Finite Group Databasebeta

← Q8×#14519801 #30560583

Subgroups — Q8×#14519801 #30560583

Order 15616.

2010 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#11758224C211yesmaximal, characteristic, complemented
2D4×Q8×D61C211yesmaximal, characteristic, complemented
3#21458745C213yesmaximal, complemented
4#21458744C213yesmaximal, complemented
5#21458722C213yesmaximal, complemented
6#21458716C213yesmaximal, complemented
7Q8×#11758190C211yesmaximal, characteristic, complemented
8Q8×#11758200C211yesmaximal, characteristic, complemented
9Q8×#11758199C211yesmaximal, characteristic, complemented
10#21456362C213yesmaximal, complemented
11#21455627C213yesmaximal, complemented
12#21455611C213yesmaximal, complemented
13Q8×#11758189C211yesmaximal, characteristic, complemented
14C4×#14519801C213yesmaximal
15Q8×D8×C61C211yesmaximal, characteristic, complemented
16C22×Q8×D61—21no—
17C2×#14519801C2211yescharacteristic
18Q8×Dic122C2211yescharacteristic, complemented
19#16840487C2213yescomplemented
20#16840470—23no—
21#16840469C2213yescomplemented
22Q8×D244C2211yescharacteristic
23D4×Dic122C2213yes—
24#16840427C2213yes—
25Q8×C22⋊D61—21no—
26Q8×C22⋊D61—21no—
27#16840425—23no—
28#16840317C2213yes—
29#16840314—23no—
30C4⋊Q8×D61C2213yes—
31C22⋊Q8×D61—23no—
32C4×#11758224C2213yes—
33C2×Q8×Dic61—21no—
34C4×Q8×D61C2211yescharacteristic
35C4×D4×D61C2213yes—
36#16840170C2216yescomplemented
37#16840170C2216yescomplemented
38#16840165C2213yescomplemented
39#16840160C2213yescomplemented
40#16840158C2213yescomplemented
41#16840065C2216yes—
42#16840064C2216yes—
43#16840056C2216yescomplemented
44#16840053C2213yes—
45#16840052C2213yes—
46#16840050C2216yescomplemented
47#16840048C2213yescomplemented
48#16840044C2213yescomplemented
49#16840042C2213yescomplemented
50#16840039C2213yes—
51#16839892C2213yescomplemented
52#16839889C2213yescomplemented
53Q8×#92418C2211yescharacteristic, complemented
54#16839874C2213yes—
55#16839871C2213yes—
56#16839861C2213yes—
57#16839859C2213yes—
58C4×#11758190C2213yes—
59C4×#11758200C2213yes—
60C4×#11758199C2213yes—
61#16839803C2216yes—
62#16839803C2213yes—
63#16839784C2213yes—
64#16839765C2213yes—
65#16839760C2213yes—
66C4×#11758189C2213yes—
67D4×Q8×C61C2211yescharacteristic
68D4×Q8×C61C2211yescharacteristic, complemented
69C61×#500C2213yescomplemented
70C61×#474C2213yes—
71C61×#474C2213yescomplemented
72C8×Q8×C61C2211yescharacteristic, complemented
73C4×D8×C61C2213yes—
74C2×#11758224C2311yescharacteristic
75C22×Dic122—23no—
76C2×Q8×D61—21no—
77C2×Q8×D61—23no—
78C2×Q8×D61—23no—
79C2×Q8×D61—48no—
80C2×Q8×D61D411yescharacteristic
81C2×D4×D61C2311yescharacteristic
82C22×C4×D61—23no—
83#14519801Q818yesdirect product factor
