Finite Group Databasebeta

← #36525468

Subgroups — #36525468

Order 22784.

670 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26030301C211yesmaximal, characteristic
2#26029256C211yesmaximal, characteristic, complemented
3M16×D356C211yesmaximal, characteristic, complemented
4#26019167C211yesmaximal, characteristic, complemented
5#26019144C211yesmaximal, characteristic, complemented
6#26019132C211yesmaximal, characteristic, complemented
7#26019118C211yesmaximal, characteristic, complemented
8#26019090C211yesmaximal, characteristic, complemented
9#26019083C211yesmaximal, characteristic, complemented
10#26019066C211yesmaximal, characteristic, complemented
11C8×D712C211yesmaximal, characteristic, complemented
12#26019041C211yesmaximal, characteristic, complemented
13#26018718C211yesmaximal, characteristic, complemented
14#26018644C211yesmaximal, characteristic, complemented
15C89×#1453C211yesmaximal, characteristic, complemented
16C2×M16×D89—21no—
17#19152132C2211yescharacteristic
18C2×C4×D356C2211yescharacteristic
19#19151934C411yescharacteristic
20#19151917C411yescharacteristic
21#19151899C411yescharacteristic
22#19151899C411yescharacteristic
23#19151814C411yescharacteristic
24#19151814C411yescharacteristic
25#19151751C411yescharacteristic
26#19151734C411yescharacteristic
27#19151728C2211yescharacteristic, complemented
28C8×Dic178C2211yescharacteristic, complemented
29C4×#13083563C2211yescharacteristic
30C4×#13083563C2211yescharacteristic
31C4×D712C2211yescharacteristic
32C4×D712C2211yescharacteristic
33#19151677C2211yescharacteristic
34#19151677C2211yescharacteristic
35#19151677C2211yescharacteristic, complemented
36#19151674—21no—
37#19151671—21no—
38#19151671—21no—
39#19151667—21no—
40C8×D356C2211yescharacteristic
41C8×D356C2211yescharacteristic
42C8×D356C2211yescharacteristic, complemented
43#19151660—21no—
44#19151651C2211yescharacteristic
45#19151622C2211yescharacteristic
46#19151561—21no—
47#19151555—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
67C22×D356C2×C411yescharacteristic
68C22×C4×D89—21no—
69C2×#13083563C2×C411yescharacteristic
70C2×#13083563C2×C411yescharacteristic
71C2×D712C2×C411yescharacteristic
72C2×D712C2×C411yescharacteristic
73C2×#13083558—21no—
74C2×#13083557—21no—
75C2×#13083557—21no—
76C2×#13083557—21no—
77M16×D89—44no—
78C2×C8×D89—21no—
79C2×C8×D89—21no—
80C2×C8×D89—21no—
81#15668006—21no—
82#15668004C2×C411yescharacteristic
83#15668000C2×C411yescharacteristic
84#15667995—21no—
85#15667991C2×C411yescharacteristic
86#15667986C2×C411yescharacteristic
87C4×Dic178C2311yescharacteristic
88C4×D356—21no—
89C4×D356—21no—
90C4×D356C2×C411yescharacteristic
91C4×D356C2×C411yescharacteristic
92C4×D356C2311yescharacteristic
93C4×D356C2311yescharacteristic
94C4×D356C2311yescharacteristic
95C4×C22⋊D89—21no—
96#15667969—21no—
97C2×C4⋊Dic89C2×C411yescharacteristic
