Finite Group Databasebeta

← #36526024

Subgroups — #36526024

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#26029373C211yesmaximal, characteristic, complemented
2#26029286C211yesmaximal, characteristic, complemented
3C4×#15668040C211yesmaximal, characteristic
4#26019156C211yesmaximal, characteristic, complemented
5#26019144C211yesmaximal, characteristic, complemented
6#26019118C211yesmaximal, characteristic, complemented
7#26019111C211yesmaximal, characteristic, complemented
8#26019090C211yesmaximal, characteristic, complemented
9#26019083C211yesmaximal, characteristic, complemented
10#26019077C211yesmaximal, characteristic, complemented
11C8×D712C211yesmaximal, characteristic, complemented
12#26019019C211yesmaximal, characteristic, complemented
13#26018687C211yesmaximal, characteristic, complemented
14#26018678C211yesmaximal, characteristic, complemented
15C89×#1453C211yesmaximal, characteristic, complemented
16C2×#15668040C411yescharacteristic
17C4×#13083574C2211yescharacteristic
18C2×C4×D356C2211yescharacteristic
19#19151728C2211yescharacteristic, complemented
20#19151728C2211yescharacteristic, complemented
21#19151711C411yescharacteristic
22#19151711C411yescharacteristic
23#19151700C411yescharacteristic
24#19151700C411yescharacteristic
25C4×#13083563C2211yescharacteristic
26C4×#13083563C2211yescharacteristic
27C4×D712C2211yescharacteristic
28C4×D712C2211yescharacteristic
29#19151677C2211yescharacteristic
30#19151677C2211yescharacteristic
31#19151672—21no—
32#19151672—21no—
33#19151668—21no—
34#19151668—21no—
35C8×D356C2211yescharacteristic
36C8×D356C2211yescharacteristic
37C8×D356C2211yescharacteristic, complemented
38C8×D356C2211yescharacteristic, complemented
39#19151655C2211yescharacteristic
40#19151654—21no—
41#19151640C411yescharacteristic
42#19151622C2211yescharacteristic
43C4⋊C8×D89—21no—
44#19151604C411yescharacteristic
45#19151600C411yescharacteristic
46#19151560—21no—
47C22⋊C8×D89—21no—
48#19151351C2211yescharacteristic, complemented
49#19151351C2211yescharacteristic, complemented
50#19151349C2211yescharacteristic, complemented
51#19151349C2211yescharacteristic, complemented
52#19151330C2211yescharacteristic, complemented
53#19151330C2211yescharacteristic, complemented
54#19151327C2211yescharacteristic
55#19151327C2211yescharacteristic
56#19151327C2211yescharacteristic
57#19151327C2211yescharacteristic
58#19151327C2211yescharacteristic, complemented
59#19151327C2211yescharacteristic, complemented
60#431×C89C2211yescharacteristic
61#431×C89C2211yescharacteristic
62C4×M16×C89C2211yescharacteristic
63#322×C89C2211yescharacteristic, complemented
64#322×C89C2211yescharacteristic, complemented
65#322×C89C2211yescharacteristic, complemented
66C82×C89C2211yescharacteristic, complemented
67C2×#13083574C2×C411yescharacteristic
68C22×D356C2×C411yescharacteristic
69C22×C4×D89—21no—
70#15668040—24no—
71C2×#13083563C2×C411yescharacteristic
72C2×#13083563C2×C411yescharacteristic
73C2×D712C2×C411yescharacteristic
74C2×D712C2×C411yescharacteristic
75C2×#13083558—21no—
76C2×#13083557—21no—
77C2×#13083557—21no—
78C2×C8×D89—21no—
79C2×C8×D89—21no—
80C2×C8×D89—21no—
81C2×C8×D89—21no—
82C4×Dic178C2×C411yescharacteristic
83C4×Dic178C2311yescharacteristic
84C4×D356—21no—
85C4×D356—21no—
86C4×D356C2×C411yescharacteristic
87C4×D356C2×C411yescharacteristic
88C4×D356C2×C411yescharacteristic
89C4×D356C2311yescharacteristic
90C4×D356C2311yescharacteristic
91C4×D356C2311yescharacteristic
92C4×C22⋊D89—21no—
93C4×C22⋊D89—21no—
94#15667971C2×C411yescharacteristic
95#15667969—21no—
96C2×C4⋊Dic89C2×C411yescharacteristic
97C2×#13083551—21no—
98C42×D89—21no—
