Finite Group Databasebeta

← Q8×#14579181 #35168732

Subgroups — Q8×#14579181 #35168732

Order 20224.

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×#11929624C211yesmaximal, characteristic, complemented
2#23776287C213yesmaximal, complemented
3Q8×#11929614C211yesmaximal, characteristic, complemented
4#23776275C213yesmaximal, complemented
5#23776271C213yesmaximal, complemented
6#23776263C213yesmaximal, complemented
7Q8×#11929612C211yesmaximal, characteristic, complemented
8Q8×#11929611C211yesmaximal, characteristic, complemented
9Q8×#11929609C211yesmaximal, characteristic, complemented
10#23773910C213yesmaximal, complemented
11#23773188C213yesmaximal, complemented
12#23773182C213yesmaximal, complemented
13C2×Q8×C79⋊C8C211yesmaximal, characteristic, complemented
14C4×#14579181C213yesmaximal
15Q8×#41×C79C211yesmaximal, characteristic, complemented
16C2×#14579181C2211yescharacteristic
17Q8×Dic158C2211yescharacteristic
18#19099291C2213yes—
19#19099273C2213yes—
20#19099268—23no—
21Q8×D316C2211yescharacteristic
22Q8×C22⋊D79—21no—
23#19099120C2213yes—
24#19099117—23no—
25C4×#11929624C2213yes—
26C4×Q8×D79—21no—
27#19098966C2213yes—
28#19098966C2213yes—
29#19098962C2216yes—
30#19098962C2216yes—
31#19098959C2213yes—
32#19098959C2213yes—
33#19098870C2216yes—
34#19098867C2216yes—
35#19098863C2216yes—
36#19098860C2216yes—
37#19098858C2213yes—
38#19098855C2213yes—
39#19098853C2213yes—
40#19098850C2213yes—
41#19098848C2213yes—
42#19098845C2213yes—
43#19098843C2213yes—
44#19098840C2213yes—
45Q8×C79⋊C8C2211yescharacteristic
46Q8×C79⋊C8C2213yes—
47Q8×C79⋊C8C2213yes—
48Q8×C79⋊C8C2211yescharacteristic
49C4×#11929614C2213yes—
50C4×#11929614C2213yes—
51C4×#11929611C2213yes—
52C4×#11929611C2213yes—
53C4×#11929612C2213yes—
54C4×#11929612C2213yes—
55C4×#11929609C2213yes—
56C4×#11929609C2213yes—
57#19098599C2216yes—
58#19098599C2213yes—
59C2×C4⋊(C79⋊C8)C2213yes—
60#19098559C2213yes—
61#19098557C2213yes—
62C2×C4×C79⋊C8C2213yes—
63Q82×C79C2211yescharacteristic
64#557×C79C2213yes—
65#554×C79—23no—
66#554×C79C2213yes—
67D4×Q8×C79—21no—
68D4×Q8×C79C2211yescharacteristic
69#533×C79—23no—
70#533×C79C2213yes—
71C4×#41×C79C2213yes—
72C2×C4×Q8×C79—21no—
73C2×C4×Q8×C79C2211yescharacteristic
74C2×#11929624C2311yescharacteristic
75C2×Q8×D79—21no—
76#14579181Q818yesdirect product factor
77C2×#11929614C2311yescharacteristic
78C2×#11929612C2311yescharacteristic
79C2×#11929611C2311yescharacteristic
