Finite Group Databasebeta

← #36525460

Subgroups — #36525460

Order 22784.

670 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26030295C211yesmaximal, characteristic
2M16×D356C211yesmaximal, characteristic, complemented
3#26019164C211yesmaximal, characteristic, complemented
4#26019142C211yesmaximal, characteristic, complemented
5#26019130C211yesmaximal, characteristic, complemented
6#26019114C211yesmaximal, characteristic, complemented
7#26019086C211yesmaximal, characteristic, complemented
8#26019081C211yesmaximal, characteristic, complemented
9#26019062C211yesmaximal, characteristic, complemented
10C8×#13083559C211yesmaximal, characteristic, complemented
11#26019040C211yesmaximal, characteristic, complemented
12#26018717C211yesmaximal, characteristic, complemented
13#26018642C211yesmaximal, characteristic, complemented
14#26018549C211yesmaximal, characteristic, complemented
15C89×#2969C211yesmaximal, characteristic, complemented
16C2×M16×D89—21no—
17C2×C4×D356C2211yescharacteristic
18#19151931C411yescharacteristic
19#19151914C411yescharacteristic
20#19151895C411yescharacteristic
21#19151895C411yescharacteristic
22#19151810C411yescharacteristic
23#19151810C411yescharacteristic
24#19151748C411yescharacteristic
25#19151731C411yescharacteristic
26C4×#13083561C2211yescharacteristic
27C4×#13083561C2211yescharacteristic
28C4×#13083559C2211yescharacteristic
29C4×#13083559C2211yescharacteristic
30#19151677C2211yescharacteristic
31#19151677C2211yescharacteristic
32#19151671—21no—
33#19151671—21no—
34C8×D356C2211yescharacteristic
35C8×D356C2211yescharacteristic
36#19151651C2211yescharacteristic
37#19151651C2211yescharacteristic
38#19151622C2211yescharacteristic
39#19151561—21no—
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
56#529×C89C2211yescharacteristic
57#446×C89C2211yescharacteristic, complemented
58C8×Q8×C89C2211yescharacteristic, complemented
59#436×C89—21no—
60#436×C89C2211yescharacteristic, complemented
61#435×C89—21no—
62C8×D4×C89C2211yescharacteristic, complemented
63#433×C89—21no—
64#431×C89C2211yescharacteristic
65#405×C89—21no—
66C4×M16×C89C2211yescharacteristic
67C22×D356C2×C411yescharacteristic
68C22×C4×D89—21no—
69C2×#13083561C2×C411yescharacteristic
70C2×#13083561C2×C411yescharacteristic
71C2×#13083559C2×C411yescharacteristic
72C2×#13083559C2×C411yescharacteristic
73C2×#13083558—21no—
74C2×#13083557—21no—
75C2×#13083557—21no—
76M16×D89—44no—
77C2×C8×D89—21no—
78C2×C8×D89—21no—
79C4×D356—21no—
80C4×D356—21no—
81C4×D356C2×C411yescharacteristic
82C4×D356C2×C411yescharacteristic
83C4×D356C2311yescharacteristic
84C4×D356C2311yescharacteristic
85C2×C4⋊Dic89C2×C411yescharacteristic
86C2×#13083551—21no—
87#15667945C2×C411yescharacteristic
88#15667945C2×C411yescharacteristic
89#15667943C2×C411yescharacteristic
90#15667943C2×C411yescharacteristic
91C4⋊#10623494—21no—
92C4⋊#10623494—21no—
93C4⋊#10623494C2×C411yescharacteristic
94C4⋊#10623494C2×C411yescharacteristic
