Finite Group Databasebeta

← #36526022

Subgroups — #36526022

Order 22784.

676 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26029377C211yesmaximal, characteristic, complemented
2#26029287C211yesmaximal, characteristic, complemented
3#26019155C211yesmaximal, characteristic, complemented
4#26019143C211yesmaximal, characteristic, complemented
5#26019117C211yesmaximal, characteristic, complemented
6#26019110C211yesmaximal, characteristic, complemented
7#26019089C211yesmaximal, characteristic, complemented
8#26019082C211yesmaximal, characteristic, complemented
9#26019076C211yesmaximal, characteristic, complemented
10D8×#10623494C211yesmaximal, characteristic, complemented
11#26019019C211yesmaximal, characteristic, complemented
12#26018686C211yesmaximal, characteristic, complemented
13#26018677C211yesmaximal, characteristic, complemented
14#26018549C211yesmaximal, characteristic, complemented
15C4×AGL(1,ℤ/8)×C89C211yesmaximal, characteristic
16#19151729C2211yescharacteristic, complemented
17#19151729C2211yescharacteristic, complemented
18#19151676C2211yescharacteristic
19#19151676C2211yescharacteristic
20#19151670—21no—
21#19151670—21no—
22#19151670—21no—
23#19151670—21no—
24D4×#10623494C2211yescharacteristic
25D4×#10623494C2211yescharacteristic
26D4×#10623494C2211yescharacteristic, complemented
27D4×#10623494C2211yescharacteristic, complemented
28#19151655—21no—
29#19151655C2211yescharacteristic
30#19151651C2211yescharacteristic
31#19151651C2211yescharacteristic
32#19151622C2211yescharacteristic
33C2×C4⋊#10623494—21no—
34#19151562—21no—
35C2×C22⋊#10623494—21no—
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
51C8×#10623494C2211yescharacteristic, complemented
52C2×AGL(1,ℤ/8)×C89C411yescharacteristic
53C4×#41×C89C2211yescharacteristic
54C2×C4×D4×C89C2211yescharacteristic
55#442×C89C411yescharacteristic
56#442×C89C411yescharacteristic
57#440×C89C411yescharacteristic
58#440×C89C411yescharacteristic
59C4×SD16×C89C2211yescharacteristic
60C4×SD16×C89C2211yescharacteristic
61C4×D8×C89C2211yescharacteristic
62C4×D8×C89C2211yescharacteristic
63#428×C89C411yescharacteristic
64#418×C89C411yescharacteristic
65#417×C89C411yescharacteristic
66C4×M16×C89C2211yescharacteristic
67C2×#13083558—21no—
68C2×#13083558—21no—
69C2×#13083558—21no—
70C22×#10623494—21no—
71C22×#10623494—21no—
72C22×#10623494—21no—
73C22×#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⋊#10623494—21no—
88C22⋊#10623494—21no—
89C22⋊#10623494—21no—
90C22⋊#10623494—21no—
91C22⋊#10623494—21no—
92C22⋊#10623494D411yescharacteristic
93C22⋊#10623494D411yescharacteristic
94#15667923—21no—
95#15667923—21no—
96#15667923—21no—
97C4×#10623494C2311yescharacteristic
98C4×#10623494C2311yescharacteristic
99C4×#10623494C2311yescharacteristic
100C4×#10623494C2311yescharacteristic
101C2×#41×C89C2×C411yescharacteristic
102C22×D4×C89C2×C411yescharacteristic
103C23×C4×C89—21no—
104AGL(1,ℤ/8)×C89—24no—
105C2×SD16×C89C2×C411yescharacteristic
106C2×SD16×C89C2×C411yescharacteristic
