Finite Group Databasebeta

← #36562281

Subgroups — #36562281

Order 22784.

730 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26029193C211yesmaximal, characteristic, complemented
2#26028680C211yesmaximal, characteristic
3#26021197C211yesmaximal, characteristic, complemented
4#26021028C211yesmaximal, characteristic, complemented
5#26020853C211yesmaximal, characteristic, complemented
6#26020853C211yesmaximal, characteristic, complemented
7#26020529C211yesmaximal, characteristic
8#26020529C211yesmaximal, characteristic
9#26020272C211yesmaximal, characteristic
10#26020138C211yesmaximal, characteristic, complemented
11#26020133C211yesmaximal, characteristic, complemented
12#26020129C211yesmaximal, characteristic, complemented
13#26020099C211yesmaximal, characteristic
14#26020092C211yesmaximal, characteristic
15C89×#3076C211yesmaximal, characteristic
16C2×#15668019C2211yescharacteristic
17C2×#15668019C2211yescharacteristic
18#19151729—21no—
19#19151729—21no—
20#19151729—21no—
21#19151729—21no—
22C2×#15667948C2211yescharacteristic
23C2×#15667948C2211yescharacteristic
24#19151625C2211yescharacteristic
25#19151625C2211yescharacteristic
26#19151622C2211yescharacteristic
27C2×#15667939C2211yescharacteristic
28C2×#15667939C2211yescharacteristic
29#19151554—21no—
30#19151554—21no—
31#19151554C2211yescharacteristic
32#19151554C2211yescharacteristic
33#19151554C2211yescharacteristic
34#19151554C2211yescharacteristic
35C4×#13083558—21no—
36C4×#13083558C2211yescharacteristic
37C2×#15667923—21no—
38C2×#15667923C2211yescharacteristic
39C2×#15667922C2211yescharacteristic
40C2×C8×Dic89C2211yescharacteristic
41C4×C4⋊Dic89C2211yescharacteristic
42#19151363C2211yescharacteristic
43#19151363C2211yescharacteristic
44#19151357C2211yescharacteristic
45#19151357C2211yescharacteristic
46#19151357C2211yescharacteristic
47#19151357C2211yescharacteristic
48#19151357C2211yescharacteristic
49#19151357C2211yescharacteristic
50#19151352C2211yescharacteristic
51#19151352C2211yescharacteristic
52#533×C89C2211yescharacteristic
53C4×#41×C89C2211yescharacteristic
54#479×C89C411yescharacteristic, complemented
55#479×C89C411yescharacteristic, complemented
56#478×C89C411yescharacteristic, complemented
57#478×C89C411yescharacteristic, complemented
58#477×C89C411yescharacteristic, complemented
59#476×C89C411yescharacteristic, complemented
60#475×C89C411yescharacteristic, complemented
61#474×C89C411yescharacteristic, complemented
62#427×C89C2211yescharacteristic
63#427×C89C2211yescharacteristic
64C2×C4⋊C8×C89C2211yescharacteristic
65#416×C89C2211yescharacteristic
66#416×C89C2211yescharacteristic
67#15668019—22no—
68#15668019—22no—
69#15668019—22no—
70#15668019Q814yescomplemented
71C2×#13083558—21no—
72C2×#13083558—21no—
73C2×#13083558C2311yescharacteristic
74C2×#13083558C2311yescharacteristic
