Finite Group Databasebeta

← Q8×#14519797 #30560476

Subgroups — Q8×#14519797 #30560476

Order 15616.

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×#11758222C211yesmaximal, characteristic, complemented
2#21458706C213yesmaximal, complemented
3Q8×#11758202C211yesmaximal, characteristic, complemented
4#21458694C213yesmaximal, complemented
5#21458690C213yesmaximal, complemented
6#21458682C213yesmaximal, complemented
7Q8×#11758200C211yesmaximal, characteristic, complemented
8Q8×#11758201C211yesmaximal, characteristic, complemented
9Q8×#11758199C211yesmaximal, characteristic, complemented
10#21456329C213yesmaximal, complemented
11#21455607C213yesmaximal, complemented
12#21455601C213yesmaximal, complemented
13C2×Q8×#92418C211yesmaximal, characteristic, complemented
14C4×#14519797C213yesmaximal
15Q8×#41×C61C211yesmaximal, characteristic, complemented
16C2×#14519797C2211yescharacteristic
17Q8×Dic122C2211yescharacteristic
18#16840490C2213yes—
19#16840472C2213yes—
20#16840467—23no—
21Q8×D244C2211yescharacteristic
22Q8×C22⋊D61—21no—
23#16840319C2213yes—
24#16840316—23no—
25C4×#11758222C2213yes—
26C4×Q8×D61—21no—
27#16840165C2213yes—
28#16840165C2213yes—
29#16840161C2216yes—
30#16840161C2216yes—
31#16840158C2213yes—
32#16840158C2213yes—
33#16840069C2216yes—
34#16840066C2216yes—
35#16840062C2216yes—
36#16840059C2216yes—
37#16840057C2213yes—
38#16840054C2213yes—
39#16840052C2213yes—
40#16840049C2213yes—
41#16840047C2213yes—
42#16840044C2213yes—
43#16840042C2213yes—
44#16840039C2213yes—
45Q8×#92418C2213yes—
46Q8×#92418C2211yescharacteristic
47Q8×#92418C2213yes—
48Q8×#92418C2211yescharacteristic
49C4×#11758202C2213yes—
50C4×#11758202C2213yes—
51C4×#11758201C2213yes—
52C4×#11758201C2213yes—
53C4×#11758200C2213yes—
54C4×#11758200C2213yes—
55C4×#11758199C2213yes—
56C4×#11758199C2213yes—
57#16839798C2213yes—
58#16839798C2216yes—
59C2×C4⋊#92418C2213yes—
60#16839758C2213yes—
61#16839756C2213yes—
62C2×C4×#92418C2213yes—
63Q82×C61C2211yescharacteristic
64C61×#557C2213yes—
65C61×#554—23no—
66C61×#554C2213yes—
67D4×Q8×C61—21no—
68D4×Q8×C61C2211yescharacteristic
69C61×#533—23no—
70C61×#533C2213yes—
71C4×#41×C61C2213yes—
72C2×C4×Q8×C61—21no—
73C2×C4×Q8×C61C2211yescharacteristic
74C2×#11758222C2311yescharacteristic
75C2×Q8×D61—21no—
76#14519797Q818yesdirect product factor
77C2×#11758202C2311yescharacteristic
78C2×#11758200C2311yescharacteristic
79C2×#11758201C2311yescharacteristic
80C2×#11758199C2311yescharacteristic
81C22×#92418C2313yes—
82C22×#92418C2311yescharacteristic
83#14519777C2313yes—
84#14519777C2313yes—
85#14519776—23no—
86#14519775—23no—
87#14519769C2316yes—
88#14519767—26no—
89#14519757—23no—
90#14519756—23no—
91#14519756—23no—
92#14519755—26no—
93#14519753—23no—
94Q8×Dic61—21no—
95C4×Dic122—23no—
96C4×Dic122—23no—
97C4×Dic122C2313yes—
98C4×Dic122C2313yes—
99#14519742—23no—
100C4×D244C2313yes—
101C4×D244C2313yes—
102C4×C22⋊D61—23no—
103C4×C22⋊D61—23no—
104#14519736C2313yes—
105#14519734—23no—
106C4⋊C4×D61—23no—
107C42×D61—23no—
108#14519710C2316yes—
109#14519710C2316yes—
110#14519710C2313yes—
111#14519710C2313yes—
112#14519708C2316yes—
113#14519708C2316yes—
114#14519708C2313yes—
115#14519708C2313yes—
116C4⋊#92418C2313yes—
117C4⋊#92418C2313yes—
118C4⋊#92418C2316yes—
119#14519701C2313yes—
120#14519701C2313yes—
121#14519700C2313yes—
122#14519700C2313yes—
