Finite Group Databasebeta

← #36525697

Subgroups — #36525697

Order 22784.

592 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26029412C211yesmaximal, characteristic
2#26029341C211yesmaximal, characteristic, complemented
3#26019161C211yesmaximal, characteristic, complemented
4#26019155C211yesmaximal, characteristic, complemented
5#26019103C211yesmaximal, characteristic, complemented
6#26019103C211yesmaximal, characteristic, complemented
7#26019096C211yesmaximal, characteristic, complemented
8#26019089C211yesmaximal, characteristic, complemented
9#26019071C211yesmaximal, characteristic, complemented
10SD16⋊#10623494C211yesmaximal, characteristic, complemented
11#26019037C211yesmaximal, characteristic, complemented
12#26018712C211yesmaximal, characteristic
13C2×Q8⋊#10623494C211yesmaximal, characteristic, complemented
14#26018545C211yesmaximal, characteristic, complemented
15C89×#3108C211yesmaximal, characteristic
16#19151729—21no—
17#19151729—21no—
18#19151729C2211yescharacteristic
19#19151729C2211yescharacteristic
20#19151729C2211yescharacteristic
21Q8×#10623494C2211yescharacteristic
22Q8×#10623494C2211yescharacteristic
23Q8×#10623494C2211yescharacteristic
24#19151676C2211yescharacteristic
25#19151670—21no—
26D4×#10623494—21no—
27D4×#10623494C2211yescharacteristic
28#19151659—21no—
29#19151655—21no—
30#19151655C2211yescharacteristic
31#19151651—21no—
32#19151651C2211yescharacteristic
33#19151625—21no—
34C2×C4⋊#10623494C2211yescharacteristic
35C2×C4×#10623494C2211yescharacteristic
36#19151351C2211yescharacteristic
37#19151351C2211yescharacteristic
38#19151349C2211yescharacteristic
39#19151349C2211yescharacteristic
40Q8⋊#10623494C2211yescharacteristic
41Q8⋊#10623494C2211yescharacteristic
42Q8⋊#10623494C2211yescharacteristic
43Q8⋊#10623494C2211yescharacteristic
44Q8⋊#10623494C2211yescharacteristic
45Q8⋊#10623494C2211yescharacteristic
46D4⋊#10623494C2211yescharacteristic
47D4⋊#10623494C2211yescharacteristic
48#19151320C2211yescharacteristic
49#19151320C2211yescharacteristic
50#19151319C2211yescharacteristic
51#19151319C2211yescharacteristic
52#529×C89C2211yescharacteristic
53C2×C4×Q8×C89C2211yescharacteristic
54#483×C89C411yescharacteristic
55#480×C89C411yescharacteristic
56#479×C89C411yescharacteristic
57#479×C89C411yescharacteristic
58#464×C89C411yescharacteristic
59#464×C89C411yescharacteristic
60#451×C89C411yescharacteristic
61#448×C89C411yescharacteristic
62C4×Q16×C89C2211yescharacteristic
63C4×Q16×C89C2211yescharacteristic
64C4×SD16×C89C2211yescharacteristic
65C4×SD16×C89C2211yescharacteristic
66#431×C89C2211yescharacteristic
67C2×#13083558—21no—
68C22×#10623494—21no—
69C22×#10623494—21no—
70C22×#10623494—21no—
71C4⋊#10623494—21no—
72C4⋊#10623494—21no—
73C4⋊#10623494—21no—
74C4⋊#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⋊#10623494D411yescharacteristic
88C22⋊#10623494D411yescharacteristic
89#15667923—21no—
90#15667923—21no—
91#15667923—21no—
92#15667923—21no—
93C4×#10623494—21no—
94C4×#10623494—21no—
95C4×#10623494—21no—
96C4×#10623494—21no—
97C4×#10623494C2311yescharacteristic
98C4×#10623494C2311yescharacteristic
99C4×#10623494C2311yescharacteristic
100C4×#10623494C2311yescharacteristic
101C22×Q8×C89C2×C411yescharacteristic
102C2×Q16×C89C2×C411yescharacteristic
103C2×Q16×C89C2×C411yescharacteristic
104C2×SD16×C89C2×C411yescharacteristic