95C4⋊#10623494C2311yescharacteristic
96C4⋊#10623494C2311yescharacteristic
97#15667936C2×C411yescharacteristic
98#15667936C2×C411yescharacteristic
99#15667933C2×C411yescharacteristic
100#15667933C2×C411yescharacteristic
101#15667932C2×C411yescharacteristic
102#15667932C2×C411yescharacteristic
103#15667932C2×C411yescharacteristic
104#15667932C2×C411yescharacteristic
105C22⋊#10623494—21no—
106C22⋊#10623494C2×C411yescharacteristic
107C22⋊#10623494C2×C411yescharacteristic
108#15667924—21no—
109#15667924—21no—
110#15667924—21no—
111#15667924—21no—
112C4×#10623494C2311yescharacteristic
113C4×#10623494C2311yescharacteristic
114C4×#10623494C2311yescharacteristic
115C4×#10623494C2311yescharacteristic
116C2×M16×C89—21no—
117C2×M16×C89D411yescharacteristic
118C2×M16×C89D411yescharacteristic
119C22×C8×C89—21no—
120#126×C89—21no—
121#125×C89C2×C411yescharacteristic
122#124×C89C2×C411yescharacteristic
123#123×C89—21no—
124C22⋊Q8×C89C2×C411yescharacteristic
125#121×C89C2×C411yescharacteristic
126C4×Q8×C89C2311yescharacteristic
127C4×D4×C89—21no—
128C4×D4×C89C2311yescharacteristic
129#117×C89—21no—
130C2×C42×C89C2311yescharacteristic
131C4⋊C8×C89—21no—
132C4⋊C8×C89—21no—
133C4⋊C8×C89C2311yescharacteristic
134C4⋊C8×C89C2311yescharacteristic
135C22⋊C8×C89—21no—
136C22⋊C8×C89—21no—
137C22⋊C8×C89—21no—
138#97×C89—21no—
139#97×C89D411yescharacteristic
140#97×C89D411yescharacteristic
141C4×C8×C89—21no—
142C4×C8×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—42no—
157C2×C4×D89—42no—
158#13083561—22no—
159#13083561—41no—
160#13083559—22no—
161#13083559—41no—
162#13083558—42no—
163#13083557—42no—
164#13083557—42no—
165C2×#10623494—21no—
166C2×#10623494—21no—
167C2×#10623494—21no—
168C2×#10623494—21no—
169C2×#10623494C22×C411yescharacteristic
170C2×#10623494C22×C411yescharacteristic
171C2×#10623494C22×C411yescharacteristic
172C2×#10623494C22×C411yescharacteristic
173C8×D89—42no—
174C8×D89—42no—
175C4⋊Dic89—21no—
176C4⋊Dic89C22×C411yescharacteristic
177C4⋊Dic89C22×C411yescharacteristic
178#13083551—21no—
179#13083551—21no—
180#13083551—21no—
181#13083551—21no—
182C2×Q8×C89C22×C411yescharacteristic
183C2×D4×C89—21no—
184C2×D4×C89C22×C411yescharacteristic
185C22×C4×C89—21no—
186C22×C4×C89C22×C411yescharacteristic
187C22×C4×C89C2×D411yescharacteristic
188C22×C4×C89#4111yescharacteristic
189M16×C89—22no—
190M16×C89—41no—
191M16×C89—42no—
192C2×C8×C89—21no—
193C2×C8×C89—21no—
194C2×C8×C89—42no—
195C2×C8×C89C2×D411yescharacteristic
196C2×C8×C89C2×D411yescharacteristic
197C2×C8×C89C2×D411yescharacteristic
198C2×C8×C89C2×D411yescharacteristic
199C4⋊C4×C89—21no—
200C4⋊C4×C89—21no—
201C4⋊C4×C89—21no—
202C4⋊C4×C89—21no—
203C4⋊C4×C89C22×C411yescharacteristic
204C4⋊C4×C89C22×C411yescharacteristic
205C22⋊C4×C89—21no—
206C22⋊C4×C89—21no—
207C22⋊C4×C89—21no—
208C22⋊C4×C89—21no—
209C22⋊C4×C89—21no—
210C42×C89—21no—
211C42×C89C22×C411yescharacteristic
212C42×C89C2×D411yescharacteristic
213C42×C89#4111yescharacteristic
