Finite Group Databasebeta

← #36525794

Subgroups — #36525794

Order 22784.

606 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26029341C211yesmaximal, characteristic, complemented
2#26029319C211yesmaximal, characteristic
3#26019155C211yesmaximal, characteristic, complemented
4#26019143C211yesmaximal, characteristic, complemented
5#26019131C211yesmaximal, characteristic, complemented
6#26019110C211yesmaximal, characteristic, complemented
7#26019096C211yesmaximal, characteristic, complemented
8#26019089C211yesmaximal, characteristic, complemented
9SD16⋊#10623494C211yesmaximal, characteristic, complemented
10SD16×#10623494C211yesmaximal, characteristic, complemented
11#26019032C211yesmaximal, characteristic, complemented
12#26018721C211yesmaximal, characteristic
13#26018647C211yesmaximal, characteristic, complemented
14#26018552C211yesmaximal, characteristic, complemented
15C89×#3103C211yesmaximal, characteristic
16#19151729—21no—
17#19151729—21no—
18#19151729C2211yescharacteristic
19#19151729C2211yescharacteristic
20Q8×#10623494C2211yescharacteristic
21Q8×#10623494C2211yescharacteristic
22#19151676—21no—
23#19151676C2211yescharacteristic
24#19151676C2211yescharacteristic
25D4×#10623494—21no—
26D4×#10623494C2211yescharacteristic
27D4×#10623494C2211yescharacteristic
28#19151655—21no—
29#19151655C2211yescharacteristic
30#19151651—21no—
31#19151651C2211yescharacteristic
32#19151651C2211yescharacteristic
33#19151651C2211yescharacteristic
34#19151565—21no—
35#19151565—21no—
36#19151351C2211yescharacteristic
37#19151351C2211yescharacteristic
38#19151349C2211yescharacteristic
39#19151349C2211yescharacteristic
40Q8⋊#10623494C2211yescharacteristic
41Q8⋊#10623494C2211yescharacteristic
42Q8⋊#10623494C2211yescharacteristic
43Q8⋊#10623494C2211yescharacteristic
44D4⋊#10623494C2211yescharacteristic
45D4⋊#10623494C2211yescharacteristic
46D4⋊#10623494C2211yescharacteristic
47D4⋊#10623494C2211yescharacteristic
48#19151320C2211yescharacteristic
49#19151320C2211yescharacteristic
50#19151319C2211yescharacteristic
51C8×#10623494C2211yescharacteristic
52#532×C89C2211yescharacteristic
53C2×C4×Q8×C89C2211yescharacteristic
54#481×C89C411yescharacteristic
55#481×C89C411yescharacteristic
56#475×C89C411yescharacteristic
57#475×C89C411yescharacteristic
58#460×C89C411yescharacteristic
59#460×C89C411yescharacteristic
60#448×C89C411yescharacteristic
61#448×C89C411yescharacteristic
62C4×SD16×C89C2211yescharacteristic
63C4×SD16×C89C2211yescharacteristic
64C4×SD16×C89C2211yescharacteristic
65C4×SD16×C89C2211yescharacteristic
66#431×C89C2211yescharacteristic
67C2×#13083558—21no—
68C2×#13083558—21no—
69C22×#10623494—21no—
70C22×#10623494—21no—
71C4⋊#10623494—21no—
72C4⋊#10623494—21no—
73C4⋊#10623494—21no—
74C4⋊#10623494—21no—
75C4⋊#10623494—21no—
76C4⋊#10623494D411yescharacteristic
77C4⋊#10623494D411yescharacteristic