99#15667946C2×C411yescharacteristic
100#15667946C2×C411yescharacteristic
101#15667944C2×C411yescharacteristic
102#15667944C2×C411yescharacteristic
103C4⋊#10623494—21no—
104C4⋊#10623494C2311yescharacteristic
105C4⋊#10623494C2311yescharacteristic
106C4⋊#10623494C2311yescharacteristic
107C4⋊#10623494C2311yescharacteristic
108#15667941—21no—
109#15667941—21no—
110#15667937C2×C411yescharacteristic
111#15667937C2×C411yescharacteristic
112#15667934C2×C411yescharacteristic
113#15667934C2×C411yescharacteristic
114#15667934C2×C411yescharacteristic
115#15667934C2×C411yescharacteristic
116#15667934C2×C411yescharacteristic
117#15667934C2×C411yescharacteristic
118C22⋊#10623494—21no—
119#15667924—21no—
120#15667924—21no—
121#15667924—21no—
122#15667924—21no—
123#15667924—21no—
124#15667924—21no—
125#15667923—21no—
126#15667922—21no—
127#15667922—21no—
128C2×M16×C89C2×C411yescharacteristic
129C2×M16×C89C2×C411yescharacteristic
130C2×C42×C89C2311yescharacteristic
131C4⋊C8×C89—21no—
132C4⋊C8×C89D411yescharacteristic
133C4⋊C8×C89D411yescharacteristic
134C22⋊C8×C89—21no—
135C22⋊C8×C89D411yescharacteristic
136C22⋊C8×C89D411yescharacteristic
137#97×C89C2×C411yescharacteristic
138#97×C89C2×C411yescharacteristic
139C4×C8×C89C2311yescharacteristic
140C4×C8×C89C2311yescharacteristic
141C4×C8×C89C2311yescharacteristic
142C4×C8×C89C2311yescharacteristic
143C4×C8×C89C2311yescharacteristic
144C4×C8×C89C2311yescharacteristic
145C23×D89—21no—
146#13083574—22no—
147#13083574—41no—
148C2×Dic178C22×C411yescharacteristic
149C2×D356—22no—
150C2×D356—21no—
151C2×D356—41no—
152C2×D356C22×C411yescharacteristic
153C2×D356C22×C411yescharacteristic
154C2×D356C22×C411yescharacteristic
155C2×C22⋊D89—21no—
156C22×Dic89—21no—
157C2×C4×D89—21no—
158C2×C4×D89—21no—
159C2×C4×D89—21no—
160C2×C4×D89—21no—
161C2×C4×D89—21no—
162C2×C4×D89—21no—
163C2×C4×D89—21no—
164C2×C4×D89—81no—
165C2×C4×D89—81no—
166#13083563—22no—
167#13083563—22no—
168D712—22no—
169D712M1614yescomplemented
170#13083558—81no—
171#13083557—81no—
172#13083557—81no—
173C2×#10623494—21no—
174C2×#10623494—21no—
175C2×#10623494—21no—
176C2×#10623494—21no—
177C8×D89—81no—
178C8×D89—81no—
179C8×D89—81no—
180C8×D89—81no—
181C4⋊Dic89—21no—
182C4⋊Dic89C22×C411yescharacteristic
183C4⋊Dic89C22×C411yescharacteristic
184C4⋊Dic89C22×C411yescharacteristic
185C4⋊Dic89C22×C411yescharacteristic
186#13083553—21no—
187#13083553—21no—
188C22⋊Dic89—21no—
189#13083551—21no—
190#13083551—21no—
191#13083551—21no—
192#13083551—21no—
193#13083551—21no—
194#13083551—21no—
195C4×Dic89—21no—
196C4×Dic89—21no—
197C4×Dic89—21no—
198C22×C4×C89C22×C411yescharacteristic
199C22×C4×C89C2×D411yescharacteristic
200C22×C4×C89#4111yescharacteristic
201M16×C89—22no—
202M16×C89—41no—
203C2×C8×C89—21no—
204C2×C8×C89—21no—
205C2×C8×C89C22×C411yescharacteristic
206C2×C8×C89C22×C411yescharacteristic
207C2×C8×C89C22×C411yescharacteristic
208C2×C8×C89C22×C411yescharacteristic
209C2×C8×C89C2×D411yescharacteristic
210C2×C8×C89C2×D411yescharacteristic
211C2×C8×C89C2×D411yescharacteristic
212C2×C8×C89C2×D411yescharacteristic
213C42×C89C22×C411yescharacteristic
214C42×C89C2×D411yescharacteristic
215C42×C89#4111yescharacteristic
216C42×C89C2411yescharacteristic
217C23×C89C4×D411yescharacteristic
218C22×D89—21no—
219C22×D89—21no—
220C22×D89—21no—
221C22×D89—42no—
222C22×D89—42no—
223C22×D89—81no—
224C2×C4×C89—21no—
225C2×C4×C89—21no—
226C2×C4×C89—21no—
227C2×C4×C89—21no—
228C2×C4×C89—41no—
229C2×C4×C89C4×D411yescharacteristic
230C2×C4×C89C4×D411yescharacteristic
231C2×C4×C89C4×D411yescharacteristic
232C2×C4×C89C23×C411yescharacteristic
233C2×C4×C89C22×D411yescharacteristic
