Finite Group Databasebeta

← #36525660

Subgroups — #36525660

Order 22784.

642 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26029412C211yesmaximal, characteristic, complemented
2#26029287C211yesmaximal, characteristic, complemented
3#26019166C211yesmaximal, characteristic, complemented
4#26019155C211yesmaximal, characteristic, complemented
5#26019096C211yesmaximal, characteristic, complemented
6#26019089C211yesmaximal, characteristic, complemented
7#26019082C211yesmaximal, characteristic, complemented
8#26019082C211yesmaximal, characteristic, complemented
9#26019076C211yesmaximal, characteristic, complemented
10SD16⋊#10623494C211yesmaximal, characteristic, complemented
11#26019037C211yesmaximal, characteristic, complemented
12#26018712C211yesmaximal, characteristic, complemented
13C2×D4⋊#10623494C211yesmaximal, characteristic, complemented
14#26018545C211yesmaximal, characteristic, complemented
15C89×#3105C211yesmaximal, characteristic
16#19151729C2211yescharacteristic, complemented
17Q8×#10623494C2211yescharacteristic, complemented
18#19151676C2211yescharacteristic
19#19151676C2211yescharacteristic
20#19151676C2211yescharacteristic, complemented
21#19151670—21no—
22#19151670—21no—
23#19151670—21no—
24D4×#10623494—21no—
25D4×#10623494C2211yescharacteristic
26D4×#10623494C2211yescharacteristic
27D4×#10623494C2211yescharacteristic, complemented
28#19151659—21no—
29#19151655C2211yescharacteristic
30#19151651C2211yescharacteristic
31#19151625—21no—
32C2×C4⋊#10623494C2211yescharacteristic
33#19151562—21no—
34C2×C22⋊#10623494—21no—
35C2×C4×#10623494C2211yescharacteristic
36#19151351C2211yescharacteristic, complemented
37#19151351C2211yescharacteristic, complemented
38#19151349C2211yescharacteristic, complemented
39#19151349C2211yescharacteristic, complemented
40Q8⋊#10623494C2211yescharacteristic, complemented
41Q8⋊#10623494C2211yescharacteristic, complemented
42D4⋊#10623494C2211yescharacteristic
43D4⋊#10623494C2211yescharacteristic
44D4⋊#10623494C2211yescharacteristic
45D4⋊#10623494C2211yescharacteristic
46D4⋊#10623494C2211yescharacteristic, complemented
47D4⋊#10623494C2211yescharacteristic, complemented
48#19151320C2211yescharacteristic, complemented
49#19151320C2211yescharacteristic, complemented
50#19151319C2211yescharacteristic, complemented
51#19151319C2211yescharacteristic, complemented
52#529×C89C2211yescharacteristic
53C2×C4×D4×C89C2211yescharacteristic
54#484×C89C411yescharacteristic
55#481×C89C411yescharacteristic
56#478×C89C411yescharacteristic
57#478×C89C411yescharacteristic
58#463×C89C411yescharacteristic
59#463×C89C411yescharacteristic
60#450×C89C411yescharacteristic
61#447×C89C411yescharacteristic
62C4×SD16×C89C2211yescharacteristic
63C4×SD16×C89C2211yescharacteristic
64C4×D8×C89C2211yescharacteristic
65C4×D8×C89C2211yescharacteristic
66#431×C89C2211yescharacteristic
67C2×#13083558—21no—
68C2×#13083558—21no—
69C2×#13083558—21no—
70C22×#10623494—21no—
71C22×#10623494—21no—
72C22×#10623494—21no—
73C22×#10623494—21no—
74C22×#10623494—21no—
75C4⋊#10623494—21no—
76C4⋊#10623494—21no—
77C4⋊#10623494—21no—
78C4⋊#10623494D411yescharacteristic
79C4⋊#10623494D411yescharacteristic
80C4⋊#10623494C2311yescharacteristic
81C4⋊#10623494C2311yescharacteristic
82C4⋊#10623494C2311yescharacteristic
83C4⋊#10623494C2311yescharacteristic
84C22⋊#10623494—21no—
85C22⋊#10623494—21no—
86C22⋊#10623494—21no—
87C22⋊#10623494—21no—
88C22⋊#10623494—21no—
89C22⋊#10623494—21no—
90C22⋊#10623494—21no—
91C22⋊#10623494D411yescharacteristic
92C22⋊#10623494D411yescharacteristic
93#15667923—21no—
94#15667923—21no—
95C4×#10623494—21no—
