Finite Group Databasebeta

← #36525664

Subgroups — #36525664

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#26030295C211yesmaximal, characteristic
2#26029412C211yesmaximal, characteristic, complemented
3#26029289C211yesmaximal, characteristic, complemented
4#26019163C211yesmaximal, characteristic, complemented
5#26019157C211yesmaximal, characteristic, complemented
6#26019092C211yesmaximal, characteristic, complemented
7#26019091C211yesmaximal, characteristic, complemented
8#26019084C211yesmaximal, characteristic, complemented
9#26019078C211yesmaximal, characteristic, complemented
10#26019073C211yesmaximal, characteristic, complemented
11#26019067C211yesmaximal, characteristic, complemented
12#26019038C211yesmaximal, characteristic, complemented
13C2×#19151327C211yesmaximal, characteristic, complemented
14#26018544C211yesmaximal, characteristic, complemented
15C89×#1494C211yesmaximal, characteristic, complemented
16C2×C4×D356C2211yescharacteristic
17#19151931C411yescharacteristic
18#19151914C411yescharacteristic
19#19151895C411yescharacteristic
20#19151895C411yescharacteristic
21#19151810C411yescharacteristic
22#19151810C411yescharacteristic
23#19151748C411yescharacteristic
24#19151731C411yescharacteristic
25#19151729C2211yescharacteristic, complemented
26Q8×#10623494C2211yescharacteristic, complemented
27C4×#13083561C2211yescharacteristic
28C4×#13083561C2211yescharacteristic
29C4×#13083559C2211yescharacteristic
30C4×#13083559C2211yescharacteristic
31#19151676C2211yescharacteristic, complemented
32#19151675C2211yescharacteristic
33#19151675C2211yescharacteristic
34#19151671—21no—
35#19151671—21no—
36#19151670—21no—
37D4×#10623494—21no—
38D4×#10623494C2211yescharacteristic, complemented
39#19151663C2211yescharacteristic
40#19151663C2211yescharacteristic
41#19151659—21no—
42#19151651C2211yescharacteristic
43#19151651C2211yescharacteristic
44#19151625—21no—
45C2×C4⋊#10623494C2211yescharacteristic
46#19151560—21no—
47C22⋊C8×D89—21no—
48#19151350C2211yescharacteristic, complemented
49#19151350C2211yescharacteristic, complemented
50#19151348C2211yescharacteristic, complemented
51#19151348C2211yescharacteristic, complemented
52#19151327C2211yescharacteristic
53#19151327C2211yescharacteristic
54#19151327C2211yescharacteristic
55#19151327C2211yescharacteristic
56#19151320C2211yescharacteristic, complemented
57#19151320C2211yescharacteristic, complemented
58#19151320C2211yescharacteristic, complemented
59#19151320C2211yescharacteristic, complemented
60#529×C89C2211yescharacteristic
61#432×C89C2211yescharacteristic
62C2×C4×C8×C89C2211yescharacteristic
63Q8⋊C8×C89C2211yescharacteristic, complemented
64Q8⋊C8×C89C2211yescharacteristic, complemented
65D4⋊C8×C89C2211yescharacteristic, complemented
66D4⋊C8×C89C2211yescharacteristic, complemented
67C22×D356C2×C411yescharacteristic
68C22×C4×D89—21no—
69C2×#13083561C2×C411yescharacteristic
70C2×#13083561C2×C411yescharacteristic
71C2×#13083559C2×C411yescharacteristic
72C2×#13083559C2×C411yescharacteristic
73C2×#13083558—21no—
74C2×#13083557—21no—
75C2×#13083557—21no—
76C22×#10623494—21no—
77C22×#10623494—21no—
78C2×C8×D89—21no—
79C2×C8×D89—21no—
80C4×D356—21no—
81C4×D356—21no—
82C4×D356C2×C411yescharacteristic
83C4×D356C2×C411yescharacteristic
84C4×D356C2311yescharacteristic
85C4×D356C2311yescharacteristic
86C2×C4⋊Dic89C2×C411yescharacteristic
87C2×#13083551—21no—
88#15667945C2×C411yescharacteristic
89#15667945C2×C411yescharacteristic
90#15667943C2×C411yescharacteristic
91#15667943C2×C411yescharacteristic
92C4⋊#10623494—21no—
93C4⋊#10623494—21no—
94C4⋊#10623494—21no—
95C4⋊#10623494C2×C411yescharacteristic
96C4⋊#10623494C2×C411yescharacteristic
97C4⋊#10623494D411yescharacteristic
98C4⋊#10623494D411yescharacteristic
99C4⋊#10623494C2311yescharacteristic
100C4⋊#10623494C2311yescharacteristic
101#15667936C2×C411yescharacteristic
102#15667936C2×C411yescharacteristic
103#15667933C2×C411yescharacteristic
104#15667933C2×C411yescharacteristic
105#15667932C2×C411yescharacteristic
106#15667932C2×C411yescharacteristic
107#15667932C2×C411yescharacteristic
108#15667932C2×C411yescharacteristic
109C22⋊#10623494—21no—
110C22⋊#10623494—21no—
111C22⋊#10623494C2×C411yescharacteristic
112C22⋊#10623494C2×C411yescharacteristic
113C22⋊#10623494D411yescharacteristic
114C22⋊#10623494D411yescharacteristic
115#15667924—21no—
116#15667924—21no—
117#15667924—21no—
