Finite Group Databasebeta

← Q8×#13063345 #25977060

Subgroups — Q8×#13063345 #25977060

Order 11008.

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×#10621898C211yesmaximal, characteristic, complemented
2#19147482C213yesmaximal, complemented
3Q8×#10621885C211yesmaximal, characteristic, complemented
4#19147470C213yesmaximal, complemented
5#19147466C213yesmaximal, complemented
6#19147458C213yesmaximal, complemented
7Q8×#10621883C211yesmaximal, characteristic, complemented
8Q8×#10621884C211yesmaximal, characteristic, complemented
9Q8×#10621882C211yesmaximal, characteristic, complemented
10#19145105C213yesmaximal, complemented
11#19144383C213yesmaximal, complemented
12#19144377C213yesmaximal, complemented
13C2×Q8×C43⋊C8C211yesmaximal, characteristic, complemented
14C4×#13063345C213yesmaximal
15Q8×#41×C43C211yesmaximal, characteristic, complemented
16C2×#13063345C2211yescharacteristic
17Q8×Dic86C2211yescharacteristic
18#14583627C2213yes—
19#14583609C2213yes—
20#14583604—23no—
21Q8×D172C2211yescharacteristic
22Q8×C22⋊D43—21no—
23#14583456C2213yes—
24#14583453—23no—
25C4×#10621898C2213yes—
26C4×Q8×D43—21no—
27#14583302C2213yes—
28#14583302C2213yes—
29#14583298C2216yes—
30#14583298C2216yes—
31#14583295C2213yes—
32#14583295C2213yes—
33#14583206C2216yes—
34#14583203C2216yes—
35#14583199C2216yes—
36#14583196C2216yes—
37#14583194C2213yes—
38#14583191C2213yes—
39#14583189C2213yes—
40#14583186C2213yes—
41#14583184C2213yes—
42#14583181C2213yes—
43#14583179C2213yes—
44#14583176C2213yes—
45Q8×C43⋊C8C2211yescharacteristic
46Q8×C43⋊C8C2211yescharacteristic
47Q8×C43⋊C8C2213yes—
48Q8×C43⋊C8C2213yes—
49C4×#10621885C2213yes—
50C4×#10621885C2213yes—
51C4×#10621884C2213yes—
52C4×#10621884C2213yes—
53C4×#10621883C2213yes—
54C4×#10621883C2213yes—
55C4×#10621882C2213yes—
56C4×#10621882C2213yes—
57#14582935C2213yes—
58#14582935C2216yes—
59C2×C4⋊(C43⋊C8)C2213yes—
60#14582895C2213yes—
61#14582893C2213yes—
62C2×C4×C43⋊C8C2213yes—
63Q82×C43C2211yescharacteristic
64C43×#557C2213yes—
65C43×#554—23no—
66C43×#554C2213yes—
67D4×Q8×C43—21no—
68D4×Q8×C43C2211yescharacteristic
69C43×#533—23no—
70C43×#533C2213yes—
71C4×#41×C43C2213yes—
72C2×C4×Q8×C43—21no—
73C2×C4×Q8×C43C2211yescharacteristic
74C2×#10621898C2311yescharacteristic
75C2×Q8×D43—21no—
76#13063345Q818yesdirect product factor
77C2×#10621885C2311yescharacteristic
78C2×#10621883C2311yescharacteristic
79C2×#10621884C2311yescharacteristic
