Finite Group Databasebeta

← #36525458

Subgroups — #36525458

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#26030299C211yesmaximal, characteristic
2#26029250C211yesmaximal, characteristic, complemented
3D4×#13083558C211yesmaximal, characteristic, complemented
4#26019163C211yesmaximal, characteristic, complemented
5#26019145C211yesmaximal, characteristic, complemented
6#26019133C211yesmaximal, characteristic, complemented
7#26019113C211yesmaximal, characteristic, complemented
8#26019085C211yesmaximal, characteristic, complemented
9#26019084C211yesmaximal, characteristic, complemented
10#26019061C211yesmaximal, characteristic, complemented
11#26019044C211yesmaximal, characteristic, complemented
12#26019040C211yesmaximal, characteristic, complemented
13#26018718C211yesmaximal, characteristic, complemented
14#26018547C211yesmaximal, characteristic, complemented
15C89×#1478C211yesmaximal, characteristic, complemented
16C22×#13083558—21no—
17#19152132C2211yescharacteristic
18#19151933C411yescharacteristic
19#19151916C411yescharacteristic
20#19151898C411yescharacteristic
21#19151898C411yescharacteristic
22#19151813C411yescharacteristic
23#19151813C411yescharacteristic
24#19151749C411yescharacteristic
25#19151732C411yescharacteristic
26#19151725C2211yescharacteristic, complemented
27#19151722C2211yescharacteristic, complemented
28C4×#13083562C2211yescharacteristic
29C4×#13083562C2211yescharacteristic
30C4×#13083559C2211yescharacteristic
31C4×#13083559C2211yescharacteristic
32#19151676C2211yescharacteristic
33#19151676C2211yescharacteristic
34#19151675C2211yescharacteristic, complemented
35#19151673—21no—
36#19151672—21no—
37#19151670—21no—
38#19151670—21no—
39D4×#10623494C2211yescharacteristic
40D4×#10623494C2211yescharacteristic
41#19151663C2211yescharacteristic, complemented
42#19151657—21no—
43#19151651C2211yescharacteristic
44#19151622C2211yescharacteristic
45#19151562—21no—
46#19151552—21no—
47C4×#13083558C2211yescharacteristic
48#19151350C2211yescharacteristic, complemented
49#19151350C2211yescharacteristic, complemented
50#19151348C2211yescharacteristic, complemented
51#19151348C2211yescharacteristic, complemented
52#19151330C2211yescharacteristic, complemented
53#19151330C2211yescharacteristic, complemented
54#19151327C2211yescharacteristic, complemented
55#19151327C2211yescharacteristic, complemented
56#19151320C2211yescharacteristic, complemented
57#19151320C2211yescharacteristic, complemented
58#19151319C2211yescharacteristic, complemented
59C8×#10623494C2211yescharacteristic, complemented
60C2×C4×D4×C89C2211yescharacteristic
61#431×C89C2211yescharacteristic
62#431×C89C2211yescharacteristic
63D4⋊C8×C89C2211yescharacteristic
64D4⋊C8×C89C2211yescharacteristic
65D4⋊C8×C89C2211yescharacteristic
66D4⋊C8×C89C2211yescharacteristic
67C2×#13083562C2×C411yescharacteristic
68C2×#13083562C2×C411yescharacteristic
69C2×#13083559C2×C411yescharacteristic
70C2×#13083559C2×C411yescharacteristic
71C2×#13083558—21no—
72C2×#13083558—21no—
73C2×#13083558—21no—
74C2×#13083558—44no—
75C2×#13083558D411yescharacteristic
76C2×#13083558D411yescharacteristic
77C2×#13083557—21no—
78C22×#10623494—21no—
79C22×#10623494—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×D356C2311yescharacteristic
89C4×C22⋊D89—21no—
90#15667969—21no—
91#15667945C2×C411yescharacteristic
92#15667945C2×C411yescharacteristic
93#15667943C2×C411yescharacteristic
94#15667943C2×C411yescharacteristic
95C4⋊#10623494—21no—
96C4⋊#10623494C2×C411yescharacteristic
97C4⋊#10623494C2×C411yescharacteristic
98C4⋊#10623494C2311yescharacteristic
99C4⋊#10623494C2311yescharacteristic
100#15667941—21no—
101#15667941—21no—
102#15667935C2×C411yescharacteristic
103#15667935C2×C411yescharacteristic
104#15667933C2×C411yescharacteristic
105#15667933C2×C411yescharacteristic
106#15667933C2×C411yescharacteristic
107#15667933C2×C411yescharacteristic
108#15667932C2×C411yescharacteristic
109#15667932C2×C411yescharacteristic
110C22⋊#10623494—21no—
111C22⋊#10623494—21no—
112C22⋊#10623494—21no—
113C22⋊#10623494—21no—
114C22⋊#10623494C2×C411yescharacteristic
115C22⋊#10623494C2×C411yescharacteristic
116#15667924—21no—
117#15667924—21no—
118#15667923D411yescharacteristic
119#15667923D411yescharacteristic
120#15667922—21no—
121C4×#10623494C2311yescharacteristic
