Finite Group Databasebeta

← #36524520

Subgroups — #36524520

Order 22784.

638 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26029363C211yesmaximal, characteristic, complemented
2#26029328C211yesmaximal, characteristic, complemented
3#26019131C211yesmaximal, characteristic, complemented
4#26019117C211yesmaximal, characteristic, complemented
5#26019096C211yesmaximal, characteristic, complemented
6#26019082C211yesmaximal, characteristic, complemented
7#26019076C211yesmaximal, characteristic, complemented
8SD16⋊#10623494C211yesmaximal, characteristic, complemented
9SD16×#10623494C211yesmaximal, characteristic, complemented
10D8×#10623494C211yesmaximal, characteristic, complemented
11#26019037C211yesmaximal, characteristic, complemented
12#26018686C211yesmaximal, characteristic, complemented
13#26018651C211yesmaximal, characteristic, complemented
14#26018552C211yesmaximal, characteristic, complemented
15C89×#3106C211yesmaximal, characteristic
16Q8×#10623494C2211yescharacteristic
17Q8×#10623494C2211yescharacteristic
18#19151676C2211yescharacteristic
19#19151676C2211yescharacteristic
20#19151670—21no—
21#19151670—21no—
22D4×#10623494—21no—
23D4×#10623494—21no—
24D4×#10623494C2211yescharacteristic
25D4×#10623494C2211yescharacteristic
26D4×#10623494C2211yescharacteristic
27D4×#10623494C2211yescharacteristic
28#19151655C2211yescharacteristic
29#19151651—21no—
30#19151651C2211yescharacteristic
31#19151651C2211yescharacteristic
32C2×C4⋊#10623494—21no—
33C2×C4⋊#10623494C2211yescharacteristic
34#19151565—21no—
35#19151565—21no—
36#19151351C2211yescharacteristic
37#19151351C2211yescharacteristic
38#19151349C2211yescharacteristic
39#19151349C2211yescharacteristic
40Q8⋊#10623494C2211yescharacteristic
41Q8⋊#10623494C2211yescharacteristic
42D4⋊#10623494C2211yescharacteristic
43D4⋊#10623494C2211yescharacteristic
44D4⋊#10623494C2211yescharacteristic
45D4⋊#10623494C2211yescharacteristic
46D4⋊#10623494C2211yescharacteristic
47D4⋊#10623494C2211yescharacteristic
48#19151320C2211yescharacteristic
49#19151320C2211yescharacteristic
50#19151319C2211yescharacteristic
51C8×#10623494C2211yescharacteristic
52#532×C89C2211yescharacteristic
53C4×#41×C89C2211yescharacteristic
54#482×C89C411yescharacteristic
55#482×C89C411yescharacteristic
56#477×C89C411yescharacteristic
57#476×C89C411yescharacteristic
58#460×C89C411yescharacteristic
59#459×C89C411yescharacteristic
60#449×C89C411yescharacteristic
61#449×C89C411yescharacteristic
62C4×SD16×C89C2211yescharacteristic
63C4×SD16×C89C2211yescharacteristic
64C4×D8×C89C2211yescharacteristic
65C4×D8×C89C2211yescharacteristic
66#431×C89C2211yescharacteristic
67C2×#13083558—21no—
68C2×#13083558—21no—
69C22×#10623494—21no—
70C22×#10623494—21no—
71C22×#10623494—21no—
72C22×#10623494—21no—
73C22×#10623494—21no—
74C4⋊#10623494—21no—
75C4⋊#10623494—21no—
76C4⋊#10623494—21no—
77C4⋊#10623494—21no—
78C4⋊#10623494D411yescharacteristic
79C4⋊#10623494D411yescharacteristic
80C4⋊#10623494C2311yescharacteristic
81C4⋊#10623494C2311yescharacteristic
82C4⋊#10623494C2311yescharacteristic
83C4⋊#10623494C2311yescharacteristic
