Finite Group Databasebeta

← #36563979

Subgroups — #36563979

Order 22784.

970 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19151659C211yesmaximal, characteristic
2#26022861C211yesmaximal, characteristic
3#26021226C211yesmaximal, characteristic, complemented
4#26021222C211yesmaximal, characteristic, complemented
5#26021218C211yesmaximal, characteristic, complemented
6#26021214C211yesmaximal, characteristic, complemented
7#26021210C211yesmaximal, characteristic, complemented
8#26021210C211yesmaximal, characteristic, complemented
9#26021205C211yesmaximal, characteristic, complemented
10#26021201C211yesmaximal, characteristic, complemented
11#26020347C211yesmaximal, characteristic
12#26020347C211yesmaximal, characteristic
13#26020342C211yesmaximal, characteristic
14#26020338C211yesmaximal, characteristic
15C2×#544×C89C211yesmaximal, characteristic
16#19151659—24no—
17C2×C4⋊#10623494C2211yescharacteristic
18C2×C4⋊#10623494C2211yescharacteristic
19C2×C22⋊#10623494C2211yescharacteristic
20C2×C22⋊#10623494C2211yescharacteristic
21C2×#15667923C2211yescharacteristic
22C2×C4×#10623494C2211yescharacteristic
23#19151491—21no—
24#19151491—21no—
25#19151484—21no—
26#19151484—21no—
27#19151468C2211yescharacteristic
28#19151464C2211yescharacteristic
29#19151458C2211yescharacteristic
30#19151458C2211yescharacteristic
31C4×C4⋊Dic89C2211yescharacteristic
32C4×C4⋊Dic89C2211yescharacteristic
33#19151360C2211yescharacteristic
34#19151360C2211yescharacteristic
35#19151360C2211yescharacteristic
36#19151360C2211yescharacteristic
37#19151360C2211yescharacteristic
38#19151360C2211yescharacteristic
39#19151360C2211yescharacteristic
40#19151360C2211yescharacteristic
41#19151360C2211yescharacteristic
42#19151360C2211yescharacteristic
43#19151360C2211yescharacteristic
44#19151360C2211yescharacteristic
45#19151360C2211yescharacteristic
46#19151360C2211yescharacteristic
47#19151360C2211yescharacteristic
48#19151360C2211yescharacteristic
49#544×C89—24no—
50#544×C89C418yescomplemented
51C2×C4⋊Q8×C89—21no—
52C2×C4⋊Q8×C89C2211yescharacteristic
53C2×C4⋊Q8×C89C2211yescharacteristic
54C2×#125×C89—21no—
55C2×#125×C89C2211yescharacteristic
56C2×#125×C89C2211yescharacteristic
57C2×C22⋊Q8×C89—21no—
58C2×C22⋊Q8×C89—21no—
59C2×#117×C89C2211yescharacteristic
60C2×#117×C89C2211yescharacteristic
61C22×C4⋊C4×C89C2211yescharacteristic
62C22×#10623494C2311yescharacteristic
63C22×#10623494C2311yescharacteristic
64C22×#10623494C2311yescharacteristic
65C22×#10623494C2311yescharacteristic
66C2×C4⋊Dic89—21no—
67C2×C4⋊Dic89—21no—
68C2×C4⋊Dic89C2311yescharacteristic
69C2×C4⋊Dic89C2311yescharacteristic
70C2×C4⋊Dic89C2311yescharacteristic
71C2×C4⋊Dic89C2311yescharacteristic
72C2×C22⋊Dic89—21no—
73C2×C22⋊Dic89—21no—
74C2×C4×Dic89—21no—
75C2×C4×Dic89—21no—
76C4⋊#10623494—22no—
77C4⋊#10623494—22no—
78C22⋊#10623494—22no—
79C22⋊#10623494—22no—
80#15667923—22no—
81C4×#10623494—22no—
82#15667919—21no—
83#15667919—21no—
