect
6. #26019109C2semidirect
7. #26019088C2semidirect
8. #26019079C2semidirect
9. #26019074C2semidirect
10. C8×#13083559C2semidirect
11. #26019017C2semidirect
12. #26018695C2semidirect
13. #26018689C2semidirect
14. #26018555C2semidirect
15. D4×M16×C89C2semidirect
18. #19151728C22semidirect
19. #19151728C22semidirect
30. C8×D356C22semidirect
31. C8×D356C22semidirect
40. #19151350C22semidirect
41. #19151350C22semidirect
42. #19151348C22semidirect
43. #19151348C22semidirect
44. #19151328C22semidirect
45. #19151328C22semidirect
50. #19151325C22semidirect
51. #19151325C22semidirect
52. #19151320C22semidirect
53. #19151320C22semidirect
54. #19151319C22semidirect
55. C8×#10623494C22semidirect
157. #13083559M16semidirect
258. D356C2×M16semidirect
270. C8×C89AGL(1,ℤ/8)semidirect
278. #10623494C2×M16semidirect
316. D4×M16D89semidirect
329. C89#18046semidirect
592. C8#15668038semidirect

Maximal subgroups

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

subgroupindex
#260292672
C4×#156680382
#260191542
#260191402
#260191142
#260191092
#260190882
#260190792
#260190742
C8×#130835592
#260190172
#260186952
#260186892
#260185552
D4×M16×C892
#1804689