122C4×#10623494C2311yescharacteristic
123C4×#10623494C2311yescharacteristic
124C4×#10623494C2311yescharacteristic
125C8×Dic89—21no—
126C22×D4×C89C2×C411yescharacteristic
127C23×C4×C89—21no—
128C4×D4×C89—21no—
129C4×D4×C89—21no—
130C4×D4×C89C2×C411yescharacteristic
131C4×D4×C89C2×C411yescharacteristic
132C4×D4×C89C2311yescharacteristic
133C4×D4×C89C2311yescharacteristic
134C2×C4⋊C4×C89C2×C411yescharacteristic
135C2×C22⋊C4×C89—21no—
136C2×C42×C89C2311yescharacteristic
137C4⋊C8×C89—21no—
138C4⋊C8×C89C2311yescharacteristic
139C4⋊C8×C89C2311yescharacteristic
140C22⋊C8×C89—21no—
141C22⋊C8×C89—21no—
142C4×C8×C89—21no—
143C4×C8×C89C2311yescharacteristic
144C4×C8×C89C2311yescharacteristic
145C2×Dic178C22×C411yescharacteristic
146C2×D356C22×C411yescharacteristic
147C2×C22⋊D89—21no—
148C2×C4×D89—21no—
149#13083562—22no—
150#13083562—41no—
151#13083559—22no—
152#13083559—41no—
153#13083558—22no—
154#13083558—41no—
155#13083558—42no—
156#13083558—42no—
157#13083558—42no—
158#13083557—42no—
159C2×#10623494—21no—
160C2×#10623494—21no—
161C2×#10623494—42no—
162C2×#10623494—42no—
163C2×#10623494C22×C411yescharacteristic
164C2×#10623494C22×C411yescharacteristic
165C2×#10623494C22×C411yescharacteristic
166C2×#10623494C22×C411yescharacteristic
167C2×#10623494C2×D411yescharacteristic
168C2×#10623494C2×D411yescharacteristic
169C2×#10623494C2×D411yescharacteristic
170C2×#10623494C2×D411yescharacteristic
171C8×D89—42no—
172C4⋊Dic89C22×C411yescharacteristic
173C4⋊Dic89C22×C411yescharacteristic
174#13083553—21no—
175#13083553—21no—
176#13083553—21no—
177#13083553—21no—
178C22⋊Dic89—21no—
179#13083551—21no—
180#13083551—21no—
181#13083551—21no—
182#13083551—21no—
183C4×Dic89—21no—
184C24×C89—21no—
185C2×D4×C89—22no—
186C2×D4×C89—21no—
187C2×D4×C89—41no—
188C2×D4×C89C22×C411yescharacteristic
189C2×D4×C89C22×C411yescharacteristic
190C22×C4×C89—21no—
191C22×C4×C89—21no—
192C22×C4×C89—21no—
193C22×C4×C89—21no—
194C22×C4×C89—21no—
195C22×C4×C89—42no—
196C22×C4×C89—42no—
197C22×C4×C89C22×C411yescharacteristic
198C22×C4×C89C2×D411yescharacteristic
199C22×C4×C89#4111yescharacteristic
200C2×C8×C89—21no—
201C2×C8×C89—21no—
202C2×C8×C89—21no—
203C2×C8×C89—21no—
204C4⋊C4×C89—21no—
205C4⋊C4×C89C22×C411yescharacteristic
206C4⋊C4×C89C22×C411yescharacteristic
207C22⋊C4×C89—21no—
208C22⋊C4×C89—21no—
209C22⋊C4×C89—21no—
210C22⋊C4×C89—21no—
211C42×C89C22×C411yescharacteristic
212C42×C89C2×D411yescharacteristic
213C42×C89#4111yescharacteristic
214C42×C89C2411yescharacteristic
215C23×C89—21no—
216C23×C89—21no—
217C23×C89—42no—
218C23×C89—42no—
219C23×C89—81no—
220C23×C89C4×D411yescharacteristic
221C22×D89—21no—
222D4×C89—21no—
223D4×C89—21no—
224D4×C89—21no—
225D4×C89—42no—
226D4×C89#13112yes—
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—42no—
235C2×C4×C89—42no—
236C2×C4×C89—42no—
237C2×C4×C89—41no—
238C2×C4×C89—41no—
239C2×C4×C89—41no—
240C2×C4×C89—41no—
241C2×C4×C89C4×D411yescharacteristic
242C2×C4×C89C4×D411yescharacteristic
243C2×C4×C89C4×D411yescharacteristic
244C2×C4×C89C23×C411yescharacteristic
245C2×C4×C89C22×D411yescharacteristic
246C2×C4×C89C2×#4111yescharacteristic
247C22⋊D89—81no—
248C2×Dic89—21no—
249C2×Dic89—21no—
250C2×Dic89—21no—
251D356—21no—
252D356—21no—
253C4×D89—41no—
254C4×D89—41no—
255Dic178—21no—
256Dic178—21no—
257C8×C89—41no—
258C8×C89—41no—
259C8×C89—41no—
260C8×C89—41no—
261#10623494—21no—
262#10623494—21no—
263#10623494—21no—
264#10623494—21no—
265#10623494—21no—
266#10623494—21no—
267#10623494—21no—
268#10623494—41no—
269#10623494—41no—
270#10623494AGL(1,ℤ/8)12yes—
271C22×C89—21no—
272C22×C89—22no—
273C22×C89—21no—
274C22×C89—21no—
275C22×C89—41no—
276C22×C89—41no—
277C22×C89—41no—
278C22×C89—41no—
279C22×C89—41no—
280C22×C89—84no—
281C22×C89C2×C4×D411yescharacteristic
282C2×D89—41no—
283C2×D89—41no—
284C2×D89—41no—
285C4×C89—21no—
286C4×C89—21no—
287C4×C89—21no—
288C4×C89—21no—
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—
297Dic89—41no—
298#18006—891nomaximal, Sylow
299C2×C89—41no—
300C2×C89—42no—
301C2×C89—81no—
302C2×C89#293311yescharacteristic
303C2×C89C4×AGL(1,ℤ/8)11yescharacteristic
304C2×C89#295711yescharacteristic
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#1478D8911yescharacteristic, complemented
320#1453—891no—
321C89#1800611yescharacteristic, complemented, Sylow
322C22×M16—1781no—
