Finite Group Databasebeta

← Q8×#13133237 #28264555

Subgroups — Q8×#13133237 #28264555

Order 13568.

1786 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#11719945C211yesmaximal, characteristic, complemented
2#20287602C213yesmaximal, complemented
3Q8×#11719925C211yesmaximal, characteristic, complemented
4#20287590C213yesmaximal, complemented
5#20287586C213yesmaximal, complemented
6#20287578C213yesmaximal, complemented
7Q8×#11719923C211yesmaximal, characteristic, complemented
8Q8×#11719924C211yesmaximal, characteristic, complemented
9Q8×#11719922C211yesmaximal, characteristic, complemented
10#20285225C213yesmaximal, complemented
11#20284503C213yesmaximal, complemented
12#20284497C213yesmaximal, complemented
13C2×Q8×#88322C211yesmaximal, characteristic, complemented
14C4×#13133237C213yesmaximal
15Q8×#41×C53C211yesmaximal, characteristic, complemented
16C2×#13133237C2211yescharacteristic
17Q8×Dic106C2211yescharacteristic
18#16811351C2213yes—
19#16811333C2213yes—
20#16811328—23no—
21Q8×D212C2211yescharacteristic
22Q8×C22⋊D53—21no—
23#16811180C2213yes—
24#16811177—23no—
25C4×#11719945C2213yes—
26C4×Q8×D53—21no—
27#16811026C2213yes—
28#16811026C2213yes—
29#16811022C2216yes—
30#16811022C2216yes—
31#16811019C2213yes—
32#16811019C2213yes—
33#16810930C2216yes—
34#16810927C2216yes—
35#16810923C2216yes—
36#16810920C2216yes—
37#16810918C2213yes—
38#16810915C2213yes—
39#16810913C2213yes—
40#16810910C2213yes—
41#16810908C2213yes—
42#16810905C2213yes—
43#16810903C2213yes—
44#16810900C2213yes—
45Q8×#88322C2211yescharacteristic
46Q8×#88322C2211yescharacteristic
47Q8×#88322C2213yes—
48Q8×#88322C2213yes—
49C4×#11719925C2213yes—
50C4×#11719925C2213yes—
51C4×#11719924C2213yes—
52C4×#11719924C2213yes—
53C4×#11719923C2213yes—
54C4×#11719923C2213yes—
55C4×#11719922C2213yes—
56C4×#11719922C2213yes—
57#16810659C2213yes—
58#16810659C2216yes—
59C2×C4⋊#88322C2213yes—
60#16810619C2213yes—
61#16810617C2213yes—
62C2×C4×#88322C2213yes—
63Q82×C53C2211yescharacteristic
64C53×#557C2213yes—
65C53×#554—23no—
66C53×#554C2213yes—
67D4×Q8×C53—21no—
68D4×Q8×C53C2211yescharacteristic
69C53×#533—23no—
70C53×#533C2213yes—
71C4×#41×C53C2213yes—
72C2×C4×Q8×C53—21no—
73C2×C4×Q8×C53C2211yescharacteristic
74C2×#11719945C2311yescharacteristic
75C2×Q8×D53—21no—
76#13133237Q818yesdirect product factor
77C2×#11719925C2311yescharacteristic
78C2×#11719923C2311yescharacteristic
79C2×#11719924C2311yescharacteristic
