Finite Group Databasebeta

← Q8×#950 #11675650

Subgroups — Q8×#950 #11675650

Order 768.

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
1C3×Q8×#41C211yesmaximal, characteristic, complemented
2Q8×#235C211yesmaximal, characteristic, complemented
3Q8×#215C211yesmaximal, characteristic, complemented
4Q8×#216C211yesmaximal, characteristic, complemented
5#83693C213yesmaximal, complemented
6#83682C213yesmaximal, complemented
7#83629C213yesmaximal, complemented
8#83628C213yesmaximal, complemented
9Q8×#214C211yesmaximal, characteristic, complemented
10Q8×#213C211yesmaximal, characteristic, complemented
11#83349C213yesmaximal, complemented
12#83339C213yesmaximal, complemented
13#82507C213yesmaximal, complemented
14C4×#950C213yesmaximal
15C2×Q8×C3⋊C8C211yesmaximal, characteristic, complemented
16Q8×#135—31nomaximal, Sylow
17C3×Q82C2211yescharacteristic
18C3×#557C2213yes—
19C3×#554—23no—
20C3×#554C2213yes—
21C3×D4×Q8—21no—
22C3×D4×Q8C2211yescharacteristic
23C3×#533—23no—
24C3×#533C2213yes—
25C3×C4×#41C2213yes—
26C2×C3×C4×Q8—21no—
27C2×C3×C4×Q8C2211yescharacteristic
28C2×#950C2211yescharacteristic
29Q8×#67—21no—
30#5579C2213yes—
31#5578—23no—
32Q8×D12C2211yescharacteristic
33C4×D3×Q8—21no—
34#5567C2213yes—
35Q8×Dic6C2211yescharacteristic
36#5511—23no—
37C4×#235C2213yes—
38#5470C2213yes—
39#5233C2213yes—
40#5094C2213yes—
41#5091C2213yes—
42#5087C2213yes—
43#5084C2213yes—
44#5080C2213yes—
45#5080C2213yes—
46#5078C2213yes—
47#5078C2213yes—
48#5066C2216yes—
49#5065C2216yes—
50#5065C2216yes—
51#5063C2216yes—
52C4×#216C2213yes—
53C4×#216C2213yes—
54C4×#215C2213yes—
55C4×#215C2213yes—
56Q8×C3⋊C8C2211yescharacteristic
57Q8×C3⋊C8C2211yescharacteristic
58Q8×C3⋊C8C2213yes—
59Q8×C3⋊C8C2213yes—
60#5022C2216yes—
61#5021C2213yes—
62#5020C2213yes—
63C4×#214C2213yes—
64C4×#214C2213yes—
65C4×#213C2213yes—
66C4×#213C2213yes—
67#5009C2216yes—
68#5008C2213yes—
69#5007C2213yes—
70#5002C2213yes—
71#4998C2213yes—
72#4998C2216yes—
73C2×C4⋊(C3⋊C8)C2213yes—
74C2×C4×C3⋊C8C2213yes—
75Q8×#41—31no—
76Q8×#41D311yescharacteristic, complemented
77#3383—33no—
78Q8×Q16—31no—
79#3381—33no—
80#3381—33no—
81#3380—33no—
82Q8×SD16—31no—
83Q8×SD16—31no—
84Q8×D8—31no—
85#3160—33no—
86#3076—33no—
87#3076—33no—
88C2×C8×Q8—31no—
89C4×#135—33no—
90C2×C3×#41C2311yescharacteristic
91C22×C3×Q8—21no—
92C22×C3×Q8D411yescharacteristic
93C2×D3×Q8—21no—
94C2×#235C2311yescharacteristic
95C3×C4⋊Q8—23no—
96C3×C4⋊Q8—23no—
97C3×C4⋊Q8C2313yes—
98C3×C4⋊Q8C2313yes—
99C3×#125—26no—
100C3×#125C2316yes—
101C3×C22⋊Q8—23no—
102C3×C22⋊Q8—23no—
103C3×C22⋊Q8—23no—
104C3×C22⋊Q8—23no—
105C3×C22⋊Q8—26no—
106C3×C4×Q8—21no—
107C3×C4×Q8—23no—
108C3×C4×Q8—23no—
109C3×C4×Q8—23no—
