Finite Group Databasebeta

← #36525470

Subgroups — #36525470

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#26030295C211yesmaximal, characteristic
2#26029257C211yesmaximal, characteristic, complemented
3#26028987C211yesmaximal, characteristic, complemented
4#26019163C211yesmaximal, characteristic, complemented
5#26019139C211yesmaximal, characteristic, complemented
6#26019127C211yesmaximal, characteristic, complemented
7#26019119C211yesmaximal, characteristic, complemented
8#26019091C211yesmaximal, characteristic, complemented
9#26019078C211yesmaximal, characteristic, complemented
10#26019067C211yesmaximal, characteristic, complemented
11#26019044C211yesmaximal, characteristic, complemented
12#26019043C211yesmaximal, characteristic, complemented
13#26018644C211yesmaximal, characteristic, complemented
14#26018547C211yesmaximal, characteristic, complemented
15C89×#1495C211yesmaximal, characteristic, complemented
16C2×M16×D89—21no—
17C2×C4×D356C2211yescharacteristic
18#19151931C411yescharacteristic
19#19151914C411yescharacteristic
20#19151895C411yescharacteristic
21#19151895C411yescharacteristic
22#19151810C411yescharacteristic
23#19151810C411yescharacteristic
24#19151748C411yescharacteristic
25#19151731C411yescharacteristic
26#19151729C2211yescharacteristic, complemented
27Q8×#10623494C2211yescharacteristic, complemented
28C4×#13083561C2211yescharacteristic
29C4×#13083561C2211yescharacteristic
30C4×#13083559C2211yescharacteristic
31C4×#13083559C2211yescharacteristic
32#19151676—21no—
33#19151676C2211yescharacteristic, complemented
34#19151675C2211yescharacteristic
35#19151675C2211yescharacteristic
36#19151670—21no—
37#19151668—21no—
38#19151668—21no—
39D4×#10623494C2211yescharacteristic, complemented
40#19151663C2211yescharacteristic
41#19151663C2211yescharacteristic
42#19151659—21no—
43#19151651C2211yescharacteristic
44#19151651C2211yescharacteristic
45#19151561—21no—
46#19151554—21no—
47C4×#13083558C2211yescharacteristic
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#19151319C2211yescharacteristic, complemented
59C8×#10623494C2211yescharacteristic, complemented
60#529×C89C2211yescharacteristic
61#431×C89C2211yescharacteristic
62#423×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×#13083558D411yescharacteristic
75C2×#13083558D411yescharacteristic
76C2×#13083557—21no—
77C2×#13083557—21no—
78M16×D89—44no—
79C22×#10623494—21no—
80C2×C8×D89—21no—
81C2×C8×D89—21no—
82C4×D356—21no—
83C4×D356—21no—
84C4×D356C2×C411yescharacteristic
85C4×D356C2×C411yescharacteristic
86C4×D356C2311yescharacteristic
87C4×D356C2311yescharacteristic
88C2×C4⋊Dic89C2×C411yescharacteristic
89C2×#13083551—21no—
90#15667945C2×C411yescharacteristic
91#15667945C2×C411yescharacteristic
92#15667943C2×C411yescharacteristic
93#15667943C2×C411yescharacteristic
94C4⋊#10623494—21no—
95C4⋊#10623494—21no—
96C4⋊#10623494C2×C411yescharacteristic
97C4⋊#10623494C2×C411yescharacteristic
98C4⋊#10623494C2311yescharacteristic
99C4⋊#10623494C2311yescharacteristic
100#15667936C2×C411yescharacteristic
101#15667936C2×C411yescharacteristic
102#15667933C2×C411yescharacteristic
103#15667933C2×C411yescharacteristic
104#15667932C2×C411yescharacteristic
105#15667932C2×C411yescharacteristic
106#15667932C2×C411yescharacteristic
107#15667932C2×C411yescharacteristic
108C22⋊#10623494—21no—
109C22⋊#10623494—21no—
110C22⋊#10623494—21no—
111C22⋊#10623494C2×C411yescharacteristic
112C22⋊#10623494C2×C411yescharacteristic
113#15667924—21no—
114#15667924—21no—
115#15667924—21no—
116#15667924—21no—
117#15667923—21no—
118#15667923D411yescharacteristic
119#15667923D411yescharacteristic
120C4×#10623494—21no—
121C4×#10623494—21no—
122C4×#10623494C2311yescharacteristic
123C4×#10623494C2311yescharacteristic
124C4×#10623494C2311yescharacteristic
