Finite Group Databasebeta

← Q8×#67003 #15634236

Subgroups — Q8×#67003 #15634236

Order 2816.

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×#4339C211yesmaximal, characteristic, complemented
2#13081503C213yesmaximal, complemented
3Q8×#4326C211yesmaximal, characteristic, complemented
4#13081491C213yesmaximal, complemented
5#13081487C213yesmaximal, complemented
6#13081479C213yesmaximal, complemented
7Q8×#4324C211yesmaximal, characteristic, complemented
8Q8×#4325C211yesmaximal, characteristic, complemented
9Q8×#4323C211yesmaximal, characteristic, complemented
10#13079126C213yesmaximal, complemented
11#13078404C213yesmaximal, complemented
12#13078398C213yesmaximal, complemented
13C2×Q8×C11⋊C8C211yesmaximal, characteristic, complemented
14C4×#67003C213yesmaximal
15Q8×C11×#41C211yesmaximal, characteristic, complemented
16Q82×C11C2211yescharacteristic
17C11×#557C2213yes—
18C11×#554—23no—
19C11×#554C2213yes—
20D4×Q8×C11—21no—
21D4×Q8×C11C2211yescharacteristic
22C11×#533—23no—
23C11×#533C2213yes—
24C4×C11×#41C2213yes—
25C2×C4×Q8×C11—21no—
26C2×C4×Q8×C11C2211yescharacteristic
27C2×#67003C2211yescharacteristic
28Q8×C22⋊D11—21no—
29#10623110C2213yes—
30#10623109—23no—
31Q8×D44C2211yescharacteristic
32C4×Q8×D11—21no—
33#10623098C2213yes—
34Q8×Dic22C2211yescharacteristic
35#10623042—23no—
36C4×#4339C2213yes—
37#10623001C2213yes—
38#10622846C2213yes—
39#10622707C2213yes—
40#10622704C2213yes—
41#10622700C2213yes—
42#10622697C2213yes—
43#10622692C2213yes—
44#10622692C2213yes—
45#10622691C2213yes—
46#10622691C2213yes—
47#10622679C2216yes—
48#10622677C2216yes—
49#10622677C2216yes—
50#10622676C2216yes—
51C4×#4326C2213yes—
52C4×#4326C2213yes—
53C4×#4325C2213yes—
54C4×#4325C2213yes—
55Q8×C11⋊C8C2211yescharacteristic
56Q8×C11⋊C8C2211yescharacteristic
57Q8×C11⋊C8C2213yes—
58Q8×C11⋊C8C2213yes—
59#10622635C2216yes—
60#10622634C2213yes—
61#10622633C2213yes—
62C4×#4324C2213yes—
63C4×#4324C2213yes—
64C4×#4323C2213yes—
65C4×#4323C2213yes—
66#10622622C2216yes—
67#10622621C2213yes—
68#10622620C2213yes—
69#10622615C2213yes—
70#10622611C2213yes—
71#10622611C2216yes—
72C2×C4⋊(C11⋊C8)C2213yes—
73C2×C4×C11⋊C8C2213yes—
74C2×C11×#41C2311yescharacteristic
75C22×Q8×C11—21no—
76C22×Q8×C11D411yescharacteristic
77C2×Q8×D11—21no—
78C2×#4339C2311yescharacteristic
79C11×C4⋊Q8—23no—
80C11×C4⋊Q8—23no—
81C11×C4⋊Q8C2313yes—
82C11×C4⋊Q8C2313yes—
83C11×#125—26no—
84C11×#125C2316yes—
85C11×C22⋊Q8—23no—
86C11×C22⋊Q8—23no—
87C11×C22⋊Q8—23no—
88C11×C22⋊Q8—23no—
89C11×C22⋊Q8—26no—
90C4×Q8×C11—21no—
91C4×Q8×C11—23no—
92C4×Q8×C11—23no—
93C4×Q8×C11—23no—
94C4×Q8×C11D411yescharacteristic
95C4×Q8×C11D413yes—
96C4×Q8×C11C2313yes—
97C4×Q8×C11C2313yes—
98C4×Q8×C11C2311yescharacteristic
99C4×Q8×C11C2313yes—
100C4×D4×C11—23no—
101C4×D4×C11—23no—
102C4×D4×C11C2313yes—
103C4×D4×C11C2313yes—
104C11×#117—23no—
105C11×#117C2313yes—
106C2×C11×C4⋊C4—23no—
107C2×C11×C4⋊C4D413yes—
108C2×C42×C11—23no—
109C2×C42×C11C2313yes—
110#67003Q818yesdirect product factor
111#66999—23no—
112Q8×Dic11—21no—
113#66997—23no—
114C2×#4326C2311yescharacteristic
115C2×#4325C2311yescharacteristic
116C2×#4324C2311yescharacteristic
117C2×#4323C2311yescharacteristic
118C4×C22⋊D11—23no—
119C4×C22⋊D11—23no—
120#66979C2313yes—
121#66977—23no—
122#66977—23no—
123C22×C11⋊C8C2313yes—
