Finite Group Databasebeta

← #12070336

Subgroups — #12070336

Order 1280.

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
1C5×D4×M16C211yesmaximal, characteristic, complemented
2#10614040C211yesmaximal, characteristic
3#10610242C211yesmaximal, characteristic, complemented
4#10602390C211yesmaximal, characteristic, complemented
5#10601659C211yesmaximal, characteristic, complemented
6#10601658C211yesmaximal, characteristic, complemented
7#10601655C211yesmaximal, characteristic, complemented
8#10601654C211yesmaximal, characteristic, complemented
9#10601637C211yesmaximal, characteristic, complemented
10#10601635C211yesmaximal, characteristic, complemented
11#10601632C211yesmaximal, characteristic, complemented
12C8×#686C211yesmaximal, characteristic, complemented
13#10601292C211yesmaximal, characteristic, complemented
14#10601288C211yesmaximal, characteristic, complemented
15#10600641C211yesmaximal, characteristic, complemented
16C22×C5×M16—21no—
17C2×C4×C5×D4C2211yescharacteristic
18#65813C2211yescharacteristic
19C5×#436C2211yescharacteristic
20C5×#436C2211yescharacteristic
21C5×#435—21no—
22C5×#435—21no—
23C5×C8×D4C2211yescharacteristic
24C5×C8×D4C2211yescharacteristic
25C5×#423C2211yescharacteristic
26C5×#407—21no—
27C4×C5×M16C2211yescharacteristic
28#65507C411yescharacteristic
29#65506C411yescharacteristic
30#65417—21no—
31#65416—21no—
32C4×#687C2211yescharacteristic
33C4×#687C2211yescharacteristic
34#65305C411yescharacteristic
35#65305C411yescharacteristic
36C4×#686C2211yescharacteristic
37C4×#686C2211yescharacteristic
38#65298C411yescharacteristic
39#65298C411yescharacteristic
40#65259C411yescharacteristic
41#65257C411yescharacteristic
42#65214C2211yescharacteristic
43#65214C2211yescharacteristic
44#65214C2211yescharacteristic
45#64998—21no—
46#64997—21no—
47#64977C2211yescharacteristic, complemented
48C8×D20C2211yescharacteristic, complemented
49#64968C2211yescharacteristic, complemented
50C8×Dic10C2211yescharacteristic, complemented
51D4⋊#168C2211yescharacteristic
52D4⋊#168C2211yescharacteristic
53D4⋊#168C2211yescharacteristic
54D4⋊#168C2211yescharacteristic
55#64704C2211yescharacteristic, complemented
56#64704C2211yescharacteristic, complemented
57#64703C2211yescharacteristic, complemented
58#64703C2211yescharacteristic, complemented
59#64700C2211yescharacteristic, complemented
60#64700C2211yescharacteristic, complemented
61#64699C2211yescharacteristic, complemented
62#64699C2211yescharacteristic, complemented
63#64675C2211yescharacteristic, complemented
64#64674C2211yescharacteristic, complemented
65#64674C2211yescharacteristic, complemented
66C8×#168C2211yescharacteristic, complemented
67#18006—51nomaximal, Sylow
68C22×C5×D4C2×C411yescharacteristic
69C23×C4×C5—21no—
70C2×C5×M16—21no—
71C2×C5×M16—21no—
72C2×C5×M16—21no—
73C2×C5×M16—44no—
74C2×C5×M16D411yescharacteristic
75C2×C5×M16D411yescharacteristic
76C22×C5×C8—21no—
77C22×C5×C8—21no—
78C4×C5×D4—21no—
79C4×C5×D4—21no—
80C4×C5×D4C2×C411yescharacteristic
81C4×C5×D4C2×C411yescharacteristic
82C4×C5×D4C2311yescharacteristic
83C4×C5×D4C2311yescharacteristic
84C2×C5×C4⋊C4C2×C411yescharacteristic
85C2×C5×C22⋊C4—21no—
86C2×C42×C5C2311yescharacteristic
87C2×#687C2×C411yescharacteristic
88C2×#687C2×C411yescharacteristic
89C2×#686C2×C411yescharacteristic
90C2×#686C2×C411yescharacteristic
91#4076C2×C411yescharacteristic
92#4075—21no—
93C4×C22⋊D5—21no—
94#4070C2×C411yescharacteristic
95C2×#676—21no—
96C2×C8×D5—21no—