80C2×#11719922C2311yescharacteristic
81C22×#88322C2313yes—
82C22×#88322C2311yescharacteristic
83#13133217C2313yes—
84#13133217C2313yes—
85#13133216—23no—
86#13133215—23no—
87#13133209C2316yes—
88#13133207—26no—
89#13133197—23no—
90#13133196—23no—
91#13133196—23no—
92#13133195—26no—
93#13133193—23no—
94Q8×Dic53—21no—
95C4×Dic106—23no—
96C4×Dic106—23no—
97C4×Dic106C2313yes—
98C4×Dic106C2313yes—
99#13133182—23no—
100C4×D212C2313yes—
101C4×D212C2313yes—
102C4×C22⋊D53—23no—
103C4×C22⋊D53—23no—
104#13133176C2313yes—
105#13133174—23no—
106C4⋊C4×D53—23no—
107C42×D53—23no—
108#13133150C2313yes—
109#13133150C2313yes—
110#13133150C2316yes—
111#13133150C2316yes—
112#13133148C2313yes—
113#13133148C2313yes—
114#13133148C2316yes—
115#13133148C2316yes—
116C4⋊#88322C2313yes—
117C4⋊#88322C2313yes—
118C4⋊#88322C2316yes—
119#13133141C2313yes—
120#13133141C2313yes—
121#13133140C2313yes—
122#13133140C2313yes—
123#13133138C2313yes—
124#13133138C2313yes—
125#13133137C2313yes—
126#13133137C2313yes—
127C4×#88322C2313yes—
128C4×#88322C2313yes—
129C4×#88322C2313yes—
130C4×#88322C2313yes—
131C2×#41×C53C2311yescharacteristic
132C22×Q8×C53—21no—
133C22×Q8×C53D411yescharacteristic
134C4⋊Q8×C53—23no—
135C4⋊Q8×C53—23no—
136C4⋊Q8×C53C2313yes—
137C4⋊Q8×C53C2313yes—
138#125×C53—26no—
139#125×C53C2316yes—
140C22⋊Q8×C53—23no—
141C22⋊Q8×C53—26no—
142C22⋊Q8×C53—23no—
143C22⋊Q8×C53—23no—
144C22⋊Q8×C53—23no—
145C4×Q8×C53—23no—
146C4×Q8×C53—23no—
147C4×Q8×C53—23no—
148C4×Q8×C53—21no—
149C4×Q8×C53D411yescharacteristic
150C4×Q8×C53D413yes—
151C4×Q8×C53C2311yescharacteristic
152C4×Q8×C53C2313yes—
153C4×Q8×C53C2313yes—
154C4×Q8×C53C2313yes—
155C4×D4×C53—23no—
156C4×D4×C53—23no—
157C4×D4×C53C2313yes—
158C4×D4×C53C2313yes—
159#117×C53—23no—
160#117×C53C2313yes—
161C2×C4⋊C4×C53—23no—
162C2×C4⋊C4×C53D413yes—
163C2×C42×C53—23no—
164C2×C42×C53C2313yes—
165#41×C53—22no—
166#41×C53C2×Q814yescomplemented
167C2×Q8×C53—24no—
168C2×Q8×C53—23no—
169C2×Q8×C53—23no—
170C2×Q8×C53—44no—
171C2×Q8×C53C2×D413yes—
172C2×Q8×C53C2×D411yescharacteristic
173C2×Q8×C53C2411yescharacteristic
174C2×D4×C53—21no—
175C2×D4×C53C2411yescharacteristic
176C22×C4×C53—21no—
177C22×C4×C53—23no—
178C22×C4×C53—23no—
179C22×C4×C53C2×D413yes—
180C22×C4×C53C2×D413yes—
181C22×C4×C53C2411yescharacteristic
182C2×C22⋊D53—21no—
183Q8×D53—44no—
184#11719945—22no—
185#11719945C2×Q814yescomplemented
186C2×D212C2411yescharacteristic
187C2×C4×D53—21no—
188C2×C4×D53—23no—