96C4×#10623494—21no—
97C4×#10623494C2311yescharacteristic
98C4×#10623494C2311yescharacteristic
99C4×#10623494C2311yescharacteristic
100C4×#10623494C2311yescharacteristic
101C22×D4×C89C2×C411yescharacteristic
102C23×C4×C89—21no—
103C2×SD16×C89C2×C411yescharacteristic
104C2×SD16×C89C2×C411yescharacteristic
105C2×D8×C89C2×C411yescharacteristic
106C2×D8×C89C2×C411yescharacteristic
107#126×C89—21no—
108#125×C89C2×C411yescharacteristic
109#124×C89C2×C411yescharacteristic
110#123×C89—21no—
111C22⋊Q8×C89C2×C411yescharacteristic
112#121×C89C2×C411yescharacteristic
113C4×Q8×C89C2311yescharacteristic
114C4×D4×C89—21no—
115C4×D4×C89—21no—
116C4×D4×C89—21no—
117C4×D4×C89C2×C411yescharacteristic
118C4×D4×C89C2×C411yescharacteristic
119C4×D4×C89C2311yescharacteristic
120C4×D4×C89C2311yescharacteristic
121C4×D4×C89C2311yescharacteristic
122#117×C89—21no—
123C2×C4⋊C4×C89C2×C411yescharacteristic
124C2×C22⋊C4×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#102×C89C2×C411yescharacteristic
135#102×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—
147C2×#10623494—21no—
148C2×#10623494—21no—
149C2×#10623494—21no—
150C2×#10623494—21no—
151C2×#10623494—21no—
152C2×#10623494—21no—
153C2×#10623494—42no—
154C2×#10623494—81no—
155C2×#10623494—81no—
156C2×#10623494—81no—
157C2×#10623494—81no—
158C2×#10623494C2×D411yescharacteristic
159C2×#10623494C2×D411yescharacteristic
160C2×#10623494C2×D411yescharacteristic
161C2×#10623494C2×D411yescharacteristic
162C24×C89—21no—
163C2×Q8×C89C22×C411yescharacteristic
164C2×D4×C89—22no—
165C2×D4×C89—21no—
166C2×D4×C89—21no—
167C2×D4×C89—41no—
168C2×D4×C89C22×C411yescharacteristic
169C2×D4×C89C22×C411yescharacteristic
170C2×D4×C89C22×C411yescharacteristic
171C22×C4×C89—21no—
172C22×C4×C89—21no—
173C22×C4×C89—21no—
174C22×C4×C89—21no—
175C22×C4×C89—21no—
176C22×C4×C89—21no—
177C22×C4×C89—81no—
178C22×C4×C89—81no—
179C22×C4×C89C22×C411yescharacteristic
180C22×C4×C89C2×D411yescharacteristic
181C22×C4×C89#4111yescharacteristic
182SD16×C89—22no—
183SD16×C89—41no—
184D8×C89—22no—
185D8×C89—41no—
186C2×C8×C89C22×C411yescharacteristic
187C2×C8×C89C22×C411yescharacteristic
188C2×C8×C89C22×C411yescharacteristic
189C2×C8×C89C22×C411yescharacteristic
190C4⋊C4×C89—21no—
191C4⋊C4×C89—21no—
192C4⋊C4×C89—21no—
193C4⋊C4×C89—21no—
194C4⋊C4×C89—21no—
195C4⋊C4×C89C22×C411yescharacteristic
196C4⋊C4×C89C22×C411yescharacteristic
197C4⋊C4×C89C22×C411yescharacteristic
198C4⋊C4×C89C22×C411yescharacteristic
199C22⋊C4×C89—21no—
200C22⋊C4×C89—21no—
201C22⋊C4×C89—21no—
202C22⋊C4×C89—21no—
203C22⋊C4×C89—21no—
204C22⋊C4×C89—21no—
205C22⋊C4×C89—21no—
206C22⋊C4×C89—21no—
207C22⋊C4×C89—21no—
208C42×C89—21no—
209C42×C89C22×C411yescharacteristic
210C42×C89C2×D411yescharacteristic
211C42×C89#4111yescharacteristic
212C42×C89C2411yescharacteristic
213C23×C89—21no—
214C23×C89—21no—
215C23×C89—21no—
216C23×C89—42no—
217C23×C89—42no—
218C23×C89—81no—
219C23×C89C4×D411yescharacteristic
220Q8×C89—21no—
221Q8×C89—21no—
222D4×C89—21no—
223D4×C89—21no—
224D4×C89—21no—
225D4×C89—21no—
226D4×C89—21no—
227D4×C89—42no—
228D4×C89—42no—
229D4×C89#13112yes—
230C2×C4×C89—22no—
231C2×C4×C89—21no—
232C2×C4×C89—21no—
233C2×C4×C89—21no—
234C2×C4×C89—21no—
235C2×C4×C89—21no—
236C2×C4×C89—21no—
237C2×C4×C89—21no—
238C2×C4×C89—21no—
239C2×C4×C89—21no—
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—81no—
247C2×C4×C89—81no—
248C2×C4×C89—81no—
249C2×C4×C89C4×D411yescharacteristic
250C2×C4×C89C4×D411yescharacteristic
251C2×C4×C89C4×D411yescharacteristic
252C2×C4×C89C23×C411yescharacteristic
253C2×C4×C89C22×D411yescharacteristic
254C2×C4×C89C2×#4111yescharacteristic