98C2×#13083551—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#15667922—21no—
126C4×#10623494—21no—
127C8×Dic89—21no—
128C2×M16×C89D411yescharacteristic
129C2×M16×C89D411yescharacteristic
130C2×C42×C89C2311yescharacteristic
131C4⋊C8×C89—21no—
132C4⋊C8×C89C2×C411yescharacteristic
133C4⋊C8×C89C2×C411yescharacteristic
134C22⋊C8×C89—21no—
135C22⋊C8×C89C2×C411yescharacteristic
136C22⋊C8×C89C2×C411yescharacteristic
137#97×C89D411yescharacteristic
138#97×C89D411yescharacteristic
139C4×C8×C89C2311yescharacteristic
140C4×C8×C89C2311yescharacteristic
141C4×C8×C89C2311yescharacteristic
142C4×C8×C89C2311yescharacteristic
143C4×C8×C89C2311yescharacteristic
144C4×C8×C89C2311yescharacteristic
145C23×D89—21no—
146C2×Dic178C22×C411yescharacteristic
147C2×D356—22no—
148C2×D356—21no—
149C2×D356—41no—
150C2×D356C22×C411yescharacteristic
151C2×D356C22×C411yescharacteristic
152C2×D356C22×C411yescharacteristic
153C2×C22⋊D89—21no—
154C22×Dic89—21no—
155C2×C4×D89—21no—
156C2×C4×D89—21no—
157C2×C4×D89—21no—
158C2×C4×D89—21no—
159C2×C4×D89—21no—
160C2×C4×D89—42no—
161C2×C4×D89—42no—
162#13083563—22no—
163#13083563—41no—
164D712—22no—
165D712—41no—
166#13083558—42no—
167#13083557—42no—
168#13083557—42no—
169#13083557—42no—
170C2×#10623494—21no—
171C2×#10623494—21no—
172C2×#10623494—21no—
173C2×#10623494—21no—
174C8×D89—42no—
175C8×D89—42no—
176C8×D89—42no—
177C4⋊Dic89—21no—
178C4⋊Dic89C22×C411yescharacteristic
179C4⋊Dic89C22×C411yescharacteristic
180C4⋊Dic89C22×C411yescharacteristic
181C4⋊Dic89C22×C411yescharacteristic
182#13083553—21no—
183#13083553—21no—
184#13083553—21no—
185#13083553—21no—
186C22⋊Dic89—21no—
187#13083551—21no—
188#13083551—21no—
189#13083551—21no—
190#13083551—21no—
191#13083551—21no—
192#13083551—21no—
193#13083551—21no—
194#13083551—21no—
195C4×Dic89—21no—
196C22×C4×C89C22×C411yescharacteristic
197C22×C4×C89C2×D411yescharacteristic
198C22×C4×C89#4111yescharacteristic
199M16×C89—22no—
200M16×C89—41no—
201C2×C8×C89—21no—
202C2×C8×C89—21no—
203C2×C8×C89C22×C411yescharacteristic
204C2×C8×C89C22×C411yescharacteristic
205C2×C8×C89C22×C411yescharacteristic
206C2×C8×C89C22×C411yescharacteristic
207C2×C8×C89C2×D411yescharacteristic
208C2×C8×C89C2×D411yescharacteristic
209C2×C8×C89C2×D411yescharacteristic
210C2×C8×C89C2×D411yescharacteristic
211C42×C89C22×C411yescharacteristic
212C42×C89C2×D411yescharacteristic
213C42×C89#4111yescharacteristic
214C42×C89C2411yescharacteristic
215C23×C89C4×D411yescharacteristic
216C22×D89—21no—
217C22×D89—21no—
218C22×D89—21no—
219C22×D89—42no—
220C22×D89—42no—
221C22×D89—81no—
222C2×C4×C89—21no—
223C2×C4×C89—21no—
224C2×C4×C89—21no—
225C2×C4×C89—21no—
226C2×C4×C89—41no—
227C2×C4×C89C4×D411yescharacteristic
228C2×C4×C89C4×D411yescharacteristic
229C2×C4×C89C4×D411yescharacteristic
230C2×C4×C89C23×C411yescharacteristic
231C2×C4×C89C22×D411yescharacteristic
232C2×C4×C89C2×#4111yescharacteristic