75C2×#13083558C2311yescharacteristic
76C2×#13083558C2311yescharacteristic
77C22×#10623494—21no—
78C22×#10623494—21no—
79C22×#10623494C2311yescharacteristic
80C22×#10623494C2311yescharacteristic
81C2×C4⋊Dic89—21no—
82C2×C4×Dic89C2311yescharacteristic
83C2×C4×Dic89C2311yescharacteristic
84#15667948—22no—
85#15667948—22no—
86C4⋊#10623494—21no—
87C4⋊#10623494—21no—
88C4⋊#10623494—21no—
89C4⋊#10623494—21no—
90C4⋊#10623494—21no—
91C4⋊#10623494—21no—
92#15667939—22no—
93#15667939—22no—
94#15667923—21no—
95#15667923—21no—
96#15667923—21no—
97#15667923—21no—
98#15667923—21no—
99#15667923D411yescharacteristic
100#15667923D411yescharacteristic
101#15667923D411yescharacteristic
102#15667923D411yescharacteristic
103#15667922—22no—
104C4×#10623494—21no—
105C4×#10623494—21no—
106C4×#10623494—21no—
107C4×#10623494—21no—
108C4×#10623494—21no—
109C8×Dic89—22no—
110#15667919—21no—
111C2×#41×C89C2311yescharacteristic
112C22×C8×C89C2311yescharacteristic
113C22×C8×C89C2311yescharacteristic
114C4⋊Q8×C89C2×C411yescharacteristic
115C4⋊Q8×C89C2×C411yescharacteristic
116#125×C89C2×C411yescharacteristic
117#125×C89C2×C411yescharacteristic
118C22⋊Q8×C89—21no—
119C22⋊Q8×C89—21no—
120C4×Q8×C89—21no—
121C4×Q8×C89—21no—
122C4×Q8×C89C2×C411yescharacteristic
123C4×Q8×C89C2×C411yescharacteristic
124C4×D4×C89—21no—
125C4×D4×C89—21no—
126C4×D4×C89C2×C411yescharacteristic
127C4×D4×C89C2×C411yescharacteristic
128#117×C89—21no—
129#117×C89C2311yescharacteristic
130#117×C89C2311yescharacteristic
131#117×C89C2311yescharacteristic
132C2×C42×C89—21no—
133C2×C42×C89C2311yescharacteristic
134#107×C89C2×C411yescharacteristic
135#107×C89C2×C411yescharacteristic
136#107×C89C2×C411yescharacteristic
137#107×C89C2×C411yescharacteristic
138#106×C89C2×C411yescharacteristic
139#106×C89C2×C411yescharacteristic
140#106×C89C2×C411yescharacteristic
141#106×C89C2×C411yescharacteristic
142C4⋊C8×C89C2×C411yescharacteristic
143C4⋊C8×C89C2×C411yescharacteristic
144C4⋊C8×C89C2×C411yescharacteristic
145C4⋊C8×C89C2×C411yescharacteristic
146#103×C89C2×C411yescharacteristic
147#103×C89C2×C411yescharacteristic
148#103×C89C2×C411yescharacteristic
149#103×C89C2×C411yescharacteristic
150#102×C89C2×C411yescharacteristic
151#102×C89C2×C411yescharacteristic
152#102×C89C2×C411yescharacteristic
153#102×C89C2×C411yescharacteristic
154C22×Dic89—21no—
155C22×Dic89—21no—
156#13083558—24no—
157#13083558—21no—
158#13083558—21no—
159#13083558—24no—
160#13083558—21no—
161#13083558—21no—
162#13083558—21no—
163#13083558—21no—
164#13083558C2×Q812yes—
165#13083558#4112yes—
166C2×#10623494—24no—
167C2×#10623494—21no—
168C2×#10623494—21no—
169C2×#10623494—21no—
170C2×#10623494—21no—
171C2×#10623494—24no—
172C2×#10623494—21no—
173C2×#10623494—21no—
174C2×#10623494C2×D411yescharacteristic
175C2×#10623494C2×D411yescharacteristic
176C2×#10623494C2×D411yescharacteristic