123#14519698C2313yes—
124#14519698C2313yes—
125#14519697C2313yes—
126#14519697C2313yes—
127C4×#92418C2313yes—
128C4×#92418C2313yes—
129C4×#92418C2313yes—
130C4×#92418C2313yes—
131C2×#41×C61C2311yescharacteristic
132C22×Q8×C61—21no—
133C22×Q8×C61D411yescharacteristic
134C4⋊Q8×C61—23no—
135C4⋊Q8×C61—23no—
136C4⋊Q8×C61C2313yes—
137C4⋊Q8×C61C2313yes—
138#125×C61—26no—
139#125×C61C2316yes—
140C22⋊Q8×C61—23no—
141C22⋊Q8×C61—23no—
142C22⋊Q8×C61—23no—
143C22⋊Q8×C61—23no—
144C22⋊Q8×C61—26no—
145C4×Q8×C61—21no—
146C4×Q8×C61—23no—
147C4×Q8×C61—23no—
148C4×Q8×C61—23no—
149C4×Q8×C61D413yes—
150C4×Q8×C61D411yescharacteristic
151C4×Q8×C61C2313yes—
152C4×Q8×C61C2313yes—
153C4×Q8×C61C2311yescharacteristic
154C4×Q8×C61C2313yes—
155C4×D4×C61—23no—
156C4×D4×C61—23no—
157C4×D4×C61C2313yes—
158C4×D4×C61C2313yes—
159#117×C61—23no—
160#117×C61C2313yes—
161C2×C4⋊C4×C61—23no—
162C2×C4⋊C4×C61D413yes—
163C2×C42×C61—23no—
164C2×C42×C61C2313yes—
165#41×C61—22no—
166#41×C61C2×Q814yescomplemented
167C2×Q8×C61—24no—
168C2×Q8×C61—23no—
169C2×Q8×C61—23no—
170C2×Q8×C61—44no—
171C2×Q8×C61C2×D413yes—
172C2×Q8×C61C2×D411yescharacteristic
173C2×Q8×C61C2411yescharacteristic
174C2×D4×C61—21no—
175C2×D4×C61C2411yescharacteristic
176C22×C4×C61—21no—
177C22×C4×C61—23no—
178C22×C4×C61—23no—
179C22×C4×C61C2×D413yes—
180C22×C4×C61C2×D413yes—
181C22×C4×C61C2411yescharacteristic
182C2×C22⋊D61—21no—
183Q8×D61—44no—
184#11758222—22no—
185#11758222C2×Q814yescomplemented
186C2×D244C2411yescharacteristic
187C2×C4×D61—21no—
188C2×C4×D61—23no—
189C2×C4×D61—23no—
190C2×Dic122—23no—
191C2×Dic122—23no—
192C2×Dic122C2411yescharacteristic
193C4⋊C4×C61—26no—
194C4⋊C4×C61—23no—
195C4⋊C4×C61—23no—
196C4⋊C4×C61—23no—
197C4⋊C4×C61—23no—
198C4⋊C4×C61—26no—
199C4⋊C4×C61C2×D413yes—
200C4⋊C4×C61C2×D416yes—
201C4⋊C4×C61C2×D413yes—
202C4⋊C4×C61C2413yes—
203C4⋊C4×C61C2413yes—
204C22⋊C4×C61—23no—
205C22⋊C4×C61—23no—
206C22⋊C4×C61—23no—
207C42×C61—23no—
208C42×C61—23no—
209C42×C61—23no—
210C42×C61C2×D413yes—
211C42×C61C2×D413yes—
212C42×C61C2413yes—
213C42×C61C2413yes—
214C22⋊Dic61—23no—
215#11758202C2×Q814yescomplemented
216#11758201C2×Q814yescomplemented
217#11758200C2×Q814yescomplemented
218#11758199C2×Q814yescomplemented
219#11758198—23no—
220#11758198—23no—
221C4⋊Dic61—23no—
222C4⋊Dic61—23no—
223C4⋊Dic61C2413yes—
224C4⋊Dic61C2413yes—
225#11758196—26no—
226#11758196—23no—
227#11758196—23no—
228#11758196—26no—
229C4×Dic61—23no—
230C4×Dic61—23no—
231C4×Dic61—23no—
232C2×#92418—26no—
233C2×#92418C2×Q814yescomplemented
234C2×#92418C2413yes—
235C2×#92418C2413yes—
236C2×#92418C2411yescharacteristic
237C2×#92418C2411yescharacteristic
238C23×C61—21no—
239C23×C61C22×D411yescharacteristic
240C22×D61—21no—
241Q8×C61—26no—
242Q8×C61—21no—
243Q8×C61—46no—
244Q8×C61—83no—
245Q8×C61#13514yescomplemented
246Q8×C61C22×Q812yes—
247D4×C61—24no—
248D4×C61—21no—
249D4×C61C22×Q812yes—
250C2×C4×C61—21no—
251C2×C4×C61—24no—
252C2×C4×C61—26no—
253C2×C4×C61—23no—
254C2×C4×C61—23no—
255C2×C4×C61—43no—
256C2×C4×C61—46no—
257C2×C4×C61—83no—
258C2×C4×C61C22×D413yes—
259C2×C4×C61C22×D413yes—
260C2×C4×C61C22×D413yes—
261C2×C4×C61C22×D411yescharacteristic
262C2×C4×C61C22×D413yes—
263C2×C4×C61C22×Q812yes—
264C2×C4×C612−1+412yes—
265C2×C4×C61C2511yescharacteristic
266C22⋊D61—24no—
267C2×Dic61—21no—
268C2×Dic61—23no—
269C2×Dic61—23no—
270D244—21no—
271D244C22×Q812yes—