84#15667919—21no—
85#15667919—21no—
86#15667919—21no—
87#15667919—21no—
88C22×Q8×C89—21no—
89C22×Q8×C89—21no—
90C23×C4×C89—21no—
91C23×C4×C89C2311yescharacteristic
92C4⋊Q8×C89—28no—
93C4⋊Q8×C89—22no—
94C4⋊Q8×C89—22no—
95C4⋊Q8×C89C2×C414yes—
96C4⋊Q8×C89C2×C414yes—
97#125×C89—28no—
98#125×C89—22no—
99#125×C89—22no—
100#125×C89C2×C414yes—
101#125×C89C2×C414yes—
102C22⋊Q8×C89—28no—
103C22⋊Q8×C89—28no—
104#117×C89—22no—
105#117×C89C2×C414yes—
106#117×C89C2×C414yes—
107#117×C89D414yes—
108C2×C4⋊C4×C89—21no—
109C2×C4⋊C4×C89—22no—
110C2×C4⋊C4×C89—21no—
111C2×C4⋊C4×C89—21no—
112C2×C4⋊C4×C89—21no—
113C2×C4⋊C4×C89—21no—
114C2×C4⋊C4×C89—21no—
115C2×C4⋊C4×C89—21no—
116C2×C4⋊C4×C89C2×C414yes—
117C2×C4⋊C4×C89D411yescharacteristic
118C2×C4⋊C4×C89D411yescharacteristic
119C2×C4⋊C4×C89C2311yescharacteristic
120C2×C4⋊C4×C89C2311yescharacteristic
121C2×C4⋊C4×C89C2311yescharacteristic
122C2×C4⋊C4×C89C2311yescharacteristic
123C2×C22⋊C4×C89—21no—
124C2×C22⋊C4×C89D411yescharacteristic
125C2×C22⋊C4×C89D411yescharacteristic
126C2×C42×C89—21no—
127C2×C42×C89C2311yescharacteristic
128C2×C42×C89C2311yescharacteristic
129C22×Dic89—21no—
130C22×Dic89—21no—
131C22×Dic89—21no—
132C22×Dic89—21no—
133C2×#10623494—21no—
134C2×#10623494—21no—
135C2×#10623494—21no—
136C2×#10623494—21no—
137C2×#10623494—21no—
138C2×#10623494—21no—
139C2×#10623494—21no—
140C2×#10623494—21no—
141C2×#10623494—21no—
142C2×#10623494—21no—
143C2×#10623494—21no—
144C2×#10623494—21no—
145C4⋊Dic89—21no—
146C4⋊Dic89—21no—
147C4⋊Dic89—21no—
148C4⋊Dic89—21no—
149C4⋊Dic89—21no—
150C4⋊Dic89—21no—
151C4⋊Dic89—21no—
152C4⋊Dic89—21no—
153C4⋊Dic89—81no—
154C4⋊Dic89—81no—
155C22⋊Dic89—81no—
156C22⋊Dic89—81no—
157C4×Dic89—81no—
158C4×Dic89—81no—
159C24×C89C2×D411yescharacteristic
160C2×Q8×C89—24no—
161C2×Q8×C89—24no—
162C2×Q8×C89—81no—
163C2×Q8×C89—81no—
164C2×Q8×C89—81no—
165C2×Q8×C89—81no—
166C22×C4×C89—21no—
167C22×C4×C89—21no—
168C22×C4×C89—21no—
169C22×C4×C89—21no—
170C22×C4×C89—21no—
171C22×C4×C89—21no—
172C22×C4×C89—21no—
173C22×C4×C89—21no—
174C22×C4×C89—24no—
175C22×C4×C89—81no—
176C22×C4×C89—81no—
177C22×C4×C89C22×C412yes—
178C22×C4×C89C2×D411yescharacteristic
179C22×C4×C89C2×D411yescharacteristic
180C22×C4×C89C2×D411yescharacteristic
181C22×C4×C89C2×D411yescharacteristic
182C22×C4×C89C2×D411yescharacteristic
183C22×C4×C89C2×D412yes—
184C22×C4×C89C2411yescharacteristic
185C4⋊C4×C89—24no—
186C4⋊C4×C89—24no—
187C4⋊C4×C89—24no—
188C4⋊C4×C89—24no—
189C4⋊C4×C89—24no—
190C4⋊C4×C89—24no—
191C4⋊C4×C89—24no—
192C4⋊C4×C89—21no—
193C4⋊C4×C89—21no—
194C4⋊C4×C89—21no—
195C4⋊C4×C89—21no—
196C4⋊C4×C89C22⋊C414yes—
197C4⋊C4×C89C22⋊C414yes—
198C4⋊C4×C89C22×C412yes—
199C4⋊C4×C89C22×C412yes—
200C4⋊C4×C89C22×C412yes—
201C4⋊C4×C89C22×C412yes—
202C22⋊C4×C89—24no—
203C22⋊C4×C89C22⋊C414yes—
204C22⋊C4×C89C22⋊C414yes—
205C42×C89—24no—
206C42×C89C22×C412yes—
207C42×C89C22×C412yes—
208C42×C89C2×D412yes—
209C42×C89C2×D412yes—
210C23×C89—21no—
211C23×C89—21no—
