Finite Group Databasebeta

← Q8×#14519806 #30560611

Subgroups — Q8×#14519806 #30560611

Order 15616.

1782 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#11758224C211yesmaximal, characteristic
2Q82×D61C211yesmaximal, characteristic, complemented
3#21458756C213yesmaximal, complemented
4#21458750C213yesmaximal, complemented
5#21458733C213yesmaximal, complemented
6#21458727C213yesmaximal, complemented
7Q8×Dic244C211yesmaximal, characteristic, complemented
8Q8×#11758202C211yesmaximal, characteristic, complemented
9Q8×#11758200C211yesmaximal, characteristic, complemented
10#21456360C213yesmaximal, complemented
11#21455631C213yesmaximal
12#21455619C213yesmaximal, complemented
13Q8×#11758189C211yesmaximal, characteristic, complemented
14C4×#14519806C213yesmaximal
15Q8×SD16×C61C211yesmaximal, characteristic, complemented
16C2×#14519806C2211yescharacteristic
17Q8×Dic122—21no—
18Q8×Dic122C2213yes—
19Q8×Dic122C2211yescharacteristic
20Q8×Dic122C2211yescharacteristic
21#16840492—23no—
22#16840492C2213yes—
23#16840487C2213yes—
24#16840470—23no—
25#16840469C2213yes—
26Q8×C22⋊D61—21no—
27#16840317C2213yes—
28#16840314—23no—
29C4⋊Q8×D61—23no—
30C4⋊Q8×D61C2213yes—
31C4×#11758224C2213yes—
32C2×Q8×Dic61—21no—
33C4×Q8×D61—21no—
34C4×Q8×D61C2211yescharacteristic
35C4×Q8×D61C2213yes—
36#16840169C2216yes—
37#16840169C2216yes—
38#16840167C2213yes—
39#16840165C2213yes—
40#16840158C2213yes—
41#16840072C2216yes—
42#16840071C2216yes—
43#16840058C2213yes—
44#16840058C2213yes—
45#16840057C2213yes—
46#16840054C2213yes—
47#16840052C2213yes—
48#16840046C2216yes—
49#16840044C2213yes—
50#16840040C2216yes—
51#16839892C2213yes—
52#16839889C2213yes—
53Q8×#92418C2211yescharacteristic
54#16839868C2213yes—
55#16839866C2213yes—
56#16839863C2213yes—
57#16839859C2213yes—
58C4×Dic244C2213yes—
59C4×#11758202C2213yes—
60C4×#11758200C2213yes—
61#16839801C2216yes—
62#16839801C2213yes—
63#16839784C2213yes—
64#16839769C2213yes—
65#16839760C2213yes—
66C4×#11758189C2213yes—
67Q82×C61C2211yescharacteristic
68D4×Q8×C61C2211yescharacteristic
69C61×#498C2213yes—
70C61×#476C2213yes—
71C61×#475C2213yes—
72C8×Q8×C61C2211yescharacteristic
73C4×SD16×C61C2213yes—
74C2×#11758224C2311yescharacteristic
75C2×Q8×D61—23no—
76C2×Q8×D61D411yescharacteristic
77C2×Q8×D61C2311yescharacteristic
78#14519806Q818yesdirect product factor
79C2×Dic244C2311yescharacteristic
80C2×#11758202C2311yescharacteristic
81C2×#11758200C2311yescharacteristic
82C2×#11758189C2313yes—
83C2×#11758189C2311yescharacteristic
84#14519777—23no—
85#14519777C2313yes—
86#14519777C2313yes—
87#14519776—23no—
88#14519776—23no—
89#14519776—23no—
90#14519776—26no—
91#14519776—23no—
92#14519776C2316yes—
93#14519775—23no—
94#14519775—26no—
95#14519775—23no—
96#14519768C2316yes—
97#14519767—26no—
98#14519757—23no—
99#14519756—23no—
100#14519754—26no—
101#14519753—23no—
102Q8×Dic61—21no—
103Q8×Dic61—21no—
104Q8×Dic61—23no—
105Q8×Dic61—23no—
106Q8×Dic61D411yescharacteristic
107Q8×Dic61C2313yes—
108C4×Dic122—23no—
109C4×Dic122—23no—
110C4×Dic122C2313yes—
111C4×Dic122C2313yes—
112C4×Dic122C2313yes—
113#14519742—23no—
114#14519742—23no—
115#14519742—23no—
116#14519742D413yes—
117#14519742C2313yes—
118#14519742C2313yes—
119D4×Dic61C2313yes—
120C4×C22⋊D61—23no—
121#14519737—23no—
122#14519735C2313yes—