177C2×#10623494C2×D411yescharacteristic
178C2×#10623494C2×Q812yes—
179C2×#10623494#4112yes—
180C4⋊Dic89—42no—
181C4×Dic89—21no—
182C4×Dic89—21no—
183C4×Dic89—21no—
184C4×Dic89—21no—
185#41×C89—21no—
186#41×C89—21no—
187#41×C89C2×Q812yes—
188#41×C89#4112yes—
189C2×Q8×C89—21no—
190C2×Q8×C89—21no—
191C2×Q8×C89C22×C411yescharacteristic
192C2×D4×C89—21no—
193C2×D4×C89C22×C411yescharacteristic
194C22×C4×C89—21no—
195C22×C4×C89—21no—
196C22×C4×C89—21no—
197C22×C4×C89#4111yescharacteristic
198C22×C4×C89#4111yescharacteristic
199C22×C4×C89C2411yescharacteristic
200C2×C8×C89—21no—
201C2×C8×C89—21no—
202C2×C8×C89—21no—
203C2×C8×C89—21no—
204C2×C8×C89C22×C411yescharacteristic
205C2×C8×C89C22×C411yescharacteristic
206C2×C8×C89C22×C411yescharacteristic
207C2×C8×C89C22×C411yescharacteristic
208C4⋊C4×C89—21no—
209C4⋊C4×C89—21no—
210C4⋊C4×C89—21no—
211C4⋊C4×C89—21no—
212C4⋊C4×C89—21no—
213C4⋊C4×C89—21no—
214C4⋊C4×C89—21no—
215C4⋊C4×C89—21no—
216C4⋊C4×C89C22×C411yescharacteristic
217C4⋊C4×C89C22×C411yescharacteristic
218C4⋊C4×C89C22×C411yescharacteristic
219C4⋊C4×C89C22×C411yescharacteristic
220C4⋊C4×C89C22×C411yescharacteristic
221C4⋊C4×C89C22×C411yescharacteristic
222C22⋊C4×C89—21no—
223C22⋊C4×C89—21no—
224C22⋊C4×C89—21no—
225C22⋊C4×C89—21no—
226C22⋊C4×C89—21no—
227C42×C89—21no—
228C42×C89—21no—
229C42×C89—21no—
230C42×C89—21no—
231C42×C89—21no—
232C42×C89C22×C411yescharacteristic
233C42×C89C22×C411yescharacteristic
234C42×C89C2×D411yescharacteristic
235C42×C89C2×D411yescharacteristic
236C23×C89—21no—
237C23×C89C2×#4111yescharacteristic
238Q8×C89—21no—
239Q8×C89—81no—
240Q8×C89—81no—
241Q8×C89C4×Q812yes—
242D4×C89—22no—
243D4×C89—22no—
244D4×C89—21no—
245D4×C89C4×Q812yes—
246C2×C4×C89—21no—
247C2×C4×C89—22no—
248C2×C4×C89—22no—
249C2×C4×C89—21no—
250C2×C4×C89—21no—
251C2×C4×C89—21no—
252C2×C4×C89—21no—
253C2×C4×C89—21no—
254C2×C4×C89—21no—
255C2×C4×C89—41no—
256C2×C4×C89—41no—
257C2×C4×C89—81no—
258C2×C4×C89—81no—
259C2×C4×C89C4×D411yescharacteristic
260C2×C4×C89C4×D411yescharacteristic
261C2×C4×C89C4×D411yescharacteristic
262C2×C4×C89C4×D411yescharacteristic
263C2×C4×C89C23×C411yescharacteristic
264C2×C4×C89C22×D411yescharacteristic
265C2×C4×C89C22×Q811yescharacteristic
266C2×C4×C89C2×#4111yescharacteristic
267C2×C4×C89C2×#4111yescharacteristic
268C2×C4×C89C2×#4111yescharacteristic
269C2×Dic89—22no—
270C2×Dic89—41no—
271C2×Dic89—41no—
272C2×Dic89—41no—
273C2×Dic89—41no—
274C2×Dic89—41no—
275C8×C89—21no—
276C8×C89—21no—
277C8×C89—21no—
278C8×C89—21no—
279#10623494—22no—
280#10623494—21no—
281#10623494—22no—
282#10623494—21no—
283#10623494—41no—
284#10623494—41no—
285#10623494C22⋊Q812yes—
286#10623494C22⋊Q812yes—
287C22×C89—21no—
288C22×C89—21no—
289C22×C89—22no—
290C22×C89—41no—
291C22×C89—41no—
292C22×C89C2×C4×D411yescharacteristic
293C22×C89#52911yescharacteristic
294C22×C89#53311yescharacteristic