84C2×#11758190C2311yescharacteristic
85C2×#11758200C2311yescharacteristic
86C2×#11758199C2311yescharacteristic
87C2×#11758189C2313yes—
88C2×#11758189C2311yescharacteristic
89#14519777C2313yes—
90#14519777C2313yes—
91#14519776—23no—
92#14519776—23no—
93#14519776C2316yes—
94#14519775—23no—
95#14519775—23no—
96#14519768C2316yes—
97#14519767—26no—
98#14519757—23no—
99#14519757—23no—
100#14519756—23no—
101#14519756—23no—
102#14519755—26no—
103#14519754—26no—
104#14519754—26no—
105#14519753—23no—
106#14519753—23no—
107Q8×Dic61—21no—
108Q8×Dic61—23no—
109Q8×Dic61—23no—
110Q8×Dic61D411yescharacteristic
111C4×Dic122C2313yes—
112C4×Dic122C2313yes—
113#14519742—23no—
114#14519742D413yes—
115#14519742C2313yes—
116C4×D244C2313yes—
117D4×Dic61C2313yes—
118D4×Dic61C2313yes—
119#14519739C2313yes—
120C4×C22⋊D61—23no—
121C4×C22⋊D61—23no—
122#14519737—23no—
123#14519737—23no—
124#14519735C2313yes—
125#14519733—23no—
126C2×C4⋊Dic61—23no—
127C4⋊C4×D61—23no—
128C4⋊C4×D61D413yes—
129C4⋊C4×D61C2313yes—
130C22⋊C4×D61—23no—
131C2×C4×Dic61—23no—
132C42×D61C2313yes—
133#14519710C2316yes—
134#14519710C2313yes—
135#14519709C2316yes—
136#14519709C2313yes—
137#14519708C2316yes—
138#14519708C2313yes—
139C4⋊#92418C2313yes—
140#14519706C2316yes—
141#14519702C2313yes—
142#14519700C2313yes—
143#14519699C2313yes—
144#14519698C2313yes—
145#14519698C2313yes—
146#14519697C2313yes—
147#14519687C2313yes—
148#14519687C2313yes—
149C4×#92418C2313yes—
150C22×Q8×C61—21no—
151C22×Q8×C61—21no—
152C2×D8×C61C2311yescharacteristic
153C4⋊Q8×C61C2313yes—
154C4⋊Q8×C61C2313yes—
155C22⋊Q8×C61—23no—
156C22⋊Q8×C61—23no—
157C4×Q8×C61C2311yescharacteristic
158C4×D4×C61C2313yes—
159C4×D4×C61C2313yes—
160#107×C61C2316yes—
161#107×C61C2313yes—
162C4⋊C8×C61C2313yes—
163#102×C61C2313yes—
164#102×C61C2313yes—
165C4×C8×C61C2313yes—
166C23×D61—21no—
167C2×Q8×C61—23no—
168C2×Q8×C61—23no—
169C2×Q8×C61—44no—
170C2×Q8×C61—44no—
171C2×Q8×C61C2×D411yescharacteristic
172C2×D4×C61C2411yescharacteristic
173C2×D4×C61C2411yescharacteristic
174C22×C4×C61—23no—
175C22×C4×C61—23no—
176C2×C22⋊D61—21no—
177C2×C22⋊D61—21no—
178C22×Dic61—21no—
179C22×Dic61—23no—
180C22×Dic61—23no—
181Q8×D61—24no—
182Q8×D61—412no—
183Q8×D61—412no—
184Q8×D61—44no—
185#11758224—22no—
186#11758224C2×Q814yescomplemented
187D4×D61—22no—
188D4×D61C2×Q814yescomplemented
189C2×D244C2411yescharacteristic
190C2×C4×D61—23no—
191C2×C4×D61—23no—
192C2×C4×D61—23no—
193C2×C4×D61—412no—
194C2×C4×D61C2×D413yes—
195C2×C4×D61C2×D413yes—
196C2×C4×D61C2411yescharacteristic
197C2×Dic122—23no—
198C2×Dic122—26no—
199C2×Dic122—23no—
200C2×Dic122—412no—
201C2×Dic122C2×D413yes—
202C2×Dic122C2411yescharacteristic
203D8×C61C2×Q814yescomplemented
204C2×C8×C61C2413yes—
