Finite Group Databasebeta

← Q8×#14273747 #29469152

Subgroups — Q8×#14273747 #29469152

Order 15104.

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×#11745685C211yesmaximal, characteristic, complemented
2#21436983C213yesmaximal, complemented
3Q8×#11745672C211yesmaximal, characteristic, complemented
4#21436971C213yesmaximal, complemented
5#21436967C213yesmaximal, complemented
6#21436959C213yesmaximal, complemented
7Q8×#11745670C211yesmaximal, characteristic, complemented
8Q8×#11745671C211yesmaximal, characteristic, complemented
9Q8×#11745669C211yesmaximal, characteristic, complemented
10#21434606C213yesmaximal, complemented
11#21433884C213yesmaximal, complemented
12#21433878C213yesmaximal, complemented
13C2×Q8×C59⋊C8C211yesmaximal, characteristic, complemented
14C4×#14273747C213yesmaximal
15Q8×#41×C59C211yesmaximal, characteristic, complemented
16C2×#14273747C2211yescharacteristic
17Q8×Dic118C2211yescharacteristic
18#16838130C2213yes—
19#16838112C2213yes—
20#16838107—23no—
21Q8×D236C2211yescharacteristic
22Q8×C22⋊D59—21no—
23#16837959C2213yes—
24#16837956—23no—
25C4×#11745685C2213yes—
26C4×Q8×D59—21no—
27#16837805C2213yes—
28#16837805C2213yes—
29#16837801C2216yes—
30#16837801C2216yes—
31#16837798C2213yes—
32#16837798C2213yes—
33#16837709C2216yes—
34#16837706C2216yes—
35#16837702C2216yes—
36#16837699C2216yes—
37#16837697C2213yes—
38#16837694C2213yes—
39#16837692C2213yes—
40#16837689C2213yes—
41#16837687C2213yes—
42#16837684C2213yes—
43#16837682C2213yes—
44#16837679C2213yes—
45Q8×C59⋊C8C2211yescharacteristic
46Q8×C59⋊C8C2211yescharacteristic
47Q8×C59⋊C8C2213yes—
48Q8×C59⋊C8C2213yes—
49C4×#11745672C2213yes—
50C4×#11745672C2213yes—
51C4×#11745671C2213yes—
52C4×#11745671C2213yes—
53C4×#11745670C2213yes—
54C4×#11745670C2213yes—
55C4×#11745669C2213yes—
56C4×#11745669C2213yes—
57#16837438C2216yes—
58#16837438C2213yes—
59C2×C4⋊(C59⋊C8)C2213yes—
60#16837398C2213yes—
61#16837396C2213yes—
62C2×C4×C59⋊C8C2213yes—
63Q82×C59C2211yescharacteristic
64C59×#557C2213yes—
65C59×#554—23no—
66C59×#554C2213yes—
67D4×Q8×C59—21no—
68D4×Q8×C59C2211yescharacteristic
69C59×#533—23no—
70C59×#533C2213yes—
71C4×#41×C59C2213yes—
72C2×C4×Q8×C59—21no—
73C2×C4×Q8×C59C2211yescharacteristic
74C2×#11745685C2311yescharacteristic
75C2×Q8×D59—21no—
76#14273747Q818yesdirect product factor
77C2×#11745672C2311yescharacteristic
78C2×#11745670C2311yescharacteristic
79C2×#11745671C2311yescharacteristic
