Finite Group Databasebeta

← #36525956

Subgroups — #36525956

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
1C4×#15668037C211yesmaximal, characteristic
2M16×D356C211yesmaximal, characteristic, complemented
3#26019152C211yesmaximal, characteristic, complemented
4#26019142C211yesmaximal, characteristic, complemented
5#26019116C211yesmaximal, characteristic, complemented
6#26019107C211yesmaximal, characteristic, complemented
7#26019086C211yesmaximal, characteristic, complemented
8#26019081C211yesmaximal, characteristic, complemented
9#26019074C211yesmaximal, characteristic, complemented
10C8×#13083559C211yesmaximal, characteristic, complemented
11#26019020C211yesmaximal, characteristic, complemented
12#26018694C211yesmaximal, characteristic, complemented
13#26018691C211yesmaximal, characteristic, complemented
14#26018555C211yesmaximal, characteristic, complemented
15C89×#2970C211yesmaximal, characteristic, complemented
16C2×#15668037C411yescharacteristic
17C2×M16×D89—21no—
18C2×C4×D356C2211yescharacteristic
19#19151708C411yescharacteristic
20#19151708C411yescharacteristic
21#19151696C411yescharacteristic
22#19151696C411yescharacteristic
23C4×#13083561C2211yescharacteristic
24C4×#13083561C2211yescharacteristic
25C4×#13083559C2211yescharacteristic
26C4×#13083559C2211yescharacteristic
27#19151677C2211yescharacteristic
28#19151677C2211yescharacteristic
29#19151671—21no—
30#19151671—21no—
31C8×D356C2211yescharacteristic
32C8×D356C2211yescharacteristic
33#19151641C411yescharacteristic
34#19151622C2211yescharacteristic
35#19151602C411yescharacteristic
36#19151599C411yescharacteristic
37#19151561—21no—
38C4×#13083558C2211yescharacteristic
39C4×#13083558C2211yescharacteristic
40#19151350C2211yescharacteristic, complemented
41#19151350C2211yescharacteristic, complemented
42#19151348C2211yescharacteristic, complemented
43#19151348C2211yescharacteristic, complemented
44Q8⋊#10623494C2211yescharacteristic, complemented
45Q8⋊#10623494C2211yescharacteristic, complemented
46D4⋊#10623494C2211yescharacteristic, complemented
47D4⋊#10623494C2211yescharacteristic, complemented
48#19151325C2211yescharacteristic
49#19151325C2211yescharacteristic
50#19151325C2211yescharacteristic
51#19151325C2211yescharacteristic
52#19151320C2211yescharacteristic, complemented
53#19151320C2211yescharacteristic, complemented
54#19151319C2211yescharacteristic, complemented
55C8×#10623494C2211yescharacteristic, complemented
56C4×#41×C89C2211yescharacteristic
57#446×C89C2211yescharacteristic, complemented
58#446×C89C2211yescharacteristic, complemented
59#435×C89—21no—
60#435×C89—21no—
61C8×D4×C89C2211yescharacteristic, complemented
62C8×D4×C89C2211yescharacteristic, complemented
63#432×C89—21no—
64#432×C89C2211yescharacteristic
65C4×M16×C89C2211yescharacteristic
66C2×#97×C89—21no—
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×#13083558C2×C411yescharacteristic
77C2×#13083558C2×C411yescharacteristic
78C2×#13083557—21no—
79C2×#13083557—21no—
80M16×D89—44no—
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⋊#10623494C2311yescharacteristic
97C4⋊#10623494C2311yescharacteristic
98#15667936C2×C411yescharacteristic
99#15667936C2×C411yescharacteristic
100#15667933C2×C411yescharacteristic
101#15667933C2×C411yescharacteristic
102#15667932C2×C411yescharacteristic
103#15667932C2×C411yescharacteristic
104#15667932C2×C411yescharacteristic
105#15667932C2×C411yescharacteristic
106#15667924—21no—
107#15667924—21no—
108#15667924—21no—
109#15667924—21no—
110#15667923—21no—
111#15667923C2×C411yescharacteristic
112#15667923C2×C411yescharacteristic
113C4×#10623494C2311yescharacteristic
114C4×#10623494C2311yescharacteristic
115C4×#10623494C2311yescharacteristic
116C4×#10623494C2311yescharacteristic
117C2×#41×C89C2×C411yescharacteristic
118C2×M16×C89D411yescharacteristic
119C2×M16×C89D411yescharacteristic
120C22×C8×C89—21no—
121C22×C8×C89—21no—
122C4×Q8×C89C2×C411yescharacteristic
123C4×Q8×C89C2311yescharacteristic
124C4×D4×C89—21no—
125C4×D4×C89—21no—
126C4×D4×C89C2×C411yescharacteristic
127C4×D4×C89C2311yescharacteristic
128#117×C89—21no—