205C2×C8×C61C2411yescharacteristic
206C4⋊C4×C61—23no—
207C4⋊C4×C61—23no—
208C4⋊C4×C61C2×D413yes—
209C4⋊C4×C61C2413yes—
210C4⋊C4×C61C2413yes—
211C22⋊C4×C61—23no—
212C22⋊C4×C61—23no—
213C42×C61C2413yes—
214C22⋊Dic61—23no—
215C22⋊Dic61—23no—
216#11758200C2×Q814yescomplemented
217#11758199C2×Q814yescomplemented
218#11758198—23no—
219#11758198—23no—
220#11758198—23no—
221C4⋊Dic61—23no—
222C4⋊Dic61—23no—
223C4⋊Dic61—26no—
224C4⋊Dic61C2×D413yes—
225C4⋊Dic61C2413yes—
226C4⋊Dic61C2413yes—
227#11758196—23no—
228#11758196—26no—
229#11758196—23no—
230#11758196—23no—
231#11758196—26no—
232#11758196C2×D416yes—
233C4×Dic61—23no—
234C4×Dic61—23no—
235C4×Dic61—23no—
236C4×Dic61C2×D413yes—
237C4×Dic61C2×D413yes—
238C4×Dic61C2413yes—
239C2×#92418C2413yes—
240C2×#92418C2411yescharacteristic
241#11758190C2×Q814yescomplemented
242#11758189—26no—
243#11758189C2×Q814yescomplemented
244C23×C61—21no—
245C23×C61—21no—
246C22×D61—21no—
247C22×D61—24no—
248C22×D61—42no—
249C22×D61—42no—
250C22×D61C22×D411yescharacteristic
251Q8×C61—46no—
252Q8×C61—46no—
253Q8×C61AGL(1,ℤ/8)14yescomplemented
254D4×C61—21no—
255D4×C61—21no—
256D4×C61C22×Q812yes—
257D4×C61C22×Q812yescomplemented
258C2×C4×C61—23no—
259C2×C4×C61—23no—
260C2×C4×C61—46no—
261C2×C4×C61—46no—
262C2×C4×C61C22×D413yes—
263C2×C4×C61C22×D413yes—
264C2×C4×C61C2511yescharacteristic
265C22⋊D61—24no—
266C22⋊D61—24no—
267C2×Dic61—21no—
268C2×Dic61—24no—
269C2×Dic61—23no—
270C2×Dic61—23no—
271C2×Dic61—46no—
272C2×Dic61—83no—
273C2×Dic61C22×D413yes—
274C2×Dic61C22×D413yes—
275C2×Dic61C22×D411yescharacteristic
276D244—21no—
277D244C22×Q812yes—
278C4×D61—26no—
279C4×D61—43no—
280C4×D61—46no—
281C4×D61—46no—
282C4×D61—46no—
283C4×D61C22×Q812yes—
284C4×D612−1+412yes—
285Dic122—26no—
286Dic122—21no—
287Dic122—412no—
288Dic122—46no—
289Dic122—83no—
290Dic122C22×Q812yescomplemented
291C8×C61—23no—
292C8×C61C22×Q812yescomplemented
293#92418—23no—
294#92418C22×Q812yescomplemented
295Q8×AGL(1,ℤ/8)—611nomaximal, Sylow
296C22×C61—22no—
297C22×C61—22no—
298C22×C61—41no—
299C22×C61—41no—
300C22×C61—41no—
301C22×C61—41no—
302C22×C61C23×D411yescharacteristic
303C2×D61—21no—
304C2×D61—21no—
305C2×D61—22no—
306C2×D61—48no—
307C2×D61—41no—
308C2×D61—41no—
309C2×D61D4×Q812yes—
310C4×C61—23no—
311C4×C61—43no—
312C4×C61—43no—
313C4×C61C2×AGL(1,ℤ/8)16yes—
314C4×C61C23×Q811yescharacteristic
315C4×C61C2×2−1+411yescharacteristic
316Dic61—23no—
317Dic61—22no—
318Dic61—23no—
319Dic61—43no—
320Dic61—43no—
321Dic61D4×Q812yes—
322Q8×#41—611no—
323C2×D4×Q8—611no—
324#3386—613no—
325#3386—613no—
326#3384—613no—
327#3384—613no—
328Q8×SD16—611no—
329Q8×SD16—611no—
330Q8×D8—611no—
331Q8×D8D6111yescharacteristic, complemented
