Finite Group Databasebeta

← #36525662

Subgroups — #36525662

Order 22784.

642 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26030301C211yesmaximal, characteristic
2#26029414C211yesmaximal, characteristic, complemented
3#26029286C211yesmaximal, characteristic, complemented
4#26019167C211yesmaximal, characteristic, complemented
5#26019156C211yesmaximal, characteristic, complemented
6#26019097C211yesmaximal, characteristic, complemented
7#26019090C211yesmaximal, characteristic, complemented
8#26019083C211yesmaximal, characteristic, complemented
9#26019083C211yesmaximal, characteristic, complemented
10#26019077C211yesmaximal, characteristic, complemented
11#26019066C211yesmaximal, characteristic, complemented
12#26019037C211yesmaximal, characteristic, complemented
13#26018713C211yesmaximal, characteristic, complemented
14C2×#19151327C211yesmaximal, characteristic, complemented
15C89×#1452C211yesmaximal, characteristic, complemented
16#19152132C2211yescharacteristic
17C2×C4×D356C2211yescharacteristic
18#19151934C411yescharacteristic
19#19151917C411yescharacteristic
20#19151899C411yescharacteristic
21#19151899C411yescharacteristic
22#19151814C411yescharacteristic
23#19151814C411yescharacteristic
24#19151751C411yescharacteristic
25#19151734C411yescharacteristic
26#19151728C2211yescharacteristic, complemented
27C8×Dic178C2211yescharacteristic, complemented
28C4×#13083563C2211yescharacteristic
29C4×#13083563C2211yescharacteristic
30C4×D712C2211yescharacteristic
31C4×D712C2211yescharacteristic
32#19151677C2211yescharacteristic
33#19151677C2211yescharacteristic
34#19151677C2211yescharacteristic, complemented
35#19151668—21no—
36#19151668—21no—
37#19151666—21no—
38C8×D356C2211yescharacteristic
39C8×D356C2211yescharacteristic
40C8×D356C2211yescharacteristic, complemented
41#19151661—21no—
42#19151658—21no—
43#19151655C2211yescharacteristic
44#19151623—21no—
45C2×C4⋊#10623494C2211yescharacteristic
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
62C2×C4×C8×C89C2211yescharacteristic
63#322×C89C2211yescharacteristic, complemented
64#322×C89C2211yescharacteristic, complemented
65#322×C89C2211yescharacteristic, complemented
66#322×C89C2211yescharacteristic, complemented
67C22×D356C2×C411yescharacteristic
68C22×C4×D89—21no—
69C2×#13083563C2×C411yescharacteristic
70C2×#13083563C2×C411yescharacteristic
71C2×D712C2×C411yescharacteristic
72C2×D712C2×C411yescharacteristic
73C2×#13083557—21no—
74C2×#13083557—21no—
75C2×#13083557—21no—
76C22×#10623494—21no—
77C2×C8×D89—21no—
78C2×C8×D89—21no—
79C2×C8×D89—21no—
80#15668006—21no—
81#15668004C2×C411yescharacteristic
82#15668000C2×C411yescharacteristic
83#15667995—21no—
84#15667991C2×C411yescharacteristic
85#15667986C2×C411yescharacteristic
86C4×Dic178C2311yescharacteristic
87C4×D356—21no—
88C4×D356—21no—
89C4×D356C2×C411yescharacteristic
90C4×D356C2×C411yescharacteristic
91C4×D356C2311yescharacteristic
92C4×D356C2311yescharacteristic
93C4×D356C2311yescharacteristic
94C4×C22⋊D89—21no—
95#15667969—21no—
96C2×C4⋊Dic89C2×C411yescharacteristic