189C2×C4×D53—23no—
190C2×Dic106—23no—
191C2×Dic106—23no—
192C2×Dic106C2411yescharacteristic
193C4⋊C4×C53—26no—
194C4⋊C4×C53—23no—
195C4⋊C4×C53—23no—
196C4⋊C4×C53—23no—
197C4⋊C4×C53—23no—
198C4⋊C4×C53—26no—
199C4⋊C4×C53C2×D413yes—
200C4⋊C4×C53C2×D416yes—
201C4⋊C4×C53C2×D413yes—
202C4⋊C4×C53C2413yes—
203C4⋊C4×C53C2413yes—
204C22⋊C4×C53—23no—
205C22⋊C4×C53—23no—
206C22⋊C4×C53—23no—
207C42×C53—23no—
208C42×C53—23no—
209C42×C53—23no—
210C42×C53C2×D413yes—
211C42×C53C2×D413yes—
212C42×C53C2413yes—
213C42×C53C2413yes—
214C22⋊Dic53—23no—
215#11719925C2×Q814yescomplemented
216#11719924C2×Q814yescomplemented
217#11719923C2×Q814yescomplemented
218#11719922C2×Q814yescomplemented
219#11719921—23no—
220#11719921—23no—
221C4⋊Dic53—23no—
222C4⋊Dic53—23no—
223C4⋊Dic53C2413yes—
224C4⋊Dic53C2413yes—
225#11719919—26no—
226#11719919—23no—
227#11719919—23no—
228#11719919—26no—
229C4×Dic53—23no—
230C4×Dic53—23no—
231C4×Dic53—23no—
232C2×#88322—26no—
233C2×#88322C2×Q814yescomplemented
234C2×#88322C2413yes—
235C2×#88322C2413yes—
236C2×#88322C2411yescharacteristic
237C2×#88322C2411yescharacteristic
238C23×C53—21no—
239C23×C53C22×D411yescharacteristic
240C22×D53—21no—
241Q8×C53—26no—
242Q8×C53—21no—
243Q8×C53—46no—
244Q8×C53—83no—
245Q8×C53#13514yescomplemented
246Q8×C53C22×Q812yes—
247D4×C53—24no—
248D4×C53—21no—
249D4×C53C22×Q812yes—
250C2×C4×C53—21no—
251C2×C4×C53—24no—
252C2×C4×C53—26no—
253C2×C4×C53—23no—
254C2×C4×C53—23no—
255C2×C4×C53—43no—
256C2×C4×C53—46no—
257C2×C4×C53—83no—
258C2×C4×C53C22×D413yes—
259C2×C4×C53C22×D413yes—
260C2×C4×C53C22×D413yes—
261C2×C4×C53C22×D413yes—
262C2×C4×C53C22×D411yescharacteristic
263C2×C4×C53C22×Q812yes—
264C2×C4×C532−1+412yes—
265C2×C4×C53C2511yescharacteristic
266C22⋊D53—24no—
267C2×Dic53—21no—
268C2×Dic53—23no—
269C2×Dic53—23no—
270D212—21no—
271D212C22×Q812yes—
272C4×D53—24no—
273C4×D53—46no—
274C4×D53—83no—
275Dic106—21no—
276Dic106—46no—
277Dic106—83no—
278Dic106C22×Q812yes—
279#88322—23no—
280#88322—23no—
281#88322C22×Q812yes—
282#88322C22×Q812yes—
283Q8×#135—531nomaximal, Sylow
284C22×C53—21no—
285C22×C53—21no—
286C22×C53—22no—
287C22×C53—41no—
288C22×C53—41no—
289C22×C53D4×Q812yes—
290C22×C53C23×D411yescharacteristic
291C2×D53—22no—
292C2×D53—41no—
293C2×D53—41no—
294C4×C53—23no—
295C4×C53—23no—
296C4×C53—22no—
297C4×C53—23no—
298C4×C53—43no—
299C4×C53—43no—
300C4×C53D4×Q812yes—
301C4×C53C2×#13516yes—
302C4×C53C23×Q811yescharacteristic
303C4×C53C2×2−1+411yescharacteristic