84C22⋊#10623494—21no—
85C22⋊#10623494—21no—
86C22⋊#10623494—21no—
87C22⋊#10623494—21no—
88C22⋊#10623494—21no—
89C22⋊#10623494—21no—
90C22⋊#10623494—21no—
91C22⋊#10623494—21no—
92C22⋊#10623494D411yescharacteristic
93C22⋊#10623494D411yescharacteristic
94#15667923—21no—
95C4×#10623494—21no—
96C4×#10623494—21no—
97C4×#10623494C2311yescharacteristic
98C4×#10623494C2311yescharacteristic
99C4×#10623494C2311yescharacteristic
100C4×#10623494C2311yescharacteristic
101C2×#41×C89—21no—
102C2×#41×C89C2×C411yescharacteristic
103C2×SD16×C89C2×C411yescharacteristic
104C2×SD16×C89C2×C411yescharacteristic
105C2×D8×C89C2×C411yescharacteristic
106C2×D8×C89C2×C411yescharacteristic
107C4⋊Q8×C89C2×C411yescharacteristic
108#127×C89C2×C411yescharacteristic
109#124×C89—21no—
110#121×C89—21no—
111#121×C89C2×C411yescharacteristic
112#121×C89C2×C411yescharacteristic
113C4×Q8×C89C2×C411yescharacteristic
114C4×Q8×C89C2311yescharacteristic
115C4×D4×C89—21no—
116C4×D4×C89—21no—
117C4×D4×C89—21no—
118C4×D4×C89C2×C411yescharacteristic
119C4×D4×C89C2311yescharacteristic
120C4×D4×C89C2311yescharacteristic
121C4×D4×C89C2311yescharacteristic
122#117×C89—21no—
123#117×C89C2×C411yescharacteristic
124C2×C42×C89—21no—
125C2×C42×C89C2311yescharacteristic
126#107×C89C2×C411yescharacteristic
127#107×C89C2×C411yescharacteristic
128#106×C89C2×C411yescharacteristic
129#106×C89C2×C411yescharacteristic
130C4⋊C8×C89C2×C411yescharacteristic
131C4⋊C8×C89C2×C411yescharacteristic
132#103×C89C2×C411yescharacteristic
133#103×C89C2×C411yescharacteristic
134#102×C89C2×C411yescharacteristic
135#102×C89C2×C411yescharacteristic
136#102×C89C2×C411yescharacteristic
137#102×C89C2×C411yescharacteristic
138#102×C89C2×C411yescharacteristic
139#102×C89C2×C411yescharacteristic
140C22⋊C8×C89C2×C411yescharacteristic
141C22⋊C8×C89C2×C411yescharacteristic
142C4×C8×C89C2311yescharacteristic
143C4×C8×C89C2311yescharacteristic
144#13083558—81no—
145#13083558—81no—
146C2×#10623494—21no—
147C2×#10623494—21no—
148C2×#10623494—21no—
149C2×#10623494—21no—
150C2×#10623494—21no—
151C2×#10623494—21no—
152C2×#10623494—81no—
153C2×#10623494—81no—
154C2×#10623494—81no—
155C2×#10623494—81no—
156C2×#10623494—81no—
157C2×#10623494C2×D411yescharacteristic
158C2×#10623494C2×D411yescharacteristic
159C2×#10623494C2×D411yescharacteristic
160C2×#10623494C2×D411yescharacteristic
161#41×C89—22no—
162#41×C89—22no—
163#41×C89—81no—
164#41×C89—81no—
165C2×Q8×C89—21no—
166C2×Q8×C89C22×C411yescharacteristic
167C2×D4×C89—21no—
168C2×D4×C89—21no—
169C2×D4×C89—21no—
170C2×D4×C89—21no—
171C2×D4×C89—21no—
172C2×D4×C89C22×C411yescharacteristic
173C2×D4×C89C22×C411yescharacteristic
174C2×D4×C89C22×C411yescharacteristic
175C22×C4×C89—21no—
176C22×C4×C89—21no—
177C22×C4×C89—21no—
178C22×C4×C89—21no—
179C22×C4×C89—21no—
180C22×C4×C89C22×C411yescharacteristic
181C22×C4×C89C2×D411yescharacteristic
182C22×C4×C89#4111yescharacteristic
183SD16×C89—22no—
184SD16×C89—22no—
185D8×C89—22no—
186D8×C89—22no—
187C2×C8×C89C22×C411yescharacteristic
188C2×C8×C89C22×C411yescharacteristic
189C2×C8×C89C22×C411yescharacteristic