107C2×D8×C89C2×C411yescharacteristic
108C2×D8×C89C2×C411yescharacteristic
109C2×M16×C89C2×C411yescharacteristic
110C2×M16×C89C2×C411yescharacteristic
111C4×Q8×C89C2×C411yescharacteristic
112C4×Q8×C89C2311yescharacteristic
113C4×D4×C89—21no—
114C4×D4×C89—21no—
115C4×D4×C89—21no—
116C4×D4×C89—21no—
117C4×D4×C89C2×C411yescharacteristic
118C4×D4×C89C2×C411yescharacteristic
119C4×D4×C89C2×C411yescharacteristic
120C4×D4×C89C2311yescharacteristic
121C4×D4×C89C2311yescharacteristic
122C4×D4×C89C2311yescharacteristic
123#117×C89—21no—
124#117×C89C2×C411yescharacteristic
125C2×C4⋊C4×C89C2×C411yescharacteristic
126C2×C22⋊C4×C89—21no—
127C2×C42×C89—21no—
128C2×C42×C89C2311yescharacteristic
129#107×C89C2×C411yescharacteristic
130#107×C89C2×C411yescharacteristic
131#106×C89C2×C411yescharacteristic
132#106×C89C2×C411yescharacteristic
133#103×C89C2×C411yescharacteristic
134#103×C89C2×C411yescharacteristic
135#102×C89C2×C411yescharacteristic
136#102×C89C2×C411yescharacteristic
137#102×C89C2×C411yescharacteristic
138#102×C89C2×C411yescharacteristic
139#102×C89C2×C411yescharacteristic
140#102×C89C2×C411yescharacteristic
141#97×C89C2×C411yescharacteristic
142#97×C89C2×C411yescharacteristic
143C4×C8×C89C2311yescharacteristic
144C4×C8×C89C2311yescharacteristic
145#13083558—81no—
146#13083558—81no—
147#13083558—81no—
148C2×#10623494—21no—
149C2×#10623494—21no—
150C2×#10623494—21no—
151C2×#10623494—21no—
152C2×#10623494—21no—
153C2×#10623494—21no—
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—
163#41×C89—22no—
164#41×C89—41no—
165C2×Q8×C89C22×C411yescharacteristic
166C2×D4×C89—22no—
167C2×D4×C89—21no—
168C2×D4×C89—21no—
169C2×D4×C89—41no—
170C2×D4×C89C22×C411yescharacteristic
171C2×D4×C89C22×C411yescharacteristic
172C2×D4×C89C22×C411yescharacteristic
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×C89—21no—
181C22×C4×C89—81no—
182C22×C4×C89—81no—
183C22×C4×C89C22×C411yescharacteristic
184C22×C4×C89C2×D411yescharacteristic
185C22×C4×C89#4111yescharacteristic
186SD16×C89—22no—
187SD16×C89—22no—
188D8×C89—22no—
189D8×C89M1614yescomplemented
190M16×C89—22no—
191M16×C89—41no—
192C2×C8×C89C22×C411yescharacteristic
193C2×C8×C89C22×C411yescharacteristic
194C2×C8×C89C22×C411yescharacteristic
195C2×C8×C89C22×C411yescharacteristic
196C4⋊C4×C89—21no—
197C4⋊C4×C89—21no—
198C4⋊C4×C89—21no—
199C4⋊C4×C89C22×C411yescharacteristic
200C4⋊C4×C89C22×C411yescharacteristic
201C4⋊C4×C89C22×C411yescharacteristic
202C4⋊C4×C89C22×C411yescharacteristic
203C22⋊C4×C89—21no—
204C22⋊C4×C89—21no—
205C22⋊C4×C89—21no—
206C22⋊C4×C89—21no—
207C22⋊C4×C89—21no—
208C22⋊C4×C89—21no—
209C22⋊C4×C89—21no—
210C42×C89—21no—
211C42×C89—21no—
212C42×C89—21no—
213C42×C89C22×C411yescharacteristic
214C42×C89C2×D411yescharacteristic
215C42×C89#4111yescharacteristic
216C42×C89C2411yescharacteristic
217C23×C89—21no—
218C23×C89—21no—
219C23×C89—21no—
220C23×C89—42no—
221C23×C89—42no—
222C23×C89—81no—
223C23×C89C4×D411yescharacteristic
224Q8×C89—21no—
225Q8×C89—21no—
