Finite Group Databasebeta

← #12070338

Subgroups — #12070338

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×#3105C211yesmaximal, characteristic
2#10614860C211yesmaximal, characteristic, complemented
3D4×#681C211yesmaximal, characteristic, complemented
4#10602546C211yesmaximal, characteristic, complemented
5#10602390C211yesmaximal, characteristic, complemented
6#10601692C211yesmaximal, characteristic, complemented
7#10601691C211yesmaximal, characteristic, complemented
8#10601688C211yesmaximal, characteristic, complemented
9SD16⋊#168C211yesmaximal, characteristic, complemented
10#10601675C211yesmaximal, characteristic, complemented
11#10601673C211yesmaximal, characteristic, complemented
12#10601672C211yesmaximal, characteristic, complemented
13D8×#168C211yesmaximal, characteristic, complemented
14#10601546C211yesmaximal, characteristic, complemented
15#10601536C211yesmaximal, characteristic, complemented
16C5×#529C2211yescharacteristic
17C2×C4×C5×D4C2211yescharacteristic
18C22×#681—21no—
19C5×#484C411yescharacteristic
20C5×#481C411yescharacteristic
21C5×#478C411yescharacteristic
22C5×#478C411yescharacteristic
23C5×#463C411yescharacteristic
24C5×#463C411yescharacteristic
25C5×#450C411yescharacteristic
26C5×#447C411yescharacteristic
27C4×C5×SD16C2211yescharacteristic
28C4×C5×SD16C2211yescharacteristic
29C4×C5×D8C2211yescharacteristic
30C4×C5×D8C2211yescharacteristic
31C5×#431C2211yescharacteristic
32#65496—21no—
33#65313C2211yescharacteristic, complemented
34Q8×#168C2211yescharacteristic, complemented
35#65301—21no—
36#65301C2211yescharacteristic
37#65301C2211yescharacteristic
38#65301C2211yescharacteristic, complemented
39#65300—21no—
40#65300—21no—
41#65300—21no—
42D4×#168C2211yescharacteristic
43D4×#168C2211yescharacteristic
44D4×#168C2211yescharacteristic, complemented
45#65289—21no—
46#65286—21no—
47#65214C2211yescharacteristic
48#65214C2211yescharacteristic
49#65213C2211yescharacteristic
50C4×#681C2211yescharacteristic
51Q8⋊#168C2211yescharacteristic, complemented
52Q8⋊#168C2211yescharacteristic, complemented
53D4⋊#168C2211yescharacteristic
54D4⋊#168C2211yescharacteristic
55D4⋊#168C2211yescharacteristic
56D4⋊#168C2211yescharacteristic
57D4⋊#168C2211yescharacteristic, complemented
58D4⋊#168C2211yescharacteristic, complemented
59#64678C2211yescharacteristic, complemented
60#64678C2211yescharacteristic, complemented
61#64677C2211yescharacteristic, complemented
62#64677C2211yescharacteristic, 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×SD16C2×C411yescharacteristic
71C2×C5×SD16C2×C411yescharacteristic
72C2×C5×D8C2×C411yescharacteristic
73C2×C5×D8C2×C411yescharacteristic
74C5×#126—21no—
75C5×#125C2×C411yescharacteristic
76C5×#124C2×C411yescharacteristic
77C5×#123—21no—
78C5×C22⋊Q8C2×C411yescharacteristic
79C5×#121C2×C411yescharacteristic
80C4×C5×Q8C2311yescharacteristic
81C4×C5×D4—21no—
82C4×C5×D4—21no—
83C4×C5×D4—21no—
84C4×C5×D4C2×C411yescharacteristic
85C4×C5×D4C2×C411yescharacteristic
86C4×C5×D4C2311yescharacteristic
87C4×C5×D4C2311yescharacteristic
88C4×C5×D4C2311yescharacteristic
89C5×#117—21no—
90C2×C5×C4⋊C4C2×C411yescharacteristic
91C2×C5×C22⋊C4—21no—
92C2×C42×C5C2311yescharacteristic
93C2×#681—21no—
94C2×#681—21no—
95C2×#681—21no—
96C2×#681—21no—
97C2×#681—44no—
98C2×#681D411yescharacteristic
99C2×#681D411yescharacteristic