124C22×C11⋊C8C2311yescharacteristic
125#66950—23no—
126#66949—26no—
127C4⋊C4×D11—23no—
128#66942—26no—
129#66941—23no—
130#66940—23no—
131C4×D44C2313yes—
132C4×D44C2313yes—
133#66925—23no—
134C42×D11—23no—
135#66923C2316yes—
136#66922C2313yes—
137#66922C2313yes—
138C4×Dic22—23no—
139C4×Dic22—23no—
140C4×Dic22C2313yes—
141C4×Dic22C2313yes—
142#66899C2313yes—
143#66899C2313yes—
144#66896C2313yes—
145#66896C2313yes—
146#66874C2313yes—
147#66874C2313yes—
148#66873C2313yes—
149#66873C2313yes—
150#66872C2316yes—
151#66872C2316yes—
152#66872C2313yes—
153#66872C2313yes—
154#66871C2316yes—
155#66871C2316yes—
156#66871C2313yes—
157#66871C2313yes—
158C4⋊(C11⋊C8)C2313yes—
159C4⋊(C11⋊C8)C2313yes—
160C4⋊(C11⋊C8)C2316yes—
161C4×C11⋊C8C2313yes—
162C4×C11⋊C8C2313yes—
163C4×C11⋊C8C2313yes—
164C4×C11⋊C8C2313yes—
165Q8×#135—111nomaximal, Sylow
166C11×#41—22no—
167C11×#41C2×Q814yescomplemented
168C2×Q8×C11—24no—
169C2×Q8×C11—23no—
170C2×Q8×C11—23no—
171C2×Q8×C11—44no—
172C2×Q8×C11C2×D413yes—
173C2×Q8×C11C2×D411yescharacteristic
174C2×Q8×C11C2411yescharacteristic
175C2×D4×C11—21no—
176C2×D4×C11C2411yescharacteristic
177C22×C4×C11—21no—
178C22×C4×C11—23no—
179C22×C4×C11—23no—
180C22×C4×C11C2×D413yes—
181C22×C4×C11C2×D413yes—
182C22×C4×C11C2411yescharacteristic
183C2×C22⋊D11—21no—
184Q8×D11—44no—
185#4339—22no—
186#4339C2×Q814yescomplemented
187C2×D44C2411yescharacteristic
188C2×C4×D11—21no—
189C2×C4×D11—23no—
190C2×C4×D11—23no—
191C2×Dic22—23no—
192C2×Dic22—23no—
193C2×Dic22C2411yescharacteristic
194C11×C4⋊C4—26no—
195C11×C4⋊C4—23no—
196C11×C4⋊C4—23no—
197C11×C4⋊C4—23no—
198C11×C4⋊C4—23no—
199C11×C4⋊C4—26no—
200C11×C4⋊C4C2×D413yes—
201C11×C4⋊C4C2×D413yes—
202C11×C4⋊C4C2×D416yes—
203C11×C4⋊C4C2413yes—
204C11×C4⋊C4C2413yes—
205C11×C22⋊C4—23no—
206C11×C22⋊C4—23no—
207C11×C22⋊C4—23no—
208C42×C11—23no—
209C42×C11—23no—
210C42×C11—23no—
211C42×C11C2×D413yes—
212C42×C11C2×D413yes—
213C42×C11C2413yes—
214C42×C11C2413yes—
215C22⋊Dic11—23no—
216#4326C2×Q814yescomplemented
217#4325C2×Q814yescomplemented
218#4324C2×Q814yescomplemented
219#4323C2×Q814yescomplemented
220(C2×D11)⋊C4—23no—
221(C2×D11)⋊C4—23no—
222C4⋊Dic11—23no—
223C4⋊Dic11—23no—
224C4⋊Dic11C2413yes—
225C4⋊Dic11C2413yes—
226Dic11⋊C4—26no—
227Dic11⋊C4—26no—
228Dic11⋊C4—23no—
229Dic11⋊C4—23no—
230C4×Dic11—23no—
231C4×Dic11—23no—
232C4×Dic11—23no—
233C2×C11⋊C8—26no—
234C2×C11⋊C8C2×Q814yescomplemented
235C2×C11⋊C8C2413yes—
236C2×C11⋊C8C2413yes—
237C2×C11⋊C8C2411yescharacteristic
238C2×C11⋊C8C2411yescharacteristic
239Q8×#41—111no—
240Q8×#41D1111yescharacteristic, complemented
241#3383—113no—
242Q8×Q16—111no—
243#3381—113no—
244#3381—113no—
245#3380—113no—
246Q8×SD16—111no—
247Q8×SD16—111no—
248Q8×D8—111no—
249#3160—113no—
250#3076—113no—
251#3076—113no—
252C2×C8×Q8—111no—
253C4×#135—113no—
254C23×C11—21no—
255C23×C11C22×D411yescharacteristic
256C22×D11—21no—
257Q8×C11—26no—
258Q8×C11—21no—
259Q8×C11—46no—
260Q8×C11—83no—
261Q8×C11#13514yescomplemented
262Q8×C11C22×Q812yes—
263D4×C11—24no—
264D4×C11—21no—
265D4×C11C22×Q812yes—
266C2×C4×C11—21no—
267C2×C4×C11—24no—
268C2×C4×C11—26no—
269C2×C4×C11—23no—
270C2×C4×C11—23no—
271C2×C4×C11—43no—
272C2×C4×C11—46no—
273C2×C4×C11—83no—
274C2×C4×C11C22×D413yes—
275C2×C4×C11C22×D413yes—
276C2×C4×C11C22×D413yes—
277C2×C4×C11C22×D413yes—
