Finite Group Databasebeta

← #36563973

Subgroups — #36563973

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×#19152226C211yesmaximal, characteristic
2C2×#19151659C211yesmaximal, characteristic
3#26021224C211yesmaximal, characteristic, complemented
4#26021220C211yesmaximal, characteristic, complemented
5#26021216C211yesmaximal, characteristic, complemented
6#26021212C211yesmaximal, characteristic, complemented
7#26021209C211yesmaximal, characteristic, complemented
8#26021209C211yesmaximal, characteristic, complemented
9#26021204C211yesmaximal, characteristic, complemented
10#26021200C211yesmaximal, characteristic, complemented
11#26020344C211yesmaximal, characteristic
12#26020344C211yesmaximal, characteristic
13#26020340C211yesmaximal, characteristic
14#26020336C211yesmaximal, characteristic
15C89×#2314C211yesmaximal, characteristic
16#19152226—24no—
17#19152226C418yescomplemented
18C2×#15668012C2211yescharacteristic
19C2×#15668012C2211yescharacteristic
20C2×#15668011—21no—
21C2×#15668004C2211yescharacteristic
22C2×#15668004C2211yescharacteristic
23C2×#15668003—21no—
24C2×#15667989—21no—
25C2×#15667989—21no—
26C2×#15667971C2211yescharacteristic
27C22×C4⋊Dic89C2211yescharacteristic
28#19151659—24no—
29C2×C4⋊#10623494C2211yescharacteristic
30C2×C4⋊#10623494C2211yescharacteristic
31C2×C22⋊#10623494C2211yescharacteristic
32C2×C22⋊#10623494C2211yescharacteristic
33C2×#15667923C2211yescharacteristic
34C2×C4×#10623494C2211yescharacteristic
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#19151360C2211yescharacteristic
50#19151360C2211yescharacteristic
51C2×#117×C89C2211yescharacteristic
52#388×C89—21no—
53#388×C89—21no—
54#387×C89—21no—
55#387×C89—21no—
56#384×C89C2211yescharacteristic
57#382×C89C2211yescharacteristic
58#380×C89C2211yescharacteristic
59#380×C89C2211yescharacteristic
60C4×C4⋊C4×C89C2211yescharacteristic
61C4×C4⋊C4×C89C2211yescharacteristic
62C22×Dic178—21no—
63C22×Dic178—21no—
64C23×Dic89—21no—
65C22×#10623494C2311yescharacteristic
66C22×#10623494C2311yescharacteristic
67C22×#10623494C2311yescharacteristic
68C22×#10623494C2311yescharacteristic
69#15668012—22no—
70#15668012—22no—
71#15668012C2×C414yes—
72#15668012C2×C414yes—
73#15668011—28no—
74#15668004—22no—
75#15668004—22no—
76#15668004C2×C414yes—
77#15668004C2×C414yes—
78#15668003—28no—
79#15667989—28no—
80#15667989—28no—
81#15667971—22no—
82#15667971C2×C414yes—
83C2×C4⋊Dic89—21no—
84C2×C4⋊Dic89—21no—
85C2×C4⋊Dic89—21no—
86C2×C4⋊Dic89—22no—
87C2×C4⋊Dic89C2×C414yes—
88C2×C4⋊Dic89C2311yescharacteristic
89C2×C4⋊Dic89C2311yescharacteristic
90C2×C4⋊Dic89C2311yescharacteristic
91C2×C4⋊Dic89C2311yescharacteristic
92C2×#13083553—21no—
93C2×#13083553—21no—
94C2×#13083553—21no—
95C2×#13083553—21no—
96C2×C22⋊Dic89—21no—
97C2×C4×Dic89—21no—
98C4⋊#10623494—22no—
99C4⋊#10623494—22no—
100C22⋊#10623494—22no—
101C22⋊#10623494—22no—
102#15667923—22no—
103C4×#10623494—22no—
104C23×C4×C89C2311yescharacteristic
105#117×C89C2×C414yes—
106#117×C89D414yes—
107C2×C4⋊C4×C89—21no—
108C2×C4⋊C4×C89—21no—