118#15667924—21no—
119#15667923—21no—
120C4×#10623494—21no—
121C4×#10623494C2311yescharacteristic
122C4×#10623494C2311yescharacteristic
123C4×#10623494C2311yescharacteristic
124C4×#10623494C2311yescharacteristic
125C22×C8×C89—21no—
126#126×C89—21no—
127#125×C89C2×C411yescharacteristic
128#124×C89C2×C411yescharacteristic
129#123×C89—21no—
130C22⋊Q8×C89C2×C411yescharacteristic
131#121×C89C2×C411yescharacteristic
132C4×Q8×C89C2311yescharacteristic
133C4×D4×C89—21no—
134C4×D4×C89C2311yescharacteristic
135#117×C89—21no—
136C2×C42×C89C2311yescharacteristic
137C4⋊C8×C89C2311yescharacteristic
138C4⋊C8×C89C2311yescharacteristic
139C22⋊C8×C89—21no—
140#97×C89—21no—
141C4×C8×C89—21no—
142C4×C8×C89C2311yescharacteristic
143C4×C8×C89C2311yescharacteristic
144C23×D89—21no—
145C2×D356—22no—
146C2×D356—21no—
147C2×D356—41no—
148C2×D356C22×C411yescharacteristic
149C2×D356C22×C411yescharacteristic
150C22×Dic89—21no—
151C2×C4×D89—21no—
152C2×C4×D89—21no—
153C2×C4×D89—21no—
154C2×C4×D89—21no—
155C2×C4×D89—81no—
156C2×C4×D89—81no—
157#13083561—22no—
158#13083561—41no—
159#13083559—22no—
160#13083559—41no—
161#13083558—81no—
162#13083557—81no—
163#13083557—81no—
164C2×#10623494—21no—
165C2×#10623494—21no—
166C2×#10623494—81no—
167C2×#10623494—81no—
168C2×#10623494C22×C411yescharacteristic
169C2×#10623494C22×C411yescharacteristic
170C2×#10623494C22×C411yescharacteristic
171C2×#10623494C22×C411yescharacteristic
172C2×#10623494C2×D411yescharacteristic
173C2×#10623494C2×D411yescharacteristic
174C2×#10623494C2×D411yescharacteristic
175C2×#10623494C2×D411yescharacteristic
176C8×D89—81no—
177C8×D89—81no—
178C4⋊Dic89—21no—
179C4⋊Dic89C22×C411yescharacteristic
180C4⋊Dic89C22×C411yescharacteristic
181#13083551—21no—
182#13083551—21no—
183#13083551—21no—
184#13083551—21no—
185C2×Q8×C89C22×C411yescharacteristic
186C2×D4×C89—21no—
187C2×D4×C89C22×C411yescharacteristic
188C22×C4×C89—21no—
189C22×C4×C89C22×C411yescharacteristic
190C22×C4×C89C2×D411yescharacteristic
191C22×C4×C89#4111yescharacteristic
192C2×C8×C89—21no—
193C2×C8×C89—21no—
194C2×C8×C89—21no—
195C2×C8×C89—21no—
196C2×C8×C89—42no—
197C4⋊C4×C89—21no—
198C4⋊C4×C89—21no—
199C4⋊C4×C89—21no—
200C4⋊C4×C89—21no—
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—
208C42×C89—21no—
209C42×C89C22×C411yescharacteristic
210C42×C89C2×D411yescharacteristic
211C42×C89#4111yescharacteristic
212C42×C89C2411yescharacteristic
213C23×C89—21no—
214C23×C89C4×D411yescharacteristic
215C22×D89—21no—
216C22×D89—21no—
217C22×D89—42no—
218C22×D89—42no—
219C22×D89—81no—
220Q8×C89—21no—
221Q8×C89—21no—
222D4×C89—21no—
223D4×C89—21no—
224D4×C89—42no—
225C2×C4×C89—22no—
226C2×C4×C89—21no—
227C2×C4×C89—21no—
228C2×C4×C89—21no—
229C2×C4×C89—21no—
230C2×C4×C89—21no—
231C2×C4×C89—21no—
232C2×C4×C89—21no—
233C2×C4×C89—41no—
234C2×C4×C89—41no—
235C2×C4×C89C4×D411yescharacteristic
236C2×C4×C89C4×D411yescharacteristic
237C2×C4×C89C4×D411yescharacteristic
238C2×C4×C89C23×C411yescharacteristic
239C2×C4×C89C22×D411yescharacteristic
240C2×C4×C89C2×#4111yescharacteristic
241C2×Dic89—21no—
242C2×Dic89—21no—
243C2×Dic89—81no—
244D356—21no—
245D356—21no—
246D356—21no—
247D356—42no—
248D356#13112yes—
249C4×D89—41no—
250C4×D89—41no—
251C4×D89—41no—
252C4×D89—41no—
253C4×D89—81no—
254C4×D89—81no—
255C8×C89—41no—
256C8×C89—41no—
257C8×C89—41no—
258C8×C89—41no—
259#10623494—21no—
260#10623494—21no—
261#10623494—21no—
262#10623494—21no—
263#10623494—21no—
264#10623494—21no—
265#10623494—21no—
266#10623494—21no—
267#10623494—41no—
268#10623494—41no—
269C22×C89—21no—
270C22×C89—21no—
271C22×C89—41no—
272C22×C89—41no—
273C22×C89—41no—
274C22×C89#44412yes—
275C22×C89C2×C4×D411yescharacteristic
276C2×D89—22no—
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×C89—41no—
288C4×C89—41no—
289C4×C89—41no—
290C4×C89C2×#13111yescharacteristic
291C4×C89#56811yescharacteristic
292C4×C89#57611yescharacteristic
293C4×C89#57711yescharacteristic
294Dic89—41no—
295Dic89—41no—
296#18020—891nomaximal, Sylow
297C2×C89—22no—
298C2×C89—81no—
299C2×C89#293311yescharacteristic
300C2×C89#294611yescharacteristic
301C2×C89C2×#44411yescharacteristic
302D89—42no—
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#1494D8911yescharacteristic, complemented
317C2×D4⋊C8—891no—
318#1452—891no—
319C89#1802011yescharacteristic, complemented, Sylow
320#529C2×D8911yescharacteristic