255C8×C89—21no—
256C8×C89—21no—
257C8×C89—21no—
258C8×C89—21no—
259#10623494—21no—
260#10623494—21no—
261#10623494—21no—
262#10623494—21no—
263#10623494—41no—
264#10623494—41no—
265#10623494—41no—
266#10623494—41no—
267#10623494—41no—
268#10623494—41no—
269C22×C89—21no—
270C22×C89—21no—
271C22×C89—22no—
272C22×C89—41no—
273C22×C89—41no—
274C22×C89—41no—
275C22×C89—41no—
276C22×C89—41no—
277C22×C89—41no—
278C22×C89—41no—
279C22×C89—41no—
280C22×C89—84no—
281C22×C89#44412yes—
282C22×C89C2×C4×D411yescharacteristic
283C4×C89—21no—
284C4×C89—21no—
285C4×C89—21no—
286C4×C89—21no—
287C4×C89—41no—
288C4×C89—41no—
289C4×C89—41no—
290C4×C89—41no—
291C4×C89—41no—
292C4×C89C2×#13111yescharacteristic
293C4×C89#56811yescharacteristic
294C4×C89#57611yescharacteristic
295C4×C89#57711yescharacteristic
296#18020—891nomaximal, Sylow
297C2×C89—22no—
298C2×C89—42no—
299C2×C89—81no—
300C2×C89—81no—
301C2×C89#293311yescharacteristic
302C2×C89#294611yescharacteristic
303C2×C89C2×#44411yescharacteristic
304#3105D8911yescharacteristic
305#2994—891no—
306#2974—891no—
307#1594—891no—
308#1592—891no—
309#1583—891no—
310#1582—891no—
311#1581—891no—
312#1581—891no—
313#1580—891no—
314SD16⋊C8—891no—
315#1573—891no—
316#1494—891no—
317C2×D4⋊C8—891no—
318#1452—891no—
319C89#1802011yescharacteristic, complemented, Sylow
320#529C2×D8911yescharacteristic
321C2×C4×D4C2×D8911yescharacteristic
322#484Dic8911yescharacteristic
323#481Dic8911yescharacteristic
324#478Dic8911yescharacteristic
325#478Dic8911yescharacteristic
326#463Dic8911yescharacteristic
327#463Dic8911yescharacteristic
328#450Dic8911yescharacteristic
329#447Dic8911yescharacteristic
330#446—891no—
331C8×Q8—891no—
332C4×SD16C2×D8911yescharacteristic
333C4×SD16C2×D8911yescharacteristic
334C4×D8C2×D8911yescharacteristic
335C4×D8C2×D8911yescharacteristic
336#436—891no—
337#436—891no—
338#436—891no—
339#435—1781no—
340#435—1781no—
341#435—1781no—
342C8×D4—891no—
343C8×D4—891no—
344C8×D4—891no—
345C8×D4—1781no—
346#433—1781no—
347#432—891no—
348#431—891no—
349#431C2×D8911yescharacteristic
350#424—1781no—
351C2×C4⋊C8—891no—
352#407—1781no—
353C2×C22⋊C8—1781no—
354C2×C4×C8—891no—
355#335—891no—
356#335—891no—
357#334—891no—
358#334—891no—
359Q8⋊C8—891no—
360Q8⋊C8—891no—
361D4⋊C8—891no—
362D4⋊C8—891no—
363D4⋊C8—891no—
364D4⋊C8—891no—
365D4⋊C8—891no—
366D4⋊C8—891no—
367#322—891no—
368#322—891no—
369#322—891no—
370#322—891no—
371C22×D4C2×Dic8911yescharacteristic
372C23×C4—21no—
373C2×SD16C2×Dic8911yescharacteristic
374C2×SD16C2×Dic8911yescharacteristic
375C2×D8C2×Dic8911yescharacteristic
376C2×D8C2×Dic8911yescharacteristic
377C2×M16—1781no—
378C2×M16—1781no—
379C2×M16—1781no—
380C22×C8—1781no—
381C22×C8—1781no—
382C22×C8—1781no—
383C22×C8—1781no—
384C22×C8—1781no—
385#126—21no—
386#125C2×Dic8911yescharacteristic
387#124C2×Dic8911yescharacteristic
388#123—21no—
389C22⋊Q8C2×Dic8911yescharacteristic
390#121C2×Dic8911yescharacteristic
391C4×Q8C22×D8911yescharacteristic
392C4×D4—21no—
393C4×D4—21no—
394C4×D4—21no—
395C4×D4C2×Dic8911yescharacteristic
396C4×D4C2×Dic8911yescharacteristic
397C4×D4C22×D8911yescharacteristic
398C4×D4C22×D8911yescharacteristic
399C4×D4C22×D8911yescharacteristic
400#117—21no—
401C2×C4⋊C4C2×Dic8911yescharacteristic
402C2×C22⋊C4—21no—
403C2×C42C22×D8911yescharacteristic
404#107C2×Dic8911yescharacteristic
405#107C2×Dic8911yescharacteristic
406#106C2×Dic8911yescharacteristic
407#106C2×Dic8911yescharacteristic
408C4⋊C8—891no—
409C4⋊C8—891no—
410C4⋊C8—891no—
411C4⋊C8—891no—
412C4⋊C8—891no—
413C4⋊C8—891no—
414C4⋊C8—1781no—
415C4⋊C8—1781no—
416C4⋊C8—1781no—
417C4⋊C8C2×Dic8911yescharacteristic
