Finite Group Databasebeta

← #36562280

Subgroups — #36562280

Order 22784.

730 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26030040C211yesmaximal, characteristic
2#26029192C211yesmaximal, characteristic, complemented
3#26021196C211yesmaximal, characteristic, complemented
4#26021027C211yesmaximal, characteristic, complemented
5#26020851C211yesmaximal, characteristic, complemented
6#26020851C211yesmaximal, characteristic, complemented
7#26020527C211yesmaximal, characteristic
8#26020527C211yesmaximal, characteristic
9#26020272C211yesmaximal, characteristic
10#26020137C211yesmaximal, characteristic, complemented
11#26020132C211yesmaximal, characteristic, complemented
12#26020124C211yesmaximal, characteristic, complemented
13#26020094C211yesmaximal, characteristic
14#26020087C211yesmaximal, characteristic
15C89×#2875C211yesmaximal, characteristic
16#19152155C2211yescharacteristic
17#19151905C411yescharacteristic, complemented
18#19151905C411yescharacteristic, complemented
19#19151898C411yescharacteristic, complemented
20#19151898C411yescharacteristic, complemented
21#19151893C411yescharacteristic, complemented
22#19151888C411yescharacteristic, complemented
23#19151883C411yescharacteristic, complemented
24#19151878C411yescharacteristic, complemented
25#19151726—21no—
26#19151726—21no—
27#19151726—21no—
28#19151726—21no—
29C2×#15667947C2211yescharacteristic
30C2×#15667947C2211yescharacteristic
31#19151634C2211yescharacteristic
32#19151634C2211yescharacteristic
33#19151624C2211yescharacteristic
34#19151624C2211yescharacteristic
35#19151622C2211yescharacteristic
36C2×C4⋊#10623494C2211yescharacteristic
37C2×#15667939C2211yescharacteristic
38C2×#15667939C2211yescharacteristic
39#19151594C2211yescharacteristic
40#19151594C2211yescharacteristic
41#19151553C2211yescharacteristic
42#19151553C2211yescharacteristic
43C2×#15667922C2211yescharacteristic
44C2×C8×Dic89C2211yescharacteristic
45C4×C4⋊Dic89C2211yescharacteristic
46#19151362C2211yescharacteristic
47#19151362C2211yescharacteristic
48#19151359C2211yescharacteristic
49#19151359C2211yescharacteristic
50#19151359C2211yescharacteristic
51#19151359C2211yescharacteristic
52#19151357C2211yescharacteristic
53#19151357C2211yescharacteristic
54#19151352C2211yescharacteristic
55#19151352C2211yescharacteristic
56C2×#131×C89C2211yescharacteristic
57C2×#131×C89C2211yescharacteristic
58C4×#41×C89C2211yescharacteristic
59#405×C89—21no—
60#405×C89—21no—
61#405×C89C2211yescharacteristic
62#405×C89C2211yescharacteristic
63C4×M16×C89—21no—
64C4×M16×C89C2211yescharacteristic
65C2×#97×C89—21no—
66C2×#97×C89C2211yescharacteristic
67C2×#13083558C2311yescharacteristic
68C2×#13083558C2311yescharacteristic
69C22×#10623494C2311yescharacteristic
70C22×#10623494C2311yescharacteristic
71#15668012C2×C411yescharacteristic
72#15668012C2×C411yescharacteristic
73#15668004C2×C411yescharacteristic
74#15668004C2×C411yescharacteristic
75#15667991—21no—
76#15667991—21no—
77C4×Dic178—21no—
78C4×Dic178—21no—
79#15667971C2311yescharacteristic
80#15667971C2311yescharacteristic
81C2×C4⋊Dic89—21no—
82C2×C4×Dic89C2311yescharacteristic
83C2×C4×Dic89C2311yescharacteristic
84#15667947—22no—
85#15667947—22no—
86#15667945C2×C411yescharacteristic
87#15667945C2×C411yescharacteristic