233C22⋊D89—81no—
234C2×Dic89—21no—
235C2×Dic89—21no—
236C2×Dic89—21no—
237C2×Dic89—21no—
238C2×Dic89—21no—
239C2×Dic89—42no—
240D356—21no—
241D356—21no—
242D356—21no—
243D356—21no—
244D356—21no—
245D356—42no—
246D356#13112yes—
247C4×D89—42no—
248C4×D89—42no—
249C4×D89—41no—
250C4×D89—41no—
251C4×D89—41no—
252C4×D89—41no—
253C4×D89—41no—
254C4×D89—41no—
255Dic178—21no—
256Dic178—21no—
257C8×C89—21no—
258C8×C89—21no—
259C8×C89—21no—
260C8×C89—21no—
261C8×C89—21no—
262C8×C89—21no—
263C8×C89—21no—
264C8×C89—41no—
265C8×C89—41no—
266C8×C89AGL(1,ℤ/8)12yes—
267#10623494—41no—
268#10623494—41no—
269#10623494—41no—
270#10623494—41no—
271C22×C89—21no—
272C22×C89—21no—
273C22×C89—21no—
274C22×C89C2×C4×D411yescharacteristic
275C2×D89—22no—
276C2×D89—41no—
277C2×D89—41no—
278C2×D89—41no—
279C2×D89—41no—
280C2×D89—41no—
281C2×D89—41no—
282C2×D89—41no—
283C2×D89—41no—
284C2×D89—84no—
285C4×C89—21no—
286C4×C89—21no—
287C4×C89—21no—
288C4×C89—21no—
289C4×C89C2×#13111yescharacteristic
290C4×C89#56811yescharacteristic
291C4×C89C2×AGL(1,ℤ/8)11yescharacteristic
292C4×C89#57511yescharacteristic
293Dic89—41no—
294Dic89—41no—
295Dic89—41no—
296Dic89—41no—
297Dic89—41no—
298#18006—891nomaximal, Sylow
299C2×C89—41no—
300C2×C89#293311yescharacteristic
301C2×C89C4×AGL(1,ℤ/8)11yescharacteristic
302C2×C89#295711yescharacteristic
303D89—42no—
304D89—81no—
305D89—81no—
306#3105—891no—
307#2969—891no—
308D4×M16—891no—
309#1594—891no—
310#1590—891no—
311#1588—891no—
312#1586—891no—
313#1582—891no—
314#1581—891no—
315SD16⋊C8—891no—
316C8×D8—891no—
317#1574—891no—
318#1495—891no—
319#1478—891no—
320#1453D8911yescharacteristic, complemented
321C89#1800611yescharacteristic, complemented, Sylow
322C22×M16—1781no—
323#529—891no—
324C2×C4×D4—891no—
325#484—891no—
326#481—891no—
327#478—891no—
328#478—891no—
329#463—891no—
330#463—891no—
331#450—891no—
332#447—891no—
333#446—891no—
334C8×Q8—891no—
335C4×SD16—891no—
336C4×SD16—891no—
337C4×D8—891no—
338C4×D8—891no—
339#436—891no—
340#436—891no—
341#436—891no—
342#436—1781no—
343#435—1781no—
344#435—1781no—
345#435—1781no—
346C8×D4—891no—
347C8×D4—891no—
348C8×D4—891no—
349#433—1781no—
350#431—891no—
351#431C2×D8911yescharacteristic
352#431C2×D8911yescharacteristic
353#423—891no—
354#407—1781no—
355#405—1781no—
356C4×M16C2×D8911yescharacteristic
357#335—891no—
358#335—891no—
359#334—891no—
360#334—891no—
361Q8⋊C8—891no—
362Q8⋊C8—891no—
363D4⋊C8—891no—
364D4⋊C8—891no—
365D4⋊C8—891no—
366D4⋊C8—891no—
367D4⋊C8—891no—
368D4⋊C8—891no—
369#322C2×D8911yescharacteristic, complemented
370#322C2×D8911yescharacteristic, complemented
371#322C2×D8911yescharacteristic, complemented
372C82C2×D8911yescharacteristic, complemented
373C22×D4—891no—
374C23×C4—1781no—
375C2×SD16—891no—
376C2×SD16—891no—
377C2×D8—891no—
378C2×D8—891no—
379C2×M16—3564no—
380C2×M16—1781no—
381C2×M16—1781no—
382C2×M16—1781no—