212C23×C89—21no—
213C23×C89—21no—
214C23×C89—21no—
215C23×C89—21no—
216C23×C89C2×C22⋊C412yes—
217C23×C89C22×D411yescharacteristic
218Q8×C89—84no—
219Q8×C89—84no—
220C2×C4×C89—21no—
221C2×C4×C89—21no—
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—22no—
231C2×C4×C89—22no—
232C2×C4×C89—22no—
233C2×C4×C89—22no—
234C2×C4×C89—22no—
235C2×C4×C89—22no—
236C2×C4×C89—22no—
237C2×C4×C89—22no—
238C2×C4×C89—41no—
239C2×C4×C89—41no—
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—84no—
247C2×C4×C89C2×C22⋊C412yes—
248C2×C4×C89C2×C22⋊C412yes—
249C2×C4×C89C2×C22⋊C412yes—
250C2×C4×C89C2×C22⋊C412yes—
251C2×C4×C89C2×C22⋊C412yes—
252C2×C4×C89C23×C411yescharacteristic
253C2×C4×C89C22×D411yescharacteristic
254C2×C4×C892−1+411yescharacteristic
255C2×C4×C892−1+411yescharacteristic
256C2×C4×C892−1+411yescharacteristic
257C2×C4×C892−1+411yescharacteristic
258C2×Dic89—41no—
259C2×Dic89—41no—
260C2×Dic89—41no—
261C2×Dic89—41no—
262C2×Dic89—41no—
263C2×Dic89—41no—
264C2×Dic89—41no—
265C2×Dic89—41no—
266C2×Dic89—41no—
267C2×Dic89—41no—
268C2×Dic89—41no—
269C2×Dic89—41no—
270#10623494—41no—
271#10623494—41no—
272#10623494—41no—
273#10623494—41no—
274C22×C89—22no—
275C22×C89—22no—
276C22×C89—22no—
277C22×C89—41no—
278C22×C89—41no—
279C22×C89—41no—
280C22×C89—41no—
281C22×C89C22×C22⋊C411yescharacteristic
282C22×C89#53611yescharacteristic
283C22×C89#53611yescharacteristic
284C22×C89#57611yescharacteristic
285C22×C89#57711yescharacteristic
286C22×C89#57711yescharacteristic
287C22×C89#57811yescharacteristic
288C4×C89—22no—
289C4×C89—22no—
290C4×C89—22no—
291C4×C89—22no—
292C4×C89—22no—
293C4×C89—42no—
294C4×C89—42no—
295C4×C89—42no—
296C4×C89—42no—
297C4×C89#51911yescharacteristic
298C4×C89#51911yescharacteristic
299C4×C89#53611yescharacteristic
300C4×C89#53611yescharacteristic
301Dic89—81no—
302Dic89—81no—
303Dic89—81no—
304Dic89—81no—
305#21843—891nomaximal, Sylow
306C2×C89—42no—
307C2×C89#231011yescharacteristic
308C2×C89#289711yescharacteristic
309C2×C89#289811yescharacteristic
310C2×C89#324111yescharacteristic
311C2×C89#324211yescharacteristic
312C2×C89#324311yescharacteristic
313C2×C89#324411yescharacteristic
314C2×#544D8911yescharacteristic
315C2×#433—891no—
316#2314—891no—
317#1993—891no—
318#1992—891no—
319#1991—891no—
320#1990—891no—
321#1989—891no—
322#1989—891no—
323#1988—891no—
324#1987—891no—
325#1825—891no—
326#1825—891no—
327#1824—891no—
328#1823—891no—
329C89#2184311yescharacteristic, complemented, Sylow
330#544—24no—
331#544Dic8918yescomplemented
332C2×C4⋊Q8—21no—
333C2×C4⋊Q8C2×D8911yescharacteristic
334C2×C4⋊Q8C2×D8911yescharacteristic
335C2×#125—21no—
336C2×#125C2×D8911yescharacteristic
337C2×#125C2×D8911yescharacteristic
338C2×C22⋊Q8—21no—
339C2×C22⋊Q8—21no—
340C2×#117C2×D8911yescharacteristic
341C2×#117C2×D8911yescharacteristic
342C22×C4⋊C4C2×D8911yescharacteristic
343#433—1784no—
344C2×C4⋊C8—891no—
345C2×C4⋊C8—891no—
346C2×C22⋊C8—891no—
347C2×C22⋊C8—891no—