80C2×#11929609C2311yescharacteristic
81C22×C79⋊C8C2313yes—
82C22×C79⋊C8C2311yescharacteristic
83#14579161C2313yes—
84#14579161C2313yes—
85#14579160—23no—
86#14579159—23no—
87#14579153C2316yes—
88#14579151—26no—
89#14579141—23no—
90#14579140—23no—
91#14579140—23no—
92#14579139—26no—
93#14579137—23no—
94Q8×Dic79—21no—
95C4×Dic158—23no—
96C4×Dic158—23no—
97C4×Dic158C2313yes—
98C4×Dic158C2313yes—
99#14579126—23no—
100C4×D316C2313yes—
101C4×D316C2313yes—
102C4×C22⋊D79—23no—
103C4×C22⋊D79—23no—
104#14579120C2313yes—
105#14579118—23no—
106C4⋊C4×D79—23no—
107C42×D79—23no—
108#14579094C2313yes—
109#14579094C2313yes—
110#14579094C2316yes—
111#14579094C2316yes—
112#14579092C2313yes—
113#14579092C2313yes—
114#14579092C2316yes—
115#14579092C2316yes—
116C4⋊(C79⋊C8)C2313yes—
117C4⋊(C79⋊C8)C2313yes—
118C4⋊(C79⋊C8)C2316yes—
119#14579085C2313yes—
120#14579085C2313yes—
121#14579084C2313yes—
122#14579084C2313yes—
123#14579082C2313yes—
124#14579082C2313yes—
125#14579081C2313yes—
126#14579081C2313yes—
127C4×C79⋊C8C2313yes—
128C4×C79⋊C8C2313yes—
129C4×C79⋊C8C2313yes—
130C4×C79⋊C8C2313yes—
131C2×#41×C79C2311yescharacteristic
132C22×Q8×C79—21no—
133C22×Q8×C79D411yescharacteristic
134C4⋊Q8×C79—23no—
135C4⋊Q8×C79—23no—
136C4⋊Q8×C79C2313yes—
137C4⋊Q8×C79C2313yes—
138#125×C79—26no—
139#125×C79C2316yes—
140C22⋊Q8×C79—23no—
141C22⋊Q8×C79—23no—
142C22⋊Q8×C79—23no—
143C22⋊Q8×C79—23no—
144C22⋊Q8×C79—26no—
145C4×Q8×C79—21no—
146C4×Q8×C79—23no—
147C4×Q8×C79—23no—
148C4×Q8×C79—23no—
149C4×Q8×C79D411yescharacteristic
150C4×Q8×C79D413yes—
151C4×Q8×C79C2313yes—
152C4×Q8×C79C2313yes—
153C4×Q8×C79C2311yescharacteristic
154C4×Q8×C79C2313yes—
155C4×D4×C79—23no—
156C4×D4×C79—23no—
157C4×D4×C79C2313yes—
158C4×D4×C79C2313yes—
159#117×C79—23no—
160#117×C79C2313yes—
161C2×C4⋊C4×C79—23no—
162C2×C4⋊C4×C79D413yes—
163C2×C42×C79—23no—
164C2×C42×C79C2313yes—
165#11929624—22no—
166#11929624C2×Q814yescomplemented
167C2×Dic158—23no—
168C2×Dic158—23no—
169C2×Dic158C2411yescharacteristic
170Q8×D79—44no—
171C2×D316C2411yescharacteristic
172C2×C22⋊D79—21no—
173C2×C4×D79—21no—
174C2×C4×D79—23no—
175C2×C4×D79—23no—
176#11929614C2×Q814yescomplemented
177#11929612C2×Q814yescomplemented
178#11929611C2×Q814yescomplemented
179#11929609C2×Q814yescomplemented
180C2×C79⋊C8—26no—
181C2×C79⋊C8C2×Q814yescomplemented
182C2×C79⋊C8C2413yes—
183C2×C79⋊C8C2413yes—
184C2×C79⋊C8C2411yescharacteristic
185C2×C79⋊C8C2411yescharacteristic