278C2×C4×C11C22×D411yescharacteristic
279C2×C4×C11C22×Q812yes—
280C2×C4×C112−1+412yes—
281C2×C4×C11C2511yescharacteristic
282C22⋊D11—24no—
283C2×Dic11—21no—
284C2×Dic11—23no—
285C2×Dic11—23no—
286D44—21no—
287D44C22×Q812yes—
288C4×D11—24no—
289C4×D11—46no—
290C4×D11—83no—
291Dic22—21no—
292Dic22—46no—
293Dic22—83no—
294Dic22C22×Q812yes—
295C11⋊C8—23no—
296C11⋊C8—23no—
297C11⋊C8C22×Q812yes—
298C11⋊C8C22×Q812yes—
299C2×#135—111no—
300Q82—111no—
301Q82C2×D1111yescharacteristic
302#557—113no—
303#557C2×D1113yes—
304#554—23no—
305#554—113no—
306#554—223no—
307#554C2×D1113yes—
308D4×Q8—21no—
309D4×Q8—111no—
310D4×Q8—221no—
311D4×Q8C2×D1111yescharacteristic
312#533—23no—
313#533—113no—
314#533—223no—
315#533C2×D1113yes—
316C4×#41—113no—
317C4×#41C2×D1113yes—
318C2×C4×Q8—21no—
319C2×C4×Q8—221no—
320C2×C4×Q8C2×D1111yescharacteristic
321#500—113no—
322#500—113no—
323#499—116no—
324#499—116no—
325#498—113no—
326#498—113no—
327#479—116no—
328#479—116no—
329#478—116no—
330#478—116no—
331#477—113no—
332#477—113no—
333#476—113no—
334#476—113no—
335#475—113no—
336#475—113no—
337#474—113no—
338#474—113no—
339C8×Q8—111no—
340C8×Q8—111no—
341C8×Q8—113no—
342C8×Q8—113no—
343C4×Q16—113no—
344C4×Q16—113no—
345C4×SD16—113no—
346C4×SD16—113no—
347C4×SD16—113no—
348C4×SD16—113no—
349C4×D8—113no—
350C4×D8—113no—
351#427—116no—
352#427—113no—
353C2×C4⋊C8—113no—
354#416—113no—
355#416—113no—
356C2×C4×C8—113no—
357C22×C11—21no—
358C22×C11—21no—
359C22×C11—22no—
360C22×C11—41no—
361C22×C11—41no—
362C22×C11D4×Q812yes—
363C22×C11C23×D411yescharacteristic
364C2×D11—22no—
365C2×D11—41no—
366C2×D11—41no—
367C4×C11—23no—
368C4×C11—23no—
369C4×C11—22no—
370C4×C11—23no—
371C4×C11—43no—
372C4×C11—43no—
373C4×C11D4×Q812yes—
374C4×C11C2×#13516yes—
375C4×C11C23×Q811yescharacteristic
376C4×C11C2×2−1+411yescharacteristic
377Dic11—22no—
378Dic11—43no—
379Dic11—43no—
380C2×#41—111no—
381C2×#41C22×D1111yescharacteristic
382C22×Q8—21no—
383C22×Q8—221no—
384C22×Q8C22⋊D1111yescharacteristic
385#135—118no—
386C2×Q16—111no—
387C2×SD16—111no—
388C2×SD16—111no—
389C2×D8—111no—
390C22×C8—111no—
391C22×C8—113no—
392C4⋊Q8—23no—
393C4⋊Q8—23no—
394C4⋊Q8—113no—
395C4⋊Q8—113no—
396C4⋊Q8—223no—
397C4⋊Q8—223no—
398C4⋊Q8C22×D1113yes—
399C4⋊Q8C22×D1113yes—
400#125—26no—
401#125—116no—
402#125—226no—
403#125C22×D1116yes—
404C22⋊Q8—23no—
405C22⋊Q8—23no—
406C22⋊Q8—23no—
407C22⋊Q8—23no—
408C22⋊Q8—26no—
409C22⋊Q8—223no—
410C22⋊Q8—223no—
411C22⋊Q8—223no—
412C22⋊Q8—223no—
413C22⋊Q8—226no—
414C4×Q8—21no—
415C4×Q8—23no—
416C4×Q8—23no—
417C4×Q8—23no—
418C4×Q8—113no—
419C4×Q8—113no—
420C4×Q8—221no—
421C4×Q8—223no—
422C4×Q8—223no—
423C4×Q8—223no—
424C4×Q8C22⋊D1111yescharacteristic
425C4×Q8C22⋊D1113yes—
426C4×Q8C22×D1113yes—
427C4×Q8C22×D1113yes—
428C4×Q8C22×D1111yescharacteristic
429C4×Q8C22×D1113yes—
430C4×D4—23no—
431C4×D4—23no—
432C4×D4—113no—
433C4×D4—113no—
434C4×D4—223no—
435C4×D4—223no—
436C4×D4C22×D1113yes—
437C4×D4C22×D1113yes—
438#117—23no—
439#117—113no—
440#117—223no—
441#117C22×D1113yes—
442C2×C4⋊C4—23no—
443C2×C4⋊C4—223no—
444C2×C4⋊C4C22⋊D1113yes—
445C2×C42—23no—
446C2×C42—223no—
447C2×C42C22×D1113yes—
448#107—113no—
449#107—113no—
450#107—116no—
451#107—116no—
452#106—113no—
453#106—113no—
454#106—116no—
455#106—116no—
456C4⋊C8—113no—
457C4⋊C8—113no—
