Finite Group Databasebeta

← #12260248

Subgroups — #12260248

Order 1280.

3106 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#65908C211yesmaximal, characteristic
2C2×#65908C211yesmaximal, characteristic
3C2×C4×#713C211yesmaximal, characteristic
4C5×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
5#10614192C211yesmaximal, characteristic, complemented
6#10614191C211yesmaximal, characteristic, complemented
7Q8×#683C211yesmaximal, characteristic, complemented
8#10607934C213yesmaximal, complemented
9#10607933C213yesmaximal, complemented
10#10607926C213yesmaximal, complemented
11D5×#389C213yesmaximal, complemented
12#10606091C213yesmaximal, complemented
13#10606090C213yesmaximal, complemented
14#10605501C213yesmaximal, complemented
15#10605486C213yesmaximal, complemented
16C22×#713C2211yescharacteristic
17C2×C4×C5×Q8C2211yescharacteristic
18C2×C4×C5×Q8C2211yescharacteristic
19C2×C4×C5×Q8C2211yescharacteristic
20C2×Q8×Dic5C2211yescharacteristic
21C2×Q8×Dic5C2211yescharacteristic
22C2×Q8×Dic5C2211yescharacteristic
23#65908—28no—
24#65908C4116yescomplemented
25C4×#713—28no—
26C2×#4039C2213yes—
27C2×#4039C2213yes—
28C2×#4039C2213yes—
29C2×#4038C2213yes—
30C2×D5×C4⋊C4C2213yes—
31C2×D5×C4⋊C4C2216yes—
32C2×D5×C4⋊C4C2213yes—
33C2×C4×D20C2213yes—
34C2×C4×D20C2213yes—
35C2×C4×D20C2213yes—
36C2×#4018C2213yes—
37C2×#4018C2213yes—
38C2×C42×D5C2213yes—
39C5×#389C2213yes—
40C5×#389C2213yes—
41C5×#384C2213yes—
42C4×C5×C4⋊C4C2213yes—
43#65278C2216yes—
44#65278C2216yes—
45#65278C2213yes—
46#65278C2213yes—
47#65276C2216yes—
48#65274C2213yes—
49#65274C2213yes—
50#65266C2213yes—
51#65265C2213yes—
52#65262C2213yes—
53#65262C2213yes—
54#65261C2213yes—
55#65261C2216yes—
56#65261C2211yescharacteristic
57#65225C2213yes—
58C4×C4⋊Dic5C2213yes—
59C4×#683C2213yes—
60C4×#683C2213yes—
61#65217C2213yes—
62#65217C2213yes—
63#64957C2213yes—
64#64957C2213yes—
65#64956C2216yes—
66D5×#95C2216yes—
67#64936C2213yes—
68#64936C2213yes—
69#34250—51nomaximal, Sylow
70C22×C5×Q8C2311yescharacteristic
71C2×#713C2×C418yes—
72C2×#713D418yes—
73C2×#713Q818yes—
74C22×D20C2313yes—
75C22×C4×D5C2313yes—
76C22×C4×D5C2313yes—
77C22×C4×D5C2316yes—
78C22×C4×D5C2316yes—
79C22×C4×D5C2313yes—
80C4×C5×Q8—24no—
81C4×C5×Q8—24no—
82C4×C5×Q8C2×C418yescomplemented
83C2×C5×C4⋊C4C2311yescharacteristic
84C2×C5×C4⋊C4C2313yes—
85C2×C5×C4⋊C4C2313yes—
86C2×C5×C4⋊C4C2313yes—
87C2×C5×C4⋊C4C2316yes—
88C2×C5×C4⋊C4C2313yes—
89C2×C42×C5C2313yes—
90C2×C42×C5C2313yes—
91C2×C42×C5C2313yes—
92Q8×Dic5—24no—
93Q8×Dic5—24no—
94Q8×Dic5C2×C418yescomplemented
95C2×#685C2313yes—
96C2×#685C2313yes—
97C2×#685C2316yes—
98C2×#685C2313yes—
99C2×#685C2313yes—
100C2×C4⋊Dic5C2313yes—
101C2×C4⋊Dic5C2313yes—
102C2×C4⋊Dic5C2313yes—
103C2×C4⋊Dic5C2313yes—
104C2×C4⋊Dic5C2313yes—