332#3165—613no—
333#3079—613no—
334#3077—613no—
335Q8×M16—611no—
336C4×AGL(1,ℤ/8)—613no—
337C2×C61—42no—
338C2×C61—42no—
339C2×C61C2×D4×Q811yescharacteristic
340C2×C61C22×AGL(1,ℤ/8)11yescharacteristic
341C2×C61#358611yescharacteristic
342D61—22no—
343D61—42no—
344C23×Q8—1221no—
345C2×AGL(1,ℤ/8)—611no—
346Q82—611no—
347#557—613no—
348#554—613no—
349#554—1223no—
350D4×Q8—613no—
351D4×Q8—611no—
352D4×Q8—613no—
353D4×Q8—1223no—
354D4×Q8—1221no—
355D4×Q8—1221no—
356D4×Q8C2×D6111yescharacteristic
357D4×Q8C2×D6111yescharacteristic, complemented
358#533—613no—
359#533—1223no—
360C2×C4⋊Q8—613no—
361C2×C22⋊Q8—1223no—
362C4×#41—613no—
363C2×C4×Q8—611no—
364C2×C4×Q8—1221no—
365C2×C4×D4—613no—
366#501—616no—
367#501—616no—
368#500—613no—
369#500C2×D6113yescomplemented
370#498—613no—
371#498—613no—
372#478—616no—
373#478—616no—
374#477—616no—
375#476—613no—
376#476—613no—
377#476—616no—
378#475—613no—
379#475—613no—
380#474—613no—
381#474—613no—
382#474C2×D6113yes—
383#474C2×D6113yescomplemented
384#446—613no—
385#446—613no—
386C8×Q8—611no—
387C8×Q8C2×D6111yescharacteristic, complemented
388#442—613no—
389#442—613no—
390#440—613no—
391#440—613no—
392C4×SD16—613no—
393C4×SD16—613no—
394C4×D8—613no—
395C4×D8C2×D6113yes—
396#428—616no—
397#428—613no—
398#423—613no—
399#418—613no—
400#417—613no—
401C4×M16—613no—
402C61Q8×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
403C2×#41—611no—
404C22×Q8—21no—
405C22×Q8—21no—
406C22×Q8—611no—
407C22×Q8—1223no—
408C22×Q8—1223no—
409C22×Q8—1223no—
410C22×Q8—1221no—
411C22×Q8—2448no—
412C22×D4—611no—
413C23×C4—1223no—
414AGL(1,ℤ/8)—618no—
415C2×SD16—611no—
416C2×SD16—611no—
417C2×D8—611no—
418C2×D8C22×D6111yescharacteristic
419C2×M16—611no—
420C2×M16—613no—
421C4⋊Q8—616no—
422C4⋊Q8—613no—
423C4⋊Q8—613no—
424C4⋊Q8—1223no—
425C4⋊Q8—1223no—
426C4⋊Q8—1223no—
427C4⋊Q8—1223no—
428C4⋊Q8C22×D6113yes—
429C4⋊Q8C22×D6113yes—
430#125—616no—
431#125—1226no—
432C22⋊Q8—23no—
433C22⋊Q8—23no—
434C22⋊Q8—1226no—
435C22⋊Q8—1226no—
436C22⋊Q8—1226no—
437C22⋊Q8—1223no—
438C22⋊Q8—1223no—
439C22⋊Q8—1223no—
440C22⋊Q8—1223no—
441C22⋊Q8—1223no—
442C22⋊Q8—1223no—
443C4×Q8—611no—
444C4×Q8—613no—
445C4×Q8—613no—
446C4×Q8—613no—
447C4×Q8—613no—
448C4×Q8—1223no—
449C4×Q8—1223no—
450C4×Q8—1223no—
451C4×Q8—1221no—
452C4×Q8C22×D6111yescharacteristic
453C4×D4—613no—
454C4×D4—613no—
455C4×D4—613no—
456C4×D4—613no—
457C4×D4—1223no—
458C4×D4—1223no—
459C4×D4—1223no—
460C4×D4—1223no—
461C4×D4C22×D6113yes—
462C4×D4C22×D6113yes—
463#117—613no—
464#117—1223no—
465C2×C4⋊C4—613no—
466C2×C4⋊C4—613no—
467C2×C4⋊C4—1223no—
468C2×C4⋊C4—1223no—
469C2×C22⋊C4—1223no—
470C2×C42—613no—
471C2×C42—1223no—
472#107—613no—
473#107—616no—
474#107C22×D6113yes—