100C22×#168—21no—
101C22×#168—21no—
102C22×#168—21no—
103C5×#107C2×C411yescharacteristic
104C5×#107C2×C411yescharacteristic
105C5×#106C2×C411yescharacteristic
106C5×#106C2×C411yescharacteristic
107C5×C4⋊C8C2×C411yescharacteristic
108C5×C4⋊C8C2×C411yescharacteristic
109C5×#103C2×C411yescharacteristic
110C5×#103C2×C411yescharacteristic
111C5×#102C2×C411yescharacteristic
112C5×#102C2×C411yescharacteristic
113C5×#102C2×C411yescharacteristic
114C5×#102C2×C411yescharacteristic
115C5×#102C2×C411yescharacteristic
116C5×#102C2×C411yescharacteristic
117C5×C22⋊C8C2×C411yescharacteristic
118C5×C22⋊C8C2×C411yescharacteristic
119C4×C5×C8C2311yescharacteristic
120C4×C5×C8C2311yescharacteristic
121C22⋊#168—21no—
122C22⋊#168—21no—
123C22⋊#168—21no—
124C22⋊#168—21no—
125C22⋊#168—21no—
126C22⋊#168—21no—
127C22⋊#168—21no—
128C22⋊#168—21no—
129C4⋊#168—21no—
130C4⋊#168—21no—
131C4⋊#168—21no—
132C4⋊#168—21no—
133C4⋊#168C2311yescharacteristic
134C4⋊#168C2311yescharacteristic
135C4⋊#168C2311yescharacteristic
136C4⋊#168C2311yescharacteristic
137#3935—21no—
138#3935D411yescharacteristic
139#3935D411yescharacteristic
140C4×#168—21no—
141C4×#168—21no—
142C4×#168C2311yescharacteristic
143C4×#168C2311yescharacteristic
144C4×#168C2311yescharacteristic
145C4×#168C2311yescharacteristic
146#3105D511yescharacteristic
147#2969—51no—
148D4×M16—51no—
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×Q8C22×C411yescharacteristic
163C2×C5×D4—22no—
164C2×C5×D4—21no—
165C2×C5×D4—21no—
166C2×C5×D4—41no—
167C2×C5×D4C22×C411yescharacteristic
168C2×C5×D4C22×C411yescharacteristic
169C2×C5×D4C22×C411yescharacteristic
170C22×C4×C5—21no—
171C22×C4×C5—21no—
172C22×C4×C5—21no—
173C22×C4×C5—21no—
174C22×C4×C5—21no—
175C22×C4×C5—21no—
176C22×C4×C5—42no—
177C22×C4×C5—42no—
178C22×C4×C5C22×C411yescharacteristic
179C22×C4×C5C2×D411yescharacteristic
180C22×C4×C5#4111yescharacteristic
181C5×SD16—22no—
182C5×SD16—41no—
183C5×D8—22no—
184C5×D8—41no—
185C2×C5×C8C22×C411yescharacteristic
186C2×C5×C8C22×C411yescharacteristic
187C2×C5×C8C22×C411yescharacteristic
188C2×C5×C8C22×C411yescharacteristic
189C5×C4⋊C4—21no—
190C5×C4⋊C4—21no—
191C5×C4⋊C4—21no—
192C5×C4⋊C4—21no—
193C5×C4⋊C4—21no—
194C5×C4⋊C4C22×C411yescharacteristic
195C5×C4⋊C4C22×C411yescharacteristic
196C5×C4⋊C4C22×C411yescharacteristic
197C5×C4⋊C4C22×C411yescharacteristic
198C5×C22⋊C4—21no—
199C5×C22⋊C4—21no—
200C5×C22⋊C4—21no—
201C5×C22⋊C4—21no—
202C5×C22⋊C4—21no—
203C5×C22⋊C4—21no—
204C5×C22⋊C4—21no—
205C5×C22⋊C4—21no—
206C5×C22⋊C4—21no—
207C42×C5—21no—
208C42×C5C22×C411yescharacteristic
209C42×C5C2×D411yescharacteristic
210C42×C5#4111yescharacteristic
211C42×C5C2411yescharacteristic
212#681—22no—
213#681—41no—
214#681—42no—
215#681—42no—
216#681—42no—
217#681—42no—
218C2×#168—21no—
219C2×#168—21no—
220C2×#168—21no—
221C2×#168—21no—
222C2×#168—21no—
223C2×#168—21no—
224C2×#168—42no—
225C2×#168—42no—
226C2×#168—42no—
227C2×#168C2×D411yescharacteristic
228C2×#168C2×D411yescharacteristic
229C2×#168C2×D411yescharacteristic
230C2×#168C2×D411yescharacteristic
231C22×M16—101no—
232#529C2×D511yescharacteristic
233C2×C4×D4C2×D511yescharacteristic
234#484Dic511yescharacteristic
235#481Dic511yescharacteristic
236#478Dic511yescharacteristic
237#478Dic511yescharacteristic
238#463Dic511yescharacteristic
239#463Dic511yescharacteristic
240#450Dic511yescharacteristic