272C4×D61—24no—
273C4×D61—46no—
274C4×D61—83no—
275Dic122—21no—
276Dic122—46no—
277Dic122—83no—
278Dic122C22×Q812yes—
279#92418—23no—
280#92418—23no—
281#92418C22×Q812yes—
282#92418C22×Q812yes—
283Q8×#135—611nomaximal, Sylow
284C22×C61—21no—
285C22×C61—21no—
286C22×C61—22no—
287C22×C61—41no—
288C22×C61—41no—
289C22×C61D4×Q812yes—
290C22×C61C23×D411yescharacteristic
291C2×D61—22no—
292C2×D61—41no—
293C2×D61—41no—
294C4×C61—23no—
295C4×C61—23no—
296C4×C61—22no—
297C4×C61—23no—
298C4×C61—43no—
299C4×C61—43no—
300C4×C61D4×Q812yes—
301C4×C61C2×#13516yes—
302C4×C61C23×Q811yescharacteristic
303C4×C61C2×2−1+411yescharacteristic
304Dic61—22no—
305Dic61—43no—
306Dic61—43no—
307Q8×#41—611no—
308Q8×#41D6111yescharacteristic, complemented
309#3383—613no—
310Q8×Q16—611no—
311#3381—613no—
312#3381—613no—
313#3380—613no—
314Q8×SD16—611no—
315Q8×SD16—611no—
316Q8×D8—611no—
317#3160—613no—
318#3076—613no—
319#3076—613no—
320C2×C8×Q8—611no—
321C4×#135—613no—
322C2×C61—22no—
323C2×C61—42no—
324C2×C61C2×D4×Q811yescharacteristic
325C2×C61C22×#13511yescharacteristic
326C2×C61#358411yescharacteristic
327D61—42no—
328C2×#135—611no—
329Q82—611no—
330Q82C2×D6111yescharacteristic
331#557—613no—
332#557C2×D6113yes—
333#554—23no—
334#554—613no—
335#554—1223no—
336#554C2×D6113yes—
337D4×Q8—21no—
338D4×Q8—611no—
339D4×Q8—1221no—
340D4×Q8C2×D6111yescharacteristic
341#533—23no—
342#533—613no—
343#533—1223no—
344#533C2×D6113yes—
345C4×#41—613no—
346C4×#41C2×D6113yes—
347C2×C4×Q8—21no—
348C2×C4×Q8—1221no—
349C2×C4×Q8C2×D6111yescharacteristic
350#500—613no—
351#500—613no—
352#499—616no—
353#499—616no—
354#498—613no—
355#498—613no—
356#479—616no—
357#479—616no—
358#478—616no—
359#478—616no—
360#477—613no—
361#477—613no—
362#476—613no—
363#476—613no—
364#475—613no—
365#475—613no—
366#474—613no—
367#474—613no—
368C8×Q8—611no—
369C8×Q8—611no—
370C8×Q8—613no—
371C8×Q8—613no—
372C4×Q16—613no—
373C4×Q16—613no—
374C4×SD16—613no—
375C4×SD16—613no—
376C4×SD16—613no—
377C4×SD16—613no—
378C4×D8—613no—
379C4×D8—613no—
380#427—616no—
381#427—613no—
382C2×C4⋊C8—613no—
383#416—613no—
384#416—613no—
385C2×C4×C8—613no—
386C61Q8×#13511yescharacteristic, complemented, Sylow
387C2×#41—611no—
388C2×#41C22×D6111yescharacteristic
389C22×Q8—21no—
390C22×Q8—1221no—
391C22×Q8C22⋊D6111yescharacteristic
392#135—618no—
393C2×Q16—611no—
394C2×SD16—611no—
395C2×SD16—611no—
396C2×D8—611no—
397C22×C8—611no—
398C22×C8—613no—
399C4⋊Q8—23no—
400C4⋊Q8—23no—
401C4⋊Q8—613no—
402C4⋊Q8—613no—
403C4⋊Q8—1223no—
404C4⋊Q8—1223no—
405C4⋊Q8C22×D6113yes—
406C4⋊Q8C22×D6113yes—
407#125—26no—
408#125—616no—
409#125—1226no—
410#125C22×D6116yes—
411C22⋊Q8—23no—
412C22⋊Q8—23no—
413C22⋊Q8—23no—
414C22⋊Q8—23no—
415C22⋊Q8—26no—
416C22⋊Q8—1223no—
417C22⋊Q8—1223no—
418C22⋊Q8—1223no—
419C22⋊Q8—1223no—
420C22⋊Q8—1226no—
421C4×Q8—21no—
422C4×Q8—23no—
423C4×Q8—23no—
424C4×Q8—23no—
425C4×Q8—613no—
426C4×Q8—613no—
427C4×Q8—1221no—
428C4×Q8—1223no—
429C4×Q8—1223no—
430C4×Q8—1223no—
431C4×Q8C22⋊D6111yescharacteristic
432C4×Q8C22⋊D6113yes—
433C4×Q8C22×D6113yes—
434C4×Q8C22×D6113yes—
435C4×Q8C22×D6111yescharacteristic
436C4×Q8C22×D6113yes—
437C4×D4—23no—
438C4×D4—23no—
439C4×D4—613no—
440C4×D4—613no—
441C4×D4—1223no—
442C4×D4—1223no—
443C4×D4C22×D6113yes—
444C4×D4C22×D6113yes—
445#117—23no—
446#117—613no—
447#117—1223no—
448#117C22×D6113yes—
