Finite Group Databasebeta

← #36525839

Subgroups — #36525839

Order 22784.

598 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26029412C211yesmaximal, characteristic, complemented
2#26029384C211yesmaximal, characteristic, complemented
3#26019155C211yesmaximal, characteristic, complemented
4#26019137C211yesmaximal, characteristic, complemented
5#26019131C211yesmaximal, characteristic, complemented
6#26019131C211yesmaximal, characteristic, complemented
7#26019103C211yesmaximal, characteristic, complemented
8#26019082C211yesmaximal, characteristic, complemented
9SD16⋊#10623494C211yesmaximal, characteristic, complemented
10Q16×#10623494C211yesmaximal, characteristic, complemented
11#26019027C211yesmaximal, characteristic, complemented
12#26018717C211yesmaximal, characteristic, complemented
13#26018639C211yesmaximal, characteristic, complemented
14#26018552C211yesmaximal, characteristic, complemented
15C89×#3102C211yesmaximal, characteristic
16#19151729C2211yescharacteristic
17Q8×#10623494C2211yescharacteristic
18Q8×#10623494C2211yescharacteristic
19Q8×#10623494C2211yescharacteristic
20#19151676—21no—
21#19151676—21no—
22#19151676C2211yescharacteristic
23#19151676C2211yescharacteristic
24#19151676C2211yescharacteristic
25#19151670—21no—
26D4×#10623494—21no—
27D4×#10623494C2211yescharacteristic
28#19151659—21no—
29#19151651—21no—
30#19151651C2211yescharacteristic
31#19151651C2211yescharacteristic
32#19151625—21no—
33#19151622—21no—
34#19151622C2211yescharacteristic
35C2×C4⋊#10623494C2211yescharacteristic
36#19151351C2211yescharacteristic
37#19151351C2211yescharacteristic
38#19151351C2211yescharacteristic
39#19151351C2211yescharacteristic
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#529×C89C2211yescharacteristic
53C4×#41×C89C2211yescharacteristic
54#485×C89C411yescharacteristic
55#482×C89C411yescharacteristic
56#479×C89C411yescharacteristic
57#478×C89C411yescharacteristic
58#464×C89C411yescharacteristic
59#463×C89C411yescharacteristic
60#452×C89C411yescharacteristic
61#449×C89C411yescharacteristic
62C4×Q16×C89C2211yescharacteristic
63C4×SD16×C89C2211yescharacteristic
64C4×SD16×C89C2211yescharacteristic
65C4×D8×C89C2211yescharacteristic
66#431×C89C2211yescharacteristic
67C2×#13083558—21no—
68C2×#13083558—21no—
69C2×#13083558—21no—
70C2×#13083558—21no—
71C22×#10623494—21no—
72C22×#10623494—21no—
73C4⋊#10623494—21no—
74C4⋊#10623494—21no—
75C4⋊#10623494—21no—
76C4⋊#10623494—21no—
77C4⋊#10623494—21no—
78C4⋊#10623494—21no—
79C4⋊#10623494—21no—
80C4⋊#10623494D411yescharacteristic
81C4⋊#10623494D411yescharacteristic
82C4⋊#10623494C2311yescharacteristic
83C4⋊#10623494C2311yescharacteristic
84C4⋊#10623494C2311yescharacteristic
85C4⋊#10623494C2311yescharacteristic
86C22⋊#10623494—21no—
87C22⋊#10623494—21no—
88C22⋊#10623494—21no—
89C22⋊#10623494—21no—
90C22⋊#10623494—21no—
91C22⋊#10623494D411yescharacteristic
92C22⋊#10623494D411yescharacteristic
93#15667923—21no—
94C4×#10623494—21no—
95C4×#10623494—21no—
96C4×#10623494—21no—
97C4×#10623494C2311yescharacteristic
98C4×#10623494C2311yescharacteristic
99C4×#10623494C2311yescharacteristic
100C4×#10623494C2311yescharacteristic
101C2×#41×C89C2×C411yescharacteristic
102C2×Q16×C89C2×C411yescharacteristic
103C2×SD16×C89C2×C411yescharacteristic