110C3×C4×Q8D411yescharacteristic
111C3×C4×Q8D413yes—
112C3×C4×Q8C2313yes—
113C3×C4×Q8C2313yes—
114C3×C4×Q8C2311yescharacteristic
115C3×C4×Q8C2313yes—
116C3×C4×D4—23no—
117C3×C4×D4—23no—
118C3×C4×D4C2313yes—
119C3×C4×D4C2313yes—
120C3×#117—23no—
121C3×#117C2313yes—
122C2×C3×C4⋊C4—23no—
123C2×C3×C4⋊C4D413yes—
124C2×C3×C42—23no—
125C2×C3×C42C2313yes—
126#950Q818yesdirect product factor
127#946—23no—
128Q8×Dic3—21no—
129#944—23no—
130C2×#216C2311yescharacteristic
131C2×#215C2311yescharacteristic
132C2×#214C2311yescharacteristic
133C2×#213C2311yescharacteristic
134C4×#67—23no—
135C4×#67—23no—
136#926C2313yes—
137#924—23no—
138#924—23no—
139C22×C3⋊C8C2313yes—
140C22×C3⋊C8C2311yescharacteristic
141#897—23no—
142#896—26no—
143D3×C4⋊C4—23no—
144#889—26no—
145#888—23no—
146#887—23no—
147C4×D12C2313yes—
148C4×D12C2313yes—
149#872—23no—
150C42×D3—23no—
151#870C2316yes—
152#869C2313yes—
153#869C2313yes—
154C4×Dic6—23no—
155C4×Dic6—23no—
156C4×Dic6C2313yes—
157C4×Dic6C2313yes—
158#835C2313yes—
159#835C2313yes—
160#832C2313yes—
161#832C2313yes—
162#810C2313yes—
163#810C2313yes—
164#809C2313yes—
165#809C2313yes—
166#808C2316yes—
167#808C2316yes—
168#808C2313yes—
169#808C2313yes—
170#807C2316yes—
171#807C2316yes—
172#807C2313yes—
173#807C2313yes—
174C4⋊(C3⋊C8)C2313yes—
175C4⋊(C3⋊C8)C2313yes—
176C4⋊(C3⋊C8)C2316yes—
177C4×C3⋊C8C2313yes—
178C4×C3⋊C8C2313yes—
179C4×C3⋊C8C2313yes—
180C4×C3⋊C8C2313yes—
181C2×#135—31no—
182Q82—31no—
183Q82C2×D311yescharacteristic
184#557—33no—
185#557C2×D313yes—
186#554—23no—
187#554—33no—
188#554—63no—
189#554C2×D313yes—
190D4×Q8—21no—
191D4×Q8—31no—
192D4×Q8—61no—
193D4×Q8C2×D311yescharacteristic
194#533—23no—
195#533—33no—
196#533—63no—
197#533C2×D313yes—
198C4×#41—33no—
199C4×#41C2×D313yes—
200C2×C4×Q8—21no—
201C2×C4×Q8—61no—
202C2×C4×Q8C2×D311yescharacteristic
203#500—33no—
204#500—33no—
205#499—36no—
206#499—36no—
207#498—33no—
208#498—33no—
209#479—36no—
210#479—36no—
211#478—36no—
212#478—36no—
213#477—33no—
214#477—33no—
215#476—33no—
216#476—33no—
217#475—33no—
218#475—33no—
219#474—33no—
220#474—33no—
221C8×Q8—31no—
222C8×Q8—31no—
223C8×Q8—33no—
224C8×Q8—33no—
225C4×Q16—33no—
226C4×Q16—33no—
227C4×SD16—33no—
228C4×SD16—33no—
229C4×SD16—33no—
230C4×SD16—33no—
231C4×D8—33no—
232C4×D8—33no—
233#427—36no—
234#427—33no—
235C2×C4⋊C8—33no—
236#416—33no—
237#416—33no—
238C2×C4×C8—33no—
239C3×#41—22no—
240C3×#41C2×Q814yescomplemented
241C2×C3×Q8—24no—
242C2×C3×Q8—23no—
243C2×C3×Q8—23no—
244C2×C3×Q8—44no—
245C2×C3×Q8C2×D413yes—
246C2×C3×Q8C2×D411yescharacteristic
247C2×C3×Q8C2411yescharacteristic
248C2×C3×D4—21no—
249C2×C3×D4C2411yescharacteristic
250C22×C3×C4—21no—
251C22×C3×C4—23no—