97#4022—21no—
98#4021C2×C411yescharacteristic
99C4×D20C2311yescharacteristic
100#4018—21no—
101#4016C2×C411yescharacteristic
102C4×Dic10C2311yescharacteristic
103C5×C4⋊C8—21no—
104C5×C4⋊C8C2311yescharacteristic
105C5×C4⋊C8C2311yescharacteristic
106C5×C22⋊C8—21no—
107C5×C22⋊C8—21no—
108C5×C22⋊C8—21no—
109C5×C22⋊C8—21no—
110C5×#97D411yescharacteristic
111C5×#97D411yescharacteristic
112C4×C5×C8C2311yescharacteristic
113C4×C5×C8C2311yescharacteristic
114#3964C2×C411yescharacteristic
115#3964C2×C411yescharacteristic
116#3964C2×C411yescharacteristic
117#3964C2×C411yescharacteristic
118C22⋊#168—21no—
119C22⋊#168—21no—
120C22⋊#168C2×C411yescharacteristic
121C22⋊#168C2×C411yescharacteristic
122#3952—21no—
123#3952—21no—
124#3947—21no—
125#3946—21no—
126#3946—21no—
127C8×Dic5—21no—
128#3942C2×C411yescharacteristic
129#3942C2×C411yescharacteristic
130#3941C2×C411yescharacteristic
131#3941C2×C411yescharacteristic
132#3940C2×C411yescharacteristic
133#3940C2×C411yescharacteristic
134#3939C2×C411yescharacteristic
135#3939C2×C411yescharacteristic
136C4⋊#168—21no—
137C4⋊#168C2×C411yescharacteristic
138C4⋊#168C2×C411yescharacteristic
139C4⋊#168C2311yescharacteristic
140C4⋊#168C2311yescharacteristic
141C4×#168—21no—
142C4×#168C2311yescharacteristic
143C4×#168C2311yescharacteristic
144C4×#168C2311yescharacteristic
145C4×#168C2311yescharacteristic
146#3105—51no—
147#2969—51no—
148D4×M16D511yescharacteristic, complemented
149#1594—51no—
150#1590—51no—
151#1588—51no—
152#1586—51no—
153#1582—51no—
154#1581—51no—
155SD16⋊C8—51no—
156C8×D8—51no—
157#1574—51no—
158#1495—51no—
159#1478—51no—
160#1453—51no—
161C24×C5—21no—
162C2×C5×D4—22no—
163C2×C5×D4—21no—
164C2×C5×D4—41no—
165C2×C5×D4C22×C411yescharacteristic
166C2×C5×D4C22×C411yescharacteristic
167C22×C4×C5—21no—
168C22×C4×C5—21no—
169C22×C4×C5—21no—
170C22×C4×C5—21no—
171C22×C4×C5—21no—
172C22×C4×C5—42no—
173C22×C4×C5—42no—
174C22×C4×C5C22×C411yescharacteristic
175C22×C4×C5C2×D411yescharacteristic
176C22×C4×C5#4111yescharacteristic
177C2×C22⋊D5—21no—
178C2×D20C22×C411yescharacteristic
179C2×C4×D5—21no—
180C2×Dic10C22×C411yescharacteristic
181C5×M16—22no—
182C5×M16—41no—
183C5×M16—42no—
184C5×M16—42no—
185C5×M16—42no—
186C2×C5×C8—21no—
187C2×C5×C8—21no—
188C2×C5×C8—42no—
189C2×C5×C8—42no—
190C2×C5×C8C2×D411yescharacteristic
191C2×C5×C8C2×D411yescharacteristic
192C2×C5×C8C2×D411yescharacteristic
193C2×C5×C8C2×D411yescharacteristic
194C5×C4⋊C4—21no—
195C5×C4⋊C4C22×C411yescharacteristic
196C5×C4⋊C4C22×C411yescharacteristic
197C5×C22⋊C4—21no—
198C5×C22⋊C4—21no—
199C5×C22⋊C4—21no—
200C5×C22⋊C4—21no—
201C42×C5C22×C411yescharacteristic
202C42×C5C2×D411yescharacteristic
203C42×C5#4111yescharacteristic
204C42×C5C2411yescharacteristic
205C22⋊Dic5—21no—
206#687—22no—
207#687—41no—
208#686—22no—
209#686—41no—
210#685—21no—
211#685—21no—
212#685—21no—
213#685—21no—
214C4⋊Dic5C22×C411yescharacteristic
215C4⋊Dic5C22×C411yescharacteristic
216#683—21no—
217#683—21no—
218#683—21no—
219#683—21no—
220C4×Dic5—21no—
221C2×#168—21no—
222C2×#168—21no—
223C2×#168—21no—
224C2×#168—21no—
225C2×#168C22×C411yescharacteristic
226C2×#168C22×C411yescharacteristic
227C2×#168C22×C411yescharacteristic
228C2×#168C22×C411yescharacteristic
229#676—42no—
230C8×D5—42no—
231C22×M16—21no—
232#529—51no—
233C2×C4×D4C2×D511yescharacteristic
234#484—51no—
235#481—51no—
236#478—51no—
237#478—51no—
238#463—51no—
239#463—51no—
240#450—51no—
241#447—51no—
242#446—51no—
243C8×Q8—51no—