97C2×#13083551—21no—
98#15667946C2×C411yescharacteristic
99#15667946C2×C411yescharacteristic
100#15667944C2×C411yescharacteristic
101#15667944C2×C411yescharacteristic
102C4⋊#10623494—21no—
103C4⋊#10623494C2311yescharacteristic
104C4⋊#10623494C2311yescharacteristic
105C4⋊#10623494C2311yescharacteristic
106C4⋊#10623494C2311yescharacteristic
107#15667941—21no—
108#15667941—21no—
109#15667937C2×C411yescharacteristic
110#15667937C2×C411yescharacteristic
111#15667934C2×C411yescharacteristic
112#15667934C2×C411yescharacteristic
113#15667934C2×C411yescharacteristic
114#15667934C2×C411yescharacteristic
115#15667934C2×C411yescharacteristic
116#15667934C2×C411yescharacteristic
117C22⋊#10623494—21no—
118#15667924—21no—
119#15667924—21no—
120#15667924—21no—
121#15667924—21no—
122#15667924—21no—
123#15667924—21no—
124#15667923—21no—
125#15667922—21no—
126C8×Dic89—21no—
127C22×C8×C89—21no—
128C2×C42×C89C2311yescharacteristic
129C4⋊C8×C89C2×C411yescharacteristic
130C4⋊C8×C89C2×C411yescharacteristic
131C4⋊C8×C89D411yescharacteristic
132C4⋊C8×C89D411yescharacteristic
133C22⋊C8×C89C2×C411yescharacteristic
134C22⋊C8×C89C2×C411yescharacteristic
135C22⋊C8×C89D411yescharacteristic
136C22⋊C8×C89D411yescharacteristic
137C4×C8×C89—21no—
138C4×C8×C89C2311yescharacteristic
139C4×C8×C89C2311yescharacteristic
140C4×C8×C89C2311yescharacteristic
141C4×C8×C89C2311yescharacteristic
142C4×C8×C89C2311yescharacteristic
143C4×C8×C89C2311yescharacteristic
144C23×D89—21no—
145C2×Dic178C22×C411yescharacteristic
146C2×D356—22no—
147C2×D356—21no—
148C2×D356—41no—
149C2×D356C22×C411yescharacteristic
150C2×D356C22×C411yescharacteristic
151C2×D356C22×C411yescharacteristic
152C2×C22⋊D89—21no—
153C22×Dic89—21no—
154C2×C4×D89—21no—
155C2×C4×D89—21no—
156C2×C4×D89—21no—
157C2×C4×D89—21no—
158C2×C4×D89—21no—
159C2×C4×D89—81no—
160C2×C4×D89—81no—
161#13083563—22no—
162#13083563—41no—
163D712—22no—
164D712—41no—
165#13083557—81no—
166#13083557—81no—
167#13083557—81no—
168C2×#10623494—21no—
169C2×#10623494—21no—
170C2×#10623494—21no—
171C2×#10623494—21no—
172C2×#10623494—81no—
173C8×D89—81no—
174C8×D89—81no—
175C8×D89—81no—
176C4⋊Dic89—21no—
177C4⋊Dic89C22×C411yescharacteristic
178C4⋊Dic89C22×C411yescharacteristic
179C4⋊Dic89C22×C411yescharacteristic
180C4⋊Dic89C22×C411yescharacteristic
181#13083553—21no—
182#13083553—21no—
183#13083553—21no—
184#13083553—21no—
185C22⋊Dic89—21no—
186#13083551—21no—
187#13083551—21no—
188#13083551—21no—
189#13083551—21no—
190#13083551—21no—
191#13083551—21no—
192#13083551—21no—
193#13083551—21no—
194C4×Dic89—21no—
195C22×C4×C89C22×C411yescharacteristic
196C22×C4×C89C2×D411yescharacteristic
197C22×C4×C89#4111yescharacteristic
198C2×C8×C89—21no—
199C2×C8×C89—21no—
200C2×C8×C89—42no—
201C2×C8×C89C22×C411yescharacteristic
202C2×C8×C89C22×C411yescharacteristic
203C2×C8×C89C22×C411yescharacteristic
204C2×C8×C89C22×C411yescharacteristic
205C2×C8×C89C2×D411yescharacteristic
206C2×C8×C89C2×D411yescharacteristic
207C2×C8×C89C2×D411yescharacteristic
208C2×C8×C89C2×D411yescharacteristic
209C42×C89C22×C411yescharacteristic