226D4×C89—21no—
227D4×C89—21no—
228D4×C89—21no—
229D4×C89—21no—
230D4×C89—42no—
231D4×C89—42no—
232D4×C89C2×M1612yes—
233D4×C89C2×M1612yes—
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—21no—
241C2×C4×C89—21no—
242C2×C4×C89—21no—
243C2×C4×C89—41no—
244C2×C4×C89—42no—
245C2×C4×C89—42no—
246C2×C4×C89—41no—
247C2×C4×C89—41no—
248C2×C4×C89—41no—
249C2×C4×C89—41no—
250C2×C4×C89—41no—
251C2×C4×C89—41no—
252C2×C4×C89—81no—
253C2×C4×C89—81no—
254C2×C4×C89—81no—
255C2×C4×C89C4×D411yescharacteristic
256C2×C4×C89C4×D411yescharacteristic
257C2×C4×C89C4×D411yescharacteristic
258C2×C4×C89C23×C411yescharacteristic
259C2×C4×C89C22×D411yescharacteristic
260C2×C4×C89C2×#4111yescharacteristic
261C8×C89—21no—
262C8×C89—21no—
263C8×C89—21no—
264C8×C89C2×M1612yes—
265#10623494—21no—
266#10623494—21no—
267#10623494—21no—
268#10623494—21no—
269#10623494—41no—
270#10623494—41no—
271#10623494—41no—
272#10623494—41no—
273#10623494—41no—
274#10623494—41no—
275C22×C89—21no—
276C22×C89—22no—
277C22×C89—22no—
278C22×C89—21no—
279C22×C89—21no—
280C22×C89—41no—
281C22×C89—41no—
282C22×C89—41no—
283C22×C89—41no—
284C22×C89—41no—
285C22×C89—41no—
286C22×C89—41no—
287C22×C89—84no—
288C22×C89C2×C4×D411yescharacteristic
289C4×C89—21no—
290C4×C89—21no—
291C4×C89—21no—
292C4×C89—21no—
293C4×C89—41no—
294C4×C89—41no—
295C4×C89—41no—
296C4×C89—41no—
297C4×C89—41no—
298C4×C89C22×M1611yescharacteristic
299C4×C89#56811yescharacteristic
300C4×C89#57611yescharacteristic
301C4×C89#57711yescharacteristic
302#18050—891nomaximal, Sylow
303C2×C89—41no—
304C2×C89—42no—
305C2×C89—42no—
306C2×C89—81no—
307C2×C89D4×M1611yescharacteristic
308C2×C89#294611yescharacteristic
309C2×C89#295511yescharacteristic
310#2990—891no—
311#2974—891no—
312C4×AGL(1,ℤ/8)D8911yescharacteristic
313#1592—891no—
314#1590—891no—
315#1586—891no—
316#1585—891no—
317#1582—891no—
318#1581—891no—
319#1580—891no—
320C8×D8—891no—
321#1569—891no—
322#1488—891no—
323#1486—891no—
324#1453—891no—
325C89#1805011yescharacteristic, complemented, Sylow
326C2×AGL(1,ℤ/8)Dic8911yescharacteristic
327C4×#41C2×D8911yescharacteristic
328C2×C4×D4C2×D8911yescharacteristic
329#446—891no—
330#446—891no—
331#442Dic8911yescharacteristic
332#442Dic8911yescharacteristic
333#440Dic8911yescharacteristic
334#440Dic8911yescharacteristic
335C4×SD16C2×D8911yescharacteristic
336C4×SD16C2×D8911yescharacteristic
337C4×D8C2×D8911yescharacteristic
338C4×D8C2×D8911yescharacteristic
339#436—891no—
340#436—891no—
341#435—1781no—
342#435—1781no—
343#435—1781no—
344#435—1781no—
345C8×D4—891no—
346C8×D4—891no—
347C8×D4—891no—
348C8×D4—891no—
349#432—891no—
350#432—1781no—
351#431—891no—
352#431—891no—
353#428Dic8911yescharacteristic
354#423—891no—
355C2×C4⋊C8—1781no—
356#418Dic8911yescharacteristic
357#417Dic8911yescharacteristic
358#407—1781no—
359C2×C22⋊C8—1781no—
360C4×M16C2×D8911yescharacteristic
361#335—891no—
362#335—891no—
363#334—891no—
364#334—891no—
365Q8⋊C8—891no—
366Q8⋊C8—891no—
367D4⋊C8—891no—
368D4⋊C8—891no—
369D4⋊C8—891no—