190C2×C8×C89C22×C411yescharacteristic
191C4⋊C4×C89—21no—
192C4⋊C4×C89—21no—
193C4⋊C4×C89—21no—
194C4⋊C4×C89C22×C411yescharacteristic
195C4⋊C4×C89C22×C411yescharacteristic
196C4⋊C4×C89C22×C411yescharacteristic
197C4⋊C4×C89C22×C411yescharacteristic
198C22⋊C4×C89—21no—
199C22⋊C4×C89—21no—
200C22⋊C4×C89—21no—
201C22⋊C4×C89—21no—
202C22⋊C4×C89—21no—
203C22⋊C4×C89—21no—
204C22⋊C4×C89—21no—
205C42×C89—21no—
206C42×C89—21no—
207C42×C89—21no—
208C42×C89C22×C411yescharacteristic
209C42×C89C2×D411yescharacteristic
210C42×C89#4111yescharacteristic
211C42×C89C2411yescharacteristic
212C23×C89—21no—
213C23×C89—21no—
214C23×C89—21no—
215C23×C89C4×D411yescharacteristic
216Q8×C89—21no—
217Q8×C89—81no—
218Q8×C89#13112yes—
219D4×C89—24no—
220D4×C89—21no—
221D4×C89—21no—
222D4×C89—21no—
223D4×C89—21no—
224D4×C89—21no—
225D4×C89—81no—
226D4×C89—81no—
227D4×C89—81no—
228D4×C89—81no—
229D4×C89#13112yes—
230C2×C4×C89—24no—
231C2×C4×C89—21no—
232C2×C4×C89—21no—
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—41no—
242C2×C4×C89—41no—
243C2×C4×C89—41no—
244C2×C4×C89—41no—
245C2×C4×C89—41no—
246C2×C4×C89—41no—
247C2×C4×C89—81no—
248C2×C4×C89C4×D411yescharacteristic
249C2×C4×C89C4×D411yescharacteristic
250C2×C4×C89C4×D411yescharacteristic
251C2×C4×C89C23×C411yescharacteristic
252C2×C4×C89C22×D411yescharacteristic
253C2×C4×C89C2×#4111yescharacteristic
254C8×C89—21no—
255C8×C89—21no—
256C8×C89—21no—
257C8×C89—21no—
258#10623494—21no—
259#10623494—21no—
260#10623494—21no—
261#10623494—21no—
262#10623494—41no—
263#10623494—41no—
264#10623494—41no—
265#10623494—41no—
266#10623494—41no—
267#10623494—41no—
268C22×C89—21no—
269C22×C89—22no—
270C22×C89—21no—
271C22×C89—21no—
272C22×C89—41no—
273C22×C89—41no—
274C22×C89—41no—
275C22×C89—41no—
276C22×C89—41no—
277C22×C89—41no—
278C22×C89—41no—
279C22×C89—41no—
280C22×C89C2×C4×D411yescharacteristic
281C4×C89—21no—
282C4×C89—22no—
283C4×C89—21no—
284C4×C89—21no—
285C4×C89—21no—
286C4×C89—41no—
287C4×C89—41no—
288C4×C89—41no—
289C4×C89—41no—
290C4×C89C2×#13111yescharacteristic
291C4×C89C2×#13111yescharacteristic
292C4×C89#57611yescharacteristic
293C4×C89#57711yescharacteristic
294#17920—891nomaximal, Sylow
295C2×C89—41no—
296C2×C89—42no—
297C2×C89—81no—
298C2×C89—81no—
299C2×C89#292911yescharacteristic
300C2×C89#294611yescharacteristic
301C2×C89#295711yescharacteristic
302#3106D8911yescharacteristic
303#2988—891no—
304#2981—891no—
305#1588—891no—
306#1586—891no—
307#1583—891no—
308#1581—891no—
309#1580—891no—
310SD16⋊C8—891no—
311C8×SD16—891no—
312C8×D8—891no—
313#1573—891no—
314#1488—891no—
315#1480—891no—
316#1454—891no—
317C89#1792011yescharacteristic, complemented, Sylow
318#532C2×D8911yescharacteristic
319C4×#41C2×D8911yescharacteristic
320#482Dic8911yescharacteristic
321#482Dic8911yescharacteristic
322#477Dic8911yescharacteristic
323#476Dic8911yescharacteristic
324#460Dic8911yescharacteristic
325#459Dic8911yescharacteristic
326#449Dic8911yescharacteristic
327#449Dic8911yescharacteristic
328C8×Q8—891no—
329C8×Q8—891no—
330C4×SD16C2×D8911yescharacteristic
331C4×SD16C2×D8911yescharacteristic
332C4×D8C2×D8911yescharacteristic