348C2×#97—891no—
349C2×C4×C8—891no—
350#388—1781no—
351#388—1781no—
352#387—1781no—
353#387—1781no—
354#384—891no—
355#382—891no—
356#380—891no—
357#380—891no—
358C4×C4⋊C4—891no—
359C4×C4⋊C4—891no—
360#340—891no—
361#340—891no—
362#340—891no—
363#340—891no—
364#340—891no—
365#340—891no—
366#340—891no—
367#340—891no—
368#340—891no—
369#340—891no—
370#340—891no—
371#340—891no—
372#340—891no—
373#340—891no—
374#340—891no—
375#340—891no—
376C22×Q8—21no—
377C22×Q8—21no—
378C23×C4—21no—
379C23×C4C22×D8911yescharacteristic
380C22×C8—891no—
381C22×C8—891no—
382C22×C8—891no—
383C22×C8—891no—
384C4⋊Q8—28no—
385C4⋊Q8—22no—
386C4⋊Q8—22no—
387C4⋊Q8C2×Dic8914yes—
388C4⋊Q8C2×Dic8914yes—
389#125—28no—
390#125—22no—
391#125—22no—
392#125C2×Dic8914yes—
393#125C2×Dic8914yes—
394C22⋊Q8—28no—
395C22⋊Q8—28no—
396#117—22no—
397#117C2×Dic8914yes—
398#117C2×Dic8914yes—
399#117C22⋊D8914yes—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—22no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—21no—
408C2×C4⋊C4—891no—
409C2×C4⋊C4—891no—
410C2×C4⋊C4—891no—
411C2×C4⋊C4—891no—
412C2×C4⋊C4—1781no—
413C2×C4⋊C4—1781no—
414C2×C4⋊C4C2×Dic8914yes—
415C2×C4⋊C4C22⋊D8911yescharacteristic
416C2×C4⋊C4C22⋊D8911yescharacteristic
417C2×C4⋊C4C22×D8911yescharacteristic
418C2×C4⋊C4C22×D8911yescharacteristic
419C2×C4⋊C4C22×D8911yescharacteristic
420C2×C4⋊C4C22×D8911yescharacteristic
421C2×C22⋊C4—21no—
422C2×C22⋊C4—1781no—
423C2×C22⋊C4—1781no—
424C2×C22⋊C4C22⋊D8911yescharacteristic
425C2×C22⋊C4C22⋊D8911yescharacteristic
426C2×C42—21no—
427C2×C42—1781no—
428C2×C42—1781no—
429C2×C42C22×D8911yescharacteristic
430C2×C42C22×D8911yescharacteristic
431C4⋊C8—1782no—
432C4⋊C8—1782no—
433C22⋊C8—1782no—
434C22⋊C8—1782no—
435#97—1782no—
436C4×C8—1782no—
437#95—1781no—
438#95—1781no—
439#95—1781no—
440#95—1781no—
441#95—1781no—
442#95—1781no—
443C24C2×C22⋊D8911yescharacteristic
444C2×Q8—24no—
445C2×Q8—24no—
446C2×Q8—81no—
447C2×Q8—81no—
448C2×Q8—81no—
449C2×Q8—81no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—24no—
459C22×C4—81no—
460C22×C4—81no—
461C22×C4—1781no—
462C22×C4—1781no—
463C22×C4—1781no—
464C22×C4—1781no—
465C22×C4C22×Dic8912yes—
466C22×C4C2×C22⋊D8912yes—
467C22×C4C2×C22⋊D8911yescharacteristic
468C22×C4C2×C22⋊D8911yescharacteristic
469C22×C4C2×C22⋊D8911yescharacteristic
470C22×C4C2×C22⋊D8911yescharacteristic
471C22×C4C2×C22⋊D8911yescharacteristic
472C22×C4C23×D8911yescharacteristic
473C2×C8—1781no—
474C2×C8—1781no—
475C2×C8—1781no—
476C2×C8—1781no—
477C2×C8—1781no—
478C2×C8—1781no—
479C2×C8—1781no—
480C2×C8—1781no—
481C2×C8—1781no—
482C2×C8—1781no—
483C2×C8—1781no—
484C2×C8—1781no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—24no—
495C4⋊C4—24no—
496C4⋊C4—1781no—
497C4⋊C4—1781no—
498C4⋊C4—1781no—
499C4⋊C4—1781no—
500C4⋊C4—1781no—
501C4⋊C4—1781no—
502C4⋊C4—1781no—
503C4⋊C4—1781no—
504C4⋊C4—7121no—
505C4⋊C4—7121no—
506C4⋊C4C22⋊Dic8914yes—
507C4⋊C4C22⋊Dic8914yes—
508C4⋊C4C22×Dic8912yes—