475#107C22×D6116yes—
476#106—613no—
477#106—616no—
478#106—613no—
479#106—616no—
480C4⋊C8—616no—
481C4⋊C8—613no—
482C4⋊C8C22×D6113yes—
483#103—613no—
484#103—613no—
485#102—613no—
486#102—613no—
487#102—613no—
488#102—613no—
489#102C22×D6113yes—
490#102C22×D6113yes—
491#97—613no—
492#97—613no—
493C4×C8—613no—
494C4×C8C22×D6113yes—
495C24—1221no—
496#41—614no—
497#41—1222no—
498C2×Q8—23no—
499C2×Q8—23no—
500C2×Q8—44no—
501C2×Q8—44no—
502C2×Q8—613no—
503C2×Q8—611no—
504C2×Q8—1226no—
505C2×Q8—1224no—
506C2×Q8—1223no—
507C2×Q8—1223no—
508C2×Q8—24412no—
509C2×Q8—24412no—
510C2×Q8—24412no—
511C2×Q8—2444no—
512C2×Q8D4×D6111yescharacteristic
513C2×D4—614no—
514C2×D4—611no—
515C2×D4—1222no—
516C2×D4—1221no—
517C2×D4—1221no—
518C2×D4C23×D6111yescharacteristic
519C2×D4C23×D6111yescharacteristic
520C22×C4—23no—
521C22×C4—23no—
522C22×C4—611no—
523C22×C4—613no—
524C22×C4—613no—
525C22×C4—1223no—
526C22×C4—1223no—
527C22×C4—1221no—
528C22×C4—1223no—
529C22×C4—1223no—
530C22×C4—1223no—
531C22×C4—24412no—
532SD16—614no—
533SD16—614no—
534D8—614no—
535D8Q8×D6114yescomplemented
536M16—614no—
537M16—1226no—
538C2×C8—611no—
539C2×C8—613no—
540C2×C8C23×D6113yes—
541C2×C8C23×D6111yescharacteristic
542C4⋊C4—23no—
543C4⋊C4—23no—
544C4⋊C4—613no—
545C4⋊C4—616no—
546C4⋊C4—613no—
547C4⋊C4—613no—
548C4⋊C4—1223no—
549C4⋊C4—1226no—
550C4⋊C4—1226no—
551C4⋊C4—1223no—
552C4⋊C4—1226no—
553C4⋊C4—1223no—
554C4⋊C4—1223no—
555C4⋊C4—1223no—
556C4⋊C4D4×D6113yes—
557C4⋊C4C23×D6113yes—
558C4⋊C4C23×D6113yes—
559C22⋊C4—23no—
560C22⋊C4—23no—
561C22⋊C4—1223no—
562C22⋊C4—1223no—
563C22⋊C4—1223no—
564C22⋊C4—1223no—
565C22⋊C4—1223no—
566C42—613no—
567C42—613no—
568C42—613no—
569C42—1223no—
570C42—1223no—
571C42—1223no—
572C42C23×D6113yes—
573C23—21no—
574C23—21no—
575C23—611no—
576C23—1224no—
577C23—1221no—
578C23—2442no—
579C23—2442no—
580Q8—46no—
581Q8—46no—
582Q8—612no—
583Q8—1226no—
584Q8—1221no—
585Q8—4883no—
586Q8—24412no—
587Q8—2446no—
588Q8#1451980114yesdirect product factor
589D4—21no—
590D4—21no—
591D4—612no—
592D4—1224no—
593D4—1224no—
594D4—1221no—
595D4C2×Q8×D6112yes—
596D4C2×Q8×D6112yescomplemented
597C2×C4—23no—
598C2×C4—23no—
599C2×C4—46no—
600C2×C4—46no—
601C2×C4—611no—
602C2×C4—612no—
603C2×C4—612no—
604C2×C4—613no—
605C2×C4—613no—
606C2×C4—1224no—
607C2×C4—1221no—
608C2×C4—1226no—
609C2×C4—1223no—
610C2×C4—1223no—
611C2×C4—4883no—
612C2×C4—2443no—
613C2×C4—2446no—
614C2×C4—2446no—
615C2×C4—2446no—
616C2×C4—2446no—
617C2×C4C2×D4×D6113yes—
618C2×C4C2×D4×D6113yes—
619C2×C4C24×D6111yescharacteristic
620C8—23no—
621C8—612no—
622C8—1223no—
623C8C2×Q8×D6112yescomplemented
624C22—22no—
625C22—22no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—612no—