333C4×D8C2×D8911yescharacteristic
334#436—891no—
335#436—891no—
336#435—1781no—
337#435—1781no—
338C8×D4—891no—
339C8×D4—891no—
340C8×D4—891no—
341C8×D4—891no—
342C8×D4—1781no—
343C8×D4—1781no—
344#432—891no—
345#431—891no—
346#431—891no—
347#431—1781no—
348#431C2×D8911yescharacteristic
349C2×C4⋊C8—891no—
350C2×C4⋊C8—1781no—
351#408—1781no—
352#408—1781no—
353#335—891no—
354#335—891no—
355#334—891no—
356#334—891no—
357Q8⋊C8—891no—
358Q8⋊C8—891no—
359D4⋊C8—891no—
360D4⋊C8—891no—
361D4⋊C8—891no—
362D4⋊C8—891no—
363D4⋊C8—891no—
364D4⋊C8—891no—
365#322—891no—
366#322—891no—
367#322—891no—
368C82—891no—
369C2×#41—21no—
370C2×#41C2×Dic8911yescharacteristic
371C2×SD16C2×Dic8911yescharacteristic
372C2×SD16C2×Dic8911yescharacteristic
373C2×D8C2×Dic8911yescharacteristic
374C2×D8C2×Dic8911yescharacteristic
375C2×M16—1781no—
376C2×M16—1781no—
377C22×C8—1781no—
378C22×C8—1781no—
379C22×C8—1781no—
380C22×C8—1781no—
381C22×C8—1781no—
382C4⋊Q8C2×Dic8911yescharacteristic
383#127C2×Dic8911yescharacteristic
384#124—21no—
385#121—21no—
386#121C2×Dic8911yescharacteristic
387#121C2×Dic8911yescharacteristic
388C4×Q8C2×Dic8911yescharacteristic
389C4×Q8C22×D8911yescharacteristic
390C4×D4—21no—
391C4×D4—21no—
392C4×D4—21no—
393C4×D4C2×Dic8911yescharacteristic
394C4×D4C22×D8911yescharacteristic
395C4×D4C22×D8911yescharacteristic
396C4×D4C22×D8911yescharacteristic
397#117—21no—
398#117C2×Dic8911yescharacteristic
399C2×C42—21no—
400C2×C42C22×D8911yescharacteristic
401#107C2×Dic8911yescharacteristic
402#107C2×Dic8911yescharacteristic
403#106C2×Dic8911yescharacteristic
404#106C2×Dic8911yescharacteristic
405C4⋊C8—891no—
406C4⋊C8—891no—
407C4⋊C8—891no—
408C4⋊C8—891no—
409C4⋊C8—891no—
410C4⋊C8—891no—
411C4⋊C8—1781no—
412C4⋊C8—1781no—
413C4⋊C8—1781no—
414C4⋊C8—1781no—
415C4⋊C8C2×Dic8911yescharacteristic
416C4⋊C8C2×Dic8911yescharacteristic
417#103C2×Dic8911yescharacteristic
418#103C2×Dic8911yescharacteristic
419#102C2×Dic8911yescharacteristic
420#102C2×Dic8911yescharacteristic
421#102C2×Dic8911yescharacteristic
422#102C2×Dic8911yescharacteristic
423#102C2×Dic8911yescharacteristic
424#102C2×Dic8911yescharacteristic
425C22⋊C8—891no—
426C22⋊C8—891no—
427C22⋊C8—1781no—
428C22⋊C8—1781no—
429C22⋊C8—1781no—
430C22⋊C8—1781no—
431C22⋊C8—1781no—
432C22⋊C8—1781no—
433C22⋊C8—1781no—
434C22⋊C8—1781no—
435C22⋊C8C2×Dic8911yescharacteristic
436C22⋊C8C2×Dic8911yescharacteristic
437#97—1781no—
438C4×C8—891no—
439C4×C8—891no—
440C4×C8—891no—
441C4×C8—891no—
442C4×C8—1781no—
443C4×C8—1781no—
444C4×C8C22×D8911yescharacteristic
445C4×C8C22×D8911yescharacteristic
446#41—22no—
447#41—22no—
448#41—81no—
449#41—81no—
450C2×Q8—21no—
451C2×Q8C22×Dic8911yescharacteristic
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4C22×Dic8911yescharacteristic
458C2×D4C22×Dic8911yescharacteristic
459C2×D4C22×Dic8911yescharacteristic
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4C22×Dic8911yescharacteristic
466C22×C4D4×D8911yescharacteristic
467C22×C4#1308357311yescharacteristic
468SD16—22no—
469SD16—22no—
470D8—22no—
471D8—22no—
472M16—7121no—
473M16—7121no—