104C2×SD16×C89C2×C411yescharacteristic
105C2×D8×C89C2×C411yescharacteristic
106#126×C89—21no—
107#125×C89C2×C411yescharacteristic
108#124×C89C2×C411yescharacteristic
109#123×C89—21no—
110C22⋊Q8×C89C2×C411yescharacteristic
111#121×C89C2×C411yescharacteristic
112C4×Q8×C89C2×C411yescharacteristic
113C4×Q8×C89C2311yescharacteristic
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
121#117×C89—21no—
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#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—
146#13083558—81no—
147#13083558—81no—
148C2×#10623494—21no—
149C2×#10623494—21no—
150C2×#10623494—21no—
151C2×#10623494—21no—
152C2×#10623494—21no—
153C2×#10623494—21no—
154C2×#10623494—81no—
155C2×#10623494—81no—
156C2×#10623494C2×D411yescharacteristic
157C2×#10623494C2×D411yescharacteristic
158C2×#10623494C2×D411yescharacteristic
159C2×#10623494C2×D411yescharacteristic
160#41×C89—22no—
161#41×C89—41no—
162C2×Q8×C89C22×C411yescharacteristic
163C2×Q8×C89C22×C411yescharacteristic
164C2×D4×C89—21no—
165C2×D4×C89—21no—
166C2×D4×C89C22×C411yescharacteristic
167C2×D4×C89C22×C411yescharacteristic
168C22×C4×C89—21no—
169C22×C4×C89—21no—
170C22×C4×C89—21no—
171C22×C4×C89—21no—
172C22×C4×C89C22×C411yescharacteristic
173C22×C4×C89C2×D411yescharacteristic
174C22×C4×C89#4111yescharacteristic
175Q16×C89—22no—
176SD16×C89—22no—
177SD16×C89—41no—
178D8×C89—41no—
179C2×C8×C89C22×C411yescharacteristic
180C2×C8×C89C22×C411yescharacteristic
181C2×C8×C89C22×C411yescharacteristic
182C2×C8×C89C22×C411yescharacteristic
183C4⋊C4×C89—21no—
184C4⋊C4×C89—21no—
185C4⋊C4×C89—21no—
186C4⋊C4×C89—21no—
187C4⋊C4×C89—21no—
188C4⋊C4×C89—21no—
189C4⋊C4×C89C22×C411yescharacteristic
190C4⋊C4×C89C22×C411yescharacteristic
191C4⋊C4×C89C22×C411yescharacteristic
192C4⋊C4×C89C22×C411yescharacteristic
193C22⋊C4×C89—21no—
194C22⋊C4×C89—21no—
195C22⋊C4×C89—21no—
196C22⋊C4×C89—21no—
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#13112yes—
216D4×C89—21no—
217D4×C89—21no—
218D4×C89—21no—
219D4×C89—21no—
220D4×C89—42no—
221D4×C89—42no—
222C2×C4×C89—21no—
223C2×C4×C89—21no—
224C2×C4×C89—21no—
225C2×C4×C89—21no—
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—41no—
233C2×C4×C89—42no—
234C2×C4×C89—42no—
235C2×C4×C89—41no—
236C2×C4×C89—41no—
237C2×C4×C89—41no—
238C2×C4×C89—41no—
239C2×C4×C89C4×D411yescharacteristic
240C2×C4×C89C4×D411yescharacteristic
241C2×C4×C89C4×D411yescharacteristic
242C2×C4×C89C23×C411yescharacteristic
243C2×C4×C89C22×D411yescharacteristic
244C2×C4×C89C2×#4111yescharacteristic
245C8×C89—21no—
246C8×C89—21no—
247C8×C89—21no—
248C8×C89—21no—
249#10623494—21no—
250#10623494—21no—
251#10623494—21no—
252#10623494—21no—
253#10623494—41no—
254#10623494—41no—
255#10623494—41no—
256#10623494—41no—
257#10623494—41no—
258#10623494—41no—
259C22×C89—21no—
260C22×C89—21no—
261C22×C89—21no—
262C22×C89—41no—
263C22×C89—41no—
264C22×C89—41no—
265C22×C89—41no—
266C22×C89—41no—
267C22×C89—41no—
268C22×C89C2×C4×D411yescharacteristic
269C4×C89—21no—
270C4×C89—22no—
271C4×C89—21no—
272C4×C89—21no—
273C4×C89—21no—
274C4×C89—41no—
275C4×C89—41no—
276C4×C89—41no—
277C4×C89—41no—
278C4×C89—41no—
279C4×C89C2×#13111yescharacteristic
280C4×C89#56811yescharacteristic
281C4×C89#57611yescharacteristic