105C2×SD16×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×C89—21no—
113C4×Q8×C89—21no—
114C4×Q8×C89C2×C411yescharacteristic
115C4×Q8×C89C2×C411yescharacteristic
116C4×Q8×C89C2311yescharacteristic
117C4×Q8×C89C2311yescharacteristic
118C4×Q8×C89C2311yescharacteristic
119C4×D4×C89—21no—
120C4×D4×C89C2311yescharacteristic
121#117×C89—21no—
122C2×C4⋊C4×C89—21no—
123C2×C4⋊C4×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#103×C89C2×C411yescharacteristic
137#103×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—
145C2×#10623494—21no—
146C2×#10623494—21no—
147C2×#10623494—21no—
148C2×#10623494—21no—
149C2×#10623494—21no—
150C2×#10623494—21no—
151C2×#10623494—42no—
152C2×#10623494—81no—
153C2×#10623494—81no—
154C2×#10623494C2×D411yescharacteristic
155C2×#10623494C2×D411yescharacteristic
156C2×#10623494C2×D411yescharacteristic
157C2×#10623494C2×D411yescharacteristic
158C2×Q8×C89—21no—
159C2×Q8×C89—22no—
160C2×Q8×C89—41no—
161C2×Q8×C89C22×C411yescharacteristic
162C2×Q8×C89C22×C411yescharacteristic
163C2×Q8×C89C22×C411yescharacteristic
164C2×D4×C89—21no—
165C2×D4×C89C22×C411yescharacteristic
166C22×C4×C89—21no—
167C22×C4×C89—21no—
168C22×C4×C89—21no—
169C22×C4×C89C22×C411yescharacteristic
170C22×C4×C89C2×D411yescharacteristic
171C22×C4×C89#4111yescharacteristic
172Q16×C89—22no—
173Q16×C89—41no—
174SD16×C89—22no—
175SD16×C89—41no—
176C2×C8×C89C22×C411yescharacteristic
177C2×C8×C89C22×C411yescharacteristic
178C2×C8×C89C22×C411yescharacteristic
179C2×C8×C89C22×C411yescharacteristic
180C4⋊C4×C89—21no—
181C4⋊C4×C89—21no—
182C4⋊C4×C89—21no—
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—
198C42×C89—21no—
199C42×C89—21no—
200C42×C89—21no—
201C42×C89—21no—
202C42×C89—21no—
203C42×C89C22×C411yescharacteristic
204C42×C89C2×D411yescharacteristic
205C42×C89#4111yescharacteristic
206C42×C89C2411yescharacteristic
207C23×C89—21no—
208C23×C89C4×D411yescharacteristic
209Q8×C89—21no—
210Q8×C89—21no—
211Q8×C89—21no—
212Q8×C89—21no—
213Q8×C89—21no—
214Q8×C89—42no—
215Q8×C89#13112yes—
216D4×C89—21no—
217D4×C89—21no—
218D4×C89—42no—
219C2×C4×C89—21no—
220C2×C4×C89—22no—
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—21no—
231C2×C4×C89—42no—
232C2×C4×C89—42no—
233C2×C4×C89—41no—
234C2×C4×C89—41no—
235C2×C4×C89C4×D411yescharacteristic
236C2×C4×C89C4×D411yescharacteristic
237C2×C4×C89C4×D411yescharacteristic
238C2×C4×C89C23×C411yescharacteristic
239C2×C4×C89C22×D411yescharacteristic
240C2×C4×C89C2×#4111yescharacteristic
241C8×C89—21no—
242C8×C89—21no—
243C8×C89—21no—
244C8×C89—21no—
245#10623494—21no—
246#10623494—21no—
247#10623494—21no—
248#10623494—21no—
249#10623494—41no—
250#10623494—41no—
251#10623494—41no—
252#10623494—41no—
253#10623494—41no—
254#10623494—41no—
255C22×C89—21no—
256C22×C89—21no—
257C22×C89—41no—
258C22×C89—41no—
259C22×C89—41no—
260C22×C89#44412yes—
261C22×C89C2×C4×D411yescharacteristic
262C4×C89—21no—
263C4×C89—22no—
264C4×C89—21no—
265C4×C89—21no—
266C4×C89—21no—
267C4×C89—41no—
268C4×C89—41no—
269C4×C89—41no—
270C4×C89—41no—
271C4×C89—41no—
272C4×C89—41no—
273C4×C89C2×#13111yescharacteristic
274C4×C89#56811yescharacteristic
275C4×C89#57711yescharacteristic
276C4×C89#57811yescharacteristic
277#18023—891nomaximal, Sylow
278C2×C89—22no—
279C2×C89—81no—
280C2×C89#293311yescharacteristic
281C2×C89#294711yescharacteristic
282C2×C89C2×#44411yescharacteristic