80C2×#10621882C2311yescharacteristic
81C22×C43⋊C8C2313yes—
82C22×C43⋊C8C2311yescharacteristic
83#13063325C2313yes—
84#13063325C2313yes—
85#13063324—23no—
86#13063323—23no—
87#13063317C2316yes—
88#13063315—26no—
89#13063305—23no—
90#13063304—23no—
91#13063304—23no—
92#13063303—26no—
93#13063301—23no—
94Q8×Dic43—21no—
95C4×Dic86—23no—
96C4×Dic86—23no—
97C4×Dic86C2313yes—
98C4×Dic86C2313yes—
99#13063290—23no—
100C4×D172C2313yes—
101C4×D172C2313yes—
102C4×C22⋊D43—23no—
103C4×C22⋊D43—23no—
104#13063284C2313yes—
105#13063282—23no—
106C4⋊C4×D43—23no—
107C42×D43—23no—
108#13063258C2313yes—
109#13063258C2313yes—
110#13063258C2316yes—
111#13063258C2316yes—
112#13063256C2313yes—
113#13063256C2313yes—
114#13063256C2316yes—
115#13063256C2316yes—
116C4⋊(C43⋊C8)C2313yes—
117C4⋊(C43⋊C8)C2313yes—
118C4⋊(C43⋊C8)C2316yes—
119#13063249C2313yes—
120#13063249C2313yes—
121#13063248C2313yes—
122#13063248C2313yes—
123#13063246C2313yes—
124#13063246C2313yes—
125#13063245C2313yes—
126#13063245C2313yes—
127C4×C43⋊C8C2313yes—
128C4×C43⋊C8C2313yes—
129C4×C43⋊C8C2313yes—
130C4×C43⋊C8C2313yes—
131C2×#41×C43C2311yescharacteristic
132C22×Q8×C43—21no—
133C22×Q8×C43D411yescharacteristic
134C4⋊Q8×C43—23no—
135C4⋊Q8×C43—23no—
136C4⋊Q8×C43C2313yes—
137C4⋊Q8×C43C2313yes—
138#125×C43—26no—
139#125×C43C2316yes—
140C22⋊Q8×C43—23no—
141C22⋊Q8×C43—23no—
142C22⋊Q8×C43—23no—
143C22⋊Q8×C43—23no—
144C22⋊Q8×C43—26no—
145C4×Q8×C43—21no—
146C4×Q8×C43—23no—
147C4×Q8×C43—23no—
148C4×Q8×C43—23no—
149C4×Q8×C43D411yescharacteristic
150C4×Q8×C43D413yes—
151C4×Q8×C43C2313yes—
152C4×Q8×C43C2313yes—
153C4×Q8×C43C2311yescharacteristic
154C4×Q8×C43C2313yes—
155C4×D4×C43—23no—
156C4×D4×C43—23no—
157C4×D4×C43C2313yes—
158C4×D4×C43C2313yes—
159#117×C43—23no—
160#117×C43C2313yes—
161C2×C4⋊C4×C43—23no—
162C2×C4⋊C4×C43D413yes—
163C2×C42×C43—23no—
164C2×C42×C43C2313yes—
165#41×C43—22no—
166#41×C43C2×Q814yescomplemented
167C2×Q8×C43—24no—
168C2×Q8×C43—23no—
169C2×Q8×C43—23no—
170C2×Q8×C43—44no—
171C2×Q8×C43C2×D413yes—
172C2×Q8×C43C2×D411yescharacteristic
173C2×Q8×C43C2411yescharacteristic
174C2×D4×C43—21no—
175C2×D4×C43C2411yescharacteristic
176C22×C4×C43—21no—
177C22×C4×C43—23no—
178C22×C4×C43—23no—
179C22×C4×C43C2×D413yes—
180C22×C4×C43C2×D413yes—
181C22×C4×C43C2411yescharacteristic
182C2×C22⋊D43—21no—
183Q8×D43—44no—
184#10621898—22no—
185#10621898C2×Q814yescomplemented
186C2×D172C2411yescharacteristic