509C4⋊C4C22×Dic8912yes—
510C4⋊C4C22×Dic8912yes—
511C4⋊C4C22×Dic8912yes—
512C22⋊C4—24no—
513C22⋊C4—7121no—
514C22⋊C4—7121no—
515C22⋊C4C22⋊Dic8914yes—
516C22⋊C4C22⋊Dic8914yes—
517C42—24no—
518C42—7121no—
519C42—7121no—
520C42C22×Dic8912yes—
521C42C22×Dic8912yes—
522C42C2×C22⋊D8912yes—
523C42C2×C22⋊D8912yes—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23C2×C22⋊Dic8912yes—
531C23C22×C22⋊D8911yescharacteristic
532Q8—84no—
533Q8—84no—
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—21no—
543C2×C4—21no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—84no—
561C2×C4—3561no—
562C2×C4—3561no—
563C2×C4—3561no—
564C2×C4—3561no—
565C2×C4—3561no—
566C2×C4—3561no—
567C2×C4—3561no—
568C2×C4—3561no—
569C2×C4—3561no—
570C2×C4—3561no—
571C2×C4—3561no—
572C2×C4—3561no—
573C2×C4C2×C22⋊Dic8912yes—
574C2×C4C2×C22⋊Dic8912yes—
575C2×C4C2×C22⋊Dic8912yes—
576C2×C4C2×C22⋊Dic8912yes—
577C2×C4C2×C22⋊Dic8912yes—
578C2×C4C23×Dic8911yescharacteristic
579C2×C4C22×C22⋊D8911yescharacteristic
580C2×C4#1566806011yescharacteristic
581C2×C4#1566806011yescharacteristic
582C2×C4#1566806011yescharacteristic
583C2×C4#1566806011yescharacteristic
584C8—3561no—
585C8—3561no—
586C8—3561no—
587C8—3561no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22C22×C22⋊Dic8911yescharacteristic
596C22#1915217211yescharacteristic
597C22#1915217211yescharacteristic
598C22#1915240111yescharacteristic
599C22#1915240611yescharacteristic
600C22#1915240911yescharacteristic
601C22#1915241311yescharacteristic
602C4—22no—
603C4—22no—
604C4—22no—
605C4—22no—
606C4—22no—
607C4—42no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—7121no—
612C4—7121no—
613C4—7121no—
614C4—7121no—
615C4#1915207611yescharacteristic
616C4#1915207611yescharacteristic
617C4#1915217211yescharacteristic
618C4#1915217211yescharacteristic
619C2—42no—
620C2#2602283811yescharacteristic
621C2#2602877811yescharacteristic
622C2#2602878511yescharacteristic
623C2#2603150511yescharacteristic
624C2#2603152011yescharacteristic
625C2#2603153511yescharacteristic
626C2#2603155011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D89
commutatorC2×C4×C89C23×C4
FrattiniC22×C4C23×D89
FittingC2×#544×C89C2
socleC23×C89C22×D4
prime radicalC24×C89C2×D4
cosocleC22×C4×C89C24
hypercenterC2×#544D89
hypocenterC89#21843
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#544

Complements

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

NH ≅ G/Ntype
3. #26021226C2semidirect
4. #26021222C2semidirect
5. #26021218C2semidirect
6. #26021214C2semidirect
7. #26021210C2semidirect
8. #26021210C2semidirect
9. #26021205C2semidirect
10. #26021201C2semidirect
50. #544×C89C4semidirect
329. C89#21843semidirect
331. #544Dic89semidirect

Maximal subgroups

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

subgroupindex
C2×#191516592
#260228612
#260212262
#260212222
#260212182
#260212142
#260212102
#260212052
#260212012
#260203472
#260203422
#260203382
C2×#544×C892
#2184389