283#3108D8911yescharacteristic
284#2994—891no—
285#2984—891no—
286#1593—891no—
287#1592—891no—
288#1584—891no—
289#1584—891no—
290#1583—891no—
291#1582—891no—
292#1579—891no—
293SD16⋊C8—891no—
294#1573—891no—
295#1494—891no—
296C2×Q8⋊C8—891no—
297#1452—891no—
298C89#1802311yescharacteristic, complemented, Sylow
299#529C2×D8911yescharacteristic
300C2×C4×Q8C2×D8911yescharacteristic
301#483Dic8911yescharacteristic
302#480Dic8911yescharacteristic
303#479Dic8911yescharacteristic
304#479Dic8911yescharacteristic
305#464Dic8911yescharacteristic
306#464Dic8911yescharacteristic
307#451Dic8911yescharacteristic
308#448Dic8911yescharacteristic
309#446—891no—
310#446—891no—
311#446—891no—
312#446—1781no—
313#446—1781no—
314C8×Q8—891no—
315C8×Q8—891no—
316C8×Q8—891no—
317C4×Q16C2×D8911yescharacteristic
318C4×Q16C2×D8911yescharacteristic
319C4×SD16C2×D8911yescharacteristic
320C4×SD16C2×D8911yescharacteristic
321#436—891no—
322#435—1781no—
323C8×D4—891no—
324C8×D4—1781no—
325#433—1781no—
326#432—891no—
327#432—1781no—
328#431—891no—
329#431—1781no—
330#431C2×D8911yescharacteristic
331#424—1781no—
332C2×C4⋊C8—891no—
333C2×C4×C8—891no—
334#335—891no—
335#335—891no—
336#334—891no—
337#334—891no—
338Q8⋊C8—891no—
339Q8⋊C8—891no—
340Q8⋊C8—891no—
341Q8⋊C8—891no—
342Q8⋊C8—891no—
343Q8⋊C8—891no—
344D4⋊C8—891no—
345D4⋊C8—891no—
346#322—891no—
347#322—891no—
348#322—891no—
349#322—891no—
350C22×Q8C2×Dic8911yescharacteristic
351C2×Q16C2×Dic8911yescharacteristic
352C2×Q16C2×Dic8911yescharacteristic
353C2×SD16C2×Dic8911yescharacteristic
354C2×SD16C2×Dic8911yescharacteristic
355C2×M16—1781no—
356C22×C8—1781no—
357C22×C8—1781no—
358C22×C8—1781no—
359#126—21no—
360#125C2×Dic8911yescharacteristic
361#124C2×Dic8911yescharacteristic
362#123—21no—
363C22⋊Q8C2×Dic8911yescharacteristic
364#121C2×Dic8911yescharacteristic
365C4×Q8—21no—
366C4×Q8—21no—
367C4×Q8C2×Dic8911yescharacteristic
368C4×Q8C2×Dic8911yescharacteristic
369C4×Q8C22×D8911yescharacteristic
370C4×Q8C22×D8911yescharacteristic
371C4×Q8C22×D8911yescharacteristic
372C4×D4—21no—
373C4×D4C22×D8911yescharacteristic
374#117—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4C2×Dic8911yescharacteristic
377C2×C42—21no—
378C2×C42C22×D8911yescharacteristic
379#107C2×Dic8911yescharacteristic
380#107C2×Dic8911yescharacteristic
381#106C2×Dic8911yescharacteristic
382#106C2×Dic8911yescharacteristic
383C4⋊C8—891no—
384C4⋊C8—891no—
385C4⋊C8—891no—
386C4⋊C8—891no—
387C4⋊C8—891no—
388C4⋊C8—891no—
389C4⋊C8—1781no—
390C4⋊C8—1781no—
391C4⋊C8—1781no—
392C4⋊C8—1781no—
393C4⋊C8—1781no—
394C4⋊C8—1781no—
395C4⋊C8—1781no—
396C4⋊C8C2×Dic8911yescharacteristic
397C4⋊C8C2×Dic8911yescharacteristic
398#103C2×Dic8911yescharacteristic
399#103C2×Dic8911yescharacteristic
400#103C2×Dic8911yescharacteristic
401#103C2×Dic8911yescharacteristic
402#103C2×Dic8911yescharacteristic
403#103C2×Dic8911yescharacteristic
404#102C2×Dic8911yescharacteristic
405#102C2×Dic8911yescharacteristic
406C22⋊C8—891no—
407C22⋊C8—891no—
408C22⋊C8—1781no—
409C22⋊C8—1781no—
410C22⋊C8—1781no—
411C22⋊C8C2×Dic8911yescharacteristic
412C22⋊C8C2×Dic8911yescharacteristic
413#97—1781no—
414#97—1781no—
415#97—1781no—
416#97—1781no—
417C4×C8—891no—
418C4×C8—891no—
419C4×C8—891no—
420C4×C8—891no—
421C4×C8—1781no—
422C4×C8—1781no—
423C4×C8—1781no—
424C4×C8—1781no—
425C4×C8C22×D8911yescharacteristic
426C4×C8C22×D8911yescharacteristic