80C2×#11745669C2311yescharacteristic
81C22×C59⋊C8C2313yes—
82C22×C59⋊C8C2311yescharacteristic
83#14273727C2313yes—
84#14273727C2313yes—
85#14273726—23no—
86#14273725—23no—
87#14273719C2316yes—
88#14273717—26no—
89#14273707—23no—
90#14273706—23no—
91#14273706—23no—
92#14273705—26no—
93#14273703—23no—
94Q8×Dic59—21no—
95C4×Dic118—23no—
96C4×Dic118—23no—
97C4×Dic118C2313yes—
98C4×Dic118C2313yes—
99#14273692—23no—
100C4×D236C2313yes—
101C4×D236C2313yes—
102C4×C22⋊D59—23no—
103C4×C22⋊D59—23no—
104#14273686C2313yes—
105#14273684—23no—
106C4⋊C4×D59—23no—
107C42×D59—23no—
108#14273660C2316yes—
109#14273660C2316yes—
110#14273660C2313yes—
111#14273660C2313yes—
112#14273658C2316yes—
113#14273658C2316yes—
114#14273658C2313yes—
115#14273658C2313yes—
116C4⋊(C59⋊C8)C2313yes—
117C4⋊(C59⋊C8)C2313yes—
118C4⋊(C59⋊C8)C2316yes—
119#14273651C2313yes—
120#14273651C2313yes—
121#14273650C2313yes—
122#14273650C2313yes—
123#14273648C2313yes—
124#14273648C2313yes—
125#14273647C2313yes—
126#14273647C2313yes—
127C4×C59⋊C8C2313yes—
128C4×C59⋊C8C2313yes—
129C4×C59⋊C8C2313yes—
130C4×C59⋊C8C2313yes—
131C2×#41×C59C2311yescharacteristic
132C22×Q8×C59—21no—
133C22×Q8×C59D411yescharacteristic
134C4⋊Q8×C59—23no—
135C4⋊Q8×C59—23no—
136C4⋊Q8×C59C2313yes—
137C4⋊Q8×C59C2313yes—
138#125×C59—26no—
139#125×C59C2316yes—
140C22⋊Q8×C59—23no—
141C22⋊Q8×C59—23no—
142C22⋊Q8×C59—23no—
143C22⋊Q8×C59—23no—
144C22⋊Q8×C59—26no—
145C4×Q8×C59—21no—
146C4×Q8×C59—23no—
147C4×Q8×C59—23no—
148C4×Q8×C59—23no—
149C4×Q8×C59D413yes—
150C4×Q8×C59D411yescharacteristic
151C4×Q8×C59C2313yes—
152C4×Q8×C59C2313yes—
153C4×Q8×C59C2311yescharacteristic
154C4×Q8×C59C2313yes—
155C4×D4×C59—23no—
156C4×D4×C59—23no—
157C4×D4×C59C2313yes—
158C4×D4×C59C2313yes—
159#117×C59—23no—
160#117×C59C2313yes—
161C2×C4⋊C4×C59—23no—
162C2×C4⋊C4×C59D413yes—
163C2×C42×C59—23no—
164C2×C42×C59C2313yes—
165#41×C59—22no—
166#41×C59C2×Q814yescomplemented
167C2×Q8×C59—24no—
168C2×Q8×C59—23no—
169C2×Q8×C59—23no—
170C2×Q8×C59—44no—
171C2×Q8×C59C2×D413yes—
172C2×Q8×C59C2×D411yescharacteristic
173C2×Q8×C59C2411yescharacteristic
174C2×D4×C59—21no—
175C2×D4×C59C2411yescharacteristic
176C22×C4×C59—21no—
177C22×C4×C59—23no—
178C22×C4×C59—23no—
179C22×C4×C59C2×D413yes—
180C22×C4×C59C2×D413yes—
181C22×C4×C59C2411yescharacteristic
182C2×C22⋊D59—21no—
183Q8×D59—44no—
184#11745685—22no—
185#11745685C2×Q814yescomplemented
186C2×D236C2411yescharacteristic
187C2×C4×D59—21no—
188C2×C4×D59—23no—
189C2×C4×D59—23no—