234C2×C4×C89C2×#4111yescharacteristic
235C22⋊D89—42no—
236C2×Dic89—21no—
237C2×Dic89—21no—
238C2×Dic89—21no—
239C2×Dic89—21no—
240C2×Dic89—21no—
241C2×Dic89—81no—
242D356—21no—
243D356—21no—
244D356—21no—
245D356—21no—
246D356—42no—
247D356C2×M1612yes—
248D356C2×M1612yes—
249C4×D89—42no—
250C4×D89—42no—
251C4×D89—41no—
252C4×D89—41no—
253C4×D89—41no—
254C4×D89—41no—
255C4×D89—41no—
256C4×D89—41no—
257C4×D89—81no—
258C4×D89—81no—
259Dic178—21no—
260Dic178—21no—
261C8×C89—21no—
262C8×C89—21no—
263C8×C89—21no—
264C8×C89—21no—
265C8×C89—21no—
266C8×C89—21no—
267C8×C89—21no—
268C8×C89—41no—
269C8×C89—41no—
270C8×C89C2×M1612yes—
271#10623494—41no—
272#10623494—41no—
273#10623494—41no—
274#10623494—41no—
275C22×C89—21no—
276C22×C89—21no—
277C22×C89—21no—
278C22×C89C2×C4×D411yescharacteristic
279C2×D89—22no—
280C2×D89—22no—
281C2×D89—41no—
282C2×D89—41no—
283C2×D89—41no—
284C2×D89—41no—
285C2×D89—41no—
286C2×D89—41no—
287C2×D89—41no—
288C2×D89—84no—
289C4×C89—21no—
290C4×C89—21no—
291C4×C89—21no—
292C4×C89—21no—
293C4×C89C22×M1611yescharacteristic
294C4×C89#56811yescharacteristic
295C4×C89#57611yescharacteristic
296C4×C89#57711yescharacteristic
297Dic89—41no—
298Dic89—41no—
299Dic89—41no—
300Dic89—41no—
301Dic89—41no—
302#18050—891nomaximal, Sylow
303C2×C89—41no—
304C2×C89D4×M1611yescharacteristic
305C2×C89#294611yescharacteristic
306C2×C89#295511yescharacteristic
307D89—42no—
308D89—42no—
309D89—81no—
310#2990—891no—
311#2974—891no—
312C4×AGL(1,ℤ/8)—891no—
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#1453D8911yescharacteristic, complemented
325C89#1805011yescharacteristic, complemented, Sylow
326C2×AGL(1,ℤ/8)—891no—
327C4×#41—891no—
328C2×C4×D4—891no—
329#446—891no—
330#446—891no—
331#442—891no—
332#442—891no—
333#440—891no—
334#440—891no—
335C4×SD16—891no—
336C4×SD16—891no—
337C4×D8—891no—
338C4×D8—891no—
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#431C2×D8911yescharacteristic
352#431C2×D8911yescharacteristic
353#428—891no—
354#423—891no—
355C2×C4⋊C8—1781no—
356#418—891no—
357#417—891no—
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#322C2×D8911yescharacteristic, complemented
374#322C2×D8911yescharacteristic, complemented
375#322C2×D8911yescharacteristic, complemented
376C82C2×D8911yescharacteristic, complemented
377C2×#41—891no—
378C22×D4—891no—
379C23×C4—1781no—
380AGL(1,ℤ/8)—1784no—
381C2×SD16—891no—
382C2×SD16—891no—
383C2×D8—891no—
384C2×D8—891no—
385C2×M16—1781no—
386C2×M16—1781no—
387C2×M16—1781no—
388C2×M16C4×D8911yescharacteristic
389C2×M16C4×D8911yescharacteristic
390C22×C8—1781no—
391C22×C8—1781no—
392C22×C8—1781no—
393C22×C8—1781no—
394C4×Q8—891no—
395C4×Q8—891no—
396C4×D4—891no—
397C4×D4—891no—
398C4×D4—891no—
399C4×D4—891no—
400C4×D4—891no—
401C4×D4—891no—
402C4×D4—1781no—
403C4×D4—1781no—
404C4×D4—1781no—
405C4×D4—1781no—
406#117—891no—
407#117—1781no—
408C2×C4⋊C4—891no—
409C2×C22⋊C4—1781no—
410C2×C42—1781no—
411C2×C42C22×D8911yescharacteristic
412#107—891no—
413#107—891no—
414#106—891no—
415#106—891no—
416C4⋊C8—21no—
417C4⋊C8—891no—
418C4⋊C8—891no—
419C4⋊C8—891no—
420C4⋊C8—891no—
421C4⋊C8—1781no—
422C4⋊C8—1781no—
423C4⋊C8—1781no—
424C4⋊C8D35611yescharacteristic
425C4⋊C8D35611yescharacteristic
426#103—891no—
427#103—891no—
428#102—891no—
429#102—891no—
430#102—891no—
431#102—891no—
432#102—891no—
433#102—891no—
434C22⋊C8—21no—
435C22⋊C8—1781no—
436C22⋊C8—1781no—
437C22⋊C8—1781no—
438C22⋊C8—1781no—
439C22⋊C8—1781no—
440C22⋊C8—1781no—
441C22⋊C8—1781no—
442C22⋊C8D35611yescharacteristic
443C22⋊C8D35611yescharacteristic
444#97—1781no—
445#97—1781no—
446#97—1781no—
447#97C4×D8911yescharacteristic
448#97C4×D8911yescharacteristic