304Dic53—22no—
305Dic53—43no—
306Dic53—43no—
307Q8×#41—531no—
308Q8×#41D5311yescharacteristic, complemented
309#3383—533no—
310Q8×Q16—531no—
311#3381—533no—
312#3381—533no—
313#3380—533no—
314Q8×SD16—531no—
315Q8×SD16—531no—
316Q8×D8—531no—
317#3160—533no—
318#3076—533no—
319#3076—533no—
320C2×C8×Q8—531no—
321C4×#135—533no—
322C2×C53—22no—
323C2×C53—42no—
324C2×C53C2×D4×Q811yescharacteristic
325C2×C53C22×#13511yescharacteristic
326C2×C53#358411yescharacteristic
327D53—42no—
328C2×#135—531no—
329Q82—531no—
330Q82C2×D5311yescharacteristic
331#557—533no—
332#557C2×D5313yes—
333#554—23no—
334#554—533no—
335#554—1063no—
336#554C2×D5313yes—
337D4×Q8—21no—
338D4×Q8—531no—
339D4×Q8—1061no—
340D4×Q8C2×D5311yescharacteristic
341#533—23no—
342#533—533no—
343#533—1063no—
344#533C2×D5313yes—
345C4×#41—533no—
346C4×#41C2×D5313yes—
347C2×C4×Q8—21no—
348C2×C4×Q8—1061no—
349C2×C4×Q8C2×D5311yescharacteristic
350#500—533no—
351#500—533no—
352#499—536no—
353#499—536no—
354#498—533no—
355#498—533no—
356#479—536no—
357#479—536no—
358#478—536no—
359#478—536no—
360#477—533no—
361#477—533no—
362#476—533no—
363#476—533no—
364#475—533no—
365#475—533no—
366#474—533no—
367#474—533no—
368C8×Q8—531no—
369C8×Q8—531no—
370C8×Q8—533no—
371C8×Q8—533no—
372C4×Q16—533no—
373C4×Q16—533no—
374C4×SD16—533no—
375C4×SD16—533no—
376C4×SD16—533no—
377C4×SD16—533no—
378C4×D8—533no—
379C4×D8—533no—
380#427—536no—
381#427—533no—
382C2×C4⋊C8—533no—
383#416—533no—
384#416—533no—
385C2×C4×C8—533no—
386C53Q8×#13511yescharacteristic, complemented, Sylow
387C2×#41—531no—
388C2×#41C22×D5311yescharacteristic
389C22×Q8—21no—
390C22×Q8—1061no—
391C22×Q8C22⋊D5311yescharacteristic
392#135—538no—
393C2×Q16—531no—
394C2×SD16—531no—
395C2×SD16—531no—
396C2×D8—531no—
397C22×C8—531no—
398C22×C8—533no—
399C4⋊Q8—23no—
400C4⋊Q8—23no—
401C4⋊Q8—533no—
402C4⋊Q8—533no—
403C4⋊Q8—1063no—
404C4⋊Q8—1063no—
405C4⋊Q8C22×D5313yes—
406C4⋊Q8C22×D5313yes—
407#125—26no—
408#125—536no—
409#125—1066no—
410#125C22×D5316yes—
411C22⋊Q8—23no—
412C22⋊Q8—23no—
413C22⋊Q8—23no—
414C22⋊Q8—23no—
415C22⋊Q8—26no—
416C22⋊Q8—1063no—
417C22⋊Q8—1063no—
418C22⋊Q8—1063no—
419C22⋊Q8—1063no—
420C22⋊Q8—1066no—
421C4×Q8—21no—
422C4×Q8—23no—
423C4×Q8—23no—
424C4×Q8—23no—
425C4×Q8—533no—
426C4×Q8—533no—
427C4×Q8—1061no—
428C4×Q8—1063no—
429C4×Q8—1063no—
430C4×Q8—1063no—
431C4×Q8C22⋊D5311yescharacteristic
432C4×Q8C22⋊D5313yes—
433C4×Q8C22×D5313yes—
434C4×Q8C22×D5313yes—