190C2×Dic118—23no—
191C2×Dic118—23no—
192C2×Dic118C2411yescharacteristic
193C4⋊C4×C59—26no—
194C4⋊C4×C59—23no—
195C4⋊C4×C59—23no—
196C4⋊C4×C59—23no—
197C4⋊C4×C59—23no—
198C4⋊C4×C59—26no—
199C4⋊C4×C59C2×D413yes—
200C4⋊C4×C59C2×D416yes—
201C4⋊C4×C59C2×D413yes—
202C4⋊C4×C59C2413yes—
203C4⋊C4×C59C2413yes—
204C22⋊C4×C59—23no—
205C22⋊C4×C59—23no—
206C22⋊C4×C59—23no—
207C42×C59—23no—
208C42×C59—23no—
209C42×C59—23no—
210C42×C59C2×D413yes—
211C42×C59C2×D413yes—
212C42×C59C2413yes—
213C42×C59C2413yes—
214C22⋊Dic59—23no—
215#11745672C2×Q814yescomplemented
216#11745671C2×Q814yescomplemented
217#11745670C2×Q814yescomplemented
218#11745669C2×Q814yescomplemented
219(C2×D59)⋊C4—23no—
220(C2×D59)⋊C4—23no—
221C4⋊Dic59—23no—
222C4⋊Dic59—23no—
223C4⋊Dic59C2413yes—
224C4⋊Dic59C2413yes—
225Dic59⋊C4—26no—
226Dic59⋊C4—23no—
227Dic59⋊C4—23no—
228Dic59⋊C4—26no—
229C4×Dic59—23no—
230C4×Dic59—23no—
231C4×Dic59—23no—
232C2×C59⋊C8—26no—
233C2×C59⋊C8C2×Q814yescomplemented
234C2×C59⋊C8C2413yes—
235C2×C59⋊C8C2413yes—
236C2×C59⋊C8C2411yescharacteristic
237C2×C59⋊C8C2411yescharacteristic
238C23×C59—21no—
239C23×C59C22×D411yescharacteristic
240C22×D59—21no—
241Q8×C59—26no—
242Q8×C59—21no—
243Q8×C59—46no—
244Q8×C59—83no—
245Q8×C59#13514yescomplemented
246Q8×C59C22×Q812yes—
247D4×C59—24no—
248D4×C59—21no—
249D4×C59C22×Q812yes—
250C2×C4×C59—21no—
251C2×C4×C59—24no—
252C2×C4×C59—26no—
253C2×C4×C59—23no—
254C2×C4×C59—23no—
255C2×C4×C59—43no—
256C2×C4×C59—46no—
257C2×C4×C59—83no—
258C2×C4×C59C22×D413yes—
259C2×C4×C59C22×D413yes—
260C2×C4×C59C22×D413yes—
261C2×C4×C59C22×D413yes—
262C2×C4×C59C22×D411yescharacteristic
263C2×C4×C59C22×Q812yes—
264C2×C4×C592−1+412yes—
265C2×C4×C59C2511yescharacteristic
266C22⋊D59—24no—
267C2×Dic59—21no—
268C2×Dic59—23no—
269C2×Dic59—23no—
270D236—21no—
271D236C22×Q812yes—
272C4×D59—24no—
273C4×D59—46no—
274C4×D59—83no—
275Dic118—21no—
276Dic118—46no—
277Dic118—83no—
278Dic118C22×Q812yes—
279C59⋊C8—23no—
280C59⋊C8—23no—
281C59⋊C8C22×Q812yes—
282C59⋊C8C22×Q812yes—
283Q8×#135—591nomaximal, Sylow
284C22×C59—21no—
285C22×C59—21no—
286C22×C59—22no—
287C22×C59—41no—
288C22×C59—41no—
289C22×C59D4×Q812yes—
290C22×C59C23×D411yescharacteristic
291C2×D59—22no—
292C2×D59—41no—
293C2×D59—41no—
294C4×C59—23no—
295C4×C59—23no—
296C4×C59—22no—
297C4×C59—23no—
298C4×C59—43no—
299C4×C59—43no—
300C4×C59D4×Q812yes—
301C4×C59C2×#13516yes—
302C4×C59C23×Q811yescharacteristic
303C4×C59C2×2−1+411yescharacteristic