214C42×C89C2411yescharacteristic
215C23×C89—21no—
216C23×C89C4×D411yescharacteristic
217C22×D89—21no—
218C22×D89—21no—
219C22×D89—42no—
220C22×D89—42no—
221C22×D89—81no—
222Q8×C89—21no—
223Q8×C89—21no—
224D4×C89—21no—
225D4×C89—21no—
226D4×C89—81no—
227C2×C4×C89—21no—
228C2×C4×C89—21no—
229C2×C4×C89—21no—
230C2×C4×C89—21no—
231C2×C4×C89—21no—
232C2×C4×C89—21no—
233C2×C4×C89—21no—
234C2×C4×C89—41no—
235C2×C4×C89—41no—
236C2×C4×C89—41no—
237C2×C4×C89C4×D411yescharacteristic
238C2×C4×C89C4×D411yescharacteristic
239C2×C4×C89C4×D411yescharacteristic
240C2×C4×C89C23×C411yescharacteristic
241C2×C4×C89C22×D411yescharacteristic
242C2×C4×C89C2×#4111yescharacteristic
243C2×Dic89—21no—
244C2×Dic89—21no—
245C2×Dic89—42no—
246D356—21no—
247D356—21no—
248D356—21no—
249D356—42no—
250D356#13112yes—
251C4×D89—42no—
252C4×D89—42no—
253C4×D89—41no—
254C4×D89—41no—
255C4×D89—41no—
256C4×D89—41no—
257C8×C89—21no—
258C8×C89—21no—
259C8×C89—21no—
260C8×C89—41no—
261C8×C89—41no—
262C8×C89AGL(1,ℤ/8)12yes—
263#10623494—21no—
264#10623494—21no—
265#10623494—21no—
266#10623494—21no—
267#10623494—41no—
268#10623494—41no—
269#10623494—41no—
270#10623494—41no—
271C22×C89—21no—
272C22×C89—21no—
273C22×C89—21no—
274C22×C89—41no—
275C22×C89—41no—
276C22×C89—41no—
277C22×C89C2×C4×D411yescharacteristic
278C2×D89—22no—
279C2×D89—41no—
280C2×D89—41no—
281C2×D89—41no—
282C2×D89—41no—
283C2×D89—41no—
284C2×D89—84no—
285C4×C89—21no—
286C4×C89—21no—
287C4×C89—21no—
288C4×C89—21no—
289C4×C89—41no—
290C4×C89—41no—
291C4×C89—41no—
292C4×C89C2×#13111yescharacteristic
293C4×C89#56811yescharacteristic
294C4×C89C2×AGL(1,ℤ/8)11yescharacteristic
295C4×C89#57511yescharacteristic
296Dic89—41no—
297Dic89—41no—
298#18006—891nomaximal, Sylow
299C2×C89—41no—
300C2×C89—81no—
301C2×C89#293311yescharacteristic
302C2×C89C4×AGL(1,ℤ/8)11yescharacteristic
303C2×C89#295711yescharacteristic
304D89—42no—
305D89—81no—
306#3105—891no—
307#2969D8911yescharacteristic, complemented
308D4×M16—891no—
309#1594—891no—
310#1590—891no—
311#1588—891no—
312#1586—891no—
313#1582—891no—
314#1581—891no—
315SD16⋊C8—891no—
316C8×D8—891no—
317#1574—891no—
318#1495—891no—
319#1478—891no—
320#1453—891no—
321C89#1800611yescharacteristic, complemented, Sylow
322C22×M16—1781no—
323#529C2×D8911yescharacteristic
324C2×C4×D4—891no—
325#484—891no—
326#481—891no—
327#478—891no—
328#478—891no—
329#463—891no—
330#463—891no—
331#450—891no—
332#447—891no—
333#446C2×D8911yescharacteristic, complemented
334C8×Q8C2×D8911yescharacteristic, complemented
335C4×SD16—891no—
336C4×SD16—891no—
337C4×D8—891no—
338C4×D8—891no—
339#436—21no—
340#436—891no—
341#436—891no—
342#436C2×D8911yescharacteristic, complemented
343#435—21no—
344#435—1781no—
345#435—1781no—
346C8×D4—891no—
347C8×D4—891no—
348C8×D4C2×D8911yescharacteristic, complemented
349#433—21no—
350#431—891no—
351#431—891no—
352#431C2×D8911yescharacteristic
353#423—891no—
354#407—1781no—
355#405—21no—
356C4×M16C2×D8911yescharacteristic