105C2×#683C2311yescharacteristic
106C2×#683C2313yes—
107C2×#683C2313yes—
108C2×#683C2311yescharacteristic
109C2×#683C2316yes—
110C2×#683C2313yes—
111C2×#683C2313yes—
112C2×#683C2311yescharacteristic
113C2×#683C2313yes—
114C2×C4×Dic5C2311yescharacteristic
115C2×C4×Dic5C2313yes—
116C2×C4×Dic5C2313yes—
117C2×C4×Dic5C2313yes—
118C2×C4×Dic5C2313yes—
119#4039—212no—
120#4039—212no—
121#4039C2×C4124yescomplemented
122#4038—212no—
123D5×C4⋊C4—212no—
124D5×C4⋊C4—412no—
125D5×C4⋊C4C2×C4124yescomplemented
126C4×D20—212no—
127C4×D20—212no—
128C4×D20C2×C4124yescomplemented
129#4018—212no—
130#4018C2×C4124yescomplemented
131C42×D5—212no—
132C5×#95C2313yes—
133C5×#95C2313yes—
134#3963C2313yes—
135#3963C2313yes—
136#3963C2316yes—
137#3963C2316yes—
138#3963C2311yescharacteristic
139#3963C2313yes—
140#3963C2311yescharacteristic
141#3963C2313yes—
142C2×#520—51no—
143C2×#520—51no—
144C2×C4×#41—51no—
145#2351—51no—
146#2351—51no—
147Q8×C4⋊C4—51no—
148Q8×C4⋊C4D511yescharacteristic, complemented
149#2349—53no—
150#2349—53no—
151D4×C4⋊C4—53no—
152D4×C4⋊C4—53no—
153#2317—53no—
154#2316—53no—
155#2296—53no—
156C2×#389—53no—
157C23×D5C2413yes—
158C2×C5×Q8C22×C414yes—
159C2×C5×Q8C2×D414yes—
160C2×C5×Q8C2×Q814yes—
161C22×C4×C5C2411yescharacteristic
162C22×C4×C5C2413yes—
163C22×C4×C5C2413yes—
164C22×C4×C5C2411yescharacteristic
165C22×C4×C5C2413yes—
166C22×C4×C5C2411yescharacteristic
167C22×C4×C5C2413yes—
168C22×Dic5C2413yes—
169C22×Dic5C2413yes—
170C22×Dic5C2411yescharacteristic
171C22×Dic5C2413yes—
172C22×Dic5C2411yescharacteristic
173C22×Dic5C2411yescharacteristic
174C22×Dic5C2411yescharacteristic
175#713C4⋊C4164yescomplemented
176C2×D20C22×C4112yes—
177C2×D20C2×D4112yes—
178C2×D20C2×Q8112yes—
179C2×C4×D5—26no—
180C2×C4×D5—26no—
181C2×C4×D5—26no—
182C2×C4×D5—26no—
183C2×C4×D5—212no—
184C2×C4×D5—46no—
185C2×C4×D5—43no—
186C2×C4×D5—43no—
187C2×C4×D5—46no—
188C2×C4×D5—46no—
189C2×C4×D5—46no—
190C2×C4×D5C22×C4124yes—
191C2×C4×D5C22×C4112yes—
192C2×C4×D5C2×D4112yes—
193C2×C4×D5C2×Q8112yes—
194C5×C4⋊C4—26no—
195C5×C4⋊C4—26no—
196C5×C4⋊C4—22no—
197C5×C4⋊C4—43no—
198C5×C4⋊C4—46no—
199C5×C4⋊C4C22×C4112yes—
200C42×C5—26no—
201C42×C5—26no—
202C42×C5C22×C4112yes—
203#685—26no—
204#685—26no—
205#685—212no—
206#685C22×C4112yes—
207#685C22×C4112yes—
208C4⋊Dic5—26no—
209C4⋊Dic5—26no—
210C4⋊Dic5—26no—
211C4⋊Dic5—43no—
212C4⋊Dic5C22×C4112yes—
213#683—22no—
214#683—22no—
215#683—26no—
216#683—26no—
217#683—46no—
218#683—43no—
219#683—43no—
220#683C22×C414yes—
221#683C22×C4112yes—
222C4×Dic5—26no—
223C4×Dic5—26no—
224C4×Dic5—22no—
225C4×Dic5—26no—
226C4×Dic5C22×C4112yes—
227C22×#41—51no—
228#520—516no—
229#520—108no—
230C4×#41—108no—
231C2×C4×Q8—51no—
232C2×C4×Q8—51no—
233C2×C4×Q8—51no—
234C2×C4×Q8C2×D511yescharacteristic