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#432C2×D8911yescharacteristic
348#431—891no—
349#431—891no—
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⋊C8C2×D8911yescharacteristic, complemented
360Q8⋊C8C2×D8911yescharacteristic, complemented
361D4⋊C8—891no—
362D4⋊C8—891no—
363D4⋊C8—891no—
364D4⋊C8—891no—
365D4⋊C8C2×D8911yescharacteristic, complemented
366D4⋊C8C2×D8911yescharacteristic, complemented
367#322—891no—
368#322—891no—
369#322—891no—
370#322—891no—
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—21no—
386#125C4×D8911yescharacteristic
387#124C4×D8911yescharacteristic
388#123—21no—
389C22⋊Q8C4×D8911yescharacteristic
390#121C4×D8911yescharacteristic
391C4×Q8C22×D8911yescharacteristic
392C4×D4—21no—
393C4×D4—891no—
394C4×D4—891no—
395C4×D4—891no—
396C4×D4—891no—
397C4×D4—1781no—
398C4×D4—1781no—
399C4×D4C22×D8911yescharacteristic
400#117—21no—
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—891no—
413C4⋊C8—891no—
414C4⋊C8—1781no—
415C4⋊C8—1781no—
416C4⋊C8—1781no—
417C4⋊C8C22×D8911yescharacteristic
418C4⋊C8C22×D8911yescharacteristic
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—21no—
428C22⋊C8—891no—
429C22⋊C8—891no—
430C22⋊C8—891no—
431C22⋊C8—891no—
432C22⋊C8—1781no—
433C22⋊C8—1781no—
434C22⋊C8—1781no—
435C22⋊C8—1781no—
436C22⋊C8—1781no—
437C22⋊C8—1781no—
438#97—21no—
439#97—1781no—
440C4×C8—21no—
441C4×C8—891no—
442C4×C8—891no—
443C4×C8—891no—
444C4×C8—891no—
445C4×C8—1781no—
446C4×C8C22×D8911yescharacteristic
447C4×C8C22×D8911yescharacteristic
448C24—1781no—
449C2×Q8C2×C4×D8911yescharacteristic
450C2×D4—21no—
451C2×D4—891no—
452C2×D4—891no—
453C2×D4—3561no—
454C2×D4—1782no—
455C2×D4—1781no—
456C2×D4C2×C4×D8911yescharacteristic
457C22×C4—21no—
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×C4D4×D8911yescharacteristic
467C22×C4#1308357311yescharacteristic
468SD16—3561no—
469SD16—1782no—
470D8—3561no—
471D8—1782no—
472M16—7121no—
473M16—7121no—
474M16—7121no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—42no—
480C2×C8—891no—
481C2×C8—891no—
482C2×C8—891no—
483C2×C8—891no—
484C2×C8—891no—
485C2×C8—891no—
486C2×C8—891no—
487C2×C8—891no—
488C2×C8—1781no—
489C2×C8—1781no—
490C2×C8—7121no—
491C2×C8—7121no—
492C2×C8—7121no—
493C2×C8—7121no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—891no—
499C4⋊C4—891no—
500C4⋊C4—1781no—
501C4⋊C4C2×C4×D8911yescharacteristic
502C4⋊C4C2×C4×D8911yescharacteristic
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—1781no—
509C22⋊C4—1781no—
510C22⋊C4—1781no—
511C22⋊C4—1781no—
512C42—21no—
513C42C2×C4×D8911yescharacteristic
514C42D4×D8911yescharacteristic
515C42#1308357311yescharacteristic
516C42C23×D8911yescharacteristic
517C23—21no—
518C23—3562no—
519C23—3562no—
520C23—1781no—
521C23—1781no—
522C23—7121no—
523C23#1566797211yescharacteristic
524Q8—21no—
525Q8—21no—
526D4—21no—
527D4—21no—
528D4—42no—
529D4—892no—
530D4—3562no—
531D4—1781no—
532D4—1781no—
533D4—1781no—
534C2×C4—22no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—3561no—
545C2×C4—3561no—
546C2×C4—3561no—
547C2×C4—3561no—
548C2×C4—1781no—
549C2×C4—1781no—
550C2×C4—7121no—
551C2×C4—7121no—
552C2×C4—7121no—
553C2×C4#1566797211yescharacteristic
554C2×C4#1566797211yescharacteristic
555C2×C4#1566797211yescharacteristic
556C2×C4C22×C4×D8911yescharacteristic
557C2×C4C2×D4×D8911yescharacteristic
558C2×C4C2×#1308357311yescharacteristic
559C8—41no—
560C8—41no—
561C8—41no—
562C8—41no—
563C8—3561no—
564C8—3561no—
565C8—1781no—
566C8—1781no—
567C8—1781no—
568C8—1781no—
569C8—1781no—
570C8—1781no—
571C8—1781no—
572C8—1781no—
573C22—21no—
574C22—21no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—3561no—
579C22—3561no—
580C22—3561no—
581C22—3561no—
582C22—3561no—
583C22—1782no—
584C22—7124no—
585C22#1915171712yes—
586C22C2×#1566797211yescharacteristic
587C4—21no—
588C4—21no—
589C4—21no—
590C4—21no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—3561no—
595C4—3561no—
596C4#131×D8911yescharacteristic
597C4#1915236311yescharacteristic
598C4#1915240011yescharacteristic
599C4#1915240511yescharacteristic
600C2—22no—
601C2—81no—
602C2—3562no—
603C2—7121no—
604C2#2602898211yescharacteristic
605C2#2602908711yescharacteristic
606C2C2×#1915171711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#1494C2
socleC22×C89C2×C4×D4
prime radical#19151731C4
cosocleC42×C89C24
hypercenter#1494D89
hypocenterC89#18020
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1494