449C2×C4⋊C4—23no—
450C2×C4⋊C4—1223no—
451C2×C4⋊C4C22⋊D6113yes—
452C2×C42—23no—
453C2×C42—1223no—
454C2×C42C22×D6113yes—
455#107—613no—
456#107—613no—
457#107—616no—
458#107—616no—
459#106—613no—
460#106—613no—
461#106—616no—
462#106—616no—
463C4⋊C8—613no—
464C4⋊C8—613no—
465C4⋊C8—616no—
466#103—613no—
467#103—613no—
468#103—613no—
469#103—613no—
470#102—613no—
471#102—613no—
472#102—613no—
473#102—613no—
474C4×C8—613no—
475C4×C8—613no—
476C4×C8—613no—
477C4×C8—613no—
478#41—22no—
479#41—614no—
480#41—1222no—
481#41Q8×D6114yescomplemented
482C2×Q8—24no—
483C2×Q8—23no—
484C2×Q8—23no—
485C2×Q8—44no—
486C2×Q8—611no—
487C2×Q8—1223no—
488C2×Q8—1223no—
489C2×Q8—2444no—
490C2×Q8C2×C22⋊D6113yes—
491C2×Q8C2×C22⋊D6111yescharacteristic
492C2×Q8C23×D6111yescharacteristic
493C2×D4—21no—
494C2×D4—611no—
495C2×D4—1221no—
496C2×D4C23×D6111yescharacteristic
497C22×C4—21no—
498C22×C4—23no—
499C22×C4—23no—
500C22×C4—1221no—
501C22×C4—1223no—
502C22×C4—1223no—
503C22×C4C2×C22⋊D6113yes—
504C22×C4C2×C22⋊D6113yes—
505C22×C4C23×D6111yescharacteristic
506Q16—614no—
507SD16—614no—
508SD16—614no—
509D8—614no—
510C2×C8—614no—
511C2×C8—611no—
512C2×C8—611no—
513C2×C8—613no—
514C2×C8—613no—
515C2×C8—1226no—
516C4⋊C4—26no—
517C4⋊C4—23no—
518C4⋊C4—23no—
519C4⋊C4—23no—
520C4⋊C4—23no—
521C4⋊C4—26no—
522C4⋊C4—613no—
523C4⋊C4—613no—
524C4⋊C4—1226no—
525C4⋊C4—1223no—
526C4⋊C4—1223no—
527C4⋊C4—1223no—
528C4⋊C4—1226no—
529C4⋊C4—1223no—
530C4⋊C4C2×C22⋊D6113yes—
531C4⋊C4C2×C22⋊D6116yes—
532C4⋊C4C2×C22⋊D6113yes—
533C4⋊C4C23×D6113yes—
534C4⋊C4C23×D6113yes—
535C22⋊C4—23no—
536C22⋊C4—23no—
537C22⋊C4—23no—
538C22⋊C4—1223no—
539C22⋊C4—1223no—
540C22⋊C4—1223no—
541C42—23no—
542C42—23no—
543C42—23no—
544C42—1223no—
545C42—1223no—
546C42—1223no—
547C42C2×C22⋊D6113yes—
548C42C2×C22⋊D6113yes—
549C42C23×D6113yes—
550C42C23×D6113yes—
551C23—21no—
552C23—1221no—
553C23C22×C22⋊D6111yescharacteristic
554Q8—26no—
555Q8—21no—
556Q8—46no—
557Q8—83no—
558Q8—612no—
559Q8—1221no—
560Q8—4883no—
561Q8—2446no—
562Q8#1451979714yesdirect product factor
563Q8C2×Q8×D6112yes—
564D4—24no—
565D4—21no—
566D4—612no—
567D4—1224no—
568D4—1221no—
569D4C2×Q8×D6112yes—
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—1221no—
579C2×C4—1224no—
580C2×C4—1223no—
581C2×C4—1223no—
582C2×C4—4883no—
583C2×C4—2446no—
584C2×C4C22×C22⋊D6113yes—
585C2×C4C22×C22⋊D6113yes—
586C2×C4C22×C22⋊D6113yes—
587C2×C4C22×C22⋊D6113yes—
588C2×C4C22×C22⋊D6111yescharacteristic
589C2×C4C2×Q8×D6112yes—
590C2×C4#1451982512yes—
591C2×C4C24×D6111yescharacteristic
592C8—612no—
593C8—612no—
594C8—1223no—
595C8—1223no—
596C22—21no—
597C22—21no—
598C22—22no—
599C22—41no—
600C22—41no—
601C22—1222no—
602C22—2441no—
603C22—2441no—
604C22Q8×C22⋊D6112yes—
605C22C23×C22⋊D6111yescharacteristic
606C4—23no—
607C4—22no—
608C4—23no—
609C4—23no—
610C4—43no—
611C4—43no—
612C4—1222no—
613C4—2443no—
614C4—2443no—
615C4Q8×C22⋊D6112yes—
616C4C2×#1451979716yes—
617C4C22×Q8×D6111yescharacteristic
618C4C2×#1451982511yescharacteristic
619C2—22no—
620C2—42no—
621C2—2442no—
622C2C2×Q8×C22⋊D6111yescharacteristic
623C2C22×#1451979711yescharacteristic
624C2#2146059011yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingQ8×#41×C61C2
socleC22×C61C23×D4
prime radicalC2×#14519797C22
hypercenterQ8×#41D61
hypocenterC61Q8×#135
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C4×Q8 ◁ Q8×#41