295C4×C89—21no—
296C4×C89—21no—
297C4×C89—21no—
298C4×C89—21no—
299C4×C89—22no—
300C4×C89—41no—
301C4×C89—41no—
302C4×C89—41no—
303C4×C89—41no—
304C4×C89—41no—
305C4×C89—41no—
306C4×C89C2×C4×Q811yescharacteristic
307C4×C89C4×#4111yescharacteristic
308C4×C89C2×C22⋊Q811yescharacteristic
309C4×C89#52511yescharacteristic
310Dic89—42no—
311Dic89—81no—
312#21629—891nomaximal, Sylow
313C2×C89—21no—
314C2×C89—21no—
315C2×C89—42no—
316C2×C89C4×C22⋊Q811yescharacteristic
317C2×C89#295611yescharacteristic
318C2×C89#295711yescharacteristic
319#3076D8911yescharacteristic
320#2961—891no—
321#2875—891no—
322#1986—891no—
323#1950—891no—
324#1922—891no—
325#1922—891no—
326#1866—891no—
327#1866—891no—
328#1806—891no—
329#1773—891no—
330#1772—891no—
331#1771—891no—
332#1765—891no—
333#1764—891no—
334C89#2162911yescharacteristic, complemented, Sylow
335C2×#131—891no—
336C2×#131—891no—
337#533C2×D8911yescharacteristic
338C4×#41C2×D8911yescharacteristic
339#479Dic8911yescharacteristic, complemented
340#479Dic8911yescharacteristic, complemented
341#478Dic8911yescharacteristic, complemented
342#478Dic8911yescharacteristic, complemented
343#477Dic8911yescharacteristic, complemented
344#476Dic8911yescharacteristic, complemented
345#475Dic8911yescharacteristic, complemented
346#474Dic8911yescharacteristic, complemented
347#446—1781no—
348#446—1781no—
349#446—1781no—
350#446—1781no—
351C2×#108—891no—
352C2×#108—891no—
353#427C2×D8911yescharacteristic
354#427C2×D8911yescharacteristic
355#424—891no—
356#424—891no—
357#423—891no—
358C2×C4⋊C8C2×D8911yescharacteristic
359C2×#104—891no—
360C2×#104—891no—
361#416C2×D8911yescharacteristic
362#416C2×D8911yescharacteristic
363#405—891no—
364#405—891no—
365#405—891no—
366#405—891no—
367#405—1781no—
368#405—1781no—
369C4×M16—891no—
370C4×M16—1781no—
371C2×#97—891no—
372C2×#97—891no—
373C2×#97—1781no—
374C2×C4×C8—891no—
375C4×C4⋊C4—891no—
376#341—891no—
377#341—891no—
378#339—891no—
379#339—891no—
380#339—891no—
381#339—891no—
382#339—891no—
383#339—891no—
384#336—891no—
385#336—891no—
386C2×#41C22×D8911yescharacteristic
387#131—894no—
388#131—1782no—
389#131—1782no—
390#131—1782no—
391C2×M16—891no—
392C2×M16—891no—
393C2×M16—891no—
394C2×M16—891no—
395C2×M16—1781no—
396C2×M16—1781no—
397C22×C8—891no—
398C22×C8—891no—
399C22×C8—1781no—
400C22×C8—1781no—
401C22×C8C22×D8911yescharacteristic
402C22×C8C22×D8911yescharacteristic
403C4⋊Q8C2×Dic8911yescharacteristic
404C4⋊Q8C2×Dic8911yescharacteristic
405#125C2×Dic8911yescharacteristic
406#125C2×Dic8911yescharacteristic
407C22⋊Q8—21no—
408C22⋊Q8—21no—
409C4×Q8—21no—
410C4×Q8—21no—
411C4×Q8C2×Dic8911yescharacteristic
412C4×Q8C2×Dic8911yescharacteristic
413C4×D4—21no—
414C4×D4—21no—
415C4×D4C2×Dic8911yescharacteristic
416C4×D4C2×Dic8911yescharacteristic
417#117—21no—
418#117C22×D8911yescharacteristic