186C4⋊Dic79—23no—
187C4⋊Dic79—23no—
188C4⋊Dic79C2413yes—
189C4⋊Dic79C2413yes—
190Dic79⋊C4—26no—
191Dic79⋊C4—23no—
192Dic79⋊C4—23no—
193Dic79⋊C4—26no—
194C22⋊Dic79—23no—
195(C2×D79)⋊C4—23no—
196(C2×D79)⋊C4—23no—
197C4×Dic79—23no—
198C4×Dic79—23no—
199C4×Dic79—23no—
200#41×C79—22no—
201#41×C79C2×Q814yescomplemented
202C2×Q8×C79—24no—
203C2×Q8×C79—23no—
204C2×Q8×C79—23no—
205C2×Q8×C79—44no—
206C2×Q8×C79C2×D413yes—
207C2×Q8×C79C2×D411yescharacteristic
208C2×Q8×C79C2411yescharacteristic
209C2×D4×C79—21no—
210C2×D4×C79C2411yescharacteristic
211C22×C4×C79—21no—
212C22×C4×C79—23no—
213C22×C4×C79—23no—
214C22×C4×C79C2×D413yes—
215C22×C4×C79C2×D413yes—
216C22×C4×C79C2411yescharacteristic
217C4⋊C4×C79—26no—
218C4⋊C4×C79—23no—
219C4⋊C4×C79—23no—
220C4⋊C4×C79—23no—
221C4⋊C4×C79—23no—
222C4⋊C4×C79—26no—
223C4⋊C4×C79C2×D413yes—
224C4⋊C4×C79C2×D416yes—
225C4⋊C4×C79C2×D413yes—
226C4⋊C4×C79C2413yes—
227C4⋊C4×C79C2413yes—
228C22⋊C4×C79—23no—
229C22⋊C4×C79—23no—
230C22⋊C4×C79—23no—
231C42×C79—23no—
232C42×C79—23no—
233C42×C79—23no—
234C42×C79C2×D413yes—
235C42×C79C2×D413yes—
236C42×C79C2413yes—
237C42×C79C2413yes—
238C23×C79—21no—
239C23×C79C22×D411yescharacteristic
240C22×D79—21no—
241Q8×C79—26no—
242Q8×C79—21no—
243Q8×C79—46no—
244Q8×C79—83no—
245Q8×C79#13514yescomplemented
246Q8×C79C22×Q812yes—
247D4×C79—24no—
248D4×C79—21no—
249D4×C79C22×Q812yes—
250C2×C4×C79—21no—
251C2×C4×C79—24no—
252C2×C4×C79—26no—
253C2×C4×C79—23no—
254C2×C4×C79—23no—
255C2×C4×C79—43no—
256C2×C4×C79—46no—
257C2×C4×C79—83no—
258C2×C4×C79C22×D413yes—
259C2×C4×C79C22×D413yes—
260C2×C4×C79C22×D413yes—
261C2×C4×C79C22×D413yes—
262C2×C4×C79C22×D411yescharacteristic
263C2×C4×C79C22×Q812yes—
264C2×C4×C792−1+412yes—
265C2×C4×C79C2511yescharacteristic
266C22⋊D79—24no—
267C2×Dic79—21no—
268C2×Dic79—23no—
269C2×Dic79—23no—
270D316—21no—
271D316C22×Q812yes—
272C4×D79—24no—
273C4×D79—46no—
274C4×D79—83no—
275Dic158—21no—
276Dic158—46no—
277Dic158—83no—
278Dic158C22×Q812yes—
279C79⋊C8—23no—
280C79⋊C8—23no—
281C79⋊C8C22×Q812yes—
282C79⋊C8C22×Q812yes—
283C22×C79—21no—
284C22×C79—21no—
285C22×C79—22no—
286C22×C79—41no—
287C22×C79—41no—
288C22×C79D4×Q812yes—
289C22×C79C23×D411yescharacteristic
290C2×D79—22no—
291C2×D79—41no—
292C2×D79—41no—
293C4×C79—23no—
294C4×C79—23no—
295C4×C79—22no—
296C4×C79—23no—
297C4×C79—43no—
298C4×C79—43no—
299C4×C79D4×Q812yes—
300C4×C79C2×#13516yes—
301C4×C79C23×Q811yescharacteristic
302C4×C79C2×2−1+411yescharacteristic