241#447Dic511yescharacteristic
242#446—51no—
243C8×Q8—51no—
244C4×SD16C2×D511yescharacteristic
245C4×SD16C2×D511yescharacteristic
246C4×D8C2×D511yescharacteristic
247C4×D8C2×D511yescharacteristic
248#436—51no—
249#436—51no—
250#436—51no—
251#436—101no—
252#435—101no—
253#435—101no—
254#435—101no—
255C8×D4—51no—
256C8×D4—51no—
257C8×D4—51no—
258#433—101no—
259#431—51no—
260#431—51no—
261#431C2×D511yescharacteristic
262#423—51no—
263#407—101no—
264#405—101no—
265C4×M16—51no—
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—21no—
285C23×C5—42no—
286C23×C5—42no—
287C23×C5—81no—
288C23×C5C4×D411yescharacteristic
289C5×Q8—21no—
290C5×Q8—21no—
291C5×D4—21no—
292C5×D4—21no—
293C5×D4—21no—
294C5×D4—21no—
295C5×D4—21no—
296C5×D4—42no—
297C5×D4—81no—
298C5×D4#13112yes—
299C2×C4×C5—21no—
300C2×C4×C5—21no—
301C2×C4×C5—21no—
302C2×C4×C5—21no—
303C2×C4×C5—21no—
304C2×C4×C5—21no—
305C2×C4×C5—21no—
306C2×C4×C5—21no—
307C2×C4×C5—21no—
308C2×C4×C5—41no—
309C2×C4×C5—42no—
310C2×C4×C5—42no—
311C2×C4×C5—42no—
312C2×C4×C5—41no—
313C2×C4×C5—41no—
314C2×C4×C5—41no—
315C2×C4×C5—41no—
316C2×C4×C5—41no—
317C2×C4×C5—41no—
318C2×C4×C5C4×D411yescharacteristic
319C2×C4×C5C4×D411yescharacteristic
320C2×C4×C5C4×D411yescharacteristic
321C2×C4×C5C23×C411yescharacteristic
322C2×C4×C5C22×D411yescharacteristic
323C2×C4×C5C2×#4111yescharacteristic
324C5×C8—21no—
325C5×C8—21no—
326C5×C8—21no—
327C5×C8—21no—
328#168—21no—
329#168—21no—
330#168—21no—
331#168—41no—
332#168—41no—
333#168—41no—
334#168—41no—
335#168—41no—
336#168—41no—
337#168AGL(1,ℤ/8)12yes—
338C22×D4C2×Dic511yescharacteristic
339C23×C4—21no—
340C2×SD16C2×Dic511yescharacteristic
341C2×SD16C2×Dic511yescharacteristic
342C2×D8C2×Dic511yescharacteristic
343C2×D8C2×Dic511yescharacteristic
344C2×M16—51no—
345C2×M16—51no—
346C2×M16—101no—
347C2×M16—101no—
348C2×M16—101no—
349C2×M16—101no—
350C2×M16—204no—
351C22×C8—101no—
352C22×C8—101no—
353C22×C8—101no—
354#126—21no—
355#125C2×Dic511yescharacteristic
356#124C2×Dic511yescharacteristic
357#123—21no—
358C22⋊Q8C2×Dic511yescharacteristic
359#121C2×Dic511yescharacteristic
360C4×Q8C22×D511yescharacteristic
361C4×D4—21no—
362C4×D4—21no—
363C4×D4—21no—
364C4×D4C2×Dic511yescharacteristic
365C4×D4C2×Dic511yescharacteristic
366C4×D4C22×D511yescharacteristic
367C4×D4C22×D511yescharacteristic
368C4×D4C22×D511yescharacteristic
369#117—21no—
370C2×C4⋊C4C2×Dic511yescharacteristic
371C2×C22⋊C4—21no—
372C2×C42C22×D511yescharacteristic
373#107C2×Dic511yescharacteristic
374#107C2×Dic511yescharacteristic
375#106C2×Dic511yescharacteristic
376#106C2×Dic511yescharacteristic
377C4⋊C8—51no—
378C4⋊C8—51no—
379C4⋊C8—51no—
380C4⋊C8—51no—
381C4⋊C8—101no—
382C4⋊C8—101no—
383C4⋊C8—101no—
384C4⋊C8—101no—
385C4⋊C8C2×Dic511yescharacteristic
386C4⋊C8C2×Dic511yescharacteristic
387#103C2×Dic511yescharacteristic
388#103C2×Dic511yescharacteristic
389#102C2×Dic511yescharacteristic
390#102C2×Dic511yescharacteristic
391#102C2×Dic511yescharacteristic
392#102C2×Dic511yescharacteristic
393#102C2×Dic511yescharacteristic
394#102C2×Dic511yescharacteristic
395C22⋊C8—101no—
396C22⋊C8—101no—
397C22⋊C8—101no—
398C22⋊C8—101no—
399C22⋊C8—101no—
400C22⋊C8—101no—
401C22⋊C8—101no—
402C22⋊C8—101no—
403C22⋊C8C2×Dic511yescharacteristic
404C22⋊C8C2×Dic511yescharacteristic
405#97—51no—
406#97—51no—
407#97—101no—
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—41no—