418C4⋊C8C2×Dic8911yescharacteristic
419#103C2×Dic8911yescharacteristic
420#103C2×Dic8911yescharacteristic
421#102C2×Dic8911yescharacteristic
422#102C2×Dic8911yescharacteristic
423#102C2×Dic8911yescharacteristic
424#102C2×Dic8911yescharacteristic
425#102C2×Dic8911yescharacteristic
426#102C2×Dic8911yescharacteristic
427C22⋊C8—891no—
428C22⋊C8—891no—
429C22⋊C8—1781no—
430C22⋊C8—1781no—
431C22⋊C8—1781no—
432C22⋊C8—1781no—
433C22⋊C8—1781no—
434C22⋊C8—1781no—
435C22⋊C8—1781no—
436C22⋊C8C2×Dic8911yescharacteristic
437C22⋊C8C2×Dic8911yescharacteristic
438#97—1781no—
439#97—1781no—
440C4×C8—891no—
441C4×C8—891no—
442C4×C8—891no—
443C4×C8—891no—
444C4×C8—1781no—
445C4×C8—1781no—
446C4×C8C22×D8911yescharacteristic
447C4×C8C22×D8911yescharacteristic
448C24—21no—
449C2×Q8C22×Dic8911yescharacteristic
450C2×D4—22no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—41no—
454C2×D4C22×Dic8911yescharacteristic
455C2×D4C22×Dic8911yescharacteristic
456C2×D4C22×Dic8911yescharacteristic
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—81no—
464C22×C4—81no—
465C22×C4C22×Dic8911yescharacteristic
466C22×C4D4×D8911yescharacteristic
467C22×C4#1308357311yescharacteristic
468SD16—22no—
469SD16—41no—
470D8—22no—
471D8—41no—
472M16—7121no—
473M16—7121no—
474M16—7121no—
475C2×C8—891no—
476C2×C8—891no—
477C2×C8—891no—
478C2×C8—891no—
479C2×C8—3562no—
480C2×C8—1781no—
481C2×C8—1781no—
482C2×C8—1781no—
483C2×C8—1781no—
484C2×C8—1781no—
485C2×C8—1781no—
486C2×C8—7121no—
487C2×C8—7121no—
488C2×C8—7121no—
489C2×C8—7121no—
490C2×C8C22×Dic8911yescharacteristic
491C2×C8C22×Dic8911yescharacteristic
492C2×C8C22×Dic8911yescharacteristic
493C2×C8C22×Dic8911yescharacteristic
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4C22×Dic8911yescharacteristic
500C4⋊C4C22×Dic8911yescharacteristic
501C4⋊C4C22×Dic8911yescharacteristic
502C4⋊C4C22×Dic8911yescharacteristic
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—21no—
512C42—21no—
513C42C22×Dic8911yescharacteristic
514C42D4×D8911yescharacteristic
515C42#1308357311yescharacteristic
516C42C23×D8911yescharacteristic
517C23—21no—
518C23—21no—
519C23—21no—
520C23—42no—
521C23—42no—
522C23—81no—
523C23D4×Dic8911yescharacteristic
524Q8—21no—
525Q8—21no—
526D4—21no—
527D4—21no—
528D4—21no—
529D4—21no—
530D4—21no—
531D4—42no—
532D4—42no—
533D4#1566801912yes—
534C2×C4—22no—
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—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—81no—
551C2×C4—81no—
552C2×C4—81no—
553C2×C4D4×Dic8911yescharacteristic
554C2×C4D4×Dic8911yescharacteristic
555C2×C4D4×Dic8911yescharacteristic
556C2×C4C23×Dic8911yescharacteristic
557C2×C4C2×D4×D8911yescharacteristic
558C2×C4C2×#1308357311yescharacteristic
559C8—21no—
560C8—21no—
561C8—21no—
562C8—21no—
563C8—3561no—
564C8—3561no—
565C8—3561no—
566C8—3561no—
567C8—3561no—
568C8—3561no—
569C8—1781no—
570C8—1781no—
571C8—1781no—
572C8—1781no—
573C22—21no—
574C22—21no—
575C22—22no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—84no—
585C22#1915172012yes—
586C22C2×D4×Dic8911yescharacteristic
587C4—21no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4C2×#1566801911yescharacteristic
597C4#1915236511yescharacteristic
598C4#1915240211yescharacteristic
599C4#1915240811yescharacteristic
600C2—22no—
601C2—42no—
602C2—81no—
603C2—81no—
604C2#2602897911yescharacteristic
605C2#2602908511yescharacteristic
606C2C2×#1915172011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#3105C2
socleC22×C89C2×C4×D4
prime radical#447×C89C4
cosocleC42×C89C24
hypercenter#3105D89
hypocenterC89#18020
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3105