303Dic79—22no—
304Dic79—43no—
305Dic79—43no—
306Q8×#135—791nomaximal, Sylow
307C2×C79—22no—
308C2×C79—42no—
309C2×C79C2×D4×Q811yescharacteristic
310C2×C79C22×#13511yescharacteristic
311C2×C79#358411yescharacteristic
312D79—42no—
313Q8×#41—791no—
314Q8×#41D7911yescharacteristic, complemented
315#3383—793no—
316Q8×Q16—791no—
317#3381—793no—
318#3381—793no—
319#3380—793no—
320Q8×SD16—791no—
321Q8×SD16—791no—
322Q8×D8—791no—
323#3160—793no—
324#3076—793no—
325#3076—793no—
326C2×C8×Q8—791no—
327C4×#135—793no—
328C79Q8×#13511yescharacteristic, complemented, Sylow
329C2×#135—791no—
330Q82—791no—
331Q82C2×D7911yescharacteristic
332#557—793no—
333#557C2×D7913yes—
334#554—23no—
335#554—793no—
336#554—1583no—
337#554C2×D7913yes—
338D4×Q8—21no—
339D4×Q8—791no—
340D4×Q8—1581no—
341D4×Q8C2×D7911yescharacteristic
342#533—23no—
343#533—793no—
344#533—1583no—
345#533C2×D7913yes—
346C4×#41—793no—
347C4×#41C2×D7913yes—
348C2×C4×Q8—21no—
349C2×C4×Q8—1581no—
350C2×C4×Q8C2×D7911yescharacteristic
351#500—793no—
352#500—793no—
353#499—796no—
354#499—796no—
355#498—793no—
356#498—793no—
357#479—796no—
358#479—796no—
359#478—796no—
360#478—796no—
361#477—793no—
362#477—793no—
363#476—793no—
364#476—793no—
365#475—793no—
366#475—793no—
367#474—793no—
368#474—793no—
369C8×Q8—791no—
370C8×Q8—791no—
371C8×Q8—793no—
372C8×Q8—793no—
373C4×Q16—793no—
374C4×Q16—793no—
375C4×SD16—793no—
376C4×SD16—793no—
377C4×SD16—793no—
378C4×SD16—793no—
379C4×D8—793no—
380C4×D8—793no—
381#427—796no—
382#427—793no—
383C2×C4⋊C8—793no—
384#416—793no—
385#416—793no—
386C2×C4×C8—793no—
387C2×#41—791no—
388C2×#41C22×D7911yescharacteristic
389C22×Q8—21no—
390C22×Q8—1581no—
391C22×Q8C22⋊D7911yescharacteristic
392#135—798no—
393C2×Q16—791no—
394C2×SD16—791no—
395C2×SD16—791no—
396C2×D8—791no—
397C22×C8—791no—
398C22×C8—793no—
399C4⋊Q8—23no—
400C4⋊Q8—23no—
401C4⋊Q8—793no—
402C4⋊Q8—793no—
403C4⋊Q8—1583no—
404C4⋊Q8—1583no—
405C4⋊Q8C22×D7913yes—
406C4⋊Q8C22×D7913yes—
407#125—26no—
408#125—796no—
409#125—1586no—
410#125C22×D7916yes—
411C22⋊Q8—23no—
412C22⋊Q8—23no—
413C22⋊Q8—23no—
414C22⋊Q8—23no—
415C22⋊Q8—26no—
416C22⋊Q8—1583no—
417C22⋊Q8—1583no—
418C22⋊Q8—1583no—
419C22⋊Q8—1583no—
420C22⋊Q8—1586no—
421C4×Q8—21no—
422C4×Q8—23no—
423C4×Q8—23no—
424C4×Q8—23no—
425C4×Q8—793no—
426C4×Q8—793no—
427C4×Q8—1581no—
428C4×Q8—1583no—
429C4×Q8—1583no—
430C4×Q8—1583no—
431C4×Q8C22⋊D7911yescharacteristic
432C4×Q8C22⋊D7913yes—
