Finite Group Databasebeta

← #36525479

Subgroups — #36525479

Order 22784.

662 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26030312C211yesmaximal, characteristic
2#26028998C211yesmaximal, characteristic, complemented
3#26019152C211yesmaximal, characteristic, complemented
4#26019142C211yesmaximal, characteristic, complemented
5#26019116C211yesmaximal, characteristic, complemented
6#26019107C211yesmaximal, characteristic, complemented
7#26019086C211yesmaximal, characteristic, complemented
8#26019081C211yesmaximal, characteristic, complemented
9#26019074C211yesmaximal, characteristic, complemented
10C8×#13083559C211yesmaximal, characteristic, complemented
11#26019038C211yesmaximal, characteristic, complemented
12#26018720C211yesmaximal, characteristic, complemented
13#26018639C211yesmaximal, characteristic, complemented
14#26018549C211yesmaximal, characteristic, complemented
15C89×#2970C211yesmaximal, characteristic, complemented
16C2×#15668020—21no—
17#19152148C2211yescharacteristic
18#19151922C411yescharacteristic
19#19151922C411yescharacteristic
20#19151893C411yescharacteristic
21#19151888C411yescharacteristic
22#19151798C411yescharacteristic
23#19151793C411yescharacteristic
24#19151743C411yescharacteristic
25#19151743C411yescharacteristic
26C4×#13083561C2211yescharacteristic
27C4×#13083561C2211yescharacteristic
28C4×#13083559C2211yescharacteristic
29C4×#13083559C2211yescharacteristic
30#19151677C2211yescharacteristic
31#19151677C2211yescharacteristic
32#19151671—21no—
33#19151671—21no—
34C8×D356C2211yescharacteristic
35C8×D356C2211yescharacteristic
36#19151651C2211yescharacteristic
37#19151651C2211yescharacteristic
38C2×C4⋊#10623494C2211yescharacteristic
39#19151566—21no—
40#19151350C2211yescharacteristic
41#19151350C2211yescharacteristic
42#19151348C2211yescharacteristic
43#19151348C2211yescharacteristic
44Q8⋊#10623494C2211yescharacteristic
45Q8⋊#10623494C2211yescharacteristic
46D4⋊#10623494C2211yescharacteristic
47D4⋊#10623494C2211yescharacteristic
48#19151325C2211yescharacteristic
49#19151325C2211yescharacteristic
50#19151325C2211yescharacteristic
51#19151325C2211yescharacteristic
52#19151320C2211yescharacteristic
53#19151320C2211yescharacteristic
54#19151319C2211yescharacteristic
55C8×#10623494C2211yescharacteristic
56C4×#41×C89C2211yescharacteristic
57#446×C89C2211yescharacteristic
58#446×C89C2211yescharacteristic
59#435×C89—21no—
60#435×C89—21no—
61C8×D4×C89C2211yescharacteristic
62C8×D4×C89C2211yescharacteristic
63#432×C89—21no—
64#432×C89C2211yescharacteristic
65C4×M16×C89C2211yescharacteristic
66C2×#97×C89—21no—
67C2×#13083574—21no—
68C2×#13083561C2×C411yescharacteristic
69C2×#13083561C2×C411yescharacteristic
70C2×#13083559C2×C411yescharacteristic
71C2×#13083559C2×C411yescharacteristic
72#15668020—44no—
73C2×#13083557—21no—
74C2×#13083557—21no—
75C22×#10623494—21no—
76C2×C8×D89—21no—
77C2×C8×D89—21no—
78#15668012C2×C411yescharacteristic
79#15668009C2×C411yescharacteristic
80#15668000—21no—
81#15667986—21no—
82#15667986C2×C411yescharacteristic