187C2×C4×D43—21no—
188C2×C4×D43—23no—
189C2×C4×D43—23no—
190C2×Dic86—23no—
191C2×Dic86—23no—
192C2×Dic86C2411yescharacteristic
193C4⋊C4×C43—26no—
194C4⋊C4×C43—23no—
195C4⋊C4×C43—23no—
196C4⋊C4×C43—23no—
197C4⋊C4×C43—23no—
198C4⋊C4×C43—26no—
199C4⋊C4×C43C2×D413yes—
200C4⋊C4×C43C2×D416yes—
201C4⋊C4×C43C2×D413yes—
202C4⋊C4×C43C2413yes—
203C4⋊C4×C43C2413yes—
204C22⋊C4×C43—23no—
205C22⋊C4×C43—23no—
206C22⋊C4×C43—23no—
207C42×C43—23no—
208C42×C43—23no—
209C42×C43—23no—
210C42×C43C2×D413yes—
211C42×C43C2×D413yes—
212C42×C43C2413yes—
213C42×C43C2413yes—
214C22⋊Dic43—23no—
215#10621885C2×Q814yescomplemented
216#10621884C2×Q814yescomplemented
217#10621883C2×Q814yescomplemented
218#10621882C2×Q814yescomplemented
219(C2×D43)⋊C4—23no—
220(C2×D43)⋊C4—23no—
221C4⋊Dic43—23no—
222C4⋊Dic43—23no—
223C4⋊Dic43C2413yes—
224C4⋊Dic43C2413yes—
225Dic43⋊C4—26no—
226Dic43⋊C4—23no—
227Dic43⋊C4—23no—
228Dic43⋊C4—26no—
229C4×Dic43—23no—
230C4×Dic43—23no—
231C4×Dic43—23no—
232C2×C43⋊C8—26no—
233C2×C43⋊C8C2×Q814yescomplemented
234C2×C43⋊C8C2413yes—
235C2×C43⋊C8C2413yes—
236C2×C43⋊C8C2411yescharacteristic
237C2×C43⋊C8C2411yescharacteristic
238C23×C43—21no—
239C23×C43C22×D411yescharacteristic
240C22×D43—21no—
241Q8×C43—26no—
242Q8×C43—21no—
243Q8×C43—46no—
244Q8×C43—83no—
245Q8×C43#13514yescomplemented
246Q8×C43C22×Q812yes—
247D4×C43—24no—
248D4×C43—21no—
249D4×C43C22×Q812yes—
250C2×C4×C43—21no—
251C2×C4×C43—24no—
252C2×C4×C43—26no—
253C2×C4×C43—23no—
254C2×C4×C43—23no—
255C2×C4×C43—43no—
256C2×C4×C43—46no—
257C2×C4×C43—83no—
258C2×C4×C43C22×D413yes—
259C2×C4×C43C22×D413yes—
260C2×C4×C43C22×D413yes—
261C2×C4×C43C22×D413yes—
262C2×C4×C43C22×D411yescharacteristic
263C2×C4×C43C22×Q812yes—
264C2×C4×C432−1+412yes—
265C2×C4×C43C2511yescharacteristic
266C22⋊D43—24no—
267C2×Dic43—21no—
268C2×Dic43—23no—
269C2×Dic43—23no—
270D172—21no—
271D172C22×Q812yes—
272C4×D43—24no—
273C4×D43—46no—
274C4×D43—83no—
275Dic86—21no—
276Dic86—46no—
277Dic86—83no—
278Dic86C22×Q812yes—
279C43⋊C8—23no—
280C43⋊C8—23no—
281C43⋊C8C22×Q812yes—
282C43⋊C8C22×Q812yes—
283Q8×#135—431nomaximal, Sylow
284C22×C43—21no—
285C22×C43—21no—
286C22×C43—22no—
287C22×C43—41no—
288C22×C43—41no—
289C22×C43D4×Q812yes—
290C22×C43C23×D411yescharacteristic
291C2×D43—22no—
292C2×D43—41no—
293C2×D43—41no—
294C4×C43—23no—
295C4×C43—23no—
296C4×C43—22no—
297C4×C43—23no—
298C4×C43—43no—
299C4×C43—43no—
300C4×C43D4×Q812yes—
301C4×C43C2×#13516yes—