458C4⋊C8—116no—
459#103—113no—
460#103—113no—
461#103—113no—
462#103—113no—
463#102—113no—
464#102—113no—
465#102—113no—
466#102—113no—
467C4×C8—113no—
468C4×C8—113no—
469C4×C8—113no—
470C4×C8—113no—
471C2×C11—22no—
472C2×C11—42no—
473C2×C11C2×D4×Q811yescharacteristic
474C2×C11C22×#13511yescharacteristic
475C2×C11#358411yescharacteristic
476D11—42no—
477#41—22no—
478#41—114no—
479#41—222no—
480#41Q8×D1114yescomplemented
481C2×Q8—24no—
482C2×Q8—23no—
483C2×Q8—23no—
484C2×Q8—44no—
485C2×Q8—111no—
486C2×Q8—223no—
487C2×Q8—223no—
488C2×Q8—444no—
489C2×Q8C2×C22⋊D1113yes—
490C2×Q8C2×C22⋊D1111yescharacteristic
491C2×Q8C23×D1111yescharacteristic
492C2×D4—21no—
493C2×D4—111no—
494C2×D4—221no—
495C2×D4C23×D1111yescharacteristic
496C22×C4—21no—
497C22×C4—23no—
498C22×C4—23no—
499C22×C4—221no—
500C22×C4—223no—
501C22×C4—223no—
502C22×C4C2×C22⋊D1113yes—
503C22×C4C2×C22⋊D1113yes—
504C22×C4C23×D1111yescharacteristic
505Q16—114no—
506SD16—114no—
507SD16—114no—
508D8—114no—
509C2×C8—114no—
510C2×C8—111no—
511C2×C8—111no—
512C2×C8—113no—
513C2×C8—113no—
514C2×C8—226no—
515C4⋊C4—26no—
516C4⋊C4—23no—
517C4⋊C4—23no—
518C4⋊C4—23no—
519C4⋊C4—23no—
520C4⋊C4—26no—
521C4⋊C4—113no—
522C4⋊C4—113no—
523C4⋊C4—226no—
524C4⋊C4—223no—
525C4⋊C4—223no—
526C4⋊C4—223no—
527C4⋊C4—226no—
528C4⋊C4—223no—
529C4⋊C4C2×C22⋊D1113yes—
530C4⋊C4C2×C22⋊D1116yes—
531C4⋊C4C2×C22⋊D1113yes—
532C4⋊C4C23×D1113yes—
533C4⋊C4C23×D1113yes—
534C22⋊C4—23no—
535C22⋊C4—23no—
536C22⋊C4—23no—
537C22⋊C4—223no—
538C22⋊C4—223no—
539C22⋊C4—223no—
540C42—23no—
541C42—23no—
542C42—23no—
543C42—223no—
544C42—223no—
545C42—223no—
546C42C2×C22⋊D1113yes—
547C42C2×C22⋊D1113yes—
548C42C23×D1113yes—
549C42C23×D1113yes—
550C11Q8×#13511yescharacteristic, complemented, Sylow
551C23—21no—
552C23—221no—
553C23C22×C22⋊D1111yescharacteristic
554Q8—26no—
555Q8—21no—
556Q8—46no—
557Q8—83no—
558Q8—112no—
559Q8—221no—
560Q8—446no—
561Q8—883no—
562Q8#6700314yesdirect product factor
563Q8C2×Q8×D1112yes—
564D4—24no—
565D4—21no—
566D4—112no—
567D4—224no—
568D4—221no—
569D4C2×Q8×D1112yes—
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—221no—
579C2×C4—224no—
580C2×C4—223no—
581C2×C4—223no—
582C2×C4—446no—
583C2×C4—883no—
584C2×C4C2×Q8×D1112yes—
585C2×C4#6704012yes—
586C2×C4C22×C22⋊D1113yes—
587C2×C4C22×C22⋊D1113yes—
588C2×C4C22×C22⋊D1113yes—
589C2×C4C22×C22⋊D1113yes—
590C2×C4C22×C22⋊D1111yescharacteristic
591C2×C4C24×D1111yescharacteristic
592C8—112no—
593C8—112no—
594C8—223no—
595C8—223no—
596C22—21no—
597C22—21no—
598C22—22no—
599C22—41no—
600C22—41no—
601C22—222no—
602C22—441no—
603C22—441no—
604C22Q8×C22⋊D1112yes—
605C22C23×C22⋊D1111yescharacteristic
606C4—23no—
607C4—22no—
608C4—23no—
609C4—23no—
610C4—43no—
611C4—43no—
612C4—222no—
613C4—443no—
614C4—443no—
615C4Q8×C22⋊D1112yes—
616C4C2×#6700316yes—
617C4C22×Q8×D1111yescharacteristic
618C4C2×#6704011yescharacteristic
619C2—22no—
620C2—42no—
621C2—442no—
622C2C2×Q8×C22⋊D1111yescharacteristic
623C2C22×#6700311yescharacteristic
624C2#1308338711yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D11
commutator, cosocleC2×C4×C11C25
FrattiniC2×C4C24×D11
FittingQ8×C11×#41C2
socleC22×C11C23×D4
prime radicalC2×#67003C22
hypercenterQ8×#41D11
hypocenterC11Q8×#135
LCSG ▷ C2×C4×C11 ▷ C2×C11 ▷ C11
UCSC1 ◁ C2×C4 ◁ C2×C4×Q8 ◁ Q8×#41