426C22×C5—41no—
427C22×C5—41no—
428C22×C5—84no—
429C22×C5C2×C4×D411yescharacteristic
430C4×C5—21no—
431C4×C5—21no—
432C4×C5—21no—
433C4×C5—21no—
434C4×C5—41no—
435C4×C5—41no—
436C4×C5—41no—
437C4×C5—41no—
438C4×C5—41no—
439C4×C5C2×#13111yescharacteristic
440C4×C5#56811yescharacteristic
441C4×C5C2×AGL(1,ℤ/8)11yescharacteristic
442C4×C5#57511yescharacteristic
443C24—21no—
444C2×Q8C22×Dic511yescharacteristic
445C2×D4—22no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—41no—
449C2×D4C22×Dic511yescharacteristic
450C2×D4C22×Dic511yescharacteristic
451C2×D4C22×Dic511yescharacteristic
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4D4×D511yescharacteristic
461C22×C4#71111yescharacteristic
462C22×C4C22×Dic511yescharacteristic
463SD16—22no—
464SD16—41no—
465D8—22no—
466D8—41no—
467M16—102no—
468M16—202no—
469M16—201no—
470M16—202no—
471M16—202no—
472M16—202no—
473C2×C8—51no—
474C2×C8—51no—
475C2×C8—51no—
476C2×C8—51no—
477C2×C8—101no—
478C2×C8—101no—
479C2×C8—101no—
480C2×C8—101no—
481C2×C8—101no—
482C2×C8—101no—
483C2×C8—202no—
484C2×C8—202no—
485C2×C8—202no—
486C2×C8C22×Dic511yescharacteristic
487C2×C8C22×Dic511yescharacteristic
488C2×C8C22×Dic511yescharacteristic
489C2×C8C22×Dic511yescharacteristic
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4C22×Dic511yescharacteristic
496C4⋊C4C22×Dic511yescharacteristic
497C4⋊C4C22×Dic511yescharacteristic
498C4⋊C4C22×Dic511yescharacteristic
499C22⋊C4—21no—
500C22⋊C4—21no—
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—21no—
508C42—21no—
509C42D4×D511yescharacteristic
510C42#71111yescharacteristic
511C42C22×Dic511yescharacteristic
512C42C23×D511yescharacteristic
513C2×C5—41no—
514C2×C5—42no—
515C2×C5—81no—
516C2×C5—81no—
517C2×C5#293311yescharacteristic
518C2×C5C4×AGL(1,ℤ/8)11yescharacteristic
519C2×C5#295711yescharacteristic
520C23—21no—
521C23—21no—
522C23—21no—
523C23—42no—
524C23—42no—
525C23—81no—
526C23D4×Dic511yescharacteristic
527Q8—21no—
528Q8—21no—
529D4—21no—
530D4—21no—
531D4—21no—
532D4—21no—
533D4—21no—
534D4—42no—
535D4—81no—
536D4#409412yes—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4D4×Dic511yescharacteristic
557C2×C4D4×Dic511yescharacteristic
558C2×C4D4×Dic511yescharacteristic
559C2×C4C2×D4×D511yescharacteristic
560C2×C4C2×#71111yescharacteristic
561C2×C4C23×Dic511yescharacteristic
562C8—21no—
563C8—21no—
564C8—21no—
565C8—21no—
566C8—52no—
567C8—101no—
568C8—101no—
569C8—101no—
570C8—201no—
571C8—201no—
572C8—201no—
573C8—201no—
574C8—201no—
575C8—201no—
576C5#1800611yescharacteristic, complemented, Sylow
577C22—21no—
578C22—21no—
579C22—21no—
580C22—22no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—84no—
590C22C2×D4×Dic511yescharacteristic
591C4—21no—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—41no—
599C4—41no—
600C4D5×AGL(1,ℤ/8)11yescharacteristic
601C4#6610711yescharacteristic
602C4C2×#409411yescharacteristic
603C4#6615311yescharacteristic
604C2—41no—
605C2—42no—
606C2—81no—
607C2—81no—
608C2#1061464911yescharacteristic
609C2#1061564111yescharacteristic
610C2Dic5×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D5
commutatorC2×C4×C5C23×C4
FrattiniC42C23×D5
FittingC5×#3105C2
socleC22×C5C2×C4×D4
prime radicalC5×#447C4
cosocleC42×C5C24
hypercenter#3105D5
hypocenterC5#18006
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3105