123#14519733—23no—
124C2×C4⋊Dic61—23no—
125C4⋊C4×D61—23no—
126C4⋊C4×D61—23no—
127C4⋊C4×D61D413yes—
128C4⋊C4×D61C2313yes—
129C2×C4×Dic61—23no—
130C42×D61—23no—
131C42×D61C2313yes—
132#14519711C2316yes—
133#14519711C2313yes—
134#14519710C2316yes—
135#14519710C2313yes—
136#14519708C2316yes—
137#14519708C2313yes—
138C4⋊#92418C2313yes—
139#14519706C2316yes—
140#14519702C2313yes—
141#14519702C2313yes—
142#14519701C2313yes—
143#14519700C2313yes—
144#14519700C2313yes—
145#14519698C2313yes—
146#14519687C2313yes—
147#14519687C2313yes—
148C4×#92418C2313yes—
149C22×Q8×C61—21no—
150C2×SD16×C61C2311yescharacteristic
151C4⋊Q8×C61—23no—
152C4⋊Q8×C61C2313yes—
153C4⋊Q8×C61C2313yes—
154C22⋊Q8×C61—23no—
155C4×Q8×C61—21no—
156C4×Q8×C61C2313yes—
157C4×Q8×C61C2311yescharacteristic
158C4×D4×C61C2313yes—
159#106×C61C2316yes—
160#106×C61C2313yes—
161C4⋊C8×C61C2313yes—
162#103×C61C2313yes—
163#102×C61C2313yes—
164C4×C8×C61C2313yes—
165C2×Q8×C61—23no—
166C2×Q8×C61—23no—
167C2×Q8×C61—44no—
168C2×Q8×C61C2×D411yescharacteristic
169C2×Q8×C61C2411yescharacteristic
170C2×D4×C61C2411yescharacteristic
171C22×C4×C61—23no—
172C2×C22⋊D61—21no—
173C22×Dic61—21no—
174C22×Dic61—23no—
175Q8×D61—24no—
176Q8×D61—22no—
177Q8×D61—86no—
178Q8×D61C2×Q814yescomplemented
179#11758224—22no—
180#11758224C2×Q814yes—
181C2×C4×D61—21no—
182C2×C4×D61—23no—
183C2×C4×D61C2×D413yes—
184C2×C4×D61C2×D413yes—
185C2×C4×D61C2411yescharacteristic
186C2×Dic122—21no—
187C2×Dic122—23no—
188C2×Dic122—23no—
189C2×Dic122—26no—
190C2×Dic122C2×D413yes—
191C2×Dic122C2411yescharacteristic
192C2×Dic122C2411yescharacteristic
193SD16×C61C2×Q814yescomplemented
194C2×C8×C61C2413yes—
195C2×C8×C61C2411yescharacteristic
196C4⋊C4×C61—23no—
197C4⋊C4×C61—23no—
198C4⋊C4×C61—23no—
199C4⋊C4×C61C2×D413yes—
200C4⋊C4×C61C2413yes—
201C4⋊C4×C61C2413yes—
202C22⋊C4×C61—23no—
203C42×C61—23no—
204C42×C61C2413yes—
205C22⋊Dic61—23no—
206#11758202C2×Q814yescomplemented
207#11758200C2×Q814yescomplemented
208#11758198—23no—
209C4⋊Dic61—23no—
210C4⋊Dic61—23no—
211C4⋊Dic61—23no—
212C4⋊Dic61—26no—
213C4⋊Dic61C2×D413yes—
214C4⋊Dic61C2413yes—
215C4⋊Dic61C2413yes—
216#11758196—23no—
217#11758196—23no—
218#11758196—26no—
219#11758196—23no—
220#11758196—23no—
221#11758196—23no—
222#11758196—26no—
223#11758196C2×D416yes—
224C4×Dic61—23no—
225C4×Dic61—23no—
226C4×Dic61—23no—
227C4×Dic61—23no—
228C4×Dic61—23no—
229C4×Dic61—23no—
230C4×Dic61C2×D413yes—
231C4×Dic61C2×D413yes—
232C4×Dic61C2413yes—
233C2×#92418C2413yes—
234C2×#92418C2411yescharacteristic
235Dic244C2×Q814yescomplemented
236#11758189—26no—
237#11758189C2×Q814yescomplemented
238C23×C61—21no—
239C22×D61C22×D411yescharacteristic
240Q8×C61—21no—
241Q8×C61—46no—
242Q8×C61—83no—
243Q8×C61#13714yescomplemented
244Q8×C61C22×Q812yes—
245D4×C61—21no—
246D4×C61C22×Q812yes—
247C2×C4×C61—21no—
248C2×C4×C61—23no—
249C2×C4×C61—23no—
250C2×C4×C61—46no—
251C2×C4×C61C22×D413yes—
252C2×C4×C61C22×D413yes—
253C2×C4×C61C2511yescharacteristic
254C22⋊D61—24no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—24no—
258C2×Dic61—23no—
259C2×Dic61—23no—
260C2×Dic61—83no—
261C2×Dic61C22×D413yes—
262C2×Dic61C22×D413yes—
263C2×Dic61C22×D411yescharacteristic
264C4×D61—24no—
265C4×D61—26no—
266C4×D61—43no—
267C4×D61—46no—
268C4×D61C22×Q812yes—