252C22×C3×C4—23no—
253C22×C3×C4C2×D413yes—
254C22×C3×C4C2×D413yes—
255C22×C3×C4C2411yescharacteristic
256C2×#67—21no—
257D3×Q8—44no—
258#235—22no—
259#235C2×Q814yescomplemented
260C2×D12C2411yescharacteristic
261C2×C4×D3—21no—
262C2×C4×D3—23no—
263C2×C4×D3—23no—
264C2×Dic6—23no—
265C2×Dic6—23no—
266C2×Dic6C2411yescharacteristic
267C3×C4⋊C4—26no—
268C3×C4⋊C4—23no—
269C3×C4⋊C4—23no—
270C3×C4⋊C4—23no—
271C3×C4⋊C4—23no—
272C3×C4⋊C4—26no—
273C3×C4⋊C4C2×D413yes—
274C3×C4⋊C4C2×D413yes—
275C3×C4⋊C4C2×D416yes—
276C3×C4⋊C4C2413yes—
277C3×C4⋊C4C2413yes—
278C3×C22⋊C4—23no—
279C3×C22⋊C4—23no—
280C3×C22⋊C4—23no—
281C3×C42—23no—
282C3×C42—23no—
283C3×C42—23no—
284C3×C42C2×D413yes—
285C3×C42C2×D413yes—
286C3×C42C2413yes—
287C3×C42C2413yes—
288#217—23no—
289#216C2×Q814yescomplemented
290#215C2×Q814yescomplemented
291#214C2×Q814yescomplemented
292#213C2×Q814yescomplemented
293(C2×D3)⋊C4—23no—
294(C2×D3)⋊C4—23no—
295C4⋊Dic3—23no—
296C4⋊Dic3—23no—
297C4⋊Dic3C2413yes—
298C4⋊Dic3C2413yes—
299Dic3⋊C4—26no—
300Dic3⋊C4—26no—
301Dic3⋊C4—23no—
302Dic3⋊C4—23no—
303C4×Dic3—23no—
304C4×Dic3—23no—
305C4×Dic3—23no—
306C2×C3⋊C8—26no—
307C2×C3⋊C8C2×Q814yescomplemented
308C2×C3⋊C8C2413yes—
309C2×C3⋊C8C2413yes—
310C2×C3⋊C8C2411yescharacteristic
311C2×C3⋊C8C2411yescharacteristic
312C2×#41—31no—
313C2×#41C22×D311yescharacteristic
314C22×Q8—21no—
315C22×Q8—61no—
316C22×Q8#6711yescharacteristic
317#135—38no—
318C2×Q16—31no—
319C2×SD16—31no—
320C2×SD16—31no—
321C2×D8—31no—
322C22×C8—31no—
323C22×C8—33no—
324C4⋊Q8—23no—
325C4⋊Q8—23no—
326C4⋊Q8—33no—
327C4⋊Q8—33no—
328C4⋊Q8—63no—
329C4⋊Q8—63no—
330C4⋊Q8C22×D313yes—
331C4⋊Q8C22×D313yes—
332#125—26no—
333#125—36no—
334#125—66no—
335#125C22×D316yes—
336C22⋊Q8—23no—
337C22⋊Q8—23no—
338C22⋊Q8—23no—
339C22⋊Q8—23no—
340C22⋊Q8—26no—
341C22⋊Q8—63no—
342C22⋊Q8—63no—
343C22⋊Q8—63no—
344C22⋊Q8—63no—
345C22⋊Q8—66no—
346C4×Q8—21no—
347C4×Q8—23no—
348C4×Q8—23no—
349C4×Q8—23no—
350C4×Q8—33no—
351C4×Q8—33no—
352C4×Q8—61no—
353C4×Q8—63no—
354C4×Q8—63no—
355C4×Q8—63no—
356C4×Q8#6711yescharacteristic
357C4×Q8#6713yes—
358C4×Q8C22×D313yes—
359C4×Q8C22×D313yes—
360C4×Q8C22×D311yescharacteristic
361C4×Q8C22×D313yes—
362C4×D4—23no—
363C4×D4—23no—
364C4×D4—33no—
365C4×D4—33no—
366C4×D4—63no—
367C4×D4—63no—
368C4×D4C22×D313yes—
369C4×D4C22×D313yes—
370#117—23no—
371#117—33no—
372#117—63no—
373#117C22×D313yes—
374C2×C4⋊C4—23no—
375C2×C4⋊C4—63no—
376C2×C4⋊C4#6713yes—
377C2×C42—23no—
378C2×C42—63no—
379C2×C42C22×D313yes—
380#107—33no—
381#107—33no—
382#107—36no—
383#107—36no—
384#106—33no—
385#106—33no—
386#106—36no—
387#106—36no—
388C4⋊C8—33no—
389C4⋊C8—33no—
390C4⋊C8—36no—
391#103—33no—
392#103—33no—
393#103—33no—