433C4×Q8C22×D7913yes—
434C4×Q8C22×D7913yes—
435C4×Q8C22×D7911yescharacteristic
436C4×Q8C22×D7913yes—
437C4×D4—23no—
438C4×D4—23no—
439C4×D4—793no—
440C4×D4—793no—
441C4×D4—1583no—
442C4×D4—1583no—
443C4×D4C22×D7913yes—
444C4×D4C22×D7913yes—
445#117—23no—
446#117—793no—
447#117—1583no—
448#117C22×D7913yes—
449C2×C4⋊C4—23no—
450C2×C4⋊C4—1583no—
451C2×C4⋊C4C22⋊D7913yes—
452C2×C42—23no—
453C2×C42—1583no—
454C2×C42C22×D7913yes—
455#107—793no—
456#107—793no—
457#107—796no—
458#107—796no—
459#106—793no—
460#106—793no—
461#106—796no—
462#106—796no—
463C4⋊C8—793no—
464C4⋊C8—793no—
465C4⋊C8—796no—
466#103—793no—
467#103—793no—
468#103—793no—
469#103—793no—
470#102—793no—
471#102—793no—
472#102—793no—
473#102—793no—
474C4×C8—793no—
475C4×C8—793no—
476C4×C8—793no—
477C4×C8—793no—
478#41—22no—
479#41—794no—
480#41—1582no—
481#41Q8×D7914yescomplemented
482C2×Q8—24no—
483C2×Q8—23no—
484C2×Q8—23no—
485C2×Q8—44no—
486C2×Q8—3164no—
487C2×Q8—791no—
488C2×Q8—1583no—
489C2×Q8—1583no—
490C2×Q8C2×C22⋊D7913yes—
491C2×Q8C2×C22⋊D7911yescharacteristic
492C2×Q8C23×D7911yescharacteristic
493C2×D4—21no—
494C2×D4—791no—
495C2×D4—1581no—
496C2×D4C23×D7911yescharacteristic
497C22×C4—21no—
498C22×C4—23no—
499C22×C4—23no—
500C22×C4—1583no—
501C22×C4—1583no—
502C22×C4—1581no—
503C22×C4C2×C22⋊D7913yes—
504C22×C4C2×C22⋊D7913yes—
505C22×C4C23×D7911yescharacteristic
506Q16—794no—
507SD16—794no—
508SD16—794no—
509D8—794no—
510C2×C8—794no—
511C2×C8—791no—
512C2×C8—791no—
513C2×C8—793no—
514C2×C8—793no—
515C2×C8—1586no—
516C4⋊C4—26no—
517C4⋊C4—23no—
518C4⋊C4—23no—
519C4⋊C4—23no—
520C4⋊C4—23no—
521C4⋊C4—26no—
522C4⋊C4—793no—
523C4⋊C4—793no—
524C4⋊C4—1586no—
525C4⋊C4—1583no—
526C4⋊C4—1583no—
527C4⋊C4—1586no—
528C4⋊C4—1583no—
529C4⋊C4—1583no—
530C4⋊C4C2×C22⋊D7913yes—
531C4⋊C4C2×C22⋊D7916yes—
532C4⋊C4C2×C22⋊D7913yes—
533C4⋊C4C23×D7913yes—
534C4⋊C4C23×D7913yes—
535C22⋊C4—23no—
536C22⋊C4—23no—
537C22⋊C4—23no—
538C22⋊C4—1583no—
539C22⋊C4—1583no—
540C22⋊C4—1583no—
541C42—23no—
542C42—23no—
543C42—23no—
544C42—1583no—
545C42—1583no—
546C42—1583no—
547C42C2×C22⋊D7913yes—
548C42C2×C22⋊D7913yes—
549C42C23×D7913yes—
550C42C23×D7913yes—
551C23—21no—
552C23—1581no—
553C23C22×C22⋊D7911yescharacteristic
554Q8—21no—
555Q8—26no—
556Q8—46no—
557Q8—83no—
558Q8—3166no—
559Q8—792no—
560Q8—6323no—
561Q8—1581no—
562Q8#1457918114yesdirect product factor
563Q8C2×Q8×D7912yes—
564D4—24no—
565D4—21no—
566D4—792no—
567D4—1584no—