109C2×C4⋊C4×C89D411yescharacteristic
110C2×C4⋊C4×C89D411yescharacteristic
111C2×C4⋊C4×C89C2311yescharacteristic
112C2×C4⋊C4×C89C2311yescharacteristic
113C2×C4⋊C4×C89C2311yescharacteristic
114C2×C4⋊C4×C89C2311yescharacteristic
115C2×C22⋊C4×C89—21no—
116C2×C22⋊C4×C89—21no—
117C2×C22⋊C4×C89D411yescharacteristic
118C2×C22⋊C4×C89D411yescharacteristic
119C2×C42×C89—21no—
120C2×C42×C89—21no—
121C2×C42×C89C2311yescharacteristic
122C2×C42×C89C2311yescharacteristic
123#95×C89—21no—
124#95×C89—21no—
125#95×C89—21no—
126#95×C89—21no—
127#95×C89—21no—
128#95×C89—21no—
129C2×Dic178—24no—
130C2×Dic178—24no—
131C2×Dic178—81no—
132C2×Dic178—81no—
133C2×Dic178—81no—
134C2×Dic178—81no—
135C22×Dic89—21no—
136C22×Dic89—21no—
137C22×Dic89—21no—
138C22×Dic89—21no—
139C22×Dic89—24no—
140C22×Dic89—81no—
141C22×Dic89—81no—
142C2×#10623494—21no—
143C2×#10623494—21no—
144C2×#10623494—21no—
145C2×#10623494—21no—
146C2×#10623494—21no—
147C2×#10623494—21no—
148C2×#10623494—21no—
149C2×#10623494—21no—
150C2×#10623494—21no—
151C2×#10623494—21no—
152C2×#10623494—21no—
153C2×#10623494—21no—
154C4⋊Dic89—24no—
155C4⋊Dic89—24no—
156C4⋊Dic89—24no—
157C4⋊Dic89—21no—
158C4⋊Dic89—21no—
159C4⋊Dic89—21no—
160C4⋊Dic89—21no—
161C4⋊Dic89C22×C412yes—
162C4⋊Dic89C22×C412yes—
163C4⋊Dic89C22×C412yes—
164C4⋊Dic89C22×C412yes—
165#13083553—24no—
166#13083553—24no—
167#13083553—24no—
168#13083553—24no—
169C22⋊Dic89—24no—
170C4×Dic89—24no—
171C24×C89C2×D411yescharacteristic
172C22×C4×C89—21no—
173C22×C4×C89—21no—
174C22×C4×C89—21no—
175C22×C4×C89—21no—
176C22×C4×C89—21no—
177C22×C4×C89—21no—
178C22×C4×C89—21no—
179C22×C4×C89—21no—
180C22×C4×C89C22×C412yes—
181C22×C4×C89C2×D411yescharacteristic
182C22×C4×C89C2×D411yescharacteristic
183C22×C4×C89C2×D411yescharacteristic
184C22×C4×C89C2×D411yescharacteristic
185C22×C4×C89C2×D411yescharacteristic
186C22×C4×C89C2×D412yes—
187C22×C4×C89C2411yescharacteristic
188C4⋊C4×C89—21no—
189C4⋊C4×C89—21no—
190C4⋊C4×C89—21no—
191C4⋊C4×C89—21no—
192C4⋊C4×C89—21no—
193C4⋊C4×C89—21no—
194C4⋊C4×C89—21no—
195C4⋊C4×C89—21no—
196C4⋊C4×C89—81no—
197C4⋊C4×C89—81no—
198C4⋊C4×C89C22⋊C414yes—
199C4⋊C4×C89C22⋊C414yes—
200C22⋊C4×C89—81no—
201C22⋊C4×C89—81no—
202C22⋊C4×C89C22⋊C414yes—
203C22⋊C4×C89C22⋊C414yes—
204C42×C89—81no—
205C42×C89—81no—
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
218C2×C4×C89—21no—
219C2×C4×C89—21no—
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—22no—
229C2×C4×C89—22no—
230C2×C4×C89—22no—
231C2×C4×C89—22no—
232C2×C4×C89—41no—
233C2×C4×C89—41no—
234C2×C4×C89—41no—
235C2×C4×C89—41no—
236C2×C4×C89—41no—
237C2×C4×C89—41no—
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×C89C2×C22⋊C412yes—
245C2×C4×C89C2×C22⋊C412yes—
246C2×C4×C89C2×C22⋊C412yes—
247C2×C4×C89C2×C22⋊C412yes—
248C2×C4×C89C2×C22⋊C412yes—
249C2×C4×C89C23×C411yescharacteristic
250C2×C4×C89C22×D411yescharacteristic
251C2×C4×C892−1+411yescharacteristic
252C2×C4×C892−1+411yescharacteristic
253C2×C4×C892−1+411yescharacteristic
254C2×C4×C892−1+411yescharacteristic
255C2×Dic89—22no—
256C2×Dic89—22no—
257C2×Dic89—22no—