129#117×C89C2×C411yescharacteristic
130C2×C42×C89—21no—
131C2×C42×C89C2311yescharacteristic
132C4⋊C8×C89—21no—
133C4⋊C8×C89—21no—
134C4⋊C8×C89C2311yescharacteristic
135C4⋊C8×C89C2311yescharacteristic
136C22⋊C8×C89—21no—
137C22⋊C8×C89—21no—
138C22⋊C8×C89—21no—
139#97×C89—21no—
140#97×C89—21no—
141#97×C89—21no—
142#97×C89D411yescharacteristic
143#97×C89D411yescharacteristic
144C4×C8×C89C2311yescharacteristic
145C4×C8×C89C2311yescharacteristic
146C23×D89—21no—
147C2×D356—22no—
148C2×D356—21no—
149C2×D356—41no—
150C2×D356C22×C411yescharacteristic
151C2×D356C22×C411yescharacteristic
152C22×Dic89—21no—
153C2×C4×D89—21no—
154C2×C4×D89—21no—
155C2×C4×D89—21no—
156C2×C4×D89—21no—
157C2×C4×D89—42no—
158C2×C4×D89—42no—
159#13083561—22no—
160#13083561—22no—
161#13083559—22no—
162#13083559M1614yescomplemented
163#13083558—22no—
164#13083558—41no—
165#13083558—42no—
166#13083558—42no—
167#13083557—42no—
168#13083557—42no—
169C2×#10623494—21no—
170C2×#10623494—21no—
171C2×#10623494—21no—
172C2×#10623494—21no—
173C2×#10623494C22×C411yescharacteristic
174C2×#10623494C22×C411yescharacteristic
175C2×#10623494C22×C411yescharacteristic
176C2×#10623494C22×C411yescharacteristic
177C8×D89—42no—
178C8×D89—42no—
179C4⋊Dic89—21no—
180C4⋊Dic89C22×C411yescharacteristic
181C4⋊Dic89C22×C411yescharacteristic
182#13083551—21no—
183#13083551—21no—
184#13083551—21no—
185#13083551—21no—
186#41×C89—22no—
187#41×C89—41no—
188C2×Q8×C89C22×C411yescharacteristic
189C2×D4×C89—21no—
190C2×D4×C89C22×C411yescharacteristic
191C22×C4×C89—21no—
192C22×C4×C89—21no—
193C22×C4×C89—21no—
194C22×C4×C89C22×C411yescharacteristic
195C22×C4×C89C2×D411yescharacteristic
196C22×C4×C89#4111yescharacteristic
197M16×C89—22no—
198M16×C89—41no—
199C2×C8×C89—21no—
200C2×C8×C89—21no—
201C2×C8×C89—42no—
202C2×C8×C89—42no—
203C2×C8×C89C2×D411yescharacteristic
204C2×C8×C89C2×D411yescharacteristic
205C2×C8×C89C2×D411yescharacteristic
206C2×C8×C89C2×D411yescharacteristic
207C4⋊C4×C89—21no—
208C4⋊C4×C89—21no—
209C4⋊C4×C89C22×C411yescharacteristic
210C4⋊C4×C89C22×C411yescharacteristic
211C22⋊C4×C89—21no—
212C22⋊C4×C89—21no—
213C22⋊C4×C89—21no—
214C42×C89—21no—
215C42×C89—21no—
216C42×C89—21no—
217C42×C89C22×C411yescharacteristic
218C42×C89C2×D411yescharacteristic
219C42×C89#4111yescharacteristic
220C42×C89C2411yescharacteristic
221C23×C89—21no—
222C23×C89C4×D411yescharacteristic
223C22×D89—21no—
224C22×D89—21no—
225C22×D89—42no—
226C22×D89—42no—
227C22×D89—81no—
228Q8×C89—21no—
229Q8×C89—21no—
230D4×C89—21no—
231D4×C89—42no—
232D4×C89C2×M1612yescomplemented
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—21no—
240C2×C4×C89—41no—
241C2×C4×C89—42no—
242C2×C4×C89—41no—
243C2×C4×C89—41no—
244C2×C4×C89—81no—
245C2×C4×C89C4×D411yescharacteristic
246C2×C4×C89C4×D411yescharacteristic
247C2×C4×C89C4×D411yescharacteristic
248C2×C4×C89C23×C411yescharacteristic
249C2×C4×C89C22×D411yescharacteristic
250C2×C4×C89C2×#4111yescharacteristic
251C2×Dic89—21no—
252C2×Dic89—21no—
253C2×Dic89—42no—
254D356—21no—
255D356—21no—
256D356—21no—
257D356—42no—
258D356C2×M1612yes—
259C4×D89—42no—
260C4×D89—42no—
261C4×D89—41no—
262C4×D89—41no—
263C4×D89—41no—
264C4×D89—41no—
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×C89C2×C4×D411yescharacteristic
286C2×D89—22no—
287C2×D89—41no—
288C2×D89—41no—
289C2×D89—41no—
290C2×D89—41no—
291C2×D89—41no—
292C2×D89—84no—
293C4×C89—21no—
294C4×C89—21no—
295C4×C89—21no—
296C4×C89—21no—
297C4×C89—41no—
298C4×C89—41no—
299C4×C89—41no—
300C4×C89C22×M1611yescharacteristic
301C4×C89#56811yescharacteristic
302C4×C89C2×AGL(1,ℤ/8)11yescharacteristic
303C4×C89#57511yescharacteristic
304Dic89—41no—
305Dic89—41no—
306#18046—891nomaximal, Sylow
307C2×C89—41no—
308C2×C89—42no—
309C2×C89D4×M1611yescharacteristic
310C2×C89C4×AGL(1,ℤ/8)11yescharacteristic
311C2×C89#295611yescharacteristic
312D89—42no—
313D89—81no—
314#2970D8911yescharacteristic, complemented
315C4×AGL(1,ℤ/8)—891no—
316D4×M16—891no—
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—1781no—
332C4×#41C2×D8911yescharacteristic
333C2×C4×D4—891no—
334#446C2×D8911yescharacteristic, complemented