Complements

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

NH ≅ G/Ntype
1. #26029412C2semidirect
2. #26029287C2semidirect
3. #26019166C2semidirect
4. #26019155C2semidirect
5. #26019096C2semidirect
6. #26019089C2semidirect
7. #26019082C2semidirect
8. #26019082C2semidirect
9. #26019076C2semidirect
10. SD16⋊#10623494C2semidirect
11. #26019037C2semidirect
12. #26018712C2semidirect
13. C2×D4⋊#10623494C2semidirect
14. #26018545C2semidirect
16. #19151729C22semidirect
17. Q8×#10623494C22semidirect
20. #19151676C22semidirect
27. D4×#10623494C22semidirect
36. #19151351C22semidirect
37. #19151351C22semidirect
38. #19151349C22semidirect
39. #19151349C22semidirect
40. Q8⋊#10623494C22semidirect
41. Q8⋊#10623494C22semidirect
46. D4⋊#10623494C22semidirect
47. D4⋊#10623494C22semidirect
48. #19151320C22semidirect
49. #19151320C22semidirect
50. #19151319C22semidirect
51. #19151319C22semidirect
319. C89#18020semidirect

Maximal subgroups

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

subgroupindex
#260294122
#260292872
#260191662
#260191552
#260190962
#260190892
#260190822
#260190762
SD16⋊#106234942
#260190372
#260187122
C2×D4⋊#106234942
#260185452
C89×#31052
#1802089