269C4×D612−1+412yes—
270Dic122—26no—
271Dic122—21no—
272Dic122—21no—
273Dic122—24no—
274Dic122—86no—
275Dic122—83no—
276Dic122—83no—
277Dic122C22×Q812yes—
278Dic122C22×Q812yes—
279C8×C61—23no—
280C8×C61C22×Q812yes—
281#92418—23no—
282#92418C22×Q812yes—
283Q8×#137—611nomaximal, Sylow
284C22×C61—22no—
285C22×C61—41no—
286C22×C61—41no—
287C22×C61C23×D411yescharacteristic
288C2×D61—21no—
289C2×D61—21no—
290C2×D61D4×Q812yes—
291C4×C61—23no—
292C4×C61—22no—
293C4×C61—43no—
294C4×C61—43no—
295C4×C61C2×#13716yes—
296C4×C61C23×Q811yescharacteristic
297C4×C61C2×2−1+411yescharacteristic
298Dic61—23no—
299Dic61—22no—
300Dic61—22no—
301Dic61—23no—
302Dic61—43no—
303Dic61—43no—
304Dic61D4×Q812yes—
305Q8×#41—611no—
306C2×Q82—611no—
307#3387—613no—
308#3387—613no—
309#3385—613no—
310#3385—613no—
311Q8×Q16—611no—
312Q8×Q16—611no—
313Q8×SD16—611no—
314Q8×SD16D6111yescharacteristic, complemented
315#3165—613no—
316#3079—613no—
317#3078—613no—
318Q8×M16—611no—
319C4×#137—613no—
320C2×C61—42no—
321C2×C61C2×D4×Q811yescharacteristic
322C2×C61C22×#13711yescharacteristic
323C2×C61#358511yescharacteristic
324D61—22no—
325C2×#137—611no—
326Q82—613no—
327Q82—613no—
328Q82—611no—
329Q82—611no—
330Q82—1221no—
331Q82—1223no—
332Q82C2×D6111yescharacteristic
333#557—613no—
334#554—613no—
335#554—1223no—
336D4×Q8—1221no—
337D4×Q8C2×D6111yescharacteristic
338#533—613no—
339#533—1223no—
340C2×C4⋊Q8—613no—
341C2×C4⋊Q8—1223no—
342C4×#41—613no—
343C2×C4×Q8—611no—
344C2×C4×Q8—613no—
345C2×C4×Q8—1221no—
346C2×C4×Q8—1221no—
347#501—616no—
348#501—616no—
349#500—613no—
350#500—613no—
351#498—613no—
352#498C2×D6113yes—
353#479—616no—
354#479—616no—
355#477—613no—
356#477—613no—
357#477—613no—
358#477—613no—
359#476—613no—
360#476C2×D6113yes—
361#475—613no—
362#475—616no—
363#475C2×D6113yes—
364#474—616no—
365#446—613no—
366#446—613no—
367C8×Q8—611no—
368C8×Q8C2×D6111yescharacteristic
369#441—613no—
370#441—613no—
371#440—613no—
372#440—613no—
373C4×Q16—613no—
374C4×Q16—613no—
375C4×SD16—613no—
376C4×SD16C2×D6113yes—
377#428—616no—
378#428—613no—
379#423—613no—
380#419—613no—
381#417—613no—
382C4×M16—613no—
383C61Q8×#13711yescharacteristic, complemented, Sylow
384C2×#41—611no—
385C22×Q8—21no—
386C22×Q8—611no—
387C22×Q8—611no—
388C22×Q8—1223no—
389#137—618no—
390C2×Q16—611no—
391C2×Q16—611no—
392C2×SD16—611no—
393C2×SD16C22×D6111yescharacteristic
394C2×M16—611no—
395C2×M16—613no—
396C4⋊Q8—23no—
397C4⋊Q8—616no—
398C4⋊Q8—613no—
399C4⋊Q8—613no—
400C4⋊Q8—1223no—
401C4⋊Q8—1223no—
402C4⋊Q8—1223no—
403C4⋊Q8—1226no—
404C4⋊Q8—1223no—
405C4⋊Q8—1226no—
406C4⋊Q8—1223no—
407C4⋊Q8—1223no—
408C4⋊Q8—1223no—
409C4⋊Q8C22×D6113yes—
410C4⋊Q8C22×D6113yes—
411#125—616no—
412#125—1226no—
413C22⋊Q8—23no—
414C22⋊Q8—1226no—
415C22⋊Q8—1223no—
416C22⋊Q8—1223no—
417C22⋊Q8—1223no—
418C4×Q8—21no—
419C4×Q8—611no—
420C4×Q8—613no—
421C4×Q8—613no—
422C4×Q8—613no—
423C4×Q8—613no—
424C4×Q8—613no—
425C4×Q8—613no—
426C4×Q8—613no—
427C4×Q8—1223no—
428C4×Q8—1223no—
429C4×Q8—1223no—
430C4×Q8—1223no—
431C4×Q8—1223no—
432C4×Q8—1223no—
433C4×Q8—1223no—
434C4×Q8—1221no—
435C4×Q8—1221no—
436C4×Q8C22×D6113yes—
437C4×Q8C22×D6111yescharacteristic
438C4×D4—613no—
439C4×D4—1223no—
440C4×D4—1223no—
441C4×D4C22×D6113yes—
442#117—613no—
443#117—1223no—
444C2×C4⋊C4—613no—
445C2×C4⋊C4—613no—