244C4×SD16—51no—
245C4×SD16—51no—
246C4×D8—51no—
247C4×D8—51no—
248#436—51no—
249#436—101no—
250#436C2×D511yescharacteristic
251#436C2×D511yescharacteristic
252#435—21no—
253#435—21no—
254#435—101no—
255C8×D4—51no—
256C8×D4C2×D511yescharacteristic
257C8×D4C2×D511yescharacteristic
258#433—101no—
259#431—51no—
260#431—51no—
261#431—51no—
262#423C2×D511yescharacteristic
263#407—21no—
264#405—101no—
265C4×M16C2×D511yescharacteristic
266#335—51no—
267#335—51no—
268#334—51no—
269#334—51no—
270Q8⋊C8—51no—
271Q8⋊C8—51no—
272D4⋊C8—51no—
273D4⋊C8—51no—
274D4⋊C8—51no—
275D4⋊C8—51no—
276D4⋊C8—51no—
277D4⋊C8—51no—
278#322—51no—
279#322—51no—
280#322—51no—
281C82—51no—
282C23×C5—21no—
283C23×C5—21no—
284C23×C5—42no—
285C23×C5—42no—
286C23×C5—81no—
287C23×C5C4×D411yescharacteristic
288C22×D5—21no—
289C5×D4—21no—
290C5×D4—21no—
291C5×D4—21no—
292C5×D4—42no—
293C5×D4#13112yes—
294C2×C4×C5—21no—
295C2×C4×C5—21no—
296C2×C4×C5—21no—
297C2×C4×C5—21no—
298C2×C4×C5—21no—
299C2×C4×C5—21no—
300C2×C4×C5—41no—
301C2×C4×C5—42no—
302C2×C4×C5—42no—
303C2×C4×C5—42no—
304C2×C4×C5—41no—
305C2×C4×C5—41no—
306C2×C4×C5—41no—
307C2×C4×C5—41no—
308C2×C4×C5C4×D411yescharacteristic
309C2×C4×C5C4×D411yescharacteristic
310C2×C4×C5C4×D411yescharacteristic
311C2×C4×C5C23×C411yescharacteristic
312C2×C4×C5C22×D411yescharacteristic
313C2×C4×C5C2×#4111yescharacteristic
314C22⋊D5—81no—
315C2×Dic5—21no—
316C2×Dic5—21no—
317C2×Dic5—21no—
318D20—21no—
319D20—21no—
320C4×D5—41no—
321C4×D5—41no—
322Dic10—21no—
323Dic10—21no—
324C5×C8—21no—
325C5×C8—21no—
326C5×C8—21no—
327C5×C8—41no—
328C5×C8—41no—
329C5×C8AGL(1,ℤ/8)12yes—
330#168—21no—
331#168—21no—
332#168—21no—
333#168—21no—
334#168—41no—
335#168—41no—
336#168—41no—
337#168—41no—
338C22×D4C4×D511yescharacteristic
339C23×C4—21no—
340C2×SD16—51no—
341C2×SD16—51no—
342C2×D8—51no—
343C2×D8—51no—
344C2×M16—21no—
345C2×M16—21no—
346C2×M16—21no—
347C2×M16—44no—
348C2×M16—101no—
349C2×M16C22⋊D511yescharacteristic
350C2×M16C22⋊D511yescharacteristic
351C22×C8—21no—
352C22×C8—21no—
353C22×C8—101no—
354#126—101no—
355#125—51no—
356#124—51no—
357#123—101no—
358C22⋊Q8—51no—
359#121—51no—
360C4×Q8—51no—
361C4×D4—21no—
362C4×D4—21no—
363C4×D4—51no—
364C4×D4—101no—
365C4×D4C4×D511yescharacteristic
366C4×D4C4×D511yescharacteristic
367C4×D4C22×D511yescharacteristic
368C4×D4C22×D511yescharacteristic
369#117—101no—
370C2×C4⋊C4C4×D511yescharacteristic
371C2×C22⋊C4—21no—
372C2×C42C22×D511yescharacteristic
373#107—51no—
374#107—51no—
375#106—51no—
376#106—51no—
377C4⋊C8—21no—
378C4⋊C8—51no—
379C4⋊C8—51no—
380C4⋊C8—51no—
381C4⋊C8—51no—
382C4⋊C8—101no—
383C4⋊C8—101no—
384C4⋊C8—101no—
385C4⋊C8C22×D511yescharacteristic
386C4⋊C8C22×D511yescharacteristic
387#103—51no—
388#103—51no—
389#102—51no—
390#102—51no—
391#102—51no—
392#102—51no—
393#102—51no—
394#102—51no—
395C22⋊C8—21no—
396C22⋊C8—21no—
397C22⋊C8—21no—
398C22⋊C8—21no—
399C22⋊C8—51no—
400C22⋊C8—51no—
401C22⋊C8—101no—
402C22⋊C8—101no—
403C22⋊C8—101no—
404C22⋊C8—101no—
405#97—101no—
406#97C22⋊D511yescharacteristic
407#97C22⋊D511yescharacteristic
408C4×C8—51no—
409C4×C8—51no—
410C4×C8—51no—
411C4×C8—51no—
412C4×C8—101no—
413C4×C8—101no—
414C4×C8C22×D511yescharacteristic
415C4×C8C22×D511yescharacteristic
416C22×C5—21no—
417C22×C5—22no—
418C22×C5—21no—
419C22×C5—21no—
420C22×C5—41no—
421C22×C5—41no—
422C22×C5—41no—
423C22×C5—41no—
424C22×C5—41no—
425C22×C5—84no—
426C22×C5C2×C4×D411yescharacteristic
427C2×D5—41no—
428C2×D5—41no—
429C2×D5—41no—