631C22—1221no—
632C22—1221no—
633C22—1222no—
634C22—2448no—
635C22—2441no—
636C22—2441no—
637C22C22×D4×D6111yescharacteristic
638C4—23no—
639C4—43no—
640C4—43no—
641C4—612no—
642C4—1223no—
643C4—1222no—
644C4—1223no—
645C4—2443no—
646C4—2443no—
647C4C2×#1451980116yes—
648C4C22×Q8×D6111yescharacteristic
649C42−1+4×D6111yescharacteristic
650C2—42no—
651C2—42no—
652C2—1222no—
653C2—2442no—
654C2D4×Q8×D6111yescharacteristic
655C2C22×#1451980111yescharacteristic
656C2#2146060411yescharacteristic
subgroupG/H
centerC22C22×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingQ8×D8×C61C2
socleC22×C61C23×D4
prime radicalC2×#14519801C22
hypercenterQ8×D8D61
hypocenterC61Q8×AGL(1,ℤ/8)
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×Q8 ◁ Q8×D8

Complements

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

NH ≅ G/Ntype
1. Q8×#11758224C2semidirect
2. D4×Q8×D61C2semidirect
3. #21458745C2semidirect
4. #21458744C2semidirect
5. #21458722C2semidirect
6. #21458716C2semidirect
7. Q8×#11758190C2semidirect
8. Q8×#11758200C2semidirect
9. Q8×#11758199C2semidirect
10. #21456362C2semidirect
11. #21455627C2semidirect
12. #21455611C2semidirect
13. Q8×#11758189C2semidirect
15. Q8×D8×C61C2semidirect
18. Q8×Dic122C22semidirect
19. #16840487C22semidirect
21. #16840469C22semidirect
36. #16840170C22semidirect
37. #16840170C22semidirect
38. #16840165C22semidirect
39. #16840160C22semidirect
40. #16840158C22semidirect
43. #16840056C22semidirect
46. #16840050C22semidirect
47. #16840048C22semidirect
48. #16840044C22semidirect
49. #16840042C22semidirect
51. #16839892C22semidirect
52. #16839889C22semidirect
53. Q8×#92418C22semidirect
68. D4×Q8×C61C22semidirect
69. C61×#500C22semidirect
71. C61×#474C22semidirect
72. C8×Q8×C61C22semidirect
83. #14519801Q8direct (N × H)
186. #11758224C2×Q8semidirect
188. D4×D61C2×Q8semidirect
203. D8×C61C2×Q8semidirect
216. #11758200C2×Q8semidirect
217. #11758199C2×Q8semidirect
241. #11758190C2×Q8semidirect
243. #11758189C2×Q8semidirect
253. Q8×C61AGL(1,ℤ/8)semidirect
257. D4×C61C22×Q8semidirect
290. Dic122C22×Q8semidirect
292. C8×C61C22×Q8semidirect
294. #92418C22×Q8semidirect
331. Q8×D8D61semidirect
357. D4×Q8C2×D61semidirect
369. #500C2×D61semidirect
383. #474C2×D61semidirect
387. C8×Q8C2×D61semidirect
402. C61Q8×AGL(1,ℤ/8)semidirect
535. D8Q8×D61semidirect
588. Q8#14519801direct (N × H)
596. D4C2×Q8×D61semidirect
623. C8C2×Q8×D61semidirect

Maximal subgroups

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

subgroupindex
Q8×#117582242
D4×Q8×D612
#214587452
#214587442
#214587222
#214587162
Q8×#117581902
Q8×#117582002
Q8×#117581992
#214563622
#214556272
#214556112
Q8×#117581892
C4×#145198012
Q8×D8×C612
Q8×AGL(1,ℤ/8)61