357#335—891no—
358#335—891no—
359#334—891no—
360#334—891no—
361Q8⋊C8—891no—
362Q8⋊C8—891no—
363D4⋊C8—891no—
364D4⋊C8—891no—
365D4⋊C8—891no—
366D4⋊C8—891no—
367D4⋊C8—891no—
368D4⋊C8—891no—
369#322—891no—
370#322—891no—
371#322—891no—
372C82—891no—
373C22×D4—891no—
374C23×C4—1781no—
375C2×SD16—891no—
376C2×SD16—891no—
377C2×D8—891no—
378C2×D8—891no—
379C2×M16—21no—
380C2×M16—3564no—
381C2×M16—1781no—
382C2×M16—1781no—
383C2×M16—1781no—
384C2×M16C22⋊D8911yescharacteristic
385C2×M16C22⋊D8911yescharacteristic
386C22×C8—21no—
387C22×C8—1781no—
388C22×C8—1781no—
389#126—21no—
390#125C4×D8911yescharacteristic
391#124C4×D8911yescharacteristic
392#123—21no—
393C22⋊Q8C4×D8911yescharacteristic
394#121C4×D8911yescharacteristic
395C4×Q8C22×D8911yescharacteristic
396C4×D4—21no—
397C4×D4—891no—
398C4×D4—891no—
399C4×D4—891no—
400C4×D4—891no—
401C4×D4—1781no—
402C4×D4—1781no—
403C4×D4C22×D8911yescharacteristic
404#117—21no—
405C2×C4⋊C4—891no—
406C2×C22⋊C4—1781no—
407C2×C42C22×D8911yescharacteristic
408#107—891no—
409#107—891no—
410#106—891no—
411#106—891no—
412C4⋊C8—21no—
413C4⋊C8—21no—
414C4⋊C8—891no—
415C4⋊C8—891no—
416C4⋊C8—891no—
417C4⋊C8—891no—
418C4⋊C8—1781no—
419C4⋊C8—1781no—
420C4⋊C8C22×D8911yescharacteristic
421C4⋊C8C22×D8911yescharacteristic
422#103—891no—
423#103—891no—
424#102—891no—
425#102—891no—
426#102—891no—
427#102—891no—
428#102—891no—
429#102—891no—
430C22⋊C8—21no—
431C22⋊C8—21no—
432C22⋊C8—21no—
433C22⋊C8—891no—
434C22⋊C8—891no—
435C22⋊C8—1781no—
436C22⋊C8—1781no—
437C22⋊C8—1781no—
438C22⋊C8—1781no—
439C22⋊C8—1781no—
440#97—21no—
441#97C22⋊D8911yescharacteristic
442#97C22⋊D8911yescharacteristic
443C4×C8—21no—
444C4×C8—21no—
445C4×C8—891no—
446C4×C8—891no—
447C4×C8—891no—
448C4×C8—891no—
449C4×C8C22×D8911yescharacteristic
450C4×C8C22×D8911yescharacteristic
451C24—1781no—
452C2×Q8C2×C4×D8911yescharacteristic
453C2×D4—21no—
454C2×D4—891no—
455C2×D4—891no—
456C2×D4—3561no—
457C2×D4—1782no—
458C2×D4—1781no—
459C2×D4C2×C4×D8911yescharacteristic
460C22×C4—21no—
461C22×C4—3562no—
462C22×C4—3562no—
463C22×C4—1781no—
464C22×C4—1781no—
465C22×C4—1781no—
466C22×C4—1781no—
467C22×C4—1781no—
468C22×C4C2×C4×D8911yescharacteristic
469C22×C4C2×C22⋊D8911yescharacteristic
470C22×C4#1308357411yescharacteristic
471SD16—3561no—
472SD16—1782no—
473D8—3561no—
474D8—1782no—
475M16—22no—
476M16—41no—
477M16—42no—
478M16—3562no—
479M16—3562no—
480M16—3562no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—42no—
484C2×C8—891no—
485C2×C8—891no—
486C2×C8—891no—
487C2×C8—891no—
488C2×C8—3562no—
489C2×C8—3562no—
490C2×C8—1781no—
491C2×C8—1781no—
492C2×C8—1781no—
493C2×C8—1781no—
494C2×C8C2×C22⋊D8911yescharacteristic
495C2×C8C2×C22⋊D8911yescharacteristic
496C2×C8C2×C22⋊D8911yescharacteristic
497C2×C8C2×C22⋊D8911yescharacteristic
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—891no—
503C4⋊C4—891no—
504C4⋊C4—1781no—
505C4⋊C4C2×C4×D8911yescharacteristic
506C4⋊C4C2×C4×D8911yescharacteristic