323#529—891no—
324C2×C4×D4C2×D8911yescharacteristic
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×M16—891no—
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⋊C8C2×D8911yescharacteristic
366D4⋊C8C2×D8911yescharacteristic
367D4⋊C8C2×D8911yescharacteristic
368D4⋊C8C2×D8911yescharacteristic
369#322—891no—
370#322—891no—
371#322—891no—
372C82—891no—
373C22×D4C4×D8911yescharacteristic
374C23×C4—21no—
375C2×SD16—891no—
376C2×SD16—891no—
377C2×D8—891no—
378C2×D8—891no—
379C2×M16—891no—
380C2×M16—891no—
381C2×M16—3564no—
382C2×M16—1781no—
383C2×M16—1781no—
384C2×M16—1781no—
385C2×M16—1781no—
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—21no—
397C4×D4—21no—
398C4×D4—891no—
399C4×D4—1781no—
400C4×D4C4×D8911yescharacteristic
401C4×D4C4×D8911yescharacteristic
402C4×D4C22×D8911yescharacteristic
403C4×D4C22×D8911yescharacteristic
404#117—1781no—
405C2×C4⋊C4C4×D8911yescharacteristic
406C2×C22⋊C4—21no—
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⋊C8C22×D8911yescharacteristic
421C4⋊C8C22×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—21no—
432C22⋊C8—891no—
433C22⋊C8—891no—
434C22⋊C8—1781no—
435C22⋊C8—1781no—
436C22⋊C8—1781no—
437C22⋊C8—1781no—
438C22⋊C8—1781no—
439C22⋊C8—1781no—
440#97—891no—
441#97—891no—
442#97—1781no—
443C4×C8—21no—
444C4×C8—891no—
445C4×C8—891no—
446C4×C8—891no—
447C4×C8—891no—
448C4×C8—1781no—
449C4×C8C22×D8911yescharacteristic
450C4×C8C22×D8911yescharacteristic
451C24—21no—
452C2×Q8—891no—
453C2×D4—22no—
454C2×D4—21no—
455C2×D4—41no—
456C2×D4—891no—
457C2×D4—1781no—
458C2×D4C2×C4×D8911yescharacteristic
459C2×D4C2×C4×D8911yescharacteristic
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—1781no—
468C22×C4C2×C4×D8911yescharacteristic
469C22×C4D4×D8911yescharacteristic
470C22×C4#1308357311yescharacteristic
471SD16—3561no—
472SD16—1782no—
473D8—3561no—
474D8—1782no—
475M16—3562no—
476M16—3562no—
477M16—3562no—
478M16—3562no—
479M16—3561no—
480M16—1782no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—891no—
486C2×C8—891no—
487C2×C8—891no—
488C2×C8—891no—
489C2×C8—891no—
490C2×C8—891no—
491C2×C8—891no—
492C2×C8—891no—
493C2×C8—3562no—
494C2×C8—3562no—
495C2×C8—3562no—
496C2×C8—1781no—
497C2×C8—1781no—
498C4⋊C4—21no—
499C4⋊C4—891no—
500C4⋊C4—891no—
501C4⋊C4—1781no—
502C4⋊C4—1781no—
503C4⋊C4—1781no—
504C4⋊C4—1781no—
505C4⋊C4C2×C4×D8911yescharacteristic
506C4⋊C4C2×C4×D8911yescharacteristic
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—1781no—
512C22⋊C4—1781no—
513C22⋊C4—1781no—
514C22⋊C4—1781no—
515C22⋊C4—1781no—
516C42—1781no—
517C42C2×C4×D8911yescharacteristic
518C42D4×D8911yescharacteristic
519C42#1308357311yescharacteristic
520C42C23×D8911yescharacteristic
521C23—21no—
522C23—21no—
523C23—42no—
524C23—42no—
525C23—81no—
526C23—1781no—
527C23#1566797211yescharacteristic
528Q8—1781no—
529Q8—1781no—
530D4—21no—
531D4—21no—
532D4—21no—
533D4—42no—
534D4—1781no—
535D4—1781no—
536D4—7121no—
537D4#1566801812yes—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—3561no—
553C2×C4—3561no—
554C2×C4—1781no—
555C2×C4—1781no—
556C2×C4—1781no—
557C2×C4#1566797211yescharacteristic
558C2×C4#1566797211yescharacteristic
559C2×C4#1566797211yescharacteristic
560C2×C4C22×C4×D8911yescharacteristic
561C2×C4C2×D4×D8911yescharacteristic
562C2×C4C2×#1308357311yescharacteristic
563C8—41no—
564C8—41no—
565C8—41no—
566C8—41no—
567C8—892no—
568C8—3561no—
569C8—3561no—
570C8—1781no—
571C8—1781no—
572C8—1781no—
573C8—1781no—
574C8—1781no—
575C8—1781no—
576C8—1781no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—22no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—84no—
587C22—3561no—
588C22—3561no—
589C22—3561no—
590C22C2×#1566797211yescharacteristic
591C4—21no—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—41no—
596C4—41no—
597C4—3561no—
598C4—3561no—
599C4—3561no—
600C4C2×#1566801811yescharacteristic
601C4#1915236411yescharacteristic
602C4AGL(1,ℤ/8)×D8911yescharacteristic
603C4#1915239411yescharacteristic
604C2—41no—
605C2—42no—
606C2—81no—
607C2—7121no—
608C2#2602897311yescharacteristic
609C2#2602905611yescharacteristic
610C2#2602916511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#1478C2
socleC22×C89C2×C4×D4
prime radical#19151732C4
cosocleC42×C89C24
hypercenter#1478D89
hypocenterC89#18006
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1478