125C4×#10623494C2311yescharacteristic
126C2×M16×C89—21no—
127#126×C89—21no—
128#125×C89C2×C411yescharacteristic
129#124×C89C2×C411yescharacteristic
130#123×C89—21no—
131C22⋊Q8×C89C2×C411yescharacteristic
132#121×C89C2×C411yescharacteristic
133C4×Q8×C89C2311yescharacteristic
134C4×D4×C89—21no—
135C4×D4×C89C2311yescharacteristic
136#117×C89—21no—
137C2×C42×C89C2311yescharacteristic
138C4⋊C8×C89—21no—
139C4⋊C8×C89—21no—
140C4⋊C8×C89C2311yescharacteristic
141C4⋊C8×C89C2311yescharacteristic
142C22⋊C8×C89—21no—
143C4×C8×C89C2311yescharacteristic
144C4×C8×C89C2311yescharacteristic
145C23×D89—21no—
146C2×D356—22no—
147C2×D356—21no—
148C2×D356—41no—
149C2×D356C22×C411yescharacteristic
150C2×D356C22×C411yescharacteristic
151C22×Dic89—21no—
152C2×C4×D89—21no—
153C2×C4×D89—21no—
154C2×C4×D89—21no—
155C2×C4×D89—21no—
156C2×C4×D89—42no—
157C2×C4×D89—42no—
158#13083561—22no—
159#13083561—41no—
160#13083559—22no—
161#13083559—41no—
162#13083558—22no—
163#13083558—41no—
164#13083558—42no—
165#13083557—42no—
166#13083557—42no—
167C2×#10623494—21no—
168C2×#10623494—21no—
169C2×#10623494—42no—
170C2×#10623494C22×C411yescharacteristic
171C2×#10623494C22×C411yescharacteristic
172C2×#10623494C22×C411yescharacteristic
173C2×#10623494C22×C411yescharacteristic
174C2×#10623494C2×D411yescharacteristic
175C2×#10623494C2×D411yescharacteristic
176C2×#10623494C2×D411yescharacteristic
177C2×#10623494C2×D411yescharacteristic
178C8×D89—42no—
179C8×D89—42no—
180C4⋊Dic89—21no—
181C4⋊Dic89C22×C411yescharacteristic
182C4⋊Dic89C22×C411yescharacteristic
183#13083551—21no—
184#13083551—21no—
185#13083551—21no—
186#13083551—21no—
187C2×Q8×C89C22×C411yescharacteristic
188C2×D4×C89—21no—
189C2×D4×C89C22×C411yescharacteristic
190C22×C4×C89—21no—
191C22×C4×C89C22×C411yescharacteristic
192C22×C4×C89C2×D411yescharacteristic
193C22×C4×C89#4111yescharacteristic
194M16×C89—42no—
195C2×C8×C89—21no—
196C2×C8×C89—21no—
197C2×C8×C89—21no—
198C2×C8×C89—21no—
199C4⋊C4×C89—21no—
200C4⋊C4×C89—21no—
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—
208C22⋊C4×C89—21no—
209C22⋊C4×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—
219C22×D89—42no—
220C22×D89—42no—
221C22×D89—81no—
222Q8×C89—21no—
223Q8×C89—21no—
224D4×C89—21no—
225D4×C89—21no—
226D4×C89—81no—
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—21no—
234C2×C4×C89—41no—
235C2×C4×C89—41no—
236C2×C4×C89—41no—
237C2×C4×C89C4×D411yescharacteristic
238C2×C4×C89C4×D411yescharacteristic
239C2×C4×C89C4×D411yescharacteristic
240C2×C4×C89C23×C411yescharacteristic
241C2×C4×C89C22×D411yescharacteristic
242C2×C4×C89C2×#4111yescharacteristic
243C2×Dic89—21no—
244C2×Dic89—21no—
245C2×Dic89—42no—
246D356—21no—
247D356—21no—
248D356—21no—
249D356—42no—
250D356#13112yes—
251C4×D89—42no—
252C4×D89—42no—
253C4×D89—41no—
254C4×D89—41no—
255C4×D89—41no—
256C4×D89—41no—
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—21no—
273C22×C89—21no—
274C22×C89—41no—
275C22×C89—41no—
276C22×C89—41no—
277C22×C89C2×C4×D411yescharacteristic
278C2×D89—22no—
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×C89—41no—
290C4×C89—41no—
291C4×C89—41no—
292C4×C89C2×#13111yescharacteristic
293C4×C89#56811yescharacteristic
294C4×C89C2×AGL(1,ℤ/8)11yescharacteristic
295C4×C89#57511yescharacteristic
296Dic89—41no—
297Dic89—41no—
298#18006—891nomaximal, Sylow
299C2×C89—41no—
300C2×C89—81no—
301C2×C89#293311yescharacteristic
302C2×C89C4×AGL(1,ℤ/8)11yescharacteristic
303C2×C89#295711yescharacteristic
304D89—42no—
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#1495D8911yescharacteristic, complemented
319#1478—891no—
320#1453—891no—
321C89#1800611yescharacteristic, complemented, Sylow
322C22×M16—1781no—
323#529C2×D8911yescharacteristic
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#431—891no—