78C4⋊#10623494C2311yescharacteristic
79C4⋊#10623494C2311yescharacteristic
80C4⋊#10623494C2311yescharacteristic
81C4⋊#10623494C2311yescharacteristic
82C22⋊#10623494—21no—
83C22⋊#10623494—21no—
84C22⋊#10623494—21no—
85C22⋊#10623494—21no—
86C22⋊#10623494—21no—
87C22⋊#10623494—21no—
88C22⋊#10623494—21no—
89C22⋊#10623494D411yescharacteristic
90C22⋊#10623494D411yescharacteristic
91#15667923—21no—
92#15667923—21no—
93#15667923—21no—
94C4×#10623494—21no—
95C4×#10623494—21no—
96C4×#10623494—21no—
97C4×#10623494C2311yescharacteristic
98C4×#10623494C2311yescharacteristic
99C4×#10623494C2311yescharacteristic
100C4×#10623494C2311yescharacteristic
101C2×#41×C89—21no—
102C22×Q8×C89C2×C411yescharacteristic
103C2×SD16×C89C2×C411yescharacteristic
104C2×SD16×C89C2×C411yescharacteristic
105C2×SD16×C89C2×C411yescharacteristic
106C2×SD16×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×C89—21no—
114C4×Q8×C89—21no—
115C4×Q8×C89C2×C411yescharacteristic
116C4×Q8×C89C2×C411yescharacteristic
117C4×Q8×C89C2311yescharacteristic
118C4×Q8×C89C2311yescharacteristic
119C4×D4×C89—21no—
120C4×D4×C89C2311yescharacteristic
121C4×D4×C89C2311yescharacteristic
122C2×C4⋊C4×C89—21no—
123C2×C4⋊C4×C89C2×C411yescharacteristic
124C2×C42×C89—21no—
125C2×C42×C89C2311yescharacteristic
126#106×C89C2×C411yescharacteristic
127#106×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#103×C89C2×C411yescharacteristic
135#103×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×#10623494C2×D411yescharacteristic
155C2×#10623494C2×D411yescharacteristic
156C2×#10623494C2×D411yescharacteristic
157C2×#10623494C2×D411yescharacteristic
158#41×C89—81no—
159#41×C89—81no—
160C2×Q8×C89—21no—
161C2×Q8×C89—21no—
162C2×Q8×C89—22no—
163C2×Q8×C89—41no—
164C2×Q8×C89C22×C411yescharacteristic
165C2×Q8×C89C22×C411yescharacteristic
166C2×D4×C89—21no—
167C2×D4×C89—21no—
168C2×D4×C89—21no—
169C2×D4×C89—21no—
170C2×D4×C89C22×C411yescharacteristic
171C2×D4×C89C22×C411yescharacteristic
172C22×C4×C89—21no—
173C22×C4×C89—21no—
174C22×C4×C89—21no—
175C22×C4×C89—21no—
176C22×C4×C89C22×C411yescharacteristic
177C22×C4×C89C2×D411yescharacteristic
178C22×C4×C89#4111yescharacteristic
179SD16×C89—22no—
180SD16×C89—22no—
181SD16×C89—41no—
182SD16×C89—41no—
183C2×C8×C89C22×C411yescharacteristic
184C2×C8×C89C22×C411yescharacteristic
185C2×C8×C89C22×C411yescharacteristic
186C2×C8×C89C22×C411yescharacteristic
187C4⋊C4×C89—21no—
188C4⋊C4×C89—21no—
189C4⋊C4×C89—21no—
190C4⋊C4×C89—21no—
191C4⋊C4×C89—21no—
192C4⋊C4×C89—21no—
193C4⋊C4×C89C22×C411yescharacteristic
194C4⋊C4×C89C22×C411yescharacteristic
195C4⋊C4×C89C22×C411yescharacteristic
196C4⋊C4×C89C22×C411yescharacteristic
197C22⋊C4×C89—21no—
198C22⋊C4×C89—21no—
199C22⋊C4×C89—21no—
200C22⋊C4×C89—21no—
201C42×C89—21no—
202C42×C89—21no—
203C42×C89—21no—
204C42×C89—21no—
205C42×C89C22×C411yescharacteristic
206C42×C89C2×D411yescharacteristic