474C2×C8—891no—
475C2×C8—891no—
476C2×C8—891no—
477C2×C8—891no—
478C2×C8—1781no—
479C2×C8—1781no—
480C2×C8—1781no—
481C2×C8—1781no—
482C2×C8—1781no—
483C2×C8—1781no—
484C2×C8—7121no—
485C2×C8—7121no—
486C2×C8—7121no—
487C2×C8—7121no—
488C2×C8—7121no—
489C2×C8C22×Dic8911yescharacteristic
490C2×C8C22×Dic8911yescharacteristic
491C2×C8C22×Dic8911yescharacteristic
492C2×C8C22×Dic8911yescharacteristic
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4C22×Dic8911yescharacteristic
497C4⋊C4C22×Dic8911yescharacteristic
498C4⋊C4C22×Dic8911yescharacteristic
499C4⋊C4C22×Dic8911yescharacteristic
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—21no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42C22×Dic8911yescharacteristic
511C42D4×D8911yescharacteristic
512C42#1308357311yescharacteristic
513C42C23×D8911yescharacteristic
514C23—21no—
515C23—21no—
516C23—21no—
517C23D4×Dic8911yescharacteristic
518Q8—21no—
519Q8—81no—
520Q8#1566801912yes—
521D4—24no—
522D4—21no—
523D4—21no—
524D4—21no—
525D4—21no—
526D4—21no—
527D4—81no—
528D4—81no—
529D4—81no—
530D4—81no—
531D4#1566801912yes—
532C2×C4—24no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—81no—
550C2×C4D4×Dic8911yescharacteristic
551C2×C4D4×Dic8911yescharacteristic
552C2×C4D4×Dic8911yescharacteristic
553C2×C4C23×Dic8911yescharacteristic
554C2×C4C2×D4×D8911yescharacteristic
555C2×C4C2×#1308357311yescharacteristic
556C8—21no—
557C8—21no—
558C8—21no—
559C8—21no—
560C8—3561no—
561C8—3561no—
562C8—3561no—
563C8—3561no—
564C8—3561no—
565C8—3561no—
566C8—1781no—
567C8—1781no—
568C8—1781no—
569C8—1781no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—22no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22C2×D4×Dic8911yescharacteristic
583C4—21no—
584C4—21no—
585C4—21no—
586C4—21no—
587C4—22no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4C2×#1566801911yescharacteristic
593C4C2×#1566801911yescharacteristic
594C4#1915240211yescharacteristic
595C4#1915240811yescharacteristic
596C2—41no—
597C2—42no—
598C2—81no—
599C2—81no—
600C2#2602894711yescharacteristic
601C2#2602908511yescharacteristic
602C2#2602916911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
Fitting, prime radicalC89×#3106C2
socleC22×C89C2×C4×D4
cosocleC42×C89C24
hypercenter#3106D89
hypocenterC89#17920
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3106

Complements

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

NH ≅ G/Ntype
1. #26029363C2semidirect
2. #26029328C2semidirect
3. #26019131C2semidirect
4. #26019117C2semidirect
5. #26019096C2semidirect
6. #26019082C2semidirect
7. #26019076C2semidirect
8. SD16⋊#10623494C2semidirect
9. SD16×#10623494C2semidirect
10. D8×#10623494C2semidirect
11. #26019037C2semidirect
12. #26018686C2semidirect
13. #26018651C2semidirect
14. #26018552C2semidirect
317. C89#17920semidirect

Maximal subgroups

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

subgroupindex
#260293632
#260293282
#260191312
#260191172
#260190962
#260190822
#260190762
SD16⋊#106234942
SD16×#106234942
D8×#106234942
#260190372
#260186862
#260186512
#260185522
C89×#31062
#1792089