352#431C2×D8911yescharacteristic
353#423C2×D8911yescharacteristic
354#407—1781no—
355#405—1781no—
356C4×M16—891no—
357#335—891no—
358#335—891no—
359#334—891no—
360#334—891no—
361Q8⋊C8C2×D8911yescharacteristic, complemented
362Q8⋊C8C2×D8911yescharacteristic, complemented
363D4⋊C8—891no—
364D4⋊C8—891no—
365D4⋊C8—891no—
366D4⋊C8—891no—
367D4⋊C8C2×D8911yescharacteristic, complemented
368D4⋊C8C2×D8911yescharacteristic, complemented
369#322—891no—
370#322—891no—
371#322—891no—
372C82—891no—
373C22×D4—891no—
374C23×C4—1781no—
375C2×SD16—891no—
376C2×SD16—891no—
377C2×D8—891no—
378C2×D8—891no—
379C2×M16—21no—
380C2×M16—891no—
381C2×M16—891no—
382C2×M16—3564no—
383C2×M16—1781no—
384C2×M16—1781no—
385C2×M16—1781no—
386C22×C8—1781no—
387C22×C8—1781no—
388C22×C8—1781no—
389#126—21no—
390#125C4×D8911yescharacteristic
391#124C4×D8911yescharacteristic
392#123—21no—
393C22⋊Q8C4×D8911yescharacteristic
394#121C4×D8911yescharacteristic
395C4×Q8C22×D8911yescharacteristic
396C4×D4—21no—
397C4×D4—891no—
398C4×D4—891no—
399C4×D4—891no—
400C4×D4—891no—
401C4×D4—1781no—
402C4×D4—1781no—
403C4×D4C22×D8911yescharacteristic
404#117—21no—
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—21no—
414C4⋊C8—891no—
415C4⋊C8—891no—
416C4⋊C8—891no—
417C4⋊C8—891no—
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—891no—
432C22⋊C8—891no—
433C22⋊C8—1781no—
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—891no—
444C4×C8—891no—
445C4×C8—891no—
446C4×C8—891no—
447C4×C8—1781no—
448C4×C8—1781no—
449C4×C8C22×D8911yescharacteristic
450C4×C8C22×D8911yescharacteristic
451C24—1781no—
452C2×Q8C2×C4×D8911yescharacteristic
453C2×D4—21no—
454C2×D4—891no—
455C2×D4—891no—
456C2×D4—3561no—
457C2×D4—1782no—
458C2×D4—1781no—
459C2×D4C2×C4×D8911yescharacteristic
460C22×C4—21no—
461C22×C4—3562no—
462C22×C4—3562no—
463C22×C4—1781no—
464C22×C4—1781no—
465C22×C4—1781no—
466C22×C4—1781no—
467C22×C4—1781no—
468C22×C4C2×C4×D8911yescharacteristic
469C22×C4D4×D8911yescharacteristic
470C22×C4#1308357311yescharacteristic
471SD16—3561no—
472SD16—1782no—
473D8—3561no—
474D8—1782no—
475M16—42no—
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—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—891no—
503C4⋊C4—891no—
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—21no—
512C22⋊C4—1781no—
513C22⋊C4—1781no—
514C22⋊C4—1781no—
515C22⋊C4—1781no—
516C42—21no—
517C42C2×C4×D8911yescharacteristic
518C42D4×D8911yescharacteristic
519C42#1308357311yescharacteristic
520C42C23×D8911yescharacteristic
521C23—21no—
522C23—3562no—
523C23—3562no—
524C23—1781no—
525C23—1781no—
526C23—7121no—
527C23#1566797211yescharacteristic
528Q8—21no—
529Q8—21no—
530D4—21no—
531D4—21no—
532D4—81no—
533D4—892no—
534D4—3562no—
535D4—1781no—
536D4—1781no—
537D4—1781no—
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—41no—
547C2×C4—41no—
548C2×C4—3562no—
549C2×C4—3562no—
550C2×C4—3562no—
551C2×C4—3561no—
552C2×C4—3561no—
553C2×C4—3561no—
554C2×C4—3561no—
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—41no—
581C22—41no—
582C22—41no—
583C22—3561no—
584C22—3561no—
585C22—3561no—
586C22—3561no—
587C22—3561no—
588C22—1782no—
589C22—7124no—
590C22C2×#1566797211yescharacteristic
591C4—21no—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—3561no—
599C4—3561no—
600C4#131×D8911yescharacteristic
601C4#1915236311yescharacteristic
602C4AGL(1,ℤ/8)×D8911yescharacteristic
603C4#1915239711yescharacteristic
604C2—41no—
605C2—81no—
606C2—3562no—
607C2—7121no—
608C2#2602898211yescharacteristic
609C2#2602906511yescharacteristic
610C2#2602916511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#1495C2
socleC22×C89C2×C4×D4
prime radical#19151731C4
cosocleC42×C89C24
hypercenter#1495D89
hypocenterC89#18006
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1495