207C42×C89#4111yescharacteristic
208C42×C89C2411yescharacteristic
209C23×C89—21no—
210C23×C89—21no—
211C23×C89C4×D411yescharacteristic
212Q8×C89—21no—
213Q8×C89—21no—
214Q8×C89—21no—
215Q8×C89—42no—
216Q8×C89—81no—
217Q8×C89#13112yes—
218D4×C89—21no—
219D4×C89—21no—
220D4×C89—21no—
221D4×C89—21no—
222D4×C89—42no—
223D4×C89—81no—
224D4×C89—81no—
225D4×C89—81no—
226C2×C4×C89—21no—
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—21no—
235C2×C4×C89—21no—
236C2×C4×C89—41no—
237C2×C4×C89—42no—
238C2×C4×C89—42no—
239C2×C4×C89—41no—
240C2×C4×C89—41no—
241C2×C4×C89—41no—
242C2×C4×C89—41no—
243C2×C4×C89C4×D411yescharacteristic
244C2×C4×C89C4×D411yescharacteristic
245C2×C4×C89C4×D411yescharacteristic
246C2×C4×C89C23×C411yescharacteristic
247C2×C4×C89C22×D411yescharacteristic
248C2×C4×C89C2×#4111yescharacteristic
249C8×C89—21no—
250C8×C89—21no—
251C8×C89—21no—
252C8×C89—21no—
253#10623494—21no—
254#10623494—21no—
255#10623494—21no—
256#10623494—21no—
257#10623494—41no—
258#10623494—41no—
259#10623494—41no—
260#10623494—41no—
261#10623494—41no—
262#10623494—41no—
263C22×C89—21no—
264C22×C89—21no—
265C22×C89—21no—
266C22×C89—41no—
267C22×C89—41no—
268C22×C89—41no—
269C22×C89—41no—
270C22×C89—41no—
271C22×C89—41no—
272C22×C89C2×C4×D411yescharacteristic
273C4×C89—21no—
274C4×C89—22no—
275C4×C89—21no—
276C4×C89—21no—
277C4×C89—21no—
278C4×C89—41no—
279C4×C89—41no—
280C4×C89—41no—
281C4×C89—41no—
282C4×C89—41no—
283C4×C89C2×#13111yescharacteristic
284C4×C89#56811yescharacteristic
285C4×C89#57711yescharacteristic
286C4×C89#57711yescharacteristic
287#18030—891nomaximal, Sylow
288C2×C89—41no—
289C2×C89—81no—
290C2×C89—81no—
291C2×C89#293311yescharacteristic
292C2×C89#294911yescharacteristic
293C2×C89#295511yescharacteristic
294#3103D8911yescharacteristic
295#2984—891no—
296#2979—891no—
297#1592—891no—
298#1590—891no—
299#1588—891no—
300#1585—891no—
301#1583—891no—
302#1582—891no—
303SD16⋊C8—891no—
304C8×SD16—891no—
305#1572—891no—
306#1496—891no—
307#1479—891no—
308#1454—891no—
309C89#1803011yescharacteristic, complemented, Sylow
310#532C2×D8911yescharacteristic
311C2×C4×Q8C2×D8911yescharacteristic
312#481Dic8911yescharacteristic
313#481Dic8911yescharacteristic
314#475Dic8911yescharacteristic
315#475Dic8911yescharacteristic
316#460Dic8911yescharacteristic
317#460Dic8911yescharacteristic
318#448Dic8911yescharacteristic
319#448Dic8911yescharacteristic
320#446—891no—
321#446—891no—
322#446—1781no—
323#446—1781no—
324C8×Q8—891no—
325C8×Q8—891no—
326C4×SD16C2×D8911yescharacteristic
327C4×SD16C2×D8911yescharacteristic
328C4×SD16C2×D8911yescharacteristic
329C4×SD16C2×D8911yescharacteristic
330#436—891no—
331#436—891no—
332#436—1781no—
333C8×D4—891no—
334C8×D4—891no—
335C8×D4—1781no—
336#432—891no—
337#432—1781no—
338#431—891no—
339#431—891no—
340#431—891no—
341#431—1781no—
342#431C2×D8911yescharacteristic