Complements

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

NH ≅ G/Ntype
1. Q8×#11758222C2semidirect
2. #21458706C2semidirect
3. Q8×#11758202C2semidirect
4. #21458694C2semidirect
5. #21458690C2semidirect
6. #21458682C2semidirect
7. Q8×#11758200C2semidirect
8. Q8×#11758201C2semidirect
9. Q8×#11758199C2semidirect
10. #21456329C2semidirect
11. #21455607C2semidirect
12. #21455601C2semidirect
13. C2×Q8×#92418C2semidirect
15. Q8×#41×C61C2semidirect
76. #14519797Q8direct (N × H)
166. #41×C61C2×Q8semidirect
185. #11758222C2×Q8semidirect
215. #11758202C2×Q8semidirect
216. #11758201C2×Q8semidirect
217. #11758200C2×Q8semidirect
218. #11758199C2×Q8semidirect
233. C2×#92418C2×Q8semidirect
245. Q8×C61#135semidirect
308. Q8×#41D61semidirect
386. C61Q8×#135semidirect
481. #41Q8×D61semidirect
562. Q8#14519797direct (N × H)

Maximal subgroups

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

subgroupindex
Q8×#117582222
#214587062
Q8×#117582022
#214586942
#214586902
#214586822
Q8×#117582002
Q8×#117582012
Q8×#117581992
#214563292
#214556072
#214556012
C2×Q8×#924182
C4×#145197972
Q8×#41×C612
Q8×#13561