88#15667945C2×C411yescharacteristic
89#15667945C2×C411yescharacteristic
90#15667943C2×C411yescharacteristic
91#15667943C2×C411yescharacteristic
92#15667943C2×C411yescharacteristic
93#15667943C2×C411yescharacteristic
94C4⋊#10623494—21no—
95C4⋊#10623494—21no—
96C4⋊#10623494C2×C411yescharacteristic
97C4⋊#10623494C2×C411yescharacteristic
98C4⋊#10623494C2×C411yescharacteristic
99C4⋊#10623494C2×C411yescharacteristic
100#15667941—21no—
101#15667941—21no—
102#15667941—21no—
103#15667941—21no—
104#15667939—22no—
105#15667939—22no—
106#15667936C2×C411yescharacteristic
107#15667936C2×C411yescharacteristic
108#15667936C2×C411yescharacteristic
109#15667936C2×C411yescharacteristic
110#15667933C2×C411yescharacteristic
111#15667933C2×C411yescharacteristic
112#15667933C2×C411yescharacteristic
113#15667933C2×C411yescharacteristic
114#15667922—21no—
115#15667922—21no—
116#15667922—22no—
117C8×Dic89—21no—
118C8×Dic89—21no—
119C8×Dic89—22no—
120#15667919—21no—
121C2×#41×C89C2311yescharacteristic
122#131×C89—22no—
123#131×C89—22no—
124#131×C89—22no—
125#131×C89Q814yescomplemented
126C2×M16×C89—21no—
127C2×M16×C89—21no—
128C2×M16×C89C2311yescharacteristic
129C2×M16×C89C2311yescharacteristic
130C22×C8×C89—21no—
131C22×C8×C89—21no—
132C22×C8×C89C2311yescharacteristic
133C22×C8×C89C2311yescharacteristic
134C4×Q8×C89C2×C411yescharacteristic
135C4×Q8×C89C2×C411yescharacteristic
136C4×D4×C89—21no—
137C4×D4×C89—21no—
138C4×D4×C89C2×C411yescharacteristic
139C4×D4×C89C2×C411yescharacteristic
140#117×C89—21no—
141#117×C89C2311yescharacteristic
142C2×C42×C89—21no—
143C2×C42×C89C2311yescharacteristic
144#97×C89—21no—
145#97×C89—21no—
146#97×C89—21no—
147#97×C89D411yescharacteristic
148#97×C89D411yescharacteristic
149#97×C89D411yescharacteristic
150#97×C89D411yescharacteristic
151C4×C8×C89—21no—
152C4×C8×C89—21no—
153C4×C8×C89—21no—
154C2×Dic178—21no—
155C2×Dic178—21no—
156C22×Dic89—21no—
157C22×Dic89—21no—
158#13083558—21no—
159#13083558—21no—
160#13083558—21no—
161#13083558—21no—
162C2×#10623494—21no—
163C2×#10623494—21no—
164C2×#10623494—21no—
165C2×#10623494—21no—
166C2×#10623494—21no—
167C2×#10623494—21no—
168C2×#10623494C22×C411yescharacteristic
169C2×#10623494C22×C411yescharacteristic
170C2×#10623494C22×C411yescharacteristic
171C2×#10623494C22×C411yescharacteristic
172C4⋊Dic89—21no—
173C4⋊Dic89—21no—
174C4⋊Dic89—42no—
175C4⋊Dic89C22×C411yescharacteristic
176C4⋊Dic89C22×C411yescharacteristic
177C4⋊Dic89C22×C411yescharacteristic
178C4⋊Dic89C22×C411yescharacteristic
179#13083553—21no—
180#13083553—21no—
181#13083553—21no—
182#13083553—21no—
183C22⋊Dic89—21no—
184C22⋊Dic89—21no—
185C4×Dic89—21no—
186C4×Dic89—21no—
187C4×Dic89—21no—
188C4×Dic89—21no—
189C4×Dic89—21no—
190C4×Dic89—21no—
191#41×C89—21no—
192#41×C89—21no—
193#41×C89C2×Q812yes—
194#41×C89#4112yes—
195C2×Q8×C89C22×C411yescharacteristic
196C2×D4×C89—21no—
197C2×D4×C89C22×C411yescharacteristic
198C22×C4×C89—21no—
199C22×C4×C89—21no—
200C22×C4×C89—21no—
201C22×C4×C89#4111yescharacteristic
202C22×C4×C89#4111yescharacteristic
203C22×C4×C89C2411yescharacteristic
204M16×C89—24no—
205M16×C89—21no—
206M16×C89—21no—
207M16×C89—24no—
208M16×C89C2×Q812yes—
209M16×C89#4112yes—
210C2×C8×C89—24no—
211C2×C8×C89—21no—
212C2×C8×C89—21no—
213C2×C8×C89—21no—