446C2×C4⋊C4—1223no—
447C2×C4⋊C4—1223no—
448C2×C4⋊C4—1223no—
449C2×C42—613no—
450C2×C42—1223no—
451C2×C42—1223no—
452#107—613no—
453#107—616no—
454#107—613no—
455#107—616no—
456#106—613no—
457#106—616no—
458#106C22×D6113yes—
459#106C22×D6116yes—
460C4⋊C8—616no—
461C4⋊C8—613no—
462C4⋊C8C22×D6113yes—
463#103—613no—
464#103—613no—
465#103—613no—
466#103—613no—
467#103—613no—
468#103C22×D6113yes—
469#102—613no—
470#102C22×D6113yes—
471#97—613no—
472#97—613no—
473C4×C8—613no—
474C4×C8C22×D6113yes—
475#41—614no—
476#41—1222no—
477C2×Q8—23no—
478C2×Q8—23no—
479C2×Q8—44no—
480C2×Q8—613no—
481C2×Q8—614no—
482C2×Q8—611no—
483C2×Q8—611no—
484C2×Q8—1221no—
485C2×Q8—1226no—
486C2×Q8—1222no—
487C2×Q8—1224no—
488C2×Q8—1223no—
489C2×Q8—1223no—
490C2×Q8—4886no—
491C2×Q8D4×D6111yescharacteristic
492C2×Q8C23×D6111yescharacteristic
493C2×D4—1221no—
494C2×D4C23×D6111yescharacteristic
495C22×C4—23no—
496C22×C4—611no—
497C22×C4—613no—
498C22×C4—613no—
499C22×C4—1221no—
500C22×C4—1223no—
501C22×C4—1221no—
502C22×C4—1223no—
503Q16—614no—
504Q16—614no—
505SD16—614no—
506SD16Q8×D6114yescomplemented
507M16—614no—
508M16—1226no—
509C2×C8—611no—
510C2×C8—613no—
511C2×C8C23×D6113yes—
512C2×C8C23×D6111yescharacteristic
513C4⋊C4—23no—
514C4⋊C4—23no—
515C4⋊C4—23no—
516C4⋊C4—613no—
517C4⋊C4—616no—
518C4⋊C4—613no—
519C4⋊C4—613no—
520C4⋊C4—1223no—
521C4⋊C4—1223no—
522C4⋊C4—1226no—
523C4⋊C4—1223no—
524C4⋊C4—1226no—
525C4⋊C4—1223no—
526C4⋊C4—1226no—
527C4⋊C4—1223no—
528C4⋊C4—1223no—
529C4⋊C4—1223no—
530C4⋊C4—1223no—
531C4⋊C4D4×D6113yes—
532C4⋊C4C23×D6113yes—
533C4⋊C4C23×D6113yes—
534C22⋊C4—23no—
535C22⋊C4—1223no—
536C22⋊C4—1223no—
537C42—23no—
538C42—613no—
539C42—613no—
540C42—613no—
541C42—1223no—
542C42—1223no—
543C42—1223no—
544C42—1223no—
545C42—1223no—
546C42—1223no—
547C42C23×D6113yes—
548C23—21no—
549C23—611no—
550Q8—21no—
551Q8—46no—
552Q8—83no—
553Q8—612no—
554Q8—612no—
555Q8—1224no—
556Q8—1226no—
557Q8—1221no—
558Q8—1221no—
559Q8—4883no—
560Q8—4883no—
561Q8—4886no—
562Q8#1451980614yesdirect product factor
563Q8C2×Q8×D6112yes—
564D4—21no—
565D4—1224no—
566D4C2×Q8×D6112yes—
567C2×C4—21no—
568C2×C4—23no—
569C2×C4—23no—
570C2×C4—46no—
571C2×C4—611no—
572C2×C4—612no—
573C2×C4—612no—
574C2×C4—613no—
575C2×C4—613no—
576C2×C4—1224no—
577C2×C4—1224no—
578C2×C4—1221no—
579C2×C4—1221no—
580C2×C4—1226no—
581C2×C4—1223no—
582C2×C4—1223no—
583C2×C4—4883no—
584C2×C4—2443no—
585C2×C4—2446no—
586C2×C4C2×D4×D6113yes—
587C2×C4C2×D4×D6113yes—
588C2×C4C24×D6111yescharacteristic
589C8—23no—
590C8—612no—
591C8—1223no—
592C8C2×Q8×D6112yes—
593C22—22no—
594C22—41no—
595C22—41no—
596C22—612no—
597C22—1221no—
598C22—1221no—
599C22C22×D4×D6111yescharacteristic
600C4—22no—
601C4—23no—
602C4—43no—
603C4—43no—
604C4—612no—
605C4—1223no—
606C4—1222no—
607C4—1222no—
608C4—1223no—
609C4—2443no—
610C4—2443no—
611C4C2×#1451980616yes—
612C4C22×Q8×D6111yescharacteristic
613C42−1+4×D6111yescharacteristic
614C2—42no—
615C2—1222no—
616C2D4×Q8×D6111yescharacteristic
617C2C22×#1451980611yescharacteristic
618C2#2146059711yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingQ8×SD16×C61C2
socleC22×C61C23×D4
prime radicalC2×#11758224C23
hypercenterQ8×SD16D61
hypocenterC61Q8×#137
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×Q8 ◁ Q8×SD16