419#117C22×D8911yescharacteristic
420#117C22×D8911yescharacteristic
421C2×C4⋊C4—1781no—
422C2×C42—21no—
423C2×C42—891no—
424C2×C42—891no—
425C2×C42C22×D8911yescharacteristic
426#108—1782no—
427#108—1782no—
428#107C2×Dic8911yescharacteristic
429#107C2×Dic8911yescharacteristic
430#107C2×Dic8911yescharacteristic
431#107C2×Dic8911yescharacteristic
432#106C2×Dic8911yescharacteristic
433#106C2×Dic8911yescharacteristic
434#106C2×Dic8911yescharacteristic
435#106C2×Dic8911yescharacteristic
436C4⋊C8—1781no—
437C4⋊C8—1781no—
438C4⋊C8—1781no—
439C4⋊C8—1781no—
440C4⋊C8—1781no—
441C4⋊C8—1781no—
442C4⋊C8C2×Dic8911yescharacteristic
443C4⋊C8C2×Dic8911yescharacteristic
444C4⋊C8C2×Dic8911yescharacteristic
445C4⋊C8C2×Dic8911yescharacteristic
446#104—1782no—
447#104—1782no—
448#103C2×Dic8911yescharacteristic
449#103C2×Dic8911yescharacteristic
450#103C2×Dic8911yescharacteristic
451#103C2×Dic8911yescharacteristic
452#102C2×Dic8911yescharacteristic
453#102C2×Dic8911yescharacteristic
454#102C2×Dic8911yescharacteristic
455#102C2×Dic8911yescharacteristic
456#97—891no—
457#97—891no—
458#97—891no—
459#97—891no—
460#97—1781no—
461#97—1781no—
462#97—1781no—
463#97—1781no—
464#97—1782no—
465#97—1781no—
466C4×C8—1781no—
467C4×C8—1781no—
468C4×C8—1781no—
469C4×C8—1781no—
470C4×C8—1782no—
471C4×C8—1781no—
472#95—1781no—
473#41—21no—
474#41—21no—
475#41Q8×D8912yes—
476#41#1308357512yes—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8C22×Dic8911yescharacteristic
480C2×D4—21no—
481C2×D4C22×Dic8911yescharacteristic
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—1781no—
486C22×C4—1781no—
487C22×C4#1308357311yescharacteristic
488C22×C4#1308357311yescharacteristic
489C22×C4C23×D8911yescharacteristic
490M16—892no—
491M16—892no—
492M16—1784no—
493M16—1781no—
494M16—1781no—
495M16—1781no—
496M16—1781no—
497M16—1784no—
498M16—1781no—
499M16—1781no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—891no—
505C2×C8—891no—
506C2×C8—892no—
507C2×C8—892no—
508C2×C8—891no—
509C2×C8—891no—
510C2×C8—1784no—
511C2×C8—1781no—
512C2×C8—1781no—
513C2×C8—1784no—
514C2×C8—1781no—
515C2×C8—1781no—
516C2×C8—1781no—
517C2×C8—1781no—
518C2×C8C22×Dic8911yescharacteristic
519C2×C8C22×Dic8911yescharacteristic
520C2×C8C22×Dic8911yescharacteristic
521C2×C8C22×Dic8911yescharacteristic
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—3562no—
531C4⋊C4C22×Dic8911yescharacteristic
532C4⋊C4C22×Dic8911yescharacteristic
533C4⋊C4C22×Dic8911yescharacteristic
534C4⋊C4C22×Dic8911yescharacteristic
535C4⋊C4C22×Dic8911yescharacteristic
536C4⋊C4C22×Dic8911yescharacteristic
537C22⋊C4—21no—
538C22⋊C4—21no—
539C22⋊C4—21no—
540C22⋊C4—21no—
541C22⋊C4—21no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—21no—
547C42—1781no—
548C42—1781no—
549C42—1781no—
550C42—1781no—
551C42C22×Dic8911yescharacteristic
552C42C22×Dic8911yescharacteristic