214C2×C8×C89—21no—
215C2×C8×C89—24no—
216C2×C8×C89C2×D411yescharacteristic
217C2×C8×C89C2×D411yescharacteristic
218C2×C8×C89C2×D411yescharacteristic
219C2×C8×C89C2×D411yescharacteristic
220C2×C8×C89C2×Q812yes—
221C2×C8×C89#4112yes—
222C4⋊C4×C89—21no—
223C4⋊C4×C89—21no—
224C4⋊C4×C89C22×C411yescharacteristic
225C4⋊C4×C89C22×C411yescharacteristic
226C22⋊C4×C89—21no—
227C22⋊C4×C89—21no—
228C22⋊C4×C89—21no—
229C42×C89—21no—
230C42×C89—21no—
231C42×C89—21no—
232C42×C89C22×C411yescharacteristic
233C42×C89C22×C411yescharacteristic
234C42×C89C2×D411yescharacteristic
235C42×C89C2×D411yescharacteristic
236C23×C89—21no—
237C23×C89C2×#4111yescharacteristic
238Q8×C89—21no—
239Q8×C89C4×Q812yes—
240D4×C89—22no—
241D4×C89—22no—
242D4×C89—21no—
243D4×C89C4×Q812yes—
244C2×C4×C89—21no—
245C2×C4×C89—22no—
246C2×C4×C89—22no—
247C2×C4×C89—21no—
248C2×C4×C89—21no—
249C2×C4×C89—41no—
250C2×C4×C89—41no—
251C2×C4×C89—81no—
252C2×C4×C89—81no—
253C2×C4×C89C4×D411yescharacteristic
254C2×C4×C89C4×D411yescharacteristic
255C2×C4×C89C4×D411yescharacteristic
256C2×C4×C89C4×D411yescharacteristic
257C2×C4×C89C23×C411yescharacteristic
258C2×C4×C89C22×D411yescharacteristic
259C2×C4×C89C22×Q811yescharacteristic
260C2×C4×C89C2×#4111yescharacteristic
261C2×C4×C89C2×#4111yescharacteristic
262C2×C4×C89C2×#4111yescharacteristic
263C2×Dic89—22no—
264C2×Dic89—21no—
265C2×Dic89—21no—
266C2×Dic89—21no—
267C2×Dic89—21no—
268C2×Dic89—41no—
269C2×Dic89—41no—
270C2×Dic89—41no—
271C2×Dic89—41no—
272C2×Dic89—41no—
273Dic178—81no—
274Dic178—81no—
275C8×C89—22no—
276C8×C89—21no—
277C8×C89—22no—
278C8×C89—21no—
279C8×C89C22⋊Q812yes—
280C8×C89C22⋊Q812yes—
281#10623494—21no—
282#10623494—21no—
283#10623494—21no—
284#10623494—21no—
285#10623494—41no—
286#10623494—41no—
287C22×C89—21no—
288C22×C89—21no—
289C22×C89—22no—
290C22×C89—41no—
291C22×C89—41no—
292C22×C89C2×C4×D411yescharacteristic
293C22×C89#52911yescharacteristic
294C22×C89#53311yescharacteristic
295C4×C89—21no—
296C4×C89—21no—
297C4×C89—21no—
298C4×C89—21no—
299C4×C89—22no—
300C4×C89—41no—
301C4×C89—41no—
302C4×C89C2×C4×Q811yescharacteristic
303C4×C89C4×#4111yescharacteristic
304C4×C89C2×C22⋊Q811yescharacteristic
305C4×C89#52511yescharacteristic
306Dic89—42no—
307Dic89—41no—
308Dic89—41no—
309Dic89—41no—
310Dic89—41no—
311Dic89—81no—
312#21629—891nomaximal, Sylow
313C2×C89—21no—
314C2×C89—21no—
315C2×C89—42no—
316C2×C89C4×C22⋊Q811yescharacteristic
317C2×C89#295611yescharacteristic
318C2×C89#295711yescharacteristic
319#3076—891no—
320#2961—891no—
321#2875D8911yescharacteristic
322#1986—891no—
323#1950—891no—
324#1922—891no—
325#1922—891no—
326#1866—891no—
327#1866—891no—
328#1806—891no—
329#1773—891no—
330#1772—891no—
331#1771—891no—
332#1765—891no—
333#1764—891no—
334C89#2162911yescharacteristic, complemented, Sylow
335C2×#131C2×D8911yescharacteristic
336C2×#131C2×D8911yescharacteristic
337#533—891no—
338C4×#41C2×D8911yescharacteristic
339#479—891no—
340#479—891no—
341#478—891no—
342#478—891no—
343#477—891no—
344#476—891no—
345#475—891no—
346#474—891no—
347#446—1781no—
348#446—1781no—
349#446—1781no—
350#446—1781no—
351C2×#108—891no—
352C2×#108—891no—
353#427—891no—
354#427—891no—
355#424—891no—
356#424—891no—
357#423—891no—
358C2×C4⋊C8—891no—
359C2×#104—891no—
360C2×#104—891no—