383C2×M16—1781no—
384C2×M16D35611yescharacteristic
385C2×M16D35611yescharacteristic
386C22×C8—1781no—
387C22×C8—1781no—
388C22×C8—1781no—
389#126—1781no—
390#125—891no—
391#124—891no—
392#123—1781no—
393C22⋊Q8—891no—
394#121—891no—
395C4×Q8—891no—
396C4×D4—891no—
397C4×D4—891no—
398C4×D4—891no—
399C4×D4—891no—
400C4×D4—891no—
401C4×D4—1781no—
402C4×D4—1781no—
403C4×D4—1781no—
404#117—1781no—
405C2×C4⋊C4—891no—
406C2×C22⋊C4—1781no—
407C2×C42C22×D8911yescharacteristic
408#107—891no—
409#107—891no—
410#106—891no—
411#106—891no—
412C4⋊C8—21no—
413C4⋊C8—891no—
414C4⋊C8—891no—
415C4⋊C8—891no—
416C4⋊C8—891no—
417C4⋊C8—1781no—
418C4⋊C8—1781no—
419C4⋊C8—1781no—
420C4⋊C8C4×D8911yescharacteristic
421C4⋊C8C4×D8911yescharacteristic
422#103—891no—
423#103—891no—
424#102—891no—
425#102—891no—
426#102—891no—
427#102—891no—
428#102—891no—
429#102—891no—
430C22⋊C8—21no—
431C22⋊C8—1781no—
432C22⋊C8—1781no—
433C22⋊C8—1781no—
434C22⋊C8—1781no—
435C22⋊C8—1781no—
436C22⋊C8—1781no—
437C22⋊C8—1781no—
438C22⋊C8C4×D8911yescharacteristic
439C22⋊C8C4×D8911yescharacteristic
440#97—1781no—
441#97D35611yescharacteristic
442#97D35611yescharacteristic
443C4×C8—1781no—
444C4×C8—1781no—
445C4×C8C22×D8911yescharacteristic
446C4×C8C22×D8911yescharacteristic
447C4×C8C22×D8911yescharacteristic
448C4×C8C22×D8911yescharacteristic
449C4×C8C22×D8911yescharacteristic
450C4×C8C22×D8911yescharacteristic
451C24—1781no—
452C2×Q8—891no—
453C2×D4—891no—
454C2×D4—891no—
455C2×D4—891no—
456C2×D4—3561no—
457C2×D4—1782no—
458C2×D4—1781no—
459C2×D4—1781no—
460C22×C4—3562no—
461C22×C4—3562no—
462C22×C4—1781no—
463C22×C4—1781no—
464C22×C4—1781no—
465C22×C4—1781no—
466C22×C4—1781no—
467C22×C4—1781no—
468C22×C4C2×C4×D8911yescharacteristic
469C22×C4C2×D35611yescharacteristic
470C22×C4#1308357411yescharacteristic
471SD16—3561no—
472SD16—1782no—
473D8—3561no—
474D8—1782no—
475M16—22no—
476M16—41no—
477M16—3562no—
478M16—3562no—
479M16—3562no—
480M16—3562no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—3562no—
484C2×C8—3562no—
485C2×C8—3562no—
486C2×C8—1781no—
487C2×C8—1781no—
488C2×C8—1781no—
489C2×C8—1781no—
490C2×C8C2×C4×D8911yescharacteristic
491C2×C8C2×C4×D8911yescharacteristic
492C2×C8C2×C4×D8911yescharacteristic
493C2×C8C2×C4×D8911yescharacteristic
494C2×C8C2×D35611yescharacteristic
495C2×C8C2×D35611yescharacteristic
496C2×C8C2×D35611yescharacteristic
497C2×C8C2×D35611yescharacteristic
498C4⋊C4—891no—
499C4⋊C4—891no—
500C4⋊C4—891no—
501C4⋊C4—891no—
502C4⋊C4—1781no—
503C4⋊C4—1781no—
504C4⋊C4—1781no—
505C4⋊C4—1781no—
506C4⋊C4—1781no—
507C22⋊C4—1781no—
508C22⋊C4—1781no—
509C22⋊C4—1781no—
510C22⋊C4—1781no—
511C22⋊C4—1781no—
512C22⋊C4—1781no—
513C22⋊C4—1781no—
514C22⋊C4—1781no—
515C22⋊C4—1781no—
516C42—1781no—
517C42C2×C4×D8911yescharacteristic
518C42C2×D35611yescharacteristic
519C42#1308357411yescharacteristic
520C42C23×D8911yescharacteristic