83#15667986C2×C411yescharacteristic
84C4×D356—21no—
85C4×D356C2311yescharacteristic
86C4×D356C2311yescharacteristic
87#15667945C2×C411yescharacteristic
88#15667945C2×C411yescharacteristic
89#15667943C2×C411yescharacteristic
90#15667943C2×C411yescharacteristic
91C4⋊#10623494—21no—
92C4⋊#10623494—21no—
93C4⋊#10623494C2×C411yescharacteristic
94C4⋊#10623494C2×C411yescharacteristic
95C4⋊#10623494C2311yescharacteristic
96C4⋊#10623494C2311yescharacteristic
97#15667936C2×C411yescharacteristic
98#15667936C2×C411yescharacteristic
99#15667933C2×C411yescharacteristic
100#15667933C2×C411yescharacteristic
101#15667932C2×C411yescharacteristic
102#15667932C2×C411yescharacteristic
103#15667932C2×C411yescharacteristic
104#15667932C2×C411yescharacteristic
105C22⋊#10623494—21no—
106C22⋊#10623494C2×C411yescharacteristic
107C22⋊#10623494C2×C411yescharacteristic
108#15667924—21no—
109#15667924—21no—
110#15667924—21no—
111#15667924—21no—
112C4×#10623494C2311yescharacteristic
113C4×#10623494C2311yescharacteristic
114C4×#10623494C2311yescharacteristic
115C4×#10623494C2311yescharacteristic
116C2×#41×C89C2×C411yescharacteristic
117C2×M16×C89D411yescharacteristic
118C2×M16×C89D411yescharacteristic
119C22×C8×C89—21no—
120C22×C8×C89—21no—
121C4×Q8×C89C2×C411yescharacteristic
122C4×Q8×C89C2311yescharacteristic
123C4×D4×C89—21no—
124C4×D4×C89—21no—
125C4×D4×C89C2×C411yescharacteristic
126C4×D4×C89C2311yescharacteristic
127#117×C89—21no—
128#117×C89C2×C411yescharacteristic
129C2×C42×C89—21no—
130C2×C42×C89C2311yescharacteristic
131C4⋊C8×C89—21no—
132C4⋊C8×C89—21no—
133C4⋊C8×C89C2311yescharacteristic
134C4⋊C8×C89C2311yescharacteristic
135C22⋊C8×C89—21no—
136C22⋊C8×C89—21no—
137C22⋊C8×C89—21no—
138#97×C89—21no—
139#97×C89—21no—
140#97×C89—21no—
141#97×C89D411yescharacteristic
142#97×C89D411yescharacteristic
143C4×C8×C89C2311yescharacteristic
144C4×C8×C89C2311yescharacteristic
145#13083574—42no—
146#13083574—42no—
147C2×Dic178—21no—
148C2×D356—21no—
149C2×D356—21no—
150C2×D356C22×C411yescharacteristic
151C2×D356C22×C411yescharacteristic
152C2×C22⋊D89—21no—
153C2×C22⋊D89—21no—
154C2×C4×D89—21no—
155C2×C4×D89—21no—
156#13083561—41no—
157#13083561—41no—
158#13083559—22no—
159#13083559—22no—
160#13083557—42no—
161#13083557—42no—
162C2×#10623494—21no—
163C2×#10623494—21no—
164C2×#10623494—21no—
165C2×#10623494—21no—
166C2×#10623494—42no—
167C2×#10623494C22×C411yescharacteristic
168C2×#10623494C22×C411yescharacteristic
169C2×#10623494C22×C411yescharacteristic
170C2×#10623494C22×C411yescharacteristic
171C8×D89—42no—
172C8×D89—42no—
173C4⋊Dic89—21no—
174C4⋊Dic89C22×C411yescharacteristic
175C4⋊Dic89C22×C411yescharacteristic
176#13083551—21no—
177#13083551—21no—
178#13083551—21no—
179#13083551—21no—
180#41×C89—22no—
181#41×C89—41no—
182C2×Q8×C89C22×C411yescharacteristic
183C2×D4×C89—21no—
184C2×D4×C89C22×C411yescharacteristic
185C22×C4×C89—21no—
186C22×C4×C89—21no—
187C22×C4×C89—21no—
188C22×C4×C89C22×C411yescharacteristic
189C22×C4×C89C2×D411yescharacteristic
190C22×C4×C89#4111yescharacteristic
191M16×C89—22no—
192M16×C89—41no—