343#408—1781no—
344#408—1781no—
345#335—891no—
346#335—891no—
347#334—891no—
348#334—891no—
349Q8⋊C8—891no—
350Q8⋊C8—891no—
351Q8⋊C8—891no—
352Q8⋊C8—891no—
353D4⋊C8—891no—
354D4⋊C8—891no—
355D4⋊C8—891no—
356D4⋊C8—891no—
357#322—891no—
358#322—891no—
359#322—891no—
360C82—891no—
361C2×#41—21no—
362C22×Q8C2×Dic8911yescharacteristic
363C2×SD16C2×Dic8911yescharacteristic
364C2×SD16C2×Dic8911yescharacteristic
365C2×SD16C2×Dic8911yescharacteristic
366C2×SD16C2×Dic8911yescharacteristic
367C2×M16—1781no—
368C2×M16—1781no—
369C22×C8—1781no—
370C22×C8—1781no—
371C4⋊Q8C2×Dic8911yescharacteristic
372#127C2×Dic8911yescharacteristic
373#124—21no—
374#121—21no—
375#121C2×Dic8911yescharacteristic
376#121C2×Dic8911yescharacteristic
377C4×Q8—21no—
378C4×Q8—21no—
379C4×Q8C2×Dic8911yescharacteristic
380C4×Q8C2×Dic8911yescharacteristic
381C4×Q8C22×D8911yescharacteristic
382C4×Q8C22×D8911yescharacteristic
383C4×D4—21no—
384C4×D4C22×D8911yescharacteristic
385C4×D4C22×D8911yescharacteristic
386C2×C4⋊C4—21no—
387C2×C4⋊C4C2×Dic8911yescharacteristic
388C2×C42—21no—
389C2×C42C22×D8911yescharacteristic
390#106C2×Dic8911yescharacteristic
391#106C2×Dic8911yescharacteristic
392#106C2×Dic8911yescharacteristic
393#106C2×Dic8911yescharacteristic
394C4⋊C8—891no—
395C4⋊C8—891no—
396C4⋊C8—891no—
397C4⋊C8—891no—
398C4⋊C8—891no—
399C4⋊C8—891no—
400C4⋊C8—1781no—
401C4⋊C8—1781no—
402C4⋊C8—1781no—
403C4⋊C8—1781no—
404C4⋊C8—1781no—
405C4⋊C8C2×Dic8911yescharacteristic
406C4⋊C8C2×Dic8911yescharacteristic
407#103C2×Dic8911yescharacteristic
408#103C2×Dic8911yescharacteristic
409#103C2×Dic8911yescharacteristic
410#103C2×Dic8911yescharacteristic
411#102C2×Dic8911yescharacteristic
412#102C2×Dic8911yescharacteristic
413#102C2×Dic8911yescharacteristic
414#102C2×Dic8911yescharacteristic
415C22⋊C8—891no—
416C22⋊C8—891no—
417C22⋊C8—1781no—
418C22⋊C8—1781no—
419C22⋊C8—1781no—
420C22⋊C8—1781no—
421C22⋊C8—1781no—
422C22⋊C8—1781no—
423C22⋊C8—1781no—
424C22⋊C8C2×Dic8911yescharacteristic
425C22⋊C8C2×Dic8911yescharacteristic
426#97—1781no—
427#97—1781no—
428#97—1781no—
429C4×C8—891no—
430C4×C8—891no—
431C4×C8—891no—
432C4×C8—891no—
433C4×C8—1781no—
434C4×C8—1781no—
435C4×C8—1781no—
436C4×C8C22×D8911yescharacteristic
437C4×C8C22×D8911yescharacteristic
438#41—81no—
439#41—81no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—22no—
443C2×Q8—41no—
444C2×Q8C22×Dic8911yescharacteristic
445C2×Q8C22×Dic8911yescharacteristic
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4C22×Dic8911yescharacteristic
451C2×D4C22×Dic8911yescharacteristic
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4C22×Dic8911yescharacteristic
457C22×C4D4×D8911yescharacteristic
458C22×C4#1308357311yescharacteristic
459SD16—22no—
460SD16—22no—
461SD16—41no—
462SD16—41no—
463M16—7121no—
464M16—7121no—
465C2×C8—891no—
466C2×C8—891no—
467C2×C8—891no—
468C2×C8—891no—