210C42×C89C2×D411yescharacteristic
211C42×C89#4111yescharacteristic
212C42×C89C2411yescharacteristic
213C23×C89C4×D411yescharacteristic
214C22×D89—21no—
215C22×D89—21no—
216C22×D89—21no—
217C22×D89—42no—
218C22×D89—42no—
219C22×D89—81no—
220C2×C4×C89—22no—
221C2×C4×C89—21no—
222C2×C4×C89—21no—
223C2×C4×C89—21no—
224C2×C4×C89—21no—
225C2×C4×C89C4×D411yescharacteristic
226C2×C4×C89C4×D411yescharacteristic
227C2×C4×C89C4×D411yescharacteristic
228C2×C4×C89C23×C411yescharacteristic
229C2×C4×C89C22×D411yescharacteristic
230C2×C4×C89C2×#4111yescharacteristic
231C22⋊D89—42no—
232C2×Dic89—21no—
233C2×Dic89—21no—
234C2×Dic89—21no—
235C2×Dic89—21no—
236C2×Dic89—21no—
237C2×Dic89—81no—
238D356—21no—
239D356—21no—
240D356—21no—
241D356—21no—
242D356—21no—
243D356—42no—
244D356#13112yes—
245C4×D89—41no—
246C4×D89—41no—
247C4×D89—41no—
248C4×D89—41no—
249C4×D89—41no—
250C4×D89—41no—
251C4×D89—81no—
252C4×D89—81no—
253Dic178—21no—
254Dic178—21no—
255C8×C89—21no—
256C8×C89—21no—
257C8×C89—21no—
258C8×C89—21no—
259C8×C89—21no—
260C8×C89—21no—
261C8×C89—21no—
262C8×C89—21no—
263C8×C89—41no—
264C8×C89—41no—
265#10623494—41no—
266#10623494—41no—
267#10623494—41no—
268#10623494—41no—
269C22×C89—21no—
270C22×C89—21no—
271C22×C89#44412yes—
272C22×C89C2×C4×D411yescharacteristic
273C2×D89—22no—
274C2×D89—41no—
275C2×D89—41no—
276C2×D89—41no—
277C2×D89—41no—
278C2×D89—41no—
279C2×D89—41no—
280C2×D89—41no—
281C2×D89—41no—
282C2×D89—84no—
283C4×C89—21no—
284C4×C89—21no—
285C4×C89—21no—
286C4×C89—21no—
287C4×C89C2×#13111yescharacteristic
288C4×C89#56811yescharacteristic
289C4×C89#57611yescharacteristic
290C4×C89#57711yescharacteristic
291Dic89—41no—
292Dic89—41no—
293Dic89—41no—
294Dic89—41no—
295Dic89—41no—
296#18020—891nomaximal, Sylow
297C2×C89—22no—
298C2×C89#293311yescharacteristic
299C2×C89#294611yescharacteristic
300C2×C89C2×#44411yescharacteristic
301D89—42no—
302D89—81no—
303D89—81no—
304#3105—891no—
305#2994—891no—
306#2974—891no—
307#1594—891no—
308#1592—891no—
309#1583—891no—
310#1582—891no—
311#1581—891no—
312#1581—891no—
313#1580—891no—
314SD16⋊C8—891no—
315#1573—891no—
316#1494—891no—
317C2×D4⋊C8—891no—
318#1452D8911yescharacteristic, complemented
319C89#1802011yescharacteristic, complemented, Sylow
320#529—891no—
321C2×C4×D4—891no—
322#484—891no—
323#481—891no—
324#478—891no—
325#478—891no—
326#463—891no—
327#463—891no—
328#450—891no—
329#447—891no—
330#446—891no—
331C8×Q8—891no—
332C4×SD16—891no—
333C4×SD16—891no—
334C4×D8—891no—
335C4×D8—891no—
336#436—891no—
337#436—891no—
338#436—891no—
339#435—1781no—
340#435—1781no—
341#435—1781no—
342C8×D4—891no—
343C8×D4—891no—
344C8×D4—891no—
345C8×D4—1781no—
346#433—1781no—
347#432—891no—
348#431C2×D8911yescharacteristic
349#431C2×D8911yescharacteristic
350#424—1781no—
351C2×C4⋊C8—891no—
352#407—1781no—
353C2×C22⋊C8—1781no—
354C2×C4×C8C2×D8911yescharacteristic
355#335—891no—
356#335—891no—
357#334—891no—
358#334—891no—
359Q8⋊C8—891no—
360Q8⋊C8—891no—
361D4⋊C8—891no—
362D4⋊C8—891no—
363D4⋊C8—891no—
364D4⋊C8—891no—
365D4⋊C8—891no—