258C2×Dic89—22no—
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—84no—
268Dic178—84no—
269Dic178—84no—
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—81no—
294C4×C89—81no—
295C4×C89—81no—
296C4×C89—81no—
297C4×C89#51911yescharacteristic
298C4×C89#51911yescharacteristic
299C4×C89#53611yescharacteristic
300C4×C89#53611yescharacteristic
301Dic89—42no—
302Dic89—42no—
303Dic89—42no—
304Dic89—42no—
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×#544—891no—
315C2×#433—891no—
316#2314D8911yescharacteristic
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—898no—
331#544—1784no—
332C2×C4⋊Q8—891no—
333C2×C4⋊Q8—891no—
334C2×C4⋊Q8—1781no—
335C2×#125—891no—
336C2×#125—891no—
337C2×#125—1781no—
338C2×C22⋊Q8—1781no—
339C2×C22⋊Q8—1781no—
340C2×#117—891no—
341C2×#117C2×D8911yescharacteristic
342C22×C4⋊C4—891no—
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—21no—
351#388—21no—
352#387—21no—
353#387—21no—
354#384C2×D8911yescharacteristic
355#382C2×D8911yescharacteristic
356#380C2×D8911yescharacteristic
357#380C2×D8911yescharacteristic
358C4×C4⋊C4C2×D8911yescharacteristic
359C4×C4⋊C4C2×D8911yescharacteristic
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—1781no—
377C22×Q8—1781no—
378C23×C4—1781no—
379C23×C4C22×D8911yescharacteristic
380C22×C8—891no—
381C22×C8—891no—
382C22×C8—891no—
383C22×C8—891no—
384C4⋊Q8—894no—
385C4⋊Q8—894no—
386C4⋊Q8—1788no—
387C4⋊Q8—1782no—
388C4⋊Q8—1782no—
389#125—894no—
390#125—894no—
391#125—1788no—
392#125—1782no—
393#125—1782no—
394C22⋊Q8—1788no—
395C22⋊Q8—1788no—
396#117—894no—
397#117—1782no—
398#117C4×D8914yes—
399#117D35614yes—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—894no—
403C2×C4⋊C4—891no—
404C2×C4⋊C4—891no—
405C2×C4⋊C4—891no—
406C2×C4⋊C4—891no—
407C2×C4⋊C4—1781no—
408C2×C4⋊C4—1781no—
409C2×C4⋊C4—1781no—
410C2×C4⋊C4—1781no—
411C2×C4⋊C4—1781no—
412C2×C4⋊C4—1781no—
413C2×C4⋊C4—1781no—
414C2×C4⋊C4—1782no—
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—21no—
423C2×C22⋊C4—1781no—
424C2×C22⋊C4C22⋊D8911yescharacteristic
425C2×C22⋊C4C22⋊D8911yescharacteristic
426C2×C42—21no—
427C2×C42—21no—
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—21no—
438#95—21no—
439#95—21no—
440#95—21no—
441#95—21no—
442#95—21no—
443C24C2×C22⋊D8911yescharacteristic
444C2×Q8—1784no—
445C2×Q8—1784no—
446C2×Q8—7121no—
447C2×Q8—7121no—
448C2×Q8—7121no—
449C2×Q8—7121no—
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—1781no—
459C22×C4—1781no—
460C22×C4—1781no—
461C22×C4—1781no—
462C22×C4—1784no—
463C22×C4—7121no—
464C22×C4—7121no—
465C22×C4C2×C4×D8912yes—
466C22×C4C2×C22⋊D8911yescharacteristic
467C22×C4C2×C22⋊D8911yescharacteristic
468C22×C4C2×C22⋊D8911yescharacteristic
469C22×C4C2×C22⋊D8911yescharacteristic
470C22×C4C2×C22⋊D8911yescharacteristic
471C22×C4C2×D35612yes—
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—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—81no—
494C4⋊C4—81no—
495C4⋊C4—892no—
496C4⋊C4—892no—
497C4⋊C4—892no—
498C4⋊C4—892no—
499C4⋊C4—1781no—
500C4⋊C4—1781no—
501C4⋊C4—1781no—
502C4⋊C4—1781no—