282C4×C89#57811yescharacteristic
283#18033—891nomaximal, Sylow
284C2×C89—41no—
285C2×C89—81no—
286C2×C89—81no—
287C2×C89#293311yescharacteristic
288C2×C89#295111yescharacteristic
289C2×C89#295511yescharacteristic
290#3102D8911yescharacteristic
291#2994—891no—
292#2991—891no—
293#1592—891no—
294#1589—891no—
295#1588—891no—
296#1588—891no—
297#1584—891no—
298#1581—891no—
299SD16⋊C8—891no—
300C8×Q16—891no—
301#1571—891no—
302#1495—891no—
303#1477—891no—
304#1454—891no—
305C89#1803311yescharacteristic, complemented, Sylow
306#529C2×D8911yescharacteristic
307C4×#41C2×D8911yescharacteristic
308#485Dic8911yescharacteristic
309#482Dic8911yescharacteristic
310#479Dic8911yescharacteristic
311#478Dic8911yescharacteristic
312#464Dic8911yescharacteristic
313#463Dic8911yescharacteristic
314#452Dic8911yescharacteristic
315#449Dic8911yescharacteristic
316#446—891no—
317C8×Q8—891no—
318C8×Q8—891no—
319C8×Q8—891no—
320C4×Q16C2×D8911yescharacteristic
321C4×SD16C2×D8911yescharacteristic
322C4×SD16C2×D8911yescharacteristic
323C4×D8C2×D8911yescharacteristic
324#436—891no—
325#436—891no—
326#436—891no—
327#436—1781no—
328#436—1781no—
329#435—1781no—
330C8×D4—891no—
331C8×D4—1781no—
332#433—1781no—
333#431—891no—
334#431—891no—
335#431—1781no—
336#431C2×D8911yescharacteristic
337#424—1781no—
338#423—891no—
339#423—1781no—
340C2×C4⋊C8—891no—
341#335—891no—
342#335—891no—
343#335—891no—
344#335—891no—
345Q8⋊C8—891no—
346Q8⋊C8—891no—
347Q8⋊C8—891no—
348Q8⋊C8—891no—
349D4⋊C8—891no—
350D4⋊C8—891no—
351D4⋊C8—891no—
352D4⋊C8—891no—
353#322—891no—
354#322—891no—
355#322—891no—
356C82—891no—
357C2×#41C2×Dic8911yescharacteristic
358C2×Q16C2×Dic8911yescharacteristic
359C2×SD16C2×Dic8911yescharacteristic
360C2×SD16C2×Dic8911yescharacteristic
361C2×D8C2×Dic8911yescharacteristic
362C2×M16—1781no—
363C2×M16—1781no—
364C2×M16—1781no—
365C2×M16—1781no—
366C22×C8—1781no—
367C22×C8—1781no—
368#126—21no—
369#125C2×Dic8911yescharacteristic
370#124C2×Dic8911yescharacteristic
371#123—21no—
372C22⋊Q8C2×Dic8911yescharacteristic
373#121C2×Dic8911yescharacteristic
374C4×Q8C2×Dic8911yescharacteristic
375C4×Q8C22×D8911yescharacteristic
376C4×Q8C22×D8911yescharacteristic
377C4×D4—21no—
378C4×D4—21no—
379C4×D4—21no—
380C4×D4C2×Dic8911yescharacteristic
381C4×D4C22×D8911yescharacteristic
382C4×D4C22×D8911yescharacteristic
383#117—21no—
384#117—21no—
385#117C2×Dic8911yescharacteristic
386C2×C42—21no—
387C2×C42C22×D8911yescharacteristic
388#107C2×Dic8911yescharacteristic
389#107C2×Dic8911yescharacteristic
390#106C2×Dic8911yescharacteristic
391#106C2×Dic8911yescharacteristic
392C4⋊C8—891no—
393C4⋊C8—891no—
394C4⋊C8—891no—
395C4⋊C8—891no—
396C4⋊C8—891no—
397C4⋊C8—891no—
398C4⋊C8—1781no—
399C4⋊C8—1781no—
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⋊C8C2×Dic8911yescharacteristic
423C22⋊C8C2×Dic8911yescharacteristic
424#97—1781no—
425C4×C8—891no—
426C4×C8—891no—
427C4×C8—891no—
428C4×C8—891no—
429C4×C8—1781no—
430C4×C8—1781no—
431C4×C8—1781no—
432C4×C8C22×D8911yescharacteristic
433C4×C8C22×D8911yescharacteristic
434#41—22no—
435#41—41no—
436C2×Q8C22×Dic8911yescharacteristic
437C2×Q8C22×Dic8911yescharacteristic
438C2×D4—21no—
439C2×D4—21no—
440C2×D4C22×Dic8911yescharacteristic
441C2×D4C22×Dic8911yescharacteristic