304Dic59—22no—
305Dic59—43no—
306Dic59—43no—
307Q8×#41—591no—
308Q8×#41D5911yescharacteristic, complemented
309#3383—593no—
310Q8×Q16—591no—
311#3381—593no—
312#3381—593no—
313#3380—593no—
314Q8×SD16—591no—
315Q8×SD16—591no—
316Q8×D8—591no—
317#3160—593no—
318#3076—593no—
319#3076—593no—
320C2×C8×Q8—591no—
321C4×#135—593no—
322C2×C59—22no—
323C2×C59—42no—
324C2×C59C2×D4×Q811yescharacteristic
325C2×C59C22×#13511yescharacteristic
326C2×C59#358411yescharacteristic
327D59—42no—
328C2×#135—591no—
329Q82—591no—
330Q82C2×D5911yescharacteristic
331#557—593no—
332#557C2×D5913yes—
333#554—23no—
334#554—593no—
335#554—1183no—
336#554C2×D5913yes—
337D4×Q8—21no—
338D4×Q8—591no—
339D4×Q8—1181no—
340D4×Q8C2×D5911yescharacteristic
341#533—23no—
342#533—593no—
343#533—1183no—
344#533C2×D5913yes—
345C4×#41—593no—
346C4×#41C2×D5913yes—
347C2×C4×Q8—21no—
348C2×C4×Q8—1181no—
349C2×C4×Q8C2×D5911yescharacteristic
350#500—593no—
351#500—593no—
352#499—596no—
353#499—596no—
354#498—593no—
355#498—593no—
356#479—596no—
357#479—596no—
358#478—596no—
359#478—596no—
360#477—593no—
361#477—593no—
362#476—593no—
363#476—593no—
364#475—593no—
365#475—593no—
366#474—593no—
367#474—593no—
368C8×Q8—591no—
369C8×Q8—591no—
370C8×Q8—593no—
371C8×Q8—593no—
372C4×Q16—593no—
373C4×Q16—593no—
374C4×SD16—593no—
375C4×SD16—593no—
376C4×SD16—593no—
377C4×SD16—593no—
378C4×D8—593no—
379C4×D8—593no—
380#427—596no—
381#427—593no—
382C2×C4⋊C8—593no—
383#416—593no—
384#416—593no—
385C2×C4×C8—593no—
386C59Q8×#13511yescharacteristic, complemented, Sylow
387C2×#41—591no—
388C2×#41C22×D5911yescharacteristic
389C22×Q8—21no—
390C22×Q8—1181no—
391C22×Q8C22⋊D5911yescharacteristic
392#135—598no—
393C2×Q16—591no—
394C2×SD16—591no—
395C2×SD16—591no—
396C2×D8—591no—
397C22×C8—591no—
398C22×C8—593no—
399C4⋊Q8—23no—
400C4⋊Q8—23no—
401C4⋊Q8—593no—
402C4⋊Q8—593no—
403C4⋊Q8—1183no—
404C4⋊Q8—1183no—
405C4⋊Q8C22×D5913yes—
406C4⋊Q8C22×D5913yes—
407#125—26no—
408#125—596no—
409#125—1186no—
410#125C22×D5916yes—
411C22⋊Q8—23no—
412C22⋊Q8—23no—
413C22⋊Q8—23no—
414C22⋊Q8—23no—
415C22⋊Q8—26no—
416C22⋊Q8—1183no—
417C22⋊Q8—1183no—
418C22⋊Q8—1183no—
419C22⋊Q8—1183no—
420C22⋊Q8—1186no—
421C4×Q8—21no—
422C4×Q8—23no—
423C4×Q8—23no—
424C4×Q8—23no—
425C4×Q8—593no—
426C4×Q8—593no—
427C4×Q8—1181no—
428C4×Q8—1183no—
429C4×Q8—1183no—
430C4×Q8—1183no—
431C4×Q8C22⋊D5911yescharacteristic
432C4×Q8C22⋊D5913yes—
433C4×Q8C22×D5913yes—
434C4×Q8C22×D5913yes—