568D4—1581no—
569D4C2×Q8×D7912yes—
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—3166no—
579C2×C4—6323no—
580C2×C4—1583no—
581C2×C4—1583no—
582C2×C4—1581no—
583C2×C4—1584no—
584C2×C4C22×C22⋊D7913yes—
585C2×C4C22×C22⋊D7913yes—
586C2×C4C22×C22⋊D7913yes—
587C2×C4C22×C22⋊D7913yes—
588C2×C4C22×C22⋊D7911yescharacteristic
589C2×C4C2×Q8×D7912yes—
590C2×C4#1457920912yes—
591C2×C4C24×D7911yescharacteristic
592C8—792no—
593C8—792no—
594C8—1583no—
595C8—1583no—
596C22—21no—
597C22—21no—
598C22—22no—
599C22—41no—
600C22—41no—
601C22—3161no—
602C22—3161no—
603C22—1582no—
604C22Q8×C22⋊D7912yes—
605C22C23×C22⋊D7911yescharacteristic
606C4—23no—
607C4—22no—
608C4—23no—
609C4—23no—
610C4—43no—
611C4—43no—
612C4—3163no—
613C4—3163no—
614C4—1582no—
615C4Q8×C22⋊D7912yes—
616C4C2×#1457918116yes—
617C4C22×Q8×D7911yescharacteristic
618C4C2×#1457920911yescharacteristic
619C2—22no—
620C2—42no—
621C2—3162no—
622C2C2×Q8×C22⋊D7911yescharacteristic
623C2C22×#1457918111yescharacteristic
624C2#2377817111yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D79
commutator, cosocleC2×C4×C79C25
FrattiniC2×C4C24×D79
FittingQ8×#41×C79C2
socleC22×C79C23×D4
prime radicalC2×#14579181C22
hypercenterQ8×#41D79
hypocenterC79Q8×#135
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C2×C4 ◁ C2×C4×Q8 ◁ Q8×#41

Complements

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

NH ≅ G/Ntype
1. Q8×#11929624C2semidirect
2. #23776287C2semidirect
3. Q8×#11929614C2semidirect
4. #23776275C2semidirect
5. #23776271C2semidirect
6. #23776263C2semidirect
7. Q8×#11929612C2semidirect
8. Q8×#11929611C2semidirect
9. Q8×#11929609C2semidirect
10. #23773910C2semidirect
11. #23773188C2semidirect
12. #23773182C2semidirect
13. C2×Q8×C79⋊C8C2semidirect
15. Q8×#41×C79C2semidirect
76. #14579181Q8direct (N × H)
166. #11929624C2×Q8semidirect
176. #11929614C2×Q8semidirect
177. #11929612C2×Q8semidirect
178. #11929611C2×Q8semidirect
179. #11929609C2×Q8semidirect
181. C2×C79⋊C8C2×Q8semidirect
201. #41×C79C2×Q8semidirect
245. Q8×C79#135semidirect
314. Q8×#41D79semidirect
328. C79Q8×#135semidirect
481. #41Q8×D79semidirect
562. Q8#14579181direct (N × H)

Maximal subgroups

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

subgroupindex
Q8×#119296242
#237762872
Q8×#119296142
#237762752
#237762712
#237762632
Q8×#119296122
Q8×#119296112
Q8×#119296092
#237739102
#237731882
#237731822
C2×Q8×C79⋊C82
C4×#145791812
Q8×#41×C792
Q8×#13579