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4C22×Dic8911yescharacteristic
447C22×C4D4×D8911yescharacteristic
448C22×C4#1308357311yescharacteristic
449Q16—22no—
450SD16—22no—
451SD16—41no—
452D8—41no—
453M16—7121no—
454M16—7121no—
455M16—7121no—
456M16—7121no—
457C2×C8—891no—
458C2×C8—891no—
459C2×C8—891no—
460C2×C8—891no—
461C2×C8—1781no—
462C2×C8—1781no—
463C2×C8—1781no—
464C2×C8—1781no—
465C2×C8—1781no—
466C2×C8—1781no—
467C2×C8—7121no—
468C2×C8—7121no—
469C2×C8C22×Dic8911yescharacteristic
470C2×C8C22×Dic8911yescharacteristic
471C2×C8C22×Dic8911yescharacteristic
472C2×C8C22×Dic8911yescharacteristic
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4C22×Dic8911yescharacteristic
480C4⋊C4C22×Dic8911yescharacteristic
481C4⋊C4C22×Dic8911yescharacteristic
482C4⋊C4C22×Dic8911yescharacteristic
483C22⋊C4—21no—
484C22⋊C4—21no—
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—21no—
489C22⋊C4—21no—
490C22⋊C4—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42C22×Dic8911yescharacteristic
496C42D4×D8911yescharacteristic
497C42#1308357311yescharacteristic
498C42C23×D8911yescharacteristic
499C23—21no—
500C23—21no—
501C23D4×Dic8911yescharacteristic
502Q8—21no—
503Q8—21no—
504Q8—21no—
505Q8#1566801912yes—
506D4—21no—
507D4—21no—
508D4—21no—
509D4—21no—
510D4—42no—
511D4—42no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—41no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4D4×Dic8911yescharacteristic
530C2×C4D4×Dic8911yescharacteristic
531C2×C4D4×Dic8911yescharacteristic
532C2×C4C23×Dic8911yescharacteristic
533C2×C4C2×D4×D8911yescharacteristic
534C2×C4C2×#1308357311yescharacteristic
535C8—21no—
536C8—21no—
537C8—21no—
538C8—21no—
539C8—3561no—
540C8—3561no—
541C8—3561no—
542C8—3561no—
543C8—3561no—
544C8—3561no—
545C8—1781no—
546C8—1781no—
547C8—1781no—
548C8—1781no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22C2×D4×Dic8911yescharacteristic
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—22no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4C2×#1566801911yescharacteristic
570C4#1915236511yescharacteristic
571C4#1915240011yescharacteristic
572C4#1915241211yescharacteristic
573C2—41no—
574C2—81no—
575C2—81no—
576C2#2602897911yescharacteristic
577C2#2602912811yescharacteristic
578C2#2602915511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#3102C2
socleC22×C89C2×C4×D4
prime radical#449×C89C4
cosocleC42×C89C24
hypercenter#3102D89
hypocenterC89#18033
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3102

Complements

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

NH ≅ G/Ntype
1. #26029412C2semidirect
2. #26029384C2semidirect
3. #26019155C2semidirect
4. #26019137C2semidirect
5. #26019131C2semidirect
6. #26019131C2semidirect
7. #26019103C2semidirect
8. #26019082C2semidirect
9. SD16⋊#10623494C2semidirect
10. Q16×#10623494C2semidirect
11. #26019027C2semidirect
12. #26018717C2semidirect
13. #26018639C2semidirect
14. #26018552C2semidirect
305. C89#18033semidirect

Maximal subgroups

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

subgroupindex
#260294122
#260293842
#260191552
#260191372
#260191312
#260191032
#260190822
SD16⋊#106234942
Q16×#106234942
#260190272
#260187172
#260186392
#260185522
C89×#31022
#1803389