Complements

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

NH ≅ G/Ntype
2. #26029250C2semidirect
3. D4×#13083558C2semidirect
4. #26019163C2semidirect
5. #26019145C2semidirect
6. #26019133C2semidirect
7. #26019113C2semidirect
8. #26019085C2semidirect
9. #26019084C2semidirect
10. #26019061C2semidirect
11. #26019044C2semidirect
12. #26019040C2semidirect
13. #26018718C2semidirect
14. #26018547C2semidirect
15. C89×#1478C2semidirect
26. #19151725C22semidirect
27. #19151722C22semidirect
34. #19151675C22semidirect
41. #19151663C22semidirect
48. #19151350C22semidirect
49. #19151350C22semidirect
50. #19151348C22semidirect
51. #19151348C22semidirect
52. #19151330C22semidirect
53. #19151330C22semidirect
54. #19151327C22semidirect
55. #19151327C22semidirect
56. #19151320C22semidirect
57. #19151320C22semidirect
58. #19151319C22semidirect
59. C8×#10623494C22semidirect
319. #1478D89semidirect
321. C89#18006semidirect

Maximal subgroups

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

subgroupindex
#260302992
#260292502
D4×#130835582
#260191632
#260191452
#260191332
#260191132
#260190852
#260190842
#260190612
#260190442
#260190402
#260187182
#260185472
C89×#14782
#1800689