302C4×C43C23×Q811yescharacteristic
303C4×C43C2×2−1+411yescharacteristic
304Dic43—22no—
305Dic43—43no—
306Dic43—43no—
307Q8×#41—431no—
308Q8×#41D4311yescharacteristic, complemented
309#3383—433no—
310Q8×Q16—431no—
311#3381—433no—
312#3381—433no—
313#3380—433no—
314Q8×SD16—431no—
315Q8×SD16—431no—
316Q8×D8—431no—
317#3160—433no—
318#3076—433no—
319#3076—433no—
320C2×C8×Q8—431no—
321C4×#135—433no—
322C2×C43—22no—
323C2×C43—42no—
324C2×C43C2×D4×Q811yescharacteristic
325C2×C43C22×#13511yescharacteristic
326C2×C43#358411yescharacteristic
327D43—42no—
328C2×#135—431no—
329Q82—431no—
330Q82C2×D4311yescharacteristic
331#557—433no—
332#557C2×D4313yes—
333#554—23no—
334#554—433no—
335#554—863no—
336#554C2×D4313yes—
337D4×Q8—21no—
338D4×Q8—431no—
339D4×Q8—861no—
340D4×Q8C2×D4311yescharacteristic
341#533—23no—
342#533—433no—
343#533—863no—
344#533C2×D4313yes—
345C4×#41—433no—
346C4×#41C2×D4313yes—
347C2×C4×Q8—21no—
348C2×C4×Q8—861no—
349C2×C4×Q8C2×D4311yescharacteristic
350#500—433no—
351#500—433no—
352#499—436no—
353#499—436no—
354#498—433no—
355#498—433no—
356#479—436no—
357#479—436no—
358#478—436no—
359#478—436no—
360#477—433no—
361#477—433no—
362#476—433no—
363#476—433no—
364#475—433no—
365#475—433no—
366#474—433no—
367#474—433no—
368C8×Q8—431no—
369C8×Q8—431no—
370C8×Q8—433no—
371C8×Q8—433no—
372C4×Q16—433no—
373C4×Q16—433no—
374C4×SD16—433no—
375C4×SD16—433no—
376C4×SD16—433no—
377C4×SD16—433no—
378C4×D8—433no—
379C4×D8—433no—
380#427—436no—
381#427—433no—
382C2×C4⋊C8—433no—
383#416—433no—
384#416—433no—
385C2×C4×C8—433no—
386C43Q8×#13511yescharacteristic, complemented, Sylow
387C2×#41—431no—
388C2×#41C22×D4311yescharacteristic
389C22×Q8—21no—
390C22×Q8—861no—
391C22×Q8C22⋊D4311yescharacteristic
392#135—438no—
393C2×Q16—431no—
394C2×SD16—431no—
395C2×SD16—431no—
396C2×D8—431no—
397C22×C8—431no—
398C22×C8—433no—
399C4⋊Q8—23no—
400C4⋊Q8—23no—
401C4⋊Q8—433no—
402C4⋊Q8—433no—
403C4⋊Q8—863no—
404C4⋊Q8—863no—
405C4⋊Q8C22×D4313yes—
406C4⋊Q8C22×D4313yes—
407#125—26no—
408#125—436no—
409#125—866no—
410#125C22×D4316yes—
411C22⋊Q8—23no—
412C22⋊Q8—23no—
413C22⋊Q8—23no—
414C22⋊Q8—23no—
415C22⋊Q8—26no—
416C22⋊Q8—863no—
417C22⋊Q8—863no—
418C22⋊Q8—863no—
419C22⋊Q8—863no—
420C22⋊Q8—866no—
421C4×Q8—21no—
422C4×Q8—23no—
423C4×Q8—23no—
424C4×Q8—23no—
425C4×Q8—433no—
426C4×Q8—433no—
427C4×Q8—861no—
428C4×Q8—863no—
429C4×Q8—863no—
430C4×Q8—863no—
431C4×Q8C22⋊D4311yescharacteristic