335#446C2×D8911yescharacteristic, complemented
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#436—891no—
345#436—891no—
346#435—21no—
347#435—21no—
348#435—1781no—
349#435—1781no—
350C8×D4—891no—
351C8×D4—891no—
352C8×D4C2×D8911yescharacteristic, complemented
353C8×D4C2×D8911yescharacteristic, complemented
354#432—21no—
355#432C2×D8911yescharacteristic
356#428—891no—
357#423—891no—
358#418—891no—
359#417—891no—
360#407—1781no—
361C4×M16—891no—
362C4×M16—891no—
363C4×M16C2×D8911yescharacteristic
364C2×#97—21no—
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×#41C4×D8911yescharacteristic
382C22×D4—891no—
383C23×C4—1781no—
384AGL(1,ℤ/8)—1784no—
385C2×SD16—891no—
386C2×SD16—891no—
387C2×D8—891no—
388C2×D8—891no—
389C2×M16—891no—
390C2×M16—891no—
391C2×M16—3564no—
392C2×M16—1781no—
393C2×M16—1781no—
394C2×M16—1781no—
395C2×M16—1781no—
396C2×M16C22⋊D8911yescharacteristic
397C2×M16C22⋊D8911yescharacteristic
398C22×C8—21no—
399C22×C8—21no—
400C22×C8—1781no—
401C22×C8—1781no—
402C4×Q8C4×D8911yescharacteristic
403C4×Q8C22×D8911yescharacteristic
404C4×D4—21no—
405C4×D4—21no—
406C4×D4—891no—
407C4×D4—891no—
408C4×D4—891no—
409C4×D4—891no—
410C4×D4—1781no—
411C4×D4—1781no—
412C4×D4C4×D8911yescharacteristic
413C4×D4C22×D8911yescharacteristic
414#117—21no—
415#117C4×D8911yescharacteristic
416C2×C4⋊C4—891no—
417C2×C22⋊C4—1781no—
418C2×C42—21no—
419C2×C42C22×D8911yescharacteristic
420#107—891no—
421#107—891no—
422#106—891no—
423#106—891no—
424C4⋊C8—21no—
425C4⋊C8—21no—
426C4⋊C8—891no—
427C4⋊C8—891no—
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—1781no—
443C22⋊C8—1781no—
444C22⋊C8—1781no—
445C22⋊C8—1781no—
446#97—21no—
447#97—21no—
448#97—21no—
449#97—891no—
450#97—891no—
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—1781no—
461#41—22no—
462#41—41no—
463C2×Q8C2×C4×D8911yescharacteristic
464C2×D4—21no—
465C2×D4—891no—
466C2×D4—891no—
467C2×D4—3561no—
468C2×D4—1782no—
469C2×D4—1781no—
470C2×D4C2×C4×D8911yescharacteristic
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—3562no—
475C22×C4—3562no—
476C22×C4—1781no—
477C22×C4—1781no—
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—3562no—
491M16—3562no—
492M16—3562no—
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—21no—
516C4⋊C4—891no—
517C4⋊C4—891no—
518C4⋊C4—1781no—
519C4⋊C4C2×C4×D8911yescharacteristic
520C4⋊C4C2×C4×D8911yescharacteristic
521C22⋊C4—21no—
522C22⋊C4—21no—
523C22⋊C4—21no—
524C22⋊C4—1781no—
525C22⋊C4—1781no—
526C22⋊C4—1781no—
527C22⋊C4—1781no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42C2×C4×D8911yescharacteristic
532C42C2×C22⋊D8911yescharacteristic
533C42#1308357411yescharacteristic
534C42C23×D8911yescharacteristic
535C23—21no—
536C23—3562no—
537C23—3562no—
538C23—1781no—
539C23—1781no—
540C23—7121no—
541C23C4×C22⋊D8911yescharacteristic
542Q8—21no—
543Q8—21no—
544D4—21no—
545D4—42no—
546D4—892no—
547D4—3562no—
548D4—1781no—
549D4—1781no—
550D4—1781no—
551D4M16×D8912yescomplemented
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—81no—
564C2×C4—3562no—
565C2×C4—3562no—
566C2×C4—3562no—
567C2×C4—3561no—
568C2×C4—3561no—
569C2×C4—3561no—
570C2×C4—3561no—
571C2×C4—1781no—
572C2×C4—1781no—
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#1566803712yescomplemented
593C22—21no—
594C22—21no—
595C22—21no—
596C22—22no—
597C22—41no—
598C22—41no—
599C22—3561no—
600C22—3561no—
601C22—3561no—
602C22—3561no—
603C22—3561no—
604C22—1782no—
605C22—7124no—
606C22C2×C4×C22⋊D8911yescharacteristic
607C4—21no—
608C4—21no—
609C4—21no—
610C4—21no—
611C4—41no—
612C4—41no—
613C4—41no—
614C4—3561no—
615C4—3561no—
616C4C2×M16×D8911yescharacteristic
617C4#1915236411yescharacteristic
618C4C2×#1566803711yescharacteristic
619C4#1915239611yescharacteristic
620C2—41no—
621C2—42no—
622C2—3562no—
623C2—7121no—
624C2M16×C22⋊D8911yescharacteristic
625C2C4×#1566803711yescharacteristic
626C2#2602915911yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#2970C2
socleC22×C89C2×C4×D4
prime radicalC2×#15668037C4
cosocleC42×C89C24
hypercenter#2970D89
hypocenterC89#18046
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #2970