Complements

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

NH ≅ G/Ntype
2. #26029257C2semidirect
3. #26028987C2semidirect
4. #26019163C2semidirect
5. #26019139C2semidirect
6. #26019127C2semidirect
7. #26019119C2semidirect
8. #26019091C2semidirect
9. #26019078C2semidirect
10. #26019067C2semidirect
11. #26019044C2semidirect
12. #26019043C2semidirect
13. #26018644C2semidirect
14. #26018547C2semidirect
15. C89×#1495C2semidirect
26. #19151729C22semidirect
27. Q8×#10623494C22semidirect
33. #19151676C22semidirect
39. D4×#10623494C22semidirect
48. #19151350C22semidirect
49. #19151350C22semidirect
50. #19151348C22semidirect
51. #19151348C22semidirect
56. #19151320C22semidirect
57. #19151320C22semidirect
58. #19151319C22semidirect
59. C8×#10623494C22semidirect
63. Q8⋊C8×C89C22semidirect
64. Q8⋊C8×C89C22semidirect
65. D4⋊C8×C89C22semidirect
66. D4⋊C8×C89C22semidirect
318. #1495D89semidirect
321. C89#18006semidirect
361. Q8⋊C8C2×D89semidirect
362. Q8⋊C8C2×D89semidirect
367. D4⋊C8C2×D89semidirect
368. D4⋊C8C2×D89semidirect

Maximal subgroups

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

subgroupindex
#260302952
#260292572
#260289872
#260191632
#260191392
#260191272
#260191192
#260190912
#260190782
#260190672
#260190442
#260190432
#260186442
#260185472
C89×#14952
#1800689