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—21no—
512C22⋊C4—1781no—
513C22⋊C4—1781no—
514C22⋊C4—1781no—
515C22⋊C4—1781no—
516C42—21no—
517C42C2×C4×D8911yescharacteristic
518C42C2×C22⋊D8911yescharacteristic
519C42#1308357411yescharacteristic
520C42C23×D8911yescharacteristic
521C23—21no—
522C23—3562no—
523C23—3562no—
524C23—1781no—
525C23—1781no—
526C23—7121no—
527C23C4×C22⋊D8911yescharacteristic
528Q8—21no—
529Q8—21no—
530D4—21no—
531D4—21no—
532D4—81no—
533D4—892no—
534D4—3562no—
535D4—1781no—
536D4—1781no—
537D4—1781no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—3562no—
549C2×C4—3562no—
550C2×C4—3562no—
551C2×C4—3561no—
552C2×C4—3561no—
553C2×C4—3561no—
554C2×C4—3561no—
555C2×C4—1781no—
556C2×C4—1781no—
557C2×C4C4×C22⋊D8911yescharacteristic
558C2×C4C4×C22⋊D8911yescharacteristic
559C2×C4C4×C22⋊D8911yescharacteristic
560C2×C4C22×C4×D8911yescharacteristic
561C2×C4C22×C22⋊D8911yescharacteristic
562C2×C4C2×#1308357411yescharacteristic
563C8—21no—
564C8—21no—
565C8—21no—
566C8—41no—
567C8—41no—
568C8—3561no—
569C8—3561no—
570C8—3561no—
571C8—3561no—
572C8—1781no—
573C8—1781no—
574C8—1781no—
575C8—1781no—
576C8#1566803712yes—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—3561no—
584C22—3561no—
585C22—3561no—
586C22—3561no—
587C22—3561no—
588C22—1782no—
589C22—7124no—
590C22C2×C4×C22⋊D8911yescharacteristic
591C4—21no—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—3561no—
599C4—3561no—
600C4#131×D8911yescharacteristic
601C4#1915236311yescharacteristic
602C4C2×#1566803711yescharacteristic
603C4#1915239611yescharacteristic
604C2—41no—
605C2—81no—
606C2—3562no—
607C2—7121no—
608C2#2602897411yescharacteristic
609C2C4×#1566803711yescharacteristic
610C2#2602916711yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#2969C2
socleC22×C89C2×C4×D4
prime radical#19151731C4
cosocleC42×C89C24
hypercenter#2969D89
hypocenterC89#18006
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #2969

Complements

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

NH ≅ G/Ntype
2. M16×D356C2semidirect
3. #26019164C2semidirect
4. #26019142C2semidirect
5. #26019130C2semidirect
6. #26019114C2semidirect
7. #26019086C2semidirect
8. #26019081C2semidirect
9. #26019062C2semidirect
10. C8×#13083559C2semidirect
11. #26019040C2semidirect
12. #26018717C2semidirect
13. #26018642C2semidirect
14. #26018549C2semidirect
15. C89×#2969C2semidirect
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. C8×Q8×C89C22semidirect
60. #436×C89C22semidirect
62. C8×D4×C89C22semidirect
307. #2969D89semidirect
321. C89#18006semidirect
333. #446C2×D89semidirect
334. C8×Q8C2×D89semidirect
342. #436C2×D89semidirect
348. C8×D4C2×D89semidirect

Maximal subgroups

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

subgroupindex
#260302952
M16×D3562
#260191642
#260191422
#260191302
#260191142
#260190862
#260190812
#260190622
C8×#130835592
#260190402
#260187172
#260186422
#260185492
C89×#29692
#1800689