521C23—3562no—
522C23—3562no—
523C23—1781no—
524C23—1781no—
525C23—1781no—
526C23—7121no—
527C23C4×D35611yescharacteristic
528Q8—1781no—
529Q8—1781no—
530D4—892no—
531D4—3562no—
532D4—1781no—
533D4—1781no—
534D4—1781no—
535D4—1781no—
536D4—1781no—
537D4—7121no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—3562no—
544C2×C4—3562no—
545C2×C4—3562no—
546C2×C4—3561no—
547C2×C4—3561no—
548C2×C4—3561no—
549C2×C4—3561no—
550C2×C4—3561no—
551C2×C4—3561no—
552C2×C4—1781no—
553C2×C4—1781no—
554C2×C4—1781no—
555C2×C4—1781no—
556C2×C4—1781no—
557C2×C4C4×D35611yescharacteristic
558C2×C4C4×D35611yescharacteristic
559C2×C4C4×D35611yescharacteristic
560C2×C4C22×C4×D8911yescharacteristic
561C2×C4C22×D35611yescharacteristic
562C2×C4C2×#1308357411yescharacteristic
563C8—21no—
564C8—21no—
565C8—21no—
566C8—21no—
567C8—21no—
568C8—21no—
569C8—21no—
570C8—41no—
571C8—41no—
572C8—3561no—
573C8—3561no—
574C8—3561no—
575C8—3561no—
576C8#1566804012yes—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—3561no—
581C22—3561no—
582C22—3561no—
583C22—3561no—
584C22—3561no—
585C22—3561no—
586C22—3561no—
587C22—3561no—
588C22—1782no—
589C22—7124no—
590C22C2×C4×D35611yescharacteristic
591C4—21no—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—3561no—
596C4—3561no—
597C4—3561no—
598C4—3561no—
599C4—3561no—
600C4#131×D8911yescharacteristic
601C4#1915236311yescharacteristic
602C4C2×#1566804011yescharacteristic
603C4#1915239811yescharacteristic
604C2—41no—
605C2—3562no—
606C2—7121no—
607C2—7121no—
608C2#2602898011yescharacteristic
609C2C4×#1566804011yescharacteristic
610C2#2602917011yescharacteristic
subgroupG/H
centerC2×C4C22×D356
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#1453C2
socleC22×C89C2×C4×D4
prime radical#19151734C4
cosocleC42×C89C24
hypercenter#1453D89
hypocenterC89#18006
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #1453

Complements

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

NH ≅ G/Ntype
2. #26029256C2semidirect
3. M16×D356C2semidirect
4. #26019167C2semidirect
5. #26019144C2semidirect
6. #26019132C2semidirect
7. #26019118C2semidirect
8. #26019090C2semidirect
9. #26019083C2semidirect
10. #26019066C2semidirect
11. C8×D712C2semidirect
12. #26019041C2semidirect
13. #26018718C2semidirect
14. #26018644C2semidirect
15. C89×#1453C2semidirect
27. #19151728C22semidirect
28. C8×Dic178C22semidirect
35. #19151677C22semidirect
42. 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
320. #1453D89semidirect
321. C89#18006semidirect
369. #322C2×D89semidirect
370. #322C2×D89semidirect
371. #322C2×D89semidirect
372. C82C2×D89semidirect

Maximal subgroups

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

subgroupindex
#260303012
#260292562
M16×D3562
#260191672
#260191442
#260191322
#260191182
#260190902
#260190832
#260190662
C8×D7122
#260190412
#260187182
#260186442
C89×#14532
#1800689