193C2×C8×C89—21no—
194C2×C8×C89—21no—
195C2×C8×C89—42no—
196C2×C8×C89—42no—
197C2×C8×C89C2×D411yescharacteristic
198C2×C8×C89C2×D411yescharacteristic
199C2×C8×C89C2×D411yescharacteristic
200C2×C8×C89C2×D411yescharacteristic
201C4⋊C4×C89—21no—
202C4⋊C4×C89—21no—
203C4⋊C4×C89C22×C411yescharacteristic
204C4⋊C4×C89C22×C411yescharacteristic
205C22⋊C4×C89—21no—
206C22⋊C4×C89—21no—
207C22⋊C4×C89—21no—
208C42×C89—21no—
209C42×C89—21no—
210C42×C89—21no—
211C42×C89C22×C411yescharacteristic
212C42×C89C2×D411yescharacteristic
213C42×C89#4111yescharacteristic
214C42×C89C2411yescharacteristic
215C23×C89—21no—
216C23×C89C4×D411yescharacteristic
217C22×D89—21no—
218C22×D89—21no—
219Q8×C89—21no—
220Q8×C89—21no—
221D4×C89—21no—
222D4×C89—42no—
223D4×C89#13112yescomplemented
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—41no—
232C2×C4×C89—42no—
233C2×C4×C89—41no—
234C2×C4×C89—41no—
235C2×C4×C89—81no—
236C2×C4×C89C4×D411yescharacteristic
237C2×C4×C89C4×D411yescharacteristic
238C2×C4×C89C4×D411yescharacteristic
239C2×C4×C89C23×C411yescharacteristic
240C2×C4×C89C22×D411yescharacteristic
241C2×C4×C89C2×#4111yescharacteristic
242C22⋊D89—42no—
243C22⋊D89—42no—
244C2×Dic89—21no—
245C2×Dic89—21no—
246D356—21no—
247D356—21no—
248D356—21no—
249D356—21no—
250D356—42no—
251D356—81no—
252C4×D89—41no—
253C4×D89—41no—
254C4×D89—41no—
255C4×D89—41no—
256Dic178—42no—
257C8×C89—21no—
258C8×C89—21no—
259C8×C89—21no—
260C8×C89—41no—
261C8×C89—41no—
262C8×C89AGL(1,ℤ/8)12yes—
263#10623494—21no—
264#10623494—21no—
265#10623494—21no—
266#10623494—21no—
267#10623494—41no—
268#10623494—41no—
269#10623494—41no—
270#10623494—41no—
271C22×C89—21no—
272C22×C89—22no—
273C22×C89—21no—
274C22×C89—21no—
275C22×C89—41no—
276C22×C89—41no—
277C22×C89C2×C4×D411yescharacteristic
278C2×D89—41no—
279C2×D89—41no—
280C2×D89—41no—
281C2×D89—41no—
282C2×D89—41no—
283C2×D89—41no—
284C4×C89—21no—
285C4×C89—21no—
286C4×C89—21no—
287C4×C89—21no—
288C4×C89—41no—
289C4×C89—41no—
290C4×C89—41no—
291C4×C89C2×#13111yescharacteristic
292C4×C89#56811yescharacteristic
293C4×C89C2×AGL(1,ℤ/8)11yescharacteristic
294C4×C89#57511yescharacteristic
295Dic89—41no—
296Dic89—41no—
297#18007—891nomaximal, Sylow
298C2×C89—41no—
299C2×C89—42no—
300C2×C89#293311yescharacteristic
301C2×C89C4×AGL(1,ℤ/8)11yescharacteristic
302C2×C89#295711yescharacteristic
303D89—81no—
304D89—81no—
305#3106—891no—
306#2970D8911yescharacteristic, complemented
307#2935—891no—
308#1592—891no—
309#1590—891no—
310#1586—891no—
311#1585—891no—
312#1582—891no—
313#1581—891no—
314#1580—891no—
315C8×D8—891no—
316#1573—891no—
317#1496—891no—
318#1477—891no—
319#1453—891no—
320C89#1800711yescharacteristic, complemented, Sylow
321C2×#131—1781no—
322#532—891no—
323C4×#41C2×D8911yescharacteristic
324#482—891no—
325#482—891no—
326#477—891no—
327#476—891no—
328#460—891no—
329#459—891no—
330#449—891no—
331#449—891no—
332#446C2×D8911yescharacteristic
333#446C2×D8911yescharacteristic
334C4×SD16—891no—
335C4×SD16—891no—
336C4×D8—891no—
337C4×D8—891no—
338#436—891no—
339#436—891no—
340#435—21no—