366D4⋊C8—891no—
367#322C2×D8911yescharacteristic, complemented
368#322C2×D8911yescharacteristic, complemented
369#322C2×D8911yescharacteristic, complemented
370#322C2×D8911yescharacteristic, complemented
371C22×D4—891no—
372C23×C4—1781no—
373C2×SD16—891no—
374C2×SD16—891no—
375C2×D8—891no—
376C2×D8—891no—
377C2×M16—1781no—
378C2×M16—1781no—
379C2×M16—1781no—
380C22×C8—21no—
381C22×C8—1781no—
382C22×C8—1781no—
383C22×C8—1781no—
384C22×C8—1781no—
385#126—1781no—
386#125—891no—
387#124—891no—
388#123—1781no—
389C22⋊Q8—891no—
390#121—891no—
391C4×Q8—891no—
392C4×D4—891no—
393C4×D4—891no—
394C4×D4—891no—
395C4×D4—891no—
396C4×D4—891no—
397C4×D4—1781no—
398C4×D4—1781no—
399C4×D4—1781no—
400#117—1781no—
401C2×C4⋊C4—891no—
402C2×C22⋊C4—1781no—
403C2×C42C22×D8911yescharacteristic
404#107—891no—
405#107—891no—
406#106—891no—
407#106—891no—
408C4⋊C8—891no—
409C4⋊C8—891no—
410C4⋊C8—891no—
411C4⋊C8—891no—
412C4⋊C8—1781no—
413C4⋊C8—1781no—
414C4⋊C8—1781no—
415C4⋊C8C4×D8911yescharacteristic
416C4⋊C8C4×D8911yescharacteristic
417C4⋊C8D35611yescharacteristic
418C4⋊C8D35611yescharacteristic
419#103—891no—
420#103—891no—
421#102—891no—
422#102—891no—
423#102—891no—
424#102—891no—
425#102—891no—
426#102—891no—
427C22⋊C8—1781no—
428C22⋊C8—1781no—
429C22⋊C8—1781no—
430C22⋊C8—1781no—
431C22⋊C8—1781no—
432C22⋊C8—1781no—
433C22⋊C8—1781no—
434C22⋊C8C4×D8911yescharacteristic
435C22⋊C8C4×D8911yescharacteristic
436C22⋊C8D35611yescharacteristic
437C22⋊C8D35611yescharacteristic
438#97—1781no—
439#97—1781no—
440C4×C8—21no—
441C4×C8—1781no—
442C4×C8C22×D8911yescharacteristic
443C4×C8C22×D8911yescharacteristic
444C4×C8C22×D8911yescharacteristic
445C4×C8C22×D8911yescharacteristic
446C4×C8C22×D8911yescharacteristic
447C4×C8C22×D8911yescharacteristic
448C24—1781no—
449C2×Q8—891no—
450C2×D4—891no—
451C2×D4—891no—
452C2×D4—891no—
453C2×D4—3561no—
454C2×D4—1782no—
455C2×D4—1781no—
456C2×D4—1781no—
457C22×C4—1781no—
458C22×C4—1781no—
459C22×C4—1781no—
460C22×C4—1781no—
461C22×C4—1781no—
462C22×C4—1781no—
463C22×C4—7121no—
464C22×C4—7121no—
465C22×C4C2×C4×D8911yescharacteristic
466C22×C4C2×D35611yescharacteristic
467C22×C4#1308357411yescharacteristic
468SD16—3561no—
469SD16—1782no—
470D8—3561no—
471D8—1782no—
472M16—7121no—
473M16—7121no—
474M16—7121no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—42no—
478C2×C8—1781no—
479C2×C8—1781no—
480C2×C8—1781no—
481C2×C8—1781no—
482C2×C8—7121no—
483C2×C8—7121no—
484C2×C8—7121no—
485C2×C8—7121no—
486C2×C8C2×C4×D8911yescharacteristic
487C2×C8C2×C4×D8911yescharacteristic
488C2×C8C2×C4×D8911yescharacteristic
489C2×C8C2×C4×D8911yescharacteristic
490C2×C8C2×D35611yescharacteristic
491C2×C8C2×D35611yescharacteristic
492C2×C8C2×D35611yescharacteristic
493C2×C8C2×D35611yescharacteristic
494C4⋊C4—891no—
495C4⋊C4—891no—
496C4⋊C4—891no—
497C4⋊C4—891no—
498C4⋊C4—1781no—
499C4⋊C4—1781no—
500C4⋊C4—1781no—
501C4⋊C4—1781no—
502C4⋊C4—1781no—
503C22⋊C4—1781no—
504C22⋊C4—1781no—
505C22⋊C4—1781no—
506C22⋊C4—1781no—
507C22⋊C4—1781no—
508C22⋊C4—1781no—
509C22⋊C4—1781no—
510C22⋊C4—1781no—