430C4×C5—21no—
431C4×C5—21no—
432C4×C5—21no—
433C4×C5—21no—
434C4×C5—41no—
435C4×C5—41no—
436C4×C5C2×#13111yescharacteristic
437C4×C5#56811yescharacteristic
438C4×C5C2×AGL(1,ℤ/8)11yescharacteristic
439C4×C5#57511yescharacteristic
440Dic5—41no—
441Dic5—41no—
442Dic5—41no—
443C24—21no—
444C2×Q8—51no—
445C2×D4—22no—
446C2×D4—21no—
447C2×D4—41no—
448C2×D4—51no—
449C2×D4—101no—
450C2×D4C2×C4×D511yescharacteristic
451C2×D4C2×C4×D511yescharacteristic
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—101no—
460C22×C4C2×C4×D511yescharacteristic
461C22×C4#70911yescharacteristic
462C22×C4C2×C22⋊D511yescharacteristic
463SD16—102no—
464SD16—201no—
465D8—102no—
466D8—201no—
467M16—22no—
468M16—41no—
469M16—42no—
470M16—42no—
471M16—42no—
472M16—202no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—42no—
476C2×C8—42no—
477C2×C8—51no—
478C2×C8—51no—
479C2×C8—51no—
480C2×C8—51no—
481C2×C8—101no—
482C2×C8—101no—
483C2×C8—101no—
484C2×C8—101no—
485C2×C8—202no—
486C2×C8C2×C22⋊D511yescharacteristic
487C2×C8C2×C22⋊D511yescharacteristic
488C2×C8C2×C22⋊D511yescharacteristic
489C2×C8C2×C22⋊D511yescharacteristic
490C4⋊C4—21no—
491C4⋊C4—51no—
492C4⋊C4—51no—
493C4⋊C4—101no—
494C4⋊C4—101no—
495C4⋊C4—101no—
496C4⋊C4—101no—
497C4⋊C4C2×C4×D511yescharacteristic
498C4⋊C4C2×C4×D511yescharacteristic
499C22⋊C4—21no—
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—101no—
504C22⋊C4—101no—
505C22⋊C4—101no—
506C22⋊C4—101no—
507C22⋊C4—101no—
508C42—101no—
509C42C2×C4×D511yescharacteristic
510C42#70911yescharacteristic
511C42C2×C22⋊D511yescharacteristic
512C42C23×D511yescharacteristic
513C2×C5—41no—
514C2×C5—42no—
515C2×C5—81no—
516C2×C5#293311yescharacteristic
517C2×C5C4×AGL(1,ℤ/8)11yescharacteristic
518C2×C5#295711yescharacteristic
519D5—81no—
520C23—21no—
521C23—21no—
522C23—42no—
523C23—42no—
524C23—81no—
525C23—101no—
526C23C4×C22⋊D511yescharacteristic
527Q8—101no—
528Q8—101no—
529D4—21no—
530D4—21no—
531D4—21no—
532D4—42no—
533D4—101no—
534D4—101no—
535D4—401no—
536D4#405312yes—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—101no—
552C2×C4—101no—
553C2×C4—101no—
554C2×C4—201no—
555C2×C4—201no—
556C2×C4C4×C22⋊D511yescharacteristic
557C2×C4C4×C22⋊D511yescharacteristic
558C2×C4C4×C22⋊D511yescharacteristic
559C2×C4C22×C4×D511yescharacteristic
560C2×C4C2×#70911yescharacteristic
561C2×C4C22×C22⋊D511yescharacteristic
562C8—21no—
563C8—21no—
564C8—21no—
565C8—41no—
566C8—41no—
567C8—101no—
568C8—101no—
569C8—101no—
570C8—101no—
571C8—201no—
572C8—201no—
573C8—201no—
574C8—201no—
575C8#407812yes—
576C5#1800611yescharacteristic, complemented, Sylow
577C22—21no—
578C22—21no—
579C22—22no—
580C22—21no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—84no—
587C22—201no—
588C22—201no—
589C22—201no—
590C22C2×C4×C22⋊D511yescharacteristic
591C4—21no—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—41no—
596C4—41no—
597C4—201no—
598C4—201no—
599C4—201no—
600C4C2×#405311yescharacteristic
601C4#6607911yescharacteristic
602C4C2×#407811yescharacteristic
603C4#6615611yescharacteristic
604C2—41no—
605C2—42no—
606C2—81no—
607C2—401no—
608C2C4×#407811yescharacteristic
609C2#1061548811yescharacteristic
610C2#1061554311yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D5
commutatorC2×C4×C5C23×C4
FrattiniC42C23×D5
FittingC5×D4×M16C2
socleC22×C5C2×C4×D4
prime radical#65506C4
cosocleC42×C5C24
hypercenterD4×M16D5
hypocenterC5#18006
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ D4×M16