Complements

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

NH ≅ G/Ntype
2. M16×D356C2semidirect
3. #26019152C2semidirect
4. #26019142C2semidirect
5. #26019116C2semidirect
6. #26019107C2semidirect
7. #26019086C2semidirect
8. #26019081C2semidirect
9. #26019074C2semidirect
10. C8×#13083559C2semidirect
11. #26019020C2semidirect
12. #26018694C2semidirect
13. #26018691C2semidirect
14. #26018555C2semidirect
15. C89×#2970C2semidirect
40. #19151350C22semidirect
41. #19151350C22semidirect
42. #19151348C22semidirect
43. #19151348C22semidirect
44. Q8⋊#10623494C22semidirect
45. Q8⋊#10623494C22semidirect
46. D4⋊#10623494C22semidirect
47. D4⋊#10623494C22semidirect
52. #19151320C22semidirect
53. #19151320C22semidirect
54. #19151319C22semidirect
55. C8×#10623494C22semidirect
57. #446×C89C22semidirect
58. #446×C89C22semidirect
61. C8×D4×C89C22semidirect
62. C8×D4×C89C22semidirect
162. #13083559M16semidirect
232. D4×C89C2×M16semidirect
270. C8×C89AGL(1,ℤ/8)semidirect
278. #10623494C2×M16semidirect
314. #2970D89semidirect
329. C89#18046semidirect
334. #446C2×D89semidirect
335. #446C2×D89semidirect
352. C8×D4C2×D89semidirect
353. C8×D4C2×D89semidirect
551. D4M16×D89semidirect
592. C8#15668037semidirect

Maximal subgroups

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

subgroupindex
C4×#156680372
M16×D3562
#260191522
#260191422
#260191162
#260191072
#260190862
#260190812
#260190742
C8×#130835592
#260190202
#260186942
#260186912
#260185552
C89×#29702
#1804689