361#416—891no—
362#416—891no—
363#405—21no—
364#405—21no—
365#405—891no—
366#405—891no—
367#405C2×D8911yescharacteristic
368#405C2×D8911yescharacteristic
369C4×M16—21no—
370C4×M16C2×D8911yescharacteristic
371C2×#97—21no—
372C2×#97—891no—
373C2×#97C2×D8911yescharacteristic
374C2×C4×C8—891no—
375C4×C4⋊C4—891no—
376#341—891no—
377#341—891no—
378#339—891no—
379#339—891no—
380#339—891no—
381#339—891no—
382#339—891no—
383#339—891no—
384#336—891no—
385#336—891no—
386C2×#41C22×D8911yescharacteristic
387#131—22no—
388#131—22no—
389#131—22no—
390#131Dic17814yescomplemented
391C2×M16—21no—
392C2×M16—21no—
393C2×M16—891no—
394C2×M16—891no—
395C2×M16C22×D8911yescharacteristic
396C2×M16C22×D8911yescharacteristic
397C22×C8—21no—
398C22×C8—21no—
399C22×C8—891no—
400C22×C8—891no—
401C22×C8C22×D8911yescharacteristic
402C22×C8C22×D8911yescharacteristic
403C4⋊Q8—891no—
404C4⋊Q8—891no—
405#125—891no—
406#125—891no—
407C22⋊Q8—1781no—
408C22⋊Q8—1781no—
409C4×Q8—1781no—
410C4×Q8—1781no—
411C4×Q8C4×D8911yescharacteristic
412C4×Q8C4×D8911yescharacteristic
413C4×D4—21no—
414C4×D4—21no—
415C4×D4C4×D8911yescharacteristic
416C4×D4C4×D8911yescharacteristic
417#117—21no—
418#117—891no—
419#117—891no—
420#117C22×D8911yescharacteristic
421C2×C4⋊C4—1781no—
422C2×C42—21no—
423C2×C42—891no—
424C2×C42—891no—
425C2×C42C22×D8911yescharacteristic
426#108—1782no—
427#108—1782no—
428#107—891no—
429#107—891no—
430#107—891no—
431#107—891no—
432#106—891no—
433#106—891no—
434#106—891no—
435#106—891no—
436C4⋊C8—891no—
437C4⋊C8—891no—
438C4⋊C8—891no—
439C4⋊C8—891no—
440C4⋊C8—1781no—
441C4⋊C8—1781no—
442C4⋊C8—1781no—
443C4⋊C8—1781no—
444C4⋊C8—1781no—
445C4⋊C8—1781no—
446#104—1782no—
447#104—1782no—
448#103—891no—
449#103—891no—
450#103—891no—
451#103—891no—
452#102—891no—
453#102—891no—
454#102—891no—
455#102—891no—
456#97—21no—
457#97—21no—
458#97—21no—
459#97—1781no—
460#97—1781no—
461#97—1782no—
462#97C22⋊D8911yescharacteristic
463#97C22⋊D8911yescharacteristic
464#97C22⋊D8911yescharacteristic
465#97C22⋊D8911yescharacteristic
466C4×C8—21no—
467C4×C8—21no—
468C4×C8—21no—
469C4×C8—1781no—
470C4×C8—1781no—
471C4×C8—1782no—
472#95—1781no—
473#41—21no—
474#41—21no—
475#41C2×Dic17812yes—
476#41#1308357412yes—
477C2×Q8—1781no—
478C2×Q8—1781no—
479C2×Q8C2×C4×D8911yescharacteristic
480C2×D4—21no—
481C2×D4C2×C4×D8911yescharacteristic
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—1781no—
486C22×C4—1781no—
487C22×C4#1308357411yescharacteristic
488C22×C4#1308357411yescharacteristic
489C22×C4C23×D8911yescharacteristic
490M16—24no—
491M16—21no—
492M16—21no—
493M16—24no—
494M16—1781no—
495M16—1781no—
496M16—1781no—
497M16—1781no—
498M16C2×Dic17812yes—
499M16#1308357412yes—
500C2×C8—24no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—24no—
506C2×C8—891no—
507C2×C8—891no—
508C2×C8—891no—
509C2×C8—891no—
510C2×C8—1781no—
511C2×C8—1781no—
512C2×C8—1781no—
513C2×C8—1781no—
514C2×C8—1781no—
515C2×C8—1781no—
516C2×C8C2×C22⋊D8911yescharacteristic
517C2×C8C2×C22⋊D8911yescharacteristic
518C2×C8C2×C22⋊D8911yescharacteristic
519C2×C8C2×C22⋊D8911yescharacteristic
520C2×C8C2×Dic17812yes—
521C2×C8#1308357412yes—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—891no—
525C4⋊C4—891no—
526C4⋊C4—891no—
527C4⋊C4—891no—
528C4⋊C4—3562no—