370D4⋊C8—891no—
371D4⋊C8—891no—
372D4⋊C8—891no—
373#322—891no—
374#322—891no—
375#322—891no—
376C82—891no—
377C2×#41C2×Dic8911yescharacteristic
378C22×D4C2×Dic8911yescharacteristic
379C23×C4—21no—
380AGL(1,ℤ/8)—24no—
381C2×SD16C2×Dic8911yescharacteristic
382C2×SD16C2×Dic8911yescharacteristic
383C2×D8C2×Dic8911yescharacteristic
384C2×D8C2×Dic8911yescharacteristic
385C2×M16—1781no—
386C2×M16—1781no—
387C2×M16—1781no—
388C2×M16C2×Dic8911yescharacteristic
389C2×M16C2×Dic8911yescharacteristic
390C22×C8—1781no—
391C22×C8—1781no—
392C22×C8—1781no—
393C22×C8—1781no—
394C4×Q8C2×Dic8911yescharacteristic
395C4×Q8C22×D8911yescharacteristic
396C4×D4—21no—
397C4×D4—21no—
398C4×D4—21no—
399C4×D4—21no—
400C4×D4C2×Dic8911yescharacteristic
401C4×D4C2×Dic8911yescharacteristic
402C4×D4C2×Dic8911yescharacteristic
403C4×D4C22×D8911yescharacteristic
404C4×D4C22×D8911yescharacteristic
405C4×D4C22×D8911yescharacteristic
406#117—21no—
407#117C2×Dic8911yescharacteristic
408C2×C4⋊C4C2×Dic8911yescharacteristic
409C2×C22⋊C4—21no—
410C2×C42—21no—
411C2×C42C22×D8911yescharacteristic
412#107C2×Dic8911yescharacteristic
413#107C2×Dic8911yescharacteristic
414#106C2×Dic8911yescharacteristic
415#106C2×Dic8911yescharacteristic
416C4⋊C8—891no—
417C4⋊C8—891no—
418C4⋊C8—891no—
419C4⋊C8—891no—
420C4⋊C8—891no—
421C4⋊C8—891no—
422C4⋊C8—1781no—
423C4⋊C8—1781no—
424C4⋊C8—1781no—
425C4⋊C8—1781no—
426#103C2×Dic8911yescharacteristic
427#103C2×Dic8911yescharacteristic
428#102C2×Dic8911yescharacteristic
429#102C2×Dic8911yescharacteristic
430#102C2×Dic8911yescharacteristic
431#102C2×Dic8911yescharacteristic
432#102C2×Dic8911yescharacteristic
433#102C2×Dic8911yescharacteristic
434C22⋊C8—891no—
435C22⋊C8—891no—
436C22⋊C8—1781no—
437C22⋊C8—1781no—
438C22⋊C8—1781no—
439C22⋊C8—1781no—
440C22⋊C8—1781no—
441C22⋊C8—1781no—
442C22⋊C8—1781no—
443C22⋊C8—1781no—
444#97—1781no—
445#97—1781no—
446#97—1781no—
447#97C2×Dic8911yescharacteristic
448#97C2×Dic8911yescharacteristic
449C4×C8—891no—
450C4×C8—891no—
451C4×C8—891no—
452C4×C8—891no—
453C4×C8C22×D8911yescharacteristic
454C4×C8C22×D8911yescharacteristic
455C24—21no—
456#41—22no—
457#41—41no—
458C2×Q8C22×Dic8911yescharacteristic
459C2×D4—22no—
460C2×D4—21no—
461C2×D4—21no—
462C2×D4—41no—
463C2×D4C22×Dic8911yescharacteristic
464C2×D4C22×Dic8911yescharacteristic
465C2×D4C22×Dic8911yescharacteristic
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—81no—
475C22×C4—81no—
476C22×C4C22×Dic8911yescharacteristic
477C22×C4D4×D8911yescharacteristic
478C22×C4#1308357311yescharacteristic
479SD16—22no—
480SD16—22no—
481D8—22no—
482D8#1308355814yescomplemented
483M16—22no—
484M16—41no—
485M16—7121no—
486M16—7121no—
487M16—7121no—
488C2×C8—891no—
489C2×C8—891no—
490C2×C8—891no—
491C2×C8—891no—
492C2×C8—1781no—
493C2×C8—1781no—
494C2×C8—1781no—
495C2×C8—1781no—
496C2×C8—1781no—
497C2×C8—1781no—
498C2×C8—7121no—
499C2×C8—7121no—
500C2×C8—7121no—
501C2×C8—7121no—
502C2×C8C22×Dic8911yescharacteristic
503C2×C8C22×Dic8911yescharacteristic