394#103—33no—
395#102—33no—
396#102—33no—
397#102—33no—
398#102—33no—
399C4×C8—33no—
400C4×C8—33no—
401C4×C8—33no—
402C4×C8—33no—
403C23×C3—21no—
404C23×C3C22×D411yescharacteristic
405C22×D3—21no—
406C3×Q8—26no—
407C3×Q8—21no—
408C3×Q8—46no—
409C3×Q8—83no—
410C3×Q8#13514yescomplemented
411C3×Q8C22×Q812yes—
412C3×D4—24no—
413C3×D4—21no—
414C3×D4C22×Q812yes—
415C2×C3×C4—21no—
416C2×C3×C4—24no—
417C2×C3×C4—26no—
418C2×C3×C4—23no—
419C2×C3×C4—23no—
420C2×C3×C4—43no—
421C2×C3×C4—46no—
422C2×C3×C4—83no—
423C2×C3×C4C22×D413yes—
424C2×C3×C4C22×D413yes—
425C2×C3×C4C22×D413yes—
426C2×C3×C4C22×D413yes—
427C2×C3×C4C22×D411yescharacteristic
428C2×C3×C4C22×Q812yes—
429C2×C3×C42−1+412yes—
430C2×C3×C4C2511yescharacteristic
431#67—24no—
432C2×Dic3—21no—
433C2×Dic3—23no—
434C2×Dic3—23no—
435D12—21no—
436D12C22×Q812yes—
437C4×D3—24no—
438C4×D3—46no—
439C4×D3—83no—
440Dic6—21no—
441Dic6—46no—
442Dic6—83no—
443Dic6C22×Q812yes—
444C3⋊C8—23no—
445C3⋊C8—23no—
446C3⋊C8C22×Q812yes—
447C3⋊C8C22×Q812yes—
448#41—22no—
449#41—34no—
450#41—62no—
451#41D3×Q814yescomplemented
452C2×Q8—24no—
453C2×Q8—23no—
454C2×Q8—23no—
455C2×Q8—31no—
456C2×Q8—44no—
457C2×Q8—63no—
458C2×Q8—63no—
459C2×Q8—124no—
460C2×Q8C2×#6713yes—
461C2×Q8C2×#6711yescharacteristic
462C2×Q8C23×D311yescharacteristic
463C2×D4—21no—
464C2×D4—31no—
465C2×D4—61no—
466C2×D4C23×D311yescharacteristic
467C22×C4—21no—
468C22×C4—23no—
469C22×C4—23no—
470C22×C4—61no—
471C22×C4—63no—
472C22×C4—63no—
473C22×C4C2×#6713yes—
474C22×C4C2×#6713yes—
475C22×C4C23×D311yescharacteristic
476Q16—34no—
477SD16—34no—
478SD16—34no—
479D8—34no—
480C2×C8—34no—
481C2×C8—31no—
482C2×C8—31no—
483C2×C8—33no—
484C2×C8—33no—
485C2×C8—66no—
486C4⋊C4—26no—
487C4⋊C4—23no—
488C4⋊C4—23no—
489C4⋊C4—23no—
490C4⋊C4—23no—
491C4⋊C4—26no—
492C4⋊C4—33no—
493C4⋊C4—33no—
494C4⋊C4—66no—
495C4⋊C4—63no—
496C4⋊C4—63no—
497C4⋊C4—63no—
498C4⋊C4—66no—
499C4⋊C4—63no—
500C4⋊C4C2×#6713yes—
501C4⋊C4C2×#6716yes—
502C4⋊C4C2×#6713yes—
503C4⋊C4C23×D313yes—
504C4⋊C4C23×D313yes—
505C22⋊C4—23no—
506C22⋊C4—23no—
507C22⋊C4—23no—
508C22⋊C4—63no—
509C22⋊C4—63no—
510C22⋊C4—63no—
511C42—23no—
512C42—23no—
513C42—23no—
514C42—63no—
515C42—63no—
516C42—63no—
517C42C2×#6713yes—
518C42C2×#6713yes—
519C42C23×D313yes—
520C42C23×D313yes—
521C22×C3—21no—
522C22×C3—21no—
523C22×C3—22no—
524C22×C3—41no—
525C22×C3—41no—
526C22×C3D4×Q812yes—
527C22×C3C23×D411yescharacteristic
528C2×D3—22no—
529C2×D3—41no—
530C2×D3—41no—
531C3×C4—23no—
532C3×C4—23no—
533C3×C4—22no—
534C3×C4—23no—
535C3×C4—43no—
536C3×C4—43no—
537C3×C4D4×Q812yes—
538C3×C4C2×#13516yes—
539C3×C4C23×Q811yescharacteristic
540C3×C4C2×2−1+411yescharacteristic
541Dic3—22no—
542Dic3—43no—