529C4⋊C4—1781no—
530C4⋊C4—1781no—
531C4⋊C4—1781no—
532C4⋊C4—1781no—
533C4⋊C4—1781no—
534C4⋊C4—1781no—
535C4⋊C4C2×C4×D8911yescharacteristic
536C4⋊C4C2×C4×D8911yescharacteristic
537C22⋊C4—21no—
538C22⋊C4—21no—
539C22⋊C4—21no—
540C22⋊C4—1781no—
541C22⋊C4—1781no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—1781no—
546C42—1781no—
547C42—1781no—
548C42—1781no—
549C42—1781no—
550C42—1781no—
551C42C2×C4×D8911yescharacteristic
552C42C2×C4×D8911yescharacteristic
553C42C2×C22⋊D8911yescharacteristic
554C42C2×C22⋊D8911yescharacteristic
555C23—21no—
556C23C2×#1308357411yescharacteristic
557Q8—21no—
558Q8—7121no—
559Q8—7121no—
560Q8C4×Dic17812yes—
561D4—22no—
562D4—22no—
563D4—21no—
564D4C4×Dic17812yes—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—81no—
573C2×C4—81no—
574C2×C4—3561no—
575C2×C4—3561no—
576C2×C4—3561no—
577C2×C4—3561no—
578C2×C4—3561no—
579C2×C4—1782no—
580C2×C4—1781no—
581C2×C4—1781no—
582C2×C4—1781no—
583C2×C4—1781no—
584C2×C4C4×C22⋊D8911yescharacteristic
585C2×C4C4×C22⋊D8911yescharacteristic
586C2×C4C4×C22⋊D8911yescharacteristic
587C2×C4C4×C22⋊D8911yescharacteristic
588C2×C4C22×C4×D8911yescharacteristic
589C2×C4C22×C22⋊D8911yescharacteristic
590C2×C4C22×Dic17811yescharacteristic
591C2×C4C2×#1308357411yescharacteristic
592C2×C4C2×#1308357411yescharacteristic
593C2×C4C2×#1308357411yescharacteristic
594C8—22no—
595C8—21no—
596C8—22no—
597C8—21no—
598C8—3561no—
599C8—3561no—
600C8—1781no—
601C8—1781no—
602C8—1781no—
603C8—1781no—
604C8#1566799112yes—
605C8#1566799112yes—
606C22—21no—
607C22—21no—
608C22—22no—
609C22—41no—
610C22—41no—
611C22C2×C4×C22⋊D8911yescharacteristic
612C22#1915213211yescharacteristic
613C22#1915215511yescharacteristic
614C4—21no—
615C4—21no—
616C4—21no—
617C4—21no—
618C4—22no—
619C4—41no—
620C4—41no—
621C4—3562no—
622C4—3561no—
623C4—3561no—
624C4—3561no—
625C4—3561no—
626C4—7121no—
627C4C2×C4×Dic17811yescharacteristic
628C4C4×#1308357411yescharacteristic
629C4C2×#1566799111yescharacteristic
630C4#1915210911yescharacteristic
631C2—21no—
632C2—21no—
633C2—42no—
634C2C4×#1566799111yescharacteristic
635C2#2602915911yescharacteristic
636C2#2602916711yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D89
commutatorC2×C4×C89C23×C4
FrattiniC22×C4C23×D89
FittingC89×#2875C2
socleC22×C89C2×C4×D4
prime radicalC2×#41×C89C23
cosocleC22×C4×C89C24
hypercenter#2875D89
hypocenterC89#21629
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C2×#97 ◁ #2875

Complements

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

NH ≅ G/Ntype
2. #26029192C2semidirect
3. #26021196C2semidirect
4. #26021027C2semidirect
5. #26020851C2semidirect
6. #26020851C2semidirect
10. #26020137C2semidirect
11. #26020132C2semidirect
12. #26020124C2semidirect
17. #19151905C4semidirect
18. #19151905C4semidirect
19. #19151898C4semidirect
20. #19151898C4semidirect
21. #19151893C4semidirect
22. #19151888C4semidirect
23. #19151883C4semidirect
24. #19151878C4semidirect
125. #131×C89Q8semidirect
334. C89#21629semidirect
390. #131Dic178semidirect

Maximal subgroups

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

subgroupindex
#260300402
#260291922
#260211962
#260210272
#260208512
#260205272
#260202722
#260201372
#260201322
#260201242
#260200942
#260200872
C89×#28752
#2162989