504C2×C8C22×Dic8911yescharacteristic
505C2×C8C22×Dic8911yescharacteristic
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4C22×Dic8911yescharacteristic
510C4⋊C4C22×Dic8911yescharacteristic
511C4⋊C4C22×Dic8911yescharacteristic
512C4⋊C4C22×Dic8911yescharacteristic
513C22⋊C4—21no—
514C22⋊C4—21no—
515C22⋊C4—21no—
516C22⋊C4—21no—
517C22⋊C4—21no—
518C22⋊C4—21no—
519C22⋊C4—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42C22×Dic8911yescharacteristic
524C42D4×D8911yescharacteristic
525C42#1308357311yescharacteristic
526C42C23×D8911yescharacteristic
527C23—21no—
528C23—21no—
529C23—21no—
530C23—42no—
531C23—42no—
532C23—81no—
533C23D4×Dic8911yescharacteristic
534Q8—21no—
535Q8—21no—
536D4—21no—
537D4—21no—
538D4—21no—
539D4—21no—
540D4—42no—
541D4—42no—
542D4C2×#1308355812yes—
543D4C2×#1308355812yes—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—81no—
563C2×C4—81no—
564C2×C4—81no—
565C2×C4D4×Dic8911yescharacteristic
566C2×C4D4×Dic8911yescharacteristic
567C2×C4D4×Dic8911yescharacteristic
568C2×C4C23×Dic8911yescharacteristic
569C2×C4C2×D4×D8911yescharacteristic
570C2×C4C2×#1308357311yescharacteristic
571C8—21no—
572C8—21no—
573C8—21no—
574C8—3561no—
575C8—3561no—
576C8—3561no—
577C8—3561no—
578C8—3561no—
579C8—3561no—
580C8—1781no—
581C8—1781no—
582C8—1781no—
583C8—1781no—
584C8C2×#1308355812yes—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—22no—
589C22—22no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—84no—
598C22C2×D4×Dic8911yescharacteristic
599C4—21no—
600C4—21no—
601C4—21no—
602C4—21no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4C22×#1308355811yescharacteristic
609C4#1915236511yescharacteristic
610C4#1915240211yescharacteristic
611C4#1915240811yescharacteristic
612C2—41no—
613C2—42no—
614C2—42no—
615C2—81no—
616C2D4×#1308355811yescharacteristic
617C2#2602908511yescharacteristic
618C2#2602915511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC4×AGL(1,ℤ/8)×C89C2
socleC22×C89C2×C4×D4
prime radicalC2×AGL(1,ℤ/8)×C89C4
cosocleC42×C89C24
hypercenterC4×AGL(1,ℤ/8)D89
hypocenterC89#18050
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. #26029377C2semidirect
2. #26029287C2semidirect
3. #26019155C2semidirect
4. #26019143C2semidirect
5. #26019117C2semidirect
6. #26019110C2semidirect
7. #26019089C2semidirect
8. #26019082C2semidirect
9. #26019076C2semidirect
10. D8×#10623494C2semidirect
11. #26019019C2semidirect
12. #26018686C2semidirect
13. #26018677C2semidirect
14. #26018549C2semidirect
16. #19151729C22semidirect
17. #19151729C22semidirect
26. D4×#10623494C22semidirect
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. C8×#10623494C22semidirect
189. D8×C89M16semidirect
325. C89#18050semidirect
482. D8#13083558semidirect

Maximal subgroups

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

subgroupindex
#260293772
#260292872
#260191552
#260191432
#260191172
#260191102
#260190892
#260190822
#260190762
D8×#106234942
#260190192
#260186862
#260186772
#260185492
C4×AGL(1,ℤ/8)×C892
#1805089