341#435—21no—
342#435—1781no—
343#435—1781no—
344C8×D4—891no—
345C8×D4—891no—
346C8×D4C2×D8911yescharacteristic
347C8×D4C2×D8911yescharacteristic
348#432—21no—
349#432C2×D8911yescharacteristic
350#431—891no—
351#431—891no—
352C2×C4⋊C8—891no—
353#408—1781no—
354C4×M16C2×D8911yescharacteristic
355C2×#97—21no—
356#335—891no—
357#335—891no—
358#334—891no—
359#334—891no—
360Q8⋊C8—891no—
361Q8⋊C8—891no—
362D4⋊C8—891no—
363D4⋊C8—891no—
364D4⋊C8—891no—
365D4⋊C8—891no—
366D4⋊C8—891no—
367D4⋊C8—891no—
368#322—891no—
369#322—891no—
370#322—891no—
371C82—891no—
372C2×#41—1781no—
373C2×#41C4×D8911yescharacteristic
374C2×SD16—891no—
375C2×SD16—891no—
376C2×D8—891no—
377C2×D8—891no—
378#131—3564no—
379C2×M16—1781no—
380C2×M16—1781no—
381C2×M16C22⋊D8911yescharacteristic
382C2×M16C22⋊D8911yescharacteristic
383C22×C8—21no—
384C22×C8—21no—
385C22×C8—1781no—
386C22×C8—1781no—
387C22×C8—1781no—
388C4⋊Q8—891no—
389#127—891no—
390#124—1781no—
391#121—891no—
392#121—891no—
393#121—1781no—
394C4×Q8C4×D8911yescharacteristic
395C4×Q8C22×D8911yescharacteristic
396C4×D4—21no—
397C4×D4—21no—
398C4×D4—891no—
399C4×D4—891no—
400C4×D4—1781no—
401C4×D4C4×D8911yescharacteristic
402C4×D4C22×D8911yescharacteristic
403#117—21no—
404#117C4×D8911yescharacteristic
405C2×C42—21no—
406C2×C42C22×D8911yescharacteristic
407#107—891no—
408#107—891no—
409#106—891no—
410#106—891no—
411C4⋊C8—21no—
412C4⋊C8—21no—
413C4⋊C8—891no—
414C4⋊C8—891no—
415C4⋊C8—891no—
416C4⋊C8—891no—
417C4⋊C8—1781no—
418C4⋊C8—1781no—
419C4⋊C8C22×D8911yescharacteristic
420C4⋊C8C22×D8911yescharacteristic
421#103—891no—
422#103—891no—
423#102—891no—
424#102—891no—
425#102—891no—
426#102—891no—
427#102—891no—
428#102—891no—
429C22⋊C8—21no—
430C22⋊C8—21no—
431C22⋊C8—21no—
432C22⋊C8—891no—
433C22⋊C8—891no—
434C22⋊C8—1781no—
435C22⋊C8—1781no—
436C22⋊C8—1781no—
437C22⋊C8—1781no—
438C22⋊C8—1781no—
439#97—21no—
440#97—21no—
441#97—21no—
442#97C22⋊D8911yescharacteristic
443#97C22⋊D8911yescharacteristic
444C4×C8—891no—
445C4×C8—891no—
446C4×C8—891no—
447C4×C8—891no—
448C4×C8C22×D8911yescharacteristic
449C4×C8C22×D8911yescharacteristic
450#41—22no—
451#41—41no—
452#41—3562no—
453#41—3562no—
454C2×Q8—1781no—
455C2×Q8C2×C4×D8911yescharacteristic
456C2×D4—21no—
457C2×D4—891no—
458C2×D4—891no—
459C2×D4—1781no—
460C2×D4—1781no—
461C2×D4—1781no—
462C2×D4—1781no—
463C2×D4C2×C4×D8911yescharacteristic
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—1781no—
468C22×C4—1781no—
469C22×C4C2×C4×D8911yescharacteristic
470C22×C4C2×C22⋊D8911yescharacteristic
471C22×C4#1308357411yescharacteristic
472SD16—3561no—
473SD16—3561no—
474D8—1782no—
475D8—1782no—
476M16—22no—
477M16—41no—
478M16—3562no—
479M16—3562no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—42no—
483C2×C8—42no—
484C2×C8—891no—
485C2×C8—891no—
486C2×C8—891no—
487C2×C8—891no—
488C2×C8—3562no—
489C2×C8—3562no—
490C2×C8—3562no—
491C2×C8—1781no—
492C2×C8—1781no—
493C2×C8—1781no—
494C2×C8—1781no—
495C2×C8C2×C22⋊D8911yescharacteristic
496C2×C8C2×C22⋊D8911yescharacteristic
497C2×C8C2×C22⋊D8911yescharacteristic