435C4×Q8C22×D5311yescharacteristic
436C4×Q8C22×D5313yes—
437C4×D4—23no—
438C4×D4—23no—
439C4×D4—533no—
440C4×D4—533no—
441C4×D4—1063no—
442C4×D4—1063no—
443C4×D4C22×D5313yes—
444C4×D4C22×D5313yes—
445#117—23no—
446#117—533no—
447#117—1063no—
448#117C22×D5313yes—
449C2×C4⋊C4—23no—
450C2×C4⋊C4—1063no—
451C2×C4⋊C4C22⋊D5313yes—
452C2×C42—23no—
453C2×C42—1063no—
454C2×C42C22×D5313yes—
455#107—533no—
456#107—533no—
457#107—536no—
458#107—536no—
459#106—533no—
460#106—533no—
461#106—536no—
462#106—536no—
463C4⋊C8—533no—
464C4⋊C8—533no—
465C4⋊C8—536no—
466#103—533no—
467#103—533no—
468#103—533no—
469#103—533no—
470#102—533no—
471#102—533no—
472#102—533no—
473#102—533no—
474C4×C8—533no—
475C4×C8—533no—
476C4×C8—533no—
477C4×C8—533no—
478#41—22no—
479#41—534no—
480#41—1062no—
481#41Q8×D5314yescomplemented
482C2×Q8—24no—
483C2×Q8—23no—
484C2×Q8—23no—
485C2×Q8—44no—
486C2×Q8—531no—
487C2×Q8—1063no—
488C2×Q8—1063no—
489C2×Q8—2124no—
490C2×Q8C2×C22⋊D5313yes—
491C2×Q8C2×C22⋊D5311yescharacteristic
492C2×Q8C23×D5311yescharacteristic
493C2×D4—21no—
494C2×D4—531no—
495C2×D4—1061no—
496C2×D4C23×D5311yescharacteristic
497C22×C4—21no—
498C22×C4—23no—
499C22×C4—23no—
500C22×C4—1061no—
501C22×C4—1063no—
502C22×C4—1063no—
503C22×C4C2×C22⋊D5313yes—
504C22×C4C2×C22⋊D5313yes—
505C22×C4C23×D5311yescharacteristic
506Q16—534no—
507SD16—534no—
508SD16—534no—
509D8—534no—
510C2×C8—534no—
511C2×C8—531no—
512C2×C8—531no—
513C2×C8—533no—
514C2×C8—533no—
515C2×C8—1066no—
516C4⋊C4—26no—
517C4⋊C4—23no—
518C4⋊C4—23no—
519C4⋊C4—23no—
520C4⋊C4—23no—
521C4⋊C4—26no—
522C4⋊C4—533no—
523C4⋊C4—533no—
524C4⋊C4—1066no—
525C4⋊C4—1063no—
526C4⋊C4—1063no—
527C4⋊C4—1063no—
528C4⋊C4—1066no—
529C4⋊C4—1063no—
530C4⋊C4C2×C22⋊D5313yes—
531C4⋊C4C2×C22⋊D5316yes—
532C4⋊C4C2×C22⋊D5313yes—
533C4⋊C4C23×D5313yes—
534C4⋊C4C23×D5313yes—
535C22⋊C4—23no—
536C22⋊C4—23no—
537C22⋊C4—23no—
538C22⋊C4—1063no—
539C22⋊C4—1063no—
540C22⋊C4—1063no—
541C42—23no—
542C42—23no—
543C42—23no—
544C42—1063no—
545C42—1063no—
546C42—1063no—
547C42C2×C22⋊D5313yes—
548C42C2×C22⋊D5313yes—
549C42C23×D5313yes—
550C42C23×D5313yes—
551C23—21no—
552C23—1061no—
553C23C22×C22⋊D5311yescharacteristic
554Q8—26no—
555Q8—21no—
556Q8—46no—
557Q8—83no—
558Q8—532no—
559Q8—1061no—
560Q8—4243no—
561Q8—2126no—
562Q8#1313323714yesdirect product factor
563Q8C2×Q8×D5312yes—
564D4—24no—
565D4—21no—
566D4—532no—
567D4—1064no—
568D4—1061no—
569D4C2×Q8×D5312yes—
570C2×C4—21no—