Complements

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

NH ≅ G/Ntype
1. C5×D4×M16C2semidirect
3. #10610242C2semidirect
4. #10602390C2semidirect
5. #10601659C2semidirect
6. #10601658C2semidirect
7. #10601655C2semidirect
8. #10601654C2semidirect
9. #10601637C2semidirect
10. #10601635C2semidirect
11. #10601632C2semidirect
12. C8×#686C2semidirect
13. #10601292C2semidirect
14. #10601288C2semidirect
15. #10600641C2semidirect
47. #64977C22semidirect
48. C8×D20C22semidirect
49. #64968C22semidirect
50. C8×Dic10C22semidirect
55. #64704C22semidirect
56. #64704C22semidirect
57. #64703C22semidirect
58. #64703C22semidirect
59. #64700C22semidirect
60. #64700C22semidirect
61. #64699C22semidirect
62. #64699C22semidirect
63. #64675C22semidirect
64. #64674C22semidirect
65. #64674C22semidirect
66. C8×#168C22semidirect
148. D4×M16D5semidirect
576. C5#18006semidirect

Maximal subgroups

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

subgroupindex
C5×D4×M162
#106140402
#106102422
#106023902
#106016592
#106016582
#106016552
#106016542
#106016372
#106016352
#106016322
C8×#6862
#106012922
#106012882
#106006412
#180065