435C4×Q8C22×D5911yescharacteristic
436C4×Q8C22×D5913yes—
437C4×D4—23no—
438C4×D4—23no—
439C4×D4—593no—
440C4×D4—593no—
441C4×D4—1183no—
442C4×D4—1183no—
443C4×D4C22×D5913yes—
444C4×D4C22×D5913yes—
445#117—23no—
446#117—593no—
447#117—1183no—
448#117C22×D5913yes—
449C2×C4⋊C4—23no—
450C2×C4⋊C4—1183no—
451C2×C4⋊C4C22⋊D5913yes—
452C2×C42—23no—
453C2×C42—1183no—
454C2×C42C22×D5913yes—
455#107—593no—
456#107—593no—
457#107—596no—
458#107—596no—
459#106—593no—
460#106—593no—
461#106—596no—
462#106—596no—
463C4⋊C8—593no—
464C4⋊C8—593no—
465C4⋊C8—596no—
466#103—593no—
467#103—593no—
468#103—593no—
469#103—593no—
470#102—593no—
471#102—593no—
472#102—593no—
473#102—593no—
474C4×C8—593no—
475C4×C8—593no—
476C4×C8—593no—
477C4×C8—593no—
478#41—22no—
479#41—594no—
480#41—1182no—
481#41Q8×D5914yescomplemented
482C2×Q8—24no—
483C2×Q8—23no—
484C2×Q8—23no—
485C2×Q8—44no—
486C2×Q8—591no—
487C2×Q8—1183no—
488C2×Q8—1183no—
489C2×Q8—2364no—
490C2×Q8C2×C22⋊D5913yes—
491C2×Q8C2×C22⋊D5911yescharacteristic
492C2×Q8C23×D5911yescharacteristic
493C2×D4—21no—
494C2×D4—591no—
495C2×D4—1181no—
496C2×D4C23×D5911yescharacteristic
497C22×C4—21no—
498C22×C4—23no—
499C22×C4—23no—
500C22×C4—1181no—
501C22×C4—1183no—
502C22×C4—1183no—
503C22×C4C2×C22⋊D5913yes—
504C22×C4C2×C22⋊D5913yes—
505C22×C4C23×D5911yescharacteristic
506Q16—594no—
507SD16—594no—
508SD16—594no—
509D8—594no—
510C2×C8—594no—
511C2×C8—591no—
512C2×C8—591no—
513C2×C8—593no—
514C2×C8—593no—
515C2×C8—1186no—
516C4⋊C4—26no—
517C4⋊C4—23no—
518C4⋊C4—23no—
519C4⋊C4—23no—
520C4⋊C4—23no—
521C4⋊C4—26no—
522C4⋊C4—593no—
523C4⋊C4—593no—
524C4⋊C4—1186no—
525C4⋊C4—1183no—
526C4⋊C4—1183no—
527C4⋊C4—1183no—
528C4⋊C4—1186no—
529C4⋊C4—1183no—
530C4⋊C4C2×C22⋊D5913yes—
531C4⋊C4C2×C22⋊D5916yes—
532C4⋊C4C2×C22⋊D5913yes—
533C4⋊C4C23×D5913yes—
534C4⋊C4C23×D5913yes—
535C22⋊C4—23no—
536C22⋊C4—23no—
537C22⋊C4—23no—
538C22⋊C4—1183no—
539C22⋊C4—1183no—
540C22⋊C4—1183no—
541C42—23no—
542C42—23no—
543C42—23no—
544C42—1183no—
545C42—1183no—
546C42—1183no—
547C42C2×C22⋊D5913yes—
548C42C2×C22⋊D5913yes—
549C42C23×D5913yes—
550C42C23×D5913yes—
551C23—21no—
552C23—1181no—
553C23C22×C22⋊D5911yescharacteristic
554Q8—26no—
555Q8—21no—
556Q8—46no—
557Q8—83no—
558Q8—592no—
559Q8—1181no—
560Q8—4723no—
561Q8—2366no—
562Q8#1427374714yesdirect product factor
563Q8C2×Q8×D5912yes—
564D4—24no—
565D4—21no—
566D4—592no—
567D4—1184no—
568D4—1181no—
569D4C2×Q8×D5912yes—
570C2×C4—21no—
571C2×C4—26no—
572C2×C4—24no—