235C2×C4×Q8C2×D511yescharacteristic
236C2×C4×Q8C2×D511yescharacteristic
237C2×C4×D4—53no—
238C2×C4×D4—53no—
239C2×C4×D4—53no—
240C2×C4×D4—53no—
241C2×C4×D4—53no—
242C2×C4×D4—53no—
243C2×#117—53no—
244C2×#117—53no—
245C2×#117—53no—
246C22×C4⋊C4—53no—
247C22×C4⋊C4—53no—
248C22×C4⋊C4—56no—
249C22×C42—53no—
250#389—56no—
251#389—53no—
252#389—53no—
253#389—53no—
254#389—53no—
255#389—53no—
256#389—51no—
257#389—53no—
258#389—53no—
259#389—53no—
260#389—53no—
261#389C2×D513yes—
262#389C2×D513yes—
263#385—56no—
264#385—56no—
265#385—53no—
266#385—53no—
267#385—53no—
268#385—53no—
269#384—53no—
270#384C2×D513yes—
271#382—53no—
272#382—53no—
273#380—56no—
274#380—56no—
275C4×C4⋊C4—53no—
276C4×C4⋊C4—53no—
277C4×C4⋊C4—53no—
278C4×C4⋊C4C2×D513yes—
279C2×#95—56no—
280C23×C5C2511yescharacteristic
281C22×D5—23no—
282C22×D5—23no—
283C22×D5—23no—
284C22×D5—23no—
285C22×D5C23×C416yes—
286C22×D5C22×D416yes—
287C22×D5C22×Q816yes—
288C5×Q8C2×C4⋊C4116yescomplemented
289C2×C4×C5—21no—
290C2×C4×C5—23no—
291C2×C4×C5—21no—
292C2×C4×C5—23no—
293C2×C4×C5—21no—
294C2×C4×C5—21no—
295C2×C4×C5—23no—
296C2×C4×C5—21no—
297C2×C4×C5—23no—
298C2×C4×C5—23no—
299C2×C4×C5—23no—
300C2×C4×C5—21no—
301C2×C4×C5—21no—
302C2×C4×C5—23no—
303C2×C4×C5—23no—
304C2×C4×C5C23×C412yes—
305C2×C4×C5C23×C416yes—
306C2×C4×C5C23×C416yes—
307C2×C4×C5C22×D416yes—
308C2×C4×C5C22×Q816yes—
309C2×C4×C52−1+412yes—
310C2×Dic5—21no—
311C2×Dic5—23no—
312C2×Dic5—21no—
313C2×Dic5—21no—
314C2×Dic5—23no—
315C2×Dic5—21no—
316C2×Dic5—21no—
317C2×Dic5—23no—
318C2×Dic5—23no—
319C2×Dic5—23no—
320C2×Dic5—21no—
321C2×Dic5—23no—
322C2×Dic5—23no—
323C2×Dic5—23no—
324C2×Dic5—21no—
325C2×Dic5C23×C412yes—
326C2×Dic5C23×C416yes—
327C2×Dic5C23×C412yes—
328C2×Dic5C22×D412yes—
329C2×Dic5C22×Q812yes—
330C2×Dic5H(5,𝔽2)12yes—
331D20C2×C4⋊C4148yescomplemented
332C4×D5—424no—
333C4×D5—412no—
334C4×D5—412no—
335C4×D5—812no—
336C4×D5C2×C4⋊C4148yescomplemented
337C2×#41—58no—
338C2×#41—58no—
339C2×#41—58no—
340C22×Q8C22×D511yescharacteristic
341C22×D4—53no—
342C23×C4—53no—
343C23×C4—56no—
344C23×C4—53no—
345C23×C4—53no—
346C23×C4—56no—
347C4×Q8—24no—
348C4×Q8—24no—
349C4×Q8—58no—
350C4×Q8—104no—
351C4×Q8—104no—
352C4×Q8C4×D518yescomplemented
353C4×D4—524no—
354C4×D4—524no—
355C4×D4—1012no—
356C4×D4—1012no—
357C4×D4—1012no—
358C4×D4—1012no—
359#117—524no—
360#117—1012no—
361#117—1012no—
362C2×C4⋊C4—53no—
363C2×C4⋊C4—53no—
364C2×C4⋊C4—51no—
365C2×C4⋊C4—524no—
366C2×C4⋊C4—53no—
367C2×C4⋊C4—53no—
368C2×C4⋊C4—53no—
369C2×C4⋊C4—53no—
370C2×C4⋊C4—51no—
371C2×C4⋊C4—53no—
372C2×C4⋊C4—51no—
373C2×C4⋊C4—56no—
374C2×C4⋊C4—53no—
375C2×C4⋊C4—53no—
376C2×C4⋊C4—53no—
377C2×C4⋊C4—1012no—
378C2×C4⋊C4—2012no—
379C2×C4⋊C4C22×D513yes—
380C2×C4⋊C4C22×D513yes—
381C2×C4⋊C4C22×D511yescharacteristic
382C2×C4⋊C4C22×D516yes—