Complements

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

NH ≅ G/Ntype
2. #10614860C2semidirect
3. D4×#681C2semidirect
4. #10602546C2semidirect
5. #10602390C2semidirect
6. #10601692C2semidirect
7. #10601691C2semidirect
8. #10601688C2semidirect
9. SD16⋊#168C2semidirect
10. #10601675C2semidirect
11. #10601673C2semidirect
12. #10601672C2semidirect
13. D8×#168C2semidirect
14. #10601546C2semidirect
15. #10601536C2semidirect
33. #65313C22semidirect
34. Q8×#168C22semidirect
38. #65301C22semidirect
44. D4×#168C22semidirect
51. Q8⋊#168C22semidirect
52. Q8⋊#168C22semidirect
57. D4⋊#168C22semidirect
58. D4⋊#168C22semidirect
59. #64678C22semidirect
60. #64678C22semidirect
61. #64677C22semidirect
62. #64677C22semidirect
63. #64675C22semidirect
64. #64674C22semidirect
65. #64674C22semidirect
66. C8×#168C22semidirect
576. C5#18006semidirect

Maximal subgroups

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

subgroupindex
C5×#31052
#106148602
D4×#6812
#106025462
#106023902
#106016922
#106016912
#106016882
SD16⋊#1682
#106016752
#106016732
#106016722
D8×#1682
#106015462
#106015362
#180065