543Dic3—43no—
544C23—21no—
545C23—61no—
546C23C22×#6711yescharacteristic
547Q8—21no—
548Q8—26no—
549Q8—32no—
550Q8—46no—
551Q8—61no—
552Q8—83no—
553Q8—126no—
554Q8—243no—
555Q8#95014yesdirect product factor
556Q8C2×D3×Q812yes—
557D4—24no—
558D4—21no—
559D4—32no—
560D4—64no—
561D4—61no—
562D4C2×D3×Q812yes—
563C2×C4—21no—
564C2×C4—24no—
565C2×C4—26no—
566C2×C4—23no—
567C2×C4—23no—
568C2×C4—43no—
569C2×C4—46no—
570C2×C4—61no—
571C2×C4—64no—
572C2×C4—63no—
573C2×C4—63no—
574C2×C4—83no—
575C2×C4—126no—
576C2×C4—243no—
577C2×C4C2×D3×Q812yes—
578C2×C4#100712yes—
579C2×C4C22×#6713yes—
580C2×C4C22×#6713yes—
581C2×C4C22×#6713yes—
582C2×C4C22×#6713yes—
583C2×C4C22×#6711yescharacteristic
584C2×C4C24×D311yescharacteristic
585C8—32no—
586C8—32no—
587C8—63no—
588C8—63no—
589C2×C3—22no—
590C2×C3—42no—
591C2×C3C2×D4×Q811yescharacteristic
592C2×C3C22×#13511yescharacteristic
593C2×C3#358411yescharacteristic
594D3—42no—
595C22—21no—
596C22—21no—
597C22—22no—
598C22—41no—
599C22—41no—
600C22—62no—
601C22—121no—
602C22—121no—
603C22Q8×#6712yes—
604C22C23×#6711yescharacteristic
605C4—23no—
606C4—23no—
607C4—22no—
608C4—23no—
609C4—43no—
610C4—43no—
611C4—62no—
612C4—123no—
613C4—123no—
614C4Q8×#6712yes—
615C4C2×#95016yes—
616C4C22×D3×Q811yescharacteristic
617C4C2×#100711yescharacteristic
618C3Q8×#13511yescharacteristic, complemented, Sylow
619C2—22no—
620C2—42no—
621C2—122no—
622C2C2×Q8×#6711yescharacteristic
623C2C22×#95011yescharacteristic
624C2#8729711yescharacteristic
subgroupG/H
centerC2×C4C22×#67
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×Q8×#41C2
socleC22×C3C23×D4
prime radicalC2×#950C22
hypercenterQ8×#41D3
hypocenterC3Q8×#135
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C4×Q8 ◁ Q8×#41

Complements

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

NH ≅ G/Ntype
1. C3×Q8×#41C2semidirect
2. Q8×#235C2semidirect
3. Q8×#215C2semidirect
4. Q8×#216C2semidirect
5. #83693C2semidirect
6. #83682C2semidirect
7. #83629C2semidirect
8. #83628C2semidirect
9. Q8×#214C2semidirect
10. Q8×#213C2semidirect
11. #83349C2semidirect
12. #83339C2semidirect
13. #82507C2semidirect
15. C2×Q8×C3⋊C8C2semidirect
76. Q8×#41D3semidirect
126. #950Q8direct (N × H)
240. C3×#41C2×Q8semidirect
259. #235C2×Q8semidirect
289. #216C2×Q8semidirect
290. #215C2×Q8semidirect
291. #214C2×Q8semidirect
292. #213C2×Q8semidirect
307. C2×C3⋊C8C2×Q8semidirect
410. C3×Q8#135semidirect
451. #41D3×Q8semidirect
555. Q8#950direct (N × H)
618. C3Q8×#135semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×#412
Q8×#2352
Q8×#2152
Q8×#2162
#836932
#836822
#836292
#836282
Q8×#2142
Q8×#2132
#833492
#833392
#825072
C4×#9502
C2×Q8×C3⋊C82
Q8×#1353