Complements

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

NH ≅ G/Ntype
1. Q8×#4339C2semidirect
2. #13081503C2semidirect
3. Q8×#4326C2semidirect
4. #13081491C2semidirect
5. #13081487C2semidirect
6. #13081479C2semidirect
7. Q8×#4324C2semidirect
8. Q8×#4325C2semidirect
9. Q8×#4323C2semidirect
10. #13079126C2semidirect
11. #13078404C2semidirect
12. #13078398C2semidirect
13. C2×Q8×C11⋊C8C2semidirect
15. Q8×C11×#41C2semidirect
110. #67003Q8direct (N × H)
167. C11×#41C2×Q8semidirect
186. #4339C2×Q8semidirect
216. #4326C2×Q8semidirect
217. #4325C2×Q8semidirect
218. #4324C2×Q8semidirect
219. #4323C2×Q8semidirect
234. C2×C11⋊C8C2×Q8semidirect
240. Q8×#41D11semidirect
261. Q8×C11#135semidirect
480. #41Q8×D11semidirect
550. C11Q8×#135semidirect
562. Q8#67003direct (N × H)

Maximal subgroups

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

subgroupindex
Q8×#43392
#130815032
Q8×#43262
#130814912
#130814872
#130814792
Q8×#43242
Q8×#43252
Q8×#43232
#130791262
#130784042
#130783982
C2×Q8×C11⋊C82
C4×#670032
Q8×C11×#412
Q8×#13511