427C2×Q8—21no—
428C2×Q8—22no—
429C2×Q8—41no—
430C2×Q8C22×Dic8911yescharacteristic
431C2×Q8C22×Dic8911yescharacteristic
432C2×Q8C22×Dic8911yescharacteristic
433C2×D4—21no—
434C2×D4C22×Dic8911yescharacteristic
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4C22×Dic8911yescharacteristic
439C22×C4D4×D8911yescharacteristic
440C22×C4#1308357311yescharacteristic
441Q16—22no—
442Q16—41no—
443SD16—22no—
444SD16—41no—
445M16—7121no—
446C2×C8—891no—
447C2×C8—891no—
448C2×C8—891no—
449C2×C8—891no—
450C2×C8—3562no—
451C2×C8—1781no—
452C2×C8—1781no—
453C2×C8—1781no—
454C2×C8—1781no—
455C2×C8—1781no—
456C2×C8—1781no—
457C2×C8—7121no—
458C2×C8—7121no—
459C2×C8C22×Dic8911yescharacteristic
460C2×C8C22×Dic8911yescharacteristic
461C2×C8C22×Dic8911yescharacteristic
462C2×C8C22×Dic8911yescharacteristic
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4C22×Dic8911yescharacteristic
473C4⋊C4C22×Dic8911yescharacteristic
474C4⋊C4C22×Dic8911yescharacteristic
475C4⋊C4C22×Dic8911yescharacteristic
476C22⋊C4—21no—
477C22⋊C4—21no—
478C22⋊C4—21no—
479C22⋊C4—21no—
480C22⋊C4—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42C22×Dic8911yescharacteristic
487C42D4×D8911yescharacteristic
488C42#1308357311yescharacteristic
489C42C23×D8911yescharacteristic
490C23—21no—
491C23D4×Dic8911yescharacteristic
492Q8—21no—
493Q8—21no—
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—42no—
498Q8#1566801912yes—
499D4—21no—
500D4—21no—
501D4—42no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4D4×Dic8911yescharacteristic
519C2×C4D4×Dic8911yescharacteristic
520C2×C4D4×Dic8911yescharacteristic
521C2×C4C23×Dic8911yescharacteristic
522C2×C4C2×D4×D8911yescharacteristic
523C2×C4C2×#1308357311yescharacteristic
524C8—21no—
525C8—21no—
526C8—21no—
527C8—21no—
528C8—3561no—
529C8—3561no—
530C8—3561no—
531C8—3561no—
532C8—3561no—
533C8—3561no—
534C8—1781no—
535C8—1781no—
536C8—1781no—
537C8—1781no—
538C22—21no—
539C22—21no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22#1915172012yes—
544C22C2×D4×Dic8911yescharacteristic
545C4—21no—
546C4—21no—
547C4—21no—
548C4—21no—
549C4—22no—
550C4—41no—
551C4—41no—
552C4—41no—
553C4—41no—
554C4—41no—
555C4—41no—
556C4C2×#1566801911yescharacteristic
557C4#1915236511yescharacteristic
558C4#1915240711yescharacteristic
559C4#1915241411yescharacteristic
560C2—22no—
561C2—81no—
562C2#2602897911yescharacteristic
563C2#2602909611yescharacteristic
564C2C2×#1915172011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#3108C2
socleC22×C89C2×C4×D4
prime radical#121×C89C2×C4
cosocleC42×C89C24
hypercenter#3108D89
hypocenterC89#18023
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3108

Complements

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

NH ≅ G/Ntype
2. #26029341C2semidirect
3. #26019161C2semidirect
4. #26019155C2semidirect
5. #26019103C2semidirect
6. #26019103C2semidirect
7. #26019096C2semidirect
8. #26019089C2semidirect
9. #26019071C2semidirect
10. SD16⋊#10623494C2semidirect
11. #26019037C2semidirect
13. C2×Q8⋊#10623494C2semidirect
14. #26018545C2semidirect
298. C89#18023semidirect

Maximal subgroups

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

subgroupindex
#260294122
#260293412
#260191612
#260191552
#260191032
#260190962
#260190892
#260190712
SD16⋊#106234942
#260190372
#260187122
C2×Q8⋊#106234942
#260185452
C89×#31082
#1802389