573C2×C4—23no—
574C2×C4—23no—
575C2×C4—43no—
576C2×C4—46no—
577C2×C4—83no—
578C2×C4—1181no—
579C2×C4—1184no—
580C2×C4—1183no—
581C2×C4—1183no—
582C2×C4—4723no—
583C2×C4—2366no—
584C2×C4C22×C22⋊D5913yes—
585C2×C4C22×C22⋊D5913yes—
586C2×C4C22×C22⋊D5913yes—
587C2×C4C22×C22⋊D5913yes—
588C2×C4C22×C22⋊D5911yescharacteristic
589C2×C4C2×Q8×D5912yes—
590C2×C4#1427377512yes—
591C2×C4C24×D5911yescharacteristic
592C8—592no—
593C8—592no—
594C8—1183no—
595C8—1183no—
596C22—21no—
597C22—21no—
598C22—22no—
599C22—41no—
600C22—41no—
601C22—1182no—
602C22—2361no—
603C22—2361no—
604C22Q8×C22⋊D5912yes—
605C22C23×C22⋊D5911yescharacteristic
606C4—23no—
607C4—22no—
608C4—23no—
609C4—23no—
610C4—43no—
611C4—43no—
612C4—1182no—
613C4—2363no—
614C4—2363no—
615C4Q8×C22⋊D5912yes—
616C4C2×#1427374716yes—
617C4C22×Q8×D5911yescharacteristic
618C4C2×#1427377511yescharacteristic
619C2—22no—
620C2—42no—
621C2—2362no—
622C2C2×Q8×C22⋊D5911yescharacteristic
623C2C22×#1427374711yescharacteristic
624C2#2143886711yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D59
commutator, cosocleC2×C4×C59C25
FrattiniC2×C4C24×D59
FittingQ8×#41×C59C2
socleC22×C59C23×D4
prime radicalC2×#14273747C22
hypercenterQ8×#41D59
hypocenterC59Q8×#135
LCSG ▷ C2×C4×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C2×C4 ◁ C2×C4×Q8 ◁ Q8×#41

Complements

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

NH ≅ G/Ntype
1. Q8×#11745685C2semidirect
2. #21436983C2semidirect
3. Q8×#11745672C2semidirect
4. #21436971C2semidirect
5. #21436967C2semidirect
6. #21436959C2semidirect
7. Q8×#11745670C2semidirect
8. Q8×#11745671C2semidirect
9. Q8×#11745669C2semidirect
10. #21434606C2semidirect
11. #21433884C2semidirect
12. #21433878C2semidirect
13. C2×Q8×C59⋊C8C2semidirect
15. Q8×#41×C59C2semidirect
76. #14273747Q8direct (N × H)
166. #41×C59C2×Q8semidirect
185. #11745685C2×Q8semidirect
215. #11745672C2×Q8semidirect
216. #11745671C2×Q8semidirect
217. #11745670C2×Q8semidirect
218. #11745669C2×Q8semidirect
233. C2×C59⋊C8C2×Q8semidirect
245. Q8×C59#135semidirect
308. Q8×#41D59semidirect
386. C59Q8×#135semidirect
481. #41Q8×D59semidirect
562. Q8#14273747direct (N × H)

Maximal subgroups

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

subgroupindex
Q8×#117456852
#214369832
Q8×#117456722
#214369712
#214369672
#214369592
Q8×#117456702
Q8×#117456712
Q8×#117456692
#214346062
#214338842
#214338782
C2×Q8×C59⋊C82
C4×#142737472
Q8×#41×C592
Q8×#13559