469C2×C8—1781no—
470C2×C8—1781no—
471C2×C8—1781no—
472C2×C8—1781no—
473C2×C8—1781no—
474C2×C8—1781no—
475C2×C8—7121no—
476C2×C8—7121no—
477C2×C8C22×Dic8911yescharacteristic
478C2×C8C22×Dic8911yescharacteristic
479C2×C8C22×Dic8911yescharacteristic
480C2×C8C22×Dic8911yescharacteristic
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4C22×Dic8911yescharacteristic
488C4⋊C4C22×Dic8911yescharacteristic
489C4⋊C4C22×Dic8911yescharacteristic
490C4⋊C4C22×Dic8911yescharacteristic
491C22⋊C4—21no—
492C22⋊C4—21no—
493C22⋊C4—21no—
494C22⋊C4—21no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42C22×Dic8911yescharacteristic
500C42D4×D8911yescharacteristic
501C42#1308357311yescharacteristic
502C42C23×D8911yescharacteristic
503C23—21no—
504C23—21no—
505C23D4×Dic8911yescharacteristic
506Q8—21no—
507Q8—21no—
508Q8—21no—
509Q8—42no—
510Q8—81no—
511Q8#1566801912yes—
512D4—21no—
513D4—21no—
514D4—21no—
515D4—21no—
516D4—42no—
517D4—81no—
518D4—81no—
519D4—81no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4D4×Dic8911yescharacteristic
538C2×C4D4×Dic8911yescharacteristic
539C2×C4D4×Dic8911yescharacteristic
540C2×C4C23×Dic8911yescharacteristic
541C2×C4C2×D4×D8911yescharacteristic
542C2×C4C2×#1308357311yescharacteristic
543C8—21no—
544C8—21no—
545C8—21no—
546C8—21no—
547C8—3561no—
548C8—3561no—
549C8—3561no—
550C8—3561no—
551C8—3561no—
552C8—3561no—
553C8—1781no—
554C8—1781no—
555C8—1781no—
556C8—1781no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—41no—
564C22—41no—
565C22—41no—
566C22C2×D4×Dic8911yescharacteristic
567C4—21no—
568C4—21no—
569C4—21no—
570C4—21no—
571C4—22no—
572C4—41no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4C2×#1566801911yescharacteristic
578C4#1915236511yescharacteristic
579C4#1915240411yescharacteristic
580C4#1915240411yescharacteristic
581C2—41no—
582C2—81no—
583C2—81no—
584C2#2602897911yescharacteristic
585C2#2602911211yescharacteristic
586C2#2602915511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#3103C2
socleC22×C89C2×C4×D4
prime radical#532×C89C22
cosocleC42×C89C24
hypercenter#3103D89
hypocenterC89#18030
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3103

Complements

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

NH ≅ G/Ntype
1. #26029341C2semidirect
3. #26019155C2semidirect
4. #26019143C2semidirect
5. #26019131C2semidirect
6. #26019110C2semidirect
7. #26019096C2semidirect
8. #26019089C2semidirect
9. SD16⋊#10623494C2semidirect
10. SD16×#10623494C2semidirect
11. #26019032C2semidirect
13. #26018647C2semidirect
14. #26018552C2semidirect
309. C89#18030semidirect

Maximal subgroups

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

subgroupindex
#260293412
#260293192
#260191552
#260191432
#260191312
#260191102
#260190962
#260190892
SD16⋊#106234942
SD16×#106234942
#260190322
#260187212
#260186472
#260185522
C89×#31032
#1803089