Complements

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

NH ≅ G/Ntype
2. Q82×D61C2semidirect
3. #21458756C2semidirect
4. #21458750C2semidirect
5. #21458733C2semidirect
6. #21458727C2semidirect
7. Q8×Dic244C2semidirect
8. Q8×#11758202C2semidirect
9. Q8×#11758200C2semidirect
10. #21456360C2semidirect
12. #21455619C2semidirect
13. Q8×#11758189C2semidirect
15. Q8×SD16×C61C2semidirect
78. #14519806Q8direct (N × H)
178. Q8×D61C2×Q8semidirect
193. SD16×C61C2×Q8semidirect
206. #11758202C2×Q8semidirect
207. #11758200C2×Q8semidirect
235. Dic244C2×Q8semidirect
237. #11758189C2×Q8semidirect
243. Q8×C61#137semidirect
314. Q8×SD16D61semidirect
383. C61Q8×#137semidirect
506. SD16Q8×D61semidirect
562. Q8#14519806direct (N × H)

Maximal subgroups

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

subgroupindex
Q8×#117582242
Q82×D612
#214587562
#214587502
#214587332
#214587272
Q8×Dic2442
Q8×#117582022
Q8×#117582002
#214563602
#214556312
#214556192
Q8×#117581892
C4×#145198062
Q8×SD16×C612
Q8×#13761