383C2×C4⋊C4C22×D513yes—
384C2×C4⋊C4C22×D513yes—
385C2×C22⋊C4—53no—
386C2×C22⋊C4—53no—
387C2×C22⋊C4—56no—
388C2×C22⋊C4—53no—
389C2×C22⋊C4—53no—
390C2×C42—53no—
391C2×C42—53no—
392C2×C42—53no—
393C2×C42—53no—
394C2×C42—51no—
395C2×C42—1012no—
396C2×C42C22×D513yes—
397C2×C42C22×D513yes—
398C2×C42C22×D513yes—
399#95—53no—
400#95—56no—
401#95—56no—
402#95—51no—
403#95—51no—
404#95—53no—
405#95—53no—
406#95—53no—
407#95C22×D513yes—
408#95C22×D513yes—
409C22×C5C24×C411yescharacteristic
410C22×C5C23×D411yescharacteristic
411C22×C5C23×Q811yescharacteristic
412C22×C5C2×H(5,𝔽2)11yescharacteristic
413C22×C5C2×2−1+411yescharacteristic
414C22×C5#58511yescharacteristic
415C22×C5#58511yescharacteristic
416C2×D5—26no—
417C2×D5—26no—
418C2×D5—26no—
419C2×D5—26no—
420C2×D5—26no—
421C2×D5—26no—
422C2×D5C22×C4⋊C4112yes—
423C4×C5—26no—
424C4×C5—22no—
425C4×C5—22no—
426C4×C5—43no—
427C4×C5—43no—
428C4×C5—41no—
429C4×C5C22×C4⋊C4112yes—
430Dic5—26no—
431Dic5—22no—
432Dic5—22no—
433Dic5—43no—
434Dic5—43no—
435Dic5—41no—
436Dic5C22×C4⋊C414yes—
437C24—53no—
438#41—564no—
439C2×Q8C2×C4×D514yes—
440C2×Q8D4×D514yes—
441C2×Q8Q8×D514yes—
442C2×D4—512no—
443C2×D4—512no—
444C2×D4—512no—
445C22×C4—51no—
446C22×C4—512no—
447C22×C4—512no—
448C22×C4—512no—
449C22×C4—51no—
450C22×C4—53no—
451C22×C4—51no—
452C22×C4—524no—
453C22×C4—53no—
454C22×C4—51no—
455C22×C4—53no—
456C22×C4—106no—
457C22×C4—106no—
458C22×C4—106no—
459C22×C4—1012no—
460C22×C4—106no—
461C22×C4—206no—
462C22×C4—206no—
463C22×C4—203no—
464C22×C4—203no—
465C22×C4—206no—
466C22×C4—206no—
467C22×C4C23×D513yes—
468C22×C4C23×D511yescharacteristic
469C22×C4C23×D513yes—
470C22×C4C23×D511yescharacteristic
471C22×C4C23×D513yes—
472C22×C4C23×D511yescharacteristic
473C22×C4C23×D513yes—
474C4⋊C4—26no—
475C4⋊C4—26no—
476C4⋊C4—22no—
477C4⋊C4—46no—
478C4⋊C4—43no—
479C4⋊C4—512no—
480C4⋊C4—512no—
481C4⋊C4—54no—
482C4⋊C4—106no—
483C4⋊C4—106no—
484C4⋊C4—106no—
485C4⋊C4—102no—
486C4⋊C4—106no—
487C4⋊C4—106no—
488C4⋊C4—102no—
489C4⋊C4—206no—
490C4⋊C4—203no—
491C4⋊C4—203no—
492C4⋊C4—203no—
493C4⋊C4C2×C4×D5112yes—
494C22⋊C4—512no—
495C22⋊C4—512no—
496C22⋊C4—106no—
497C22⋊C4—1012no—
498C22⋊C4—106no—
499C42—26no—
500C42—26no—
501C42—512no—
502C42—102no—
503C42—106no—
504C42—106no—
505C42—106no—
506C42C2×C4×D5112yes—
507C2×C5C23×C4⋊C411yescharacteristic
508C2×C5C4×H(5,𝔽2)11yescharacteristic
509C2×C5C4×2−1+411yescharacteristic
510C2×C5#348811yescharacteristic
511C2×C5#348911yescharacteristic
512C2×C5#350311yescharacteristic
513C2×C5#350411yescharacteristic
514D5—212no—
515C23—56no—
516C23—56no—
517C23—56no—
518C23—103no—
519C23—103no—
520C23—103no—
521C23—103no—
522C23C24×D511yescharacteristic
523Q8D5×C4⋊C4116yescomplemented
524D4—548no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—23no—