511C22⋊C4—1781no—
512C42—1781no—
513C42C2×C4×D8911yescharacteristic
514C42C2×D35611yescharacteristic
515C42#1308357411yescharacteristic
516C42C23×D8911yescharacteristic
517C23—3562no—
518C23—3562no—
519C23—1781no—
520C23—1781no—
521C23—1781no—
522C23—7121no—
523C23C4×D35611yescharacteristic
524Q8—1781no—
525Q8—1781no—
526D4—892no—
527D4—3562no—
528D4—3562no—
529D4—1781no—
530D4—1781no—
531D4—1781no—
532D4—1781no—
533D4—1781no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—3561no—
540C2×C4—3561no—
541C2×C4—3561no—
542C2×C4—3561no—
543C2×C4—3561no—
544C2×C4—3561no—
545C2×C4—1781no—
546C2×C4—1781no—
547C2×C4—1781no—
548C2×C4—1781no—
549C2×C4—1781no—
550C2×C4—7121no—
551C2×C4—7121no—
552C2×C4—7121no—
553C2×C4C4×D35611yescharacteristic
554C2×C4C4×D35611yescharacteristic
555C2×C4C4×D35611yescharacteristic
556C2×C4C22×C4×D8911yescharacteristic
557C2×C4C22×D35611yescharacteristic
558C2×C4C2×#1308357411yescharacteristic
559C8—21no—
560C8—21no—
561C8—21no—
562C8—21no—
563C8—21no—
564C8—21no—
565C8—21no—
566C8—21no—
567C8—41no—
568C8—41no—
569C8—3561no—
570C8—3561no—
571C8—3561no—
572C8—3561no—
573C22—21no—
574C22—21no—
575C22—3561no—
576C22—3561no—
577C22—3561no—
578C22—3561no—
579C22—3561no—
580C22—3561no—
581C22—3561no—
582C22—3561no—
583C22—1782no—
584C22—7124no—
585C22#1915172112yes—
586C22C2×C4×D35611yescharacteristic
587C4—21no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—3561no—
592C4—3561no—
593C4—3561no—
594C4—3561no—
595C4—3561no—
596C4#131×D8911yescharacteristic
597C4#1915236311yescharacteristic
598C4#1915240311yescharacteristic
599C4#1915241011yescharacteristic
600C2—22no—
601C2—3562no—
602C2—7121no—
603C2—7121no—
604C2#2602898011yescharacteristic
605C2#2602908611yescharacteristic
606C2C2×#1915172111yescharacteristic
subgroupG/H
centerC2×C4C22×D356
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#1452C2
socleC22×C89C2×C4×D4
prime radical#19151734C4
cosocleC42×C89C24
hypercenter#1452D89
hypocenterC89#18020
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ #431 ◁ #1452

Complements

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

NH ≅ G/Ntype
2. #26029414C2semidirect
3. #26029286C2semidirect
4. #26019167C2semidirect
5. #26019156C2semidirect
6. #26019097C2semidirect
7. #26019090C2semidirect
8. #26019083C2semidirect
9. #26019083C2semidirect
10. #26019077C2semidirect
11. #26019066C2semidirect
12. #26019037C2semidirect
13. #26018713C2semidirect
14. C2×#19151327C2semidirect
15. C89×#1452C2semidirect
26. #19151728C22semidirect
27. C8×Dic178C22semidirect
34. #19151677C22semidirect
40. 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. #322×C89C22semidirect
318. #1452D89semidirect
319. C89#18020semidirect
367. #322C2×D89semidirect
368. #322C2×D89semidirect
369. #322C2×D89semidirect
370. #322C2×D89semidirect

Maximal subgroups

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

subgroupindex
#260303012
#260294142
#260292862
#260191672
#260191562
#260190972
#260190902
#260190832
#260190772
#260190662
#260190372
#260187132
C2×#191513272
C89×#14522
#1802089