432C4×Q8C22⋊D4313yes—
433C4×Q8C22×D4313yes—
434C4×Q8C22×D4313yes—
435C4×Q8C22×D4311yescharacteristic
436C4×Q8C22×D4313yes—
437C4×D4—23no—
438C4×D4—23no—
439C4×D4—433no—
440C4×D4—433no—
441C4×D4—863no—
442C4×D4—863no—
443C4×D4C22×D4313yes—
444C4×D4C22×D4313yes—
445#117—23no—
446#117—433no—
447#117—863no—
448#117C22×D4313yes—
449C2×C4⋊C4—23no—
450C2×C4⋊C4—863no—
451C2×C4⋊C4C22⋊D4313yes—
452C2×C42—23no—
453C2×C42—863no—
454C2×C42C22×D4313yes—
455#107—433no—
456#107—433no—
457#107—436no—
458#107—436no—
459#106—433no—
460#106—433no—
461#106—436no—
462#106—436no—
463C4⋊C8—433no—
464C4⋊C8—433no—
465C4⋊C8—436no—
466#103—433no—
467#103—433no—
468#103—433no—
469#103—433no—
470#102—433no—
471#102—433no—
472#102—433no—
473#102—433no—
474C4×C8—433no—
475C4×C8—433no—
476C4×C8—433no—
477C4×C8—433no—
478#41—22no—
479#41—434no—
480#41—862no—
481#41Q8×D4314yescomplemented
482C2×Q8—24no—
483C2×Q8—23no—
484C2×Q8—23no—
485C2×Q8—44no—
486C2×Q8—431no—
487C2×Q8—863no—
488C2×Q8—863no—
489C2×Q8—1724no—
490C2×Q8C2×C22⋊D4313yes—
491C2×Q8C2×C22⋊D4311yescharacteristic
492C2×Q8C23×D4311yescharacteristic
493C2×D4—21no—
494C2×D4—431no—
495C2×D4—861no—
496C2×D4C23×D4311yescharacteristic
497C22×C4—21no—
498C22×C4—23no—
499C22×C4—23no—
500C22×C4—861no—
501C22×C4—863no—
502C22×C4—863no—
503C22×C4C2×C22⋊D4313yes—
504C22×C4C2×C22⋊D4313yes—
505C22×C4C23×D4311yescharacteristic
506Q16—434no—
507SD16—434no—
508SD16—434no—
509D8—434no—
510C2×C8—434no—
511C2×C8—431no—
512C2×C8—431no—
513C2×C8—433no—
514C2×C8—433no—
515C2×C8—866no—
516C4⋊C4—26no—
517C4⋊C4—23no—
518C4⋊C4—23no—
519C4⋊C4—23no—
520C4⋊C4—23no—
521C4⋊C4—26no—
522C4⋊C4—433no—
523C4⋊C4—433no—
524C4⋊C4—866no—
525C4⋊C4—863no—
526C4⋊C4—863no—
527C4⋊C4—863no—
528C4⋊C4—866no—
529C4⋊C4—863no—
530C4⋊C4C2×C22⋊D4313yes—
531C4⋊C4C2×C22⋊D4316yes—
532C4⋊C4C2×C22⋊D4313yes—
533C4⋊C4C23×D4313yes—
534C4⋊C4C23×D4313yes—
535C22⋊C4—23no—
536C22⋊C4—23no—
537C22⋊C4—23no—
538C22⋊C4—863no—
539C22⋊C4—863no—
540C22⋊C4—863no—
541C42—23no—
542C42—23no—
543C42—23no—
544C42—863no—
545C42—863no—
546C42—863no—
547C42C2×C22⋊D4313yes—
548C42C2×C22⋊D4313yes—
549C42C23×D4313yes—
550C42C23×D4313yes—
551C23—21no—
552C23—861no—
553C23C22×C22⋊D4311yescharacteristic
554Q8—26no—
555Q8—21no—
556Q8—46no—
557Q8—83no—
558Q8—432no—
559Q8—861no—
560Q8—3443no—
561Q8—1726no—
562Q8#1306334514yesdirect product factor
563Q8C2×Q8×D4312yes—
564D4—24no—
565D4—21no—
566D4—432no—
567D4—864no—