571C2×C4—24no—
572C2×C4—26no—
573C2×C4—23no—
574C2×C4—23no—
575C2×C4—43no—
576C2×C4—46no—
577C2×C4—83no—
578C2×C4—1061no—
579C2×C4—1064no—
580C2×C4—1063no—
581C2×C4—1063no—
582C2×C4—4243no—
583C2×C4—2126no—
584C2×C4C22×C22⋊D5313yes—
585C2×C4C22×C22⋊D5313yes—
586C2×C4C22×C22⋊D5313yes—
587C2×C4C22×C22⋊D5313yes—
588C2×C4C22×C22⋊D5311yescharacteristic
589C2×C4C2×Q8×D5312yes—
590C2×C4#1313326512yes—
591C2×C4C24×D5311yescharacteristic
592C8—532no—
593C8—532no—
594C8—1063no—
595C8—1063no—
596C22—21no—
597C22—21no—
598C22—22no—
599C22—41no—
600C22—41no—
601C22—1062no—
602C22—2121no—
603C22—2121no—
604C22Q8×C22⋊D5312yes—
605C22C23×C22⋊D5311yescharacteristic
606C4—23no—
607C4—22no—
608C4—23no—
609C4—23no—
610C4—43no—
611C4—43no—
612C4—1062no—
613C4—2123no—
614C4—2123no—
615C4Q8×C22⋊D5312yes—
616C4C2×#1313323716yes—
617C4C22×Q8×D5311yescharacteristic
618C4C2×#1313326511yescharacteristic
619C2—22no—
620C2—42no—
621C2—2122no—
622C2C2×Q8×C22⋊D5311yescharacteristic
623C2C22×#1313323711yescharacteristic
624C2#2028948611yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D53
commutator, cosocleC2×C4×C53C25
FrattiniC2×C4C24×D53
FittingQ8×#41×C53C2
socleC22×C53C23×D4
prime radicalC2×#13133237C22
hypercenterQ8×#41D53
hypocenterC53Q8×#135
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C4×Q8 ◁ Q8×#41

Complements

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

NH ≅ G/Ntype
1. Q8×#11719945C2semidirect
2. #20287602C2semidirect
3. Q8×#11719925C2semidirect
4. #20287590C2semidirect
5. #20287586C2semidirect
6. #20287578C2semidirect
7. Q8×#11719923C2semidirect
8. Q8×#11719924C2semidirect
9. Q8×#11719922C2semidirect
10. #20285225C2semidirect
11. #20284503C2semidirect
12. #20284497C2semidirect
13. C2×Q8×#88322C2semidirect
15. Q8×#41×C53C2semidirect
76. #13133237Q8direct (N × H)
166. #41×C53C2×Q8semidirect
185. #11719945C2×Q8semidirect
215. #11719925C2×Q8semidirect
216. #11719924C2×Q8semidirect
217. #11719923C2×Q8semidirect
218. #11719922C2×Q8semidirect
233. C2×#88322C2×Q8semidirect
245. Q8×C53#135semidirect
308. Q8×#41D53semidirect
386. C53Q8×#135semidirect
481. #41Q8×D53semidirect
562. Q8#13133237direct (N × H)

Maximal subgroups

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

subgroupindex
Q8×#117199452
#202876022
Q8×#117199252
#202875902
#202875862
#202875782
Q8×#117199232
Q8×#117199242
Q8×#117199222
#202852252
#202845032
#202844972
C2×Q8×#883222
C4×#131332372
Q8×#41×C532
Q8×#13553