529C2×C4—23no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—23no—
533C2×C4—23no—
534C2×C4—21no—
535C2×C4—23no—
536C2×C4—23no—
537C2×C4—23no—
538C2×C4—23no—
539C2×C4—21no—
540C2×C4—52no—
541C2×C4—52no—
542C2×C4—52no—
543C2×C4—548no—
544C2×C4—52no—
545C2×C4—56no—
546C2×C4—52no—
547C2×C4—101no—
548C2×C4—101no—
549C2×C4—101no—
550C2×C4—101no—
551C2×C4—103no—
552C2×C4—103no—
553C2×C4—101no—
554C2×C4—103no—
555C2×C4—103no—
556C2×C4—101no—
557C2×C4—103no—
558C2×C4—103no—
559C2×C4—103no—
560C2×C4—101no—
561C2×C4—103no—
562C2×C4—2024no—
563C2×C4—2012no—
564C2×C4—2012no—
565C2×C4—4012no—
566C2×C4C22×C4×D512yes—
567C2×C4C22×C4×D516yes—
568C2×C4C22×C4×D516yes—
569C2×C4C2×D4×D516yes—
570C2×C4C2×Q8×D516yes—
571C2×C4#414712yes—
572C5#3425011yescharacteristic, complemented, Sylow
573C22—512no—
574C22—106no—
575C22—106no—
576C22—106no—
577C22—106no—
578C22—106no—
579C22—106no—
580C22C23×C4×D511yescharacteristic
581C22C22×D4×D511yescharacteristic
582C22C22×Q8×D511yescharacteristic
583C22C2×#414711yescharacteristic
584C22#6628311yescharacteristic
585C22D5×H(5,𝔽2)11yescharacteristic
586C22#6628711yescharacteristic
587C4—22no—
588C4—22no—
589C4—26no—
590C4—41no—
591C4—43no—
592C4—43no—
593C4—54no—
594C4—102no—
595C4—102no—
596C4—106no—
597C4—203no—
598C4—201no—
599C4—203no—
600C4C2×D5×C4⋊C4112yes—
601C2—1012no—
602C2C22×D5×C4⋊C411yescharacteristic
603C2C4×#414711yescharacteristic
604C2#1061778711yescharacteristic
605C2#1061864611yescharacteristic
606C2#1061867011yescharacteristic
607C2#1061892011yescharacteristic
608C2#1061894211yescharacteristic
subgroupG/H
center, FrattiniC23C24×D5
commutatorC22×C5C24×C4
FittingC5×Q8×C4⋊C4C2
socle, cosocleC23×C5C25
prime radicalC22×#713C22
hypercenterQ8×C4⋊C4D5
hypocenterC5#34250
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. C5×Q8×C4⋊C4C2semidirect
5. #10614192C2semidirect
6. #10614191C2semidirect
7. Q8×#683C2semidirect
8. #10607934C2semidirect
9. #10607933C2semidirect
10. #10607926C2semidirect
11. D5×#389C2semidirect
12. #10606091C2semidirect
13. #10606090C2semidirect
14. #10605501C2semidirect
15. #10605486C2semidirect
24. #65908C4semidirect
82. C4×C5×Q8C2×C4semidirect
94. Q8×Dic5C2×C4semidirect
121. #4039C2×C4semidirect
125. D5×C4⋊C4C2×C4semidirect
128. C4×D20C2×C4semidirect
130. #4018C2×C4semidirect
148. Q8×C4⋊C4D5semidirect
175. #713C4⋊C4semidirect
288. C5×Q8C2×C4⋊C4semidirect
331. D20C2×C4⋊C4semidirect
336. C4×D5C2×C4⋊C4semidirect
352. C4×Q8C4×D5semidirect
523. Q8D5×C4⋊C4semidirect
572. C5#34250semidirect

Maximal subgroups

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

subgroupindex
C2×#659082
C2×C4×#7132
C5×Q8×C4⋊C42
#106141922
#106141912
Q8×#6832
#106079342
#106079332
#106079262
D5×#3892
#106060912
#106060902
#106055012
#106054862
#342505