Complements

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

NH ≅ G/Ntype
2. #26029412C2semidirect
3. #26029289C2semidirect
4. #26019163C2semidirect
5. #26019157C2semidirect
6. #26019092C2semidirect
7. #26019091C2semidirect
8. #26019084C2semidirect
9. #26019078C2semidirect
10. #26019073C2semidirect
11. #26019067C2semidirect
12. #26019038C2semidirect
13. C2×#19151327C2semidirect
14. #26018544C2semidirect
15. C89×#1494C2semidirect
25. #19151729C22semidirect
26. Q8×#10623494C22semidirect
31. #19151676C22semidirect
38. D4×#10623494C22semidirect
48. #19151350C22semidirect
49. #19151350C22semidirect
50. #19151348C22semidirect
51. #19151348C22semidirect
56. #19151320C22semidirect
57. #19151320C22semidirect
58. #19151320C22semidirect
59. #19151320C22semidirect
63. Q8⋊C8×C89C22semidirect
64. Q8⋊C8×C89C22semidirect
65. D4⋊C8×C89C22semidirect
66. D4⋊C8×C89C22semidirect
316. #1494D89semidirect
319. C89#18020semidirect
359. Q8⋊C8C2×D89semidirect
360. Q8⋊C8C2×D89semidirect
365. D4⋊C8C2×D89semidirect
366. D4⋊C8C2×D89semidirect

Maximal subgroups

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

subgroupindex
#260302952
#260294122
#260292892
#260191632
#260191572
#260190922
#260190912
#260190842
#260190782
#260190732
#260190672
#260190382
C2×#191513272
#260185442
C89×#14942
#1802089