568D4—861no—
569D4C2×Q8×D4312yes—
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—861no—
579C2×C4—864no—
580C2×C4—863no—
581C2×C4—863no—
582C2×C4—3443no—
583C2×C4—1726no—
584C2×C4C22×C22⋊D4313yes—
585C2×C4C22×C22⋊D4313yes—
586C2×C4C22×C22⋊D4313yes—
587C2×C4C22×C22⋊D4313yes—
588C2×C4C22×C22⋊D4311yescharacteristic
589C2×C4C2×Q8×D4312yes—
590C2×C4#1306337312yes—
591C2×C4C24×D4311yescharacteristic
592C8—432no—
593C8—432no—
594C8—863no—
595C8—863no—
596C22—21no—
597C22—21no—
598C22—22no—
599C22—41no—
600C22—41no—
601C22—862no—
602C22—1721no—
603C22—1721no—
604C22Q8×C22⋊D4312yes—
605C22C23×C22⋊D4311yescharacteristic
606C4—23no—
607C4—22no—
608C4—23no—
609C4—23no—
610C4—43no—
611C4—43no—
612C4—862no—
613C4—1723no—
614C4—1723no—
615C4Q8×C22⋊D4312yes—
616C4C2×#1306334516yes—
617C4C22×Q8×D4311yescharacteristic
618C4C2×#1306337311yescharacteristic
619C2—22no—
620C2—42no—
621C2—1722no—
622C2C2×Q8×C22⋊D4311yescharacteristic
623C2C22×#1306334511yescharacteristic
624C2#1914936611yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D43
commutator, cosocleC2×C4×C43C25
FrattiniC2×C4C24×D43
FittingQ8×#41×C43C2
socleC22×C43C23×D4
prime radicalC2×#13063345C22
hypercenterQ8×#41D43
hypocenterC43Q8×#135
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C2×C4 ◁ C2×C4×Q8 ◁ Q8×#41

Complements

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

NH ≅ G/Ntype
1. Q8×#10621898C2semidirect
2. #19147482C2semidirect
3. Q8×#10621885C2semidirect
4. #19147470C2semidirect
5. #19147466C2semidirect
6. #19147458C2semidirect
7. Q8×#10621883C2semidirect
8. Q8×#10621884C2semidirect
9. Q8×#10621882C2semidirect
10. #19145105C2semidirect
11. #19144383C2semidirect
12. #19144377C2semidirect
13. C2×Q8×C43⋊C8C2semidirect
15. Q8×#41×C43C2semidirect
76. #13063345Q8direct (N × H)
166. #41×C43C2×Q8semidirect
185. #10621898C2×Q8semidirect
215. #10621885C2×Q8semidirect
216. #10621884C2×Q8semidirect
217. #10621883C2×Q8semidirect
218. #10621882C2×Q8semidirect
233. C2×C43⋊C8C2×Q8semidirect
245. Q8×C43#135semidirect
308. Q8×#41D43semidirect
386. C43Q8×#135semidirect
481. #41Q8×D43semidirect
562. Q8#13063345direct (N × H)

Maximal subgroups

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

subgroupindex
Q8×#106218982
#191474822
Q8×#106218852
#191474702
#191474662
#191474582
Q8×#106218832
Q8×#106218842
Q8×#106218822
#191451052
#191443832
#191443772
C2×Q8×C43⋊C82
C4×#130633452
Q8×#41×C432
Q8×#13543