449C4×C8C22×D8911yescharacteristic
450C4×C8C22×D8911yescharacteristic
451C4×C8C22×D8911yescharacteristic
452C4×C8C22×D8911yescharacteristic
453C4×C8C22×D8911yescharacteristic
454C4×C8C22×D8911yescharacteristic
455C24—1781no—
456#41—3561no—
457#41—1782no—
458C2×Q8—891no—
459C2×D4—891no—
460C2×D4—891no—
461C2×D4—891no—
462C2×D4—3561no—
463C2×D4—1782no—
464C2×D4—1781no—
465C2×D4—1781no—
466C22×C4—1781no—
467C22×C4—1781no—
468C22×C4—1781no—
469C22×C4—1781no—
470C22×C4—1781no—
471C22×C4—1781no—
472C22×C4—1781no—
473C22×C4—1781no—
474C22×C4—7121no—
475C22×C4—7121no—
476C22×C4C2×C4×D8911yescharacteristic
477C22×C4C2×D35611yescharacteristic
478C22×C4#1308357411yescharacteristic
479SD16—1782no—
480SD16—1782no—
481D8—894no—
482D8—1782no—
483M16—22no—
484M16—41no—
485M16—7121no—
486M16—7121no—
487M16—7121no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—1781no—
491C2×C8—1781no—
492C2×C8—1781no—
493C2×C8—1781no—
494C2×C8—7121no—
495C2×C8—7121no—
496C2×C8—7121no—
497C2×C8—7121no—
498C2×C8C2×C4×D8911yescharacteristic
499C2×C8C2×C4×D8911yescharacteristic
500C2×C8C2×C4×D8911yescharacteristic
501C2×C8C2×C4×D8911yescharacteristic
502C2×C8C2×D35611yescharacteristic
503C2×C8C2×D35611yescharacteristic
504C2×C8C2×D35611yescharacteristic
505C2×C8C2×D35611yescharacteristic
506C4⋊C4—891no—
507C4⋊C4—891no—
508C4⋊C4—891no—
509C4⋊C4—891no—
510C4⋊C4—1781no—
511C4⋊C4—1781no—
512C4⋊C4—1781no—
513C22⋊C4—1781no—
514C22⋊C4—1781no—
515C22⋊C4—1781no—
516C22⋊C4—1781no—
517C22⋊C4—1781no—
518C22⋊C4—1781no—
519C22⋊C4—1781no—
520C42—1781no—
521C42—1781no—
522C42—1781no—
523C42C2×C4×D8911yescharacteristic
524C42C2×D35611yescharacteristic
525C42#1308357411yescharacteristic
526C42C23×D8911yescharacteristic
527C23—3562no—
528C23—3562no—
529C23—1781no—
530C23—1781no—
531C23—1781no—
532C23—7121no—
533C23C4×D35611yescharacteristic
534Q8—1781no—
535Q8—1781no—
536D4—892no—
537D4—892no—
538D4—3562no—
539D4—3562no—
540D4—1781no—
541D4—1781no—
542D4—1781no—
543D4—1781no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—41no—
549C2×C4—3562no—
550C2×C4—3562no—
551C2×C4—3561no—
552C2×C4—3561no—
553C2×C4—3561no—
554C2×C4—3561no—
555C2×C4—3561no—
556C2×C4—3561no—
557C2×C4—1781no—
558C2×C4—1781no—
559C2×C4—1781no—
560C2×C4—1781no—
561C2×C4—1781no—
562C2×C4—7121no—
563C2×C4—7121no—
564C2×C4—7121no—
565C2×C4C4×D35611yescharacteristic
566C2×C4C4×D35611yescharacteristic
567C2×C4C4×D35611yescharacteristic
568C2×C4C22×C4×D8911yescharacteristic
569C2×C4C22×D35611yescharacteristic
570C2×C4C2×#1308357411yescharacteristic
571C8—21no—
572C8—21no—
573C8—21no—
574C8—21no—
575C8—21no—
576C8—21no—
577C8—21no—
578C8—41no—
579C8—41no—
580C8—3561no—
581C8—3561no—
582C8—3561no—
583C8—3561no—
584C8M16×D8912yes—
585C22—21no—
586C22—21no—
587C22—21no—
588C22—3561no—
589C22—3561no—
590C22—3561no—
591C22—3561no—
592C22—3561no—
593C22—3561no—
594C22—3561no—
595C22—1782no—
596C22—1782no—
597C22—7124no—
598C22C2×C4×D35611yescharacteristic
599C4—21no—
600C4—21no—
601C4—21no—
602C4—21no—
603C4—3561no—
604C4—3561no—
605C4—3561no—
606C4—3561no—
607C4—3561no—
608C4C2×M16×D8911yescharacteristic
609C4#1915236411yescharacteristic
610C4#1915240311yescharacteristic
611C4#1915241011yescharacteristic
612C2—41no—
613C2—3562no—
614C2—3562no—
615C2—7121no—
616C2M16×D35611yescharacteristic
617C2#2602908611yescharacteristic
618C2#2602915611yescharacteristic
subgroupG/H
centerC2×C4C22×D356
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#1453C2
socleC22×C89C2×C4×D4
prime radicalC2×#15668040C4
cosocleC42×C89C24
hypercenter#1453D89
hypocenterC89#18050
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ #431 ◁ #1453