503C4⋊C4—1784no—
504C4⋊C4—1784no—
505C4⋊C4—1784no—
506C4⋊C4—1784no—
507C4⋊C4—1784no—
508C4⋊C4—1784no—
509C4⋊C4—1784no—
510C4⋊C4#1308355114yes—
511C4⋊C4#1308355114yes—
512C22⋊C4—81no—
513C22⋊C4—81no—
514C22⋊C4—1784no—
515C22⋊C4#1308355114yes—
516C22⋊C4#1308355114yes—
517C42—81no—
518C42—81no—
519C42—1784no—
520C42C2×C4×D8912yes—
521C42C2×C4×D8912yes—
522C42C2×D35612yes—
523C42C2×D35612yes—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23C2×#1308355112yes—
531C23C22×C22⋊D8911yescharacteristic
532Q8—7124no—
533Q8—7124no—
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—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
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—3561no—
561C2×C4—3561no—
562C2×C4—3561no—
563C2×C4—3561no—
564C2×C4—3561no—
565C2×C4—3561no—
566C2×C4—3561no—
567C2×C4—3561no—
568C2×C4—1782no—
569C2×C4—1782no—
570C2×C4—1782no—
571C2×C4—1782no—
572C2×C4—7124no—
573C2×C4C2×#1308355112yes—
574C2×C4C2×#1308355112yes—
575C2×C4C2×#1308355112yes—
576C2×C4C2×#1308355112yes—
577C2×C4C2×#1308355112yes—
578C2×C4C22×C4×D8911yescharacteristic
579C2×C4C22×D35611yescharacteristic
580C2×C4#1566805911yescharacteristic
581C2×C4#1566805911yescharacteristic
582C2×C4#1566805911yescharacteristic
583C2×C4#1566805911yescharacteristic
584C8—3561no—
585C8—3561no—
586C8—3561no—
587C8—3561no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22C22×#1308355111yescharacteristic
596C22#1915216911yescharacteristic
597C22#1915216911yescharacteristic
598C22#1915240111yescharacteristic
599C22#1915240611yescharacteristic
600C22#1915240911yescharacteristic
601C22#1915241311yescharacteristic
602C4—22no—
603C4—22no—
604C4—22no—
605C4—22no—
606C4—22no—
607C4—81no—
608C4—81no—
609C4—81no—
610C4—81no—
611C4—3562no—
612C4—3562no—
613C4—3562no—
614C4—3562no—
615C4#1915207511yescharacteristic
616C4#1915207511yescharacteristic
617C4#1915217411yescharacteristic
618C4#1915217411yescharacteristic
619C2—42no—
620C2#2602283611yescharacteristic
621C2#2602877511yescharacteristic
622C2#2602878211yescharacteristic
623C2#2603149911yescharacteristic
624C2#2603151411yescharacteristic
625C2#2603152911yescharacteristic
626C2#2603154411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D89
commutatorC2×C4×C89C23×C4
FrattiniC22×C4C23×D89
FittingC89×#2314C2
socleC23×C89C22×D4
prime radicalC24×C89C2×D4
cosocleC22×C4×C89C24
hypercenter#2314D89
hypocenterC89#21843
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2314

Complements

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

NH ≅ G/Ntype
3. #26021224C2semidirect
4. #26021220C2semidirect
5. #26021216C2semidirect
6. #26021212C2semidirect
7. #26021209C2semidirect
8. #26021209C2semidirect
9. #26021204C2semidirect
10. #26021200C2semidirect
17. #19152226C4semidirect
329. C89#21843semidirect

Maximal subgroups

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

subgroupindex
C2×#191522262
C2×#191516592
#260212242
#260212202
#260212162
#260212122
#260212092
#260212042
#260212002
#260203442
#260203402
#260203362
C89×#23142
#2184389