498C2×C8C2×C22⋊D8911yescharacteristic
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—891no—
502C4⋊C4—891no—
503C4⋊C4—1781no—
504C4⋊C4C2×C4×D8911yescharacteristic
505C4⋊C4C2×C4×D8911yescharacteristic
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—1781no—
510C22⋊C4—1781no—
511C22⋊C4—1781no—
512C22⋊C4—1781no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42C2×C4×D8911yescharacteristic
517C42C2×C22⋊D8911yescharacteristic
518C42#1308357411yescharacteristic
519C42C23×D8911yescharacteristic
520C23—21no—
521C23—1781no—
522C23—1781no—
523C23C4×C22⋊D8911yescharacteristic
524Q8—21no—
525Q8—21no—
526Q8—3562no—
527D4—21no—
528D4—42no—
529D4—3562no—
530D4—3562no—
531D4—3562no—
532D4—1781no—
533D4—1781no—
534D4—1781no—
535D4—1781no—
536D4—7121no—
537D4#1566802012yescomplemented
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—81no—
550C2×C4—3561no—
551C2×C4—3561no—
552C2×C4—3561no—
553C2×C4—3561no—
554C2×C4—1781no—
555C2×C4—1781no—
556C2×C4C4×C22⋊D8911yescharacteristic
557C2×C4C4×C22⋊D8911yescharacteristic
558C2×C4C4×C22⋊D8911yescharacteristic
559C2×C4C22×C4×D8911yescharacteristic
560C2×C4C22×C22⋊D8911yescharacteristic
561C2×C4C2×#1308357411yescharacteristic
562C8—21no—
563C8—21no—
564C8—21no—
565C8—41no—
566C8—41no—
567C8—3561no—
568C8—3561no—
569C8—3561no—
570C8—3561no—
571C8—1781no—
572C8—1781no—
573C8—1781no—
574C8—1781no—
575C8#1566803712yes—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—22no—
580C22—41no—
581C22—41no—
582C22—3561no—
583C22—3561no—
584C22—3561no—
585C22—3561no—
586C22—3561no—
587C22—3561no—
588C22C2×C4×C22⋊D8911yescharacteristic
589C4—21no—
590C4—21no—
591C4—21no—
592C4—21no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4—3561no—
597C4—3561no—
598C4C2×#1566802011yescharacteristic
599C4#1915236411yescharacteristic
600C4C2×#1566803711yescharacteristic
601C4#1915239611yescharacteristic
602C2—41no—
603C2—42no—
604C2—7121no—
605C2—7121no—
606C2#2602897711yescharacteristic
607C2C4×#1566803711yescharacteristic
608C2#2602916711yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#2970C2
socleC22×C89C2×C4×D4
prime radical#26030312C2
cosocleC42×C89C24
hypercenter#2970D89
hypocenterC89#18007
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #2970

Complements

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

NH ≅ G/Ntype
2. #26028998C2semidirect
3. #26019152C2semidirect
4. #26019142C2semidirect
5. #26019116C2semidirect
6. #26019107C2semidirect
7. #26019086C2semidirect
8. #26019081C2semidirect
9. #26019074C2semidirect
10. C8×#13083559C2semidirect
11. #26019038C2semidirect
12. #26018720C2semidirect
13. #26018639C2semidirect
14. #26018549C2semidirect
15. C89×#2970C2semidirect
223. D4×C89#131semidirect
306. #2970D89semidirect
320. C89#18007semidirect
537. D4#15668020semidirect

Maximal subgroups

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

subgroupindex
#260303122
#260289982
#260191522
#260191422
#260191162
#260191072
#260190862
#260190812
#260190742
C8×#130835592
#260190382
#260187202
#260186392
#260185492
C89×#29702
#1800789