Complements

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

NH ≅ G/Ntype
1. #26029373C2semidirect
2. #26029286C2semidirect
4. #26019156C2semidirect
5. #26019144C2semidirect
6. #26019118C2semidirect
7. #26019111C2semidirect
8. #26019090C2semidirect
9. #26019083C2semidirect
10. #26019077C2semidirect
11. C8×D712C2semidirect
12. #26019019C2semidirect
13. #26018687C2semidirect
14. #26018678C2semidirect
15. C89×#1453C2semidirect
19. #19151728C22semidirect
20. #19151728C22semidirect
37. C8×D356C22semidirect
38. C8×D356C22semidirect
48. #19151351C22semidirect
49. #19151351C22semidirect
50. #19151349C22semidirect
51. #19151349C22semidirect
52. #19151330C22semidirect
53. #19151330C22semidirect
58. #19151327C22semidirect
59. #19151327C22semidirect
63. #322×C89C22semidirect
64. #322×C89C22semidirect
65. #322×C89C22semidirect
66. C82×C89C22semidirect
169. D712M16semidirect
324. #1453D89semidirect
325. C89#18050semidirect
373. #322C2×D89semidirect
374. #322C2×D89semidirect
375. #322C2×D89semidirect
376. C82C2×D89semidirect

Maximal subgroups

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

subgroupindex
#260293732
#260292862
C4×#156680402
#260191562
#260191442
#260191182
#260191112
#260190902
#260190832
#260190772
C8×D7122
#260190192
#260186872
#260186782
C89×#14532
#1805089