553C42D4×D8911yescharacteristic
554C42D4×D8911yescharacteristic
555C23—21no—
556C23C2×#1308357311yescharacteristic
557Q8—21no—
558Q8—81no—
559Q8—81no—
560Q8Q8×Dic8912yes—
561D4—22no—
562D4—22no—
563D4—21no—
564D4Q8×Dic8912yes—
565C2×C4—21no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—81no—
577C2×C4—81no—
578C2×C4—3561no—
579C2×C4—3561no—
580C2×C4—3561no—
581C2×C4—3561no—
582C2×C4—3561no—
583C2×C4—1782no—
584C2×C4D4×Dic8911yescharacteristic
585C2×C4D4×Dic8911yescharacteristic
586C2×C4D4×Dic8911yescharacteristic
587C2×C4D4×Dic8911yescharacteristic
588C2×C4C23×Dic8911yescharacteristic
589C2×C4C2×D4×D8911yescharacteristic
590C2×C4C2×Q8×D8911yescharacteristic
591C2×C4#41×D8911yescharacteristic
592C2×C4#41×D8911yescharacteristic
593C2×C4C2×#1308357511yescharacteristic
594C8—21no—
595C8—21no—
596C8—21no—
597C8—21no—
598C8—892no—
599C8—892no—
600C8—3561no—
601C8—3561no—
602C8—1782no—
603C8—1781no—
604C8—1781no—
605C8—1782no—
606C22—21no—
607C22—21no—
608C22—22no—
609C22—41no—
610C22—41no—
611C22C2×D4×Dic8911yescharacteristic
612C22#1915213411yescharacteristic
613C22#1915215011yescharacteristic
614C4—21no—
615C4—21no—
616C4—21no—
617C4—21no—
618C4—22no—
619C4—41no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—41no—
624C4—41no—
625C4—3562no—
626C4—7121no—
627C4C2×Q8×Dic8911yescharacteristic
628C4#41×Dic8911yescharacteristic
629C4C22⋊Q8×D8911yescharacteristic
630C4#1915211111yescharacteristic
631C2—21no—
632C2—21no—
633C2—42no—
634C2C22⋊Q8×Dic8911yescharacteristic
635C2#2602916211yescharacteristic
636C2#2602916911yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D89
commutatorC2×C4×C89C23×C4
FrattiniC22×C4C23×D89
FittingC89×#3076C2
socleC22×C89C2×C4×D4
prime radicalC2×#41×C89C23
cosocleC22×C4×C89C24
hypercenter#3076D89
hypocenterC89#21629
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3076

Complements

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

NH ≅ G/Ntype
1. #26029193C2semidirect
3. #26021197C2semidirect
4. #26021028C2semidirect
5. #26020853C2semidirect
6. #26020853C2semidirect
10. #26020138C2semidirect
11. #26020133C2semidirect
12. #26020129C2semidirect
54. #479×C89C4semidirect
55. #479×C89C4semidirect
56. #478×C89C4semidirect
57. #478×C89C4semidirect
58. #477×C89C4semidirect
59. #476×C89C4semidirect
60. #475×C89C4semidirect
61. #474×C89C4semidirect
70. #15668019Q8semidirect
334. C89#21629semidirect
339. #479Dic89semidirect
340. #479Dic89semidirect
341. #478Dic89semidirect
342. #478Dic89semidirect
343. #477Dic89semidirect
344. #476Dic89semidirect
345. #475Dic89semidirect
346. #474Dic89semidirect

Maximal subgroups

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

subgroupindex
#260291932
#260286802
#260211972
#260210282
#260208532
#260205292
#260202722
#260201382
#260201332
#260201292
#260200992
#260200922
C89×#30762
#2162989