Finite Group Databasebeta

← Q8×#914 #11675766

Subgroups — Q8×#914 #11675766

Order 768.

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×#239C211yesmaximal, characteristic, complemented
2D3×D4×Q8C211yesmaximal, characteristic, complemented
3C3×Q8×SD16C211yesmaximal, characteristic, complemented
4Q8×#215C211yesmaximal, characteristic, complemented
5Q8×#213C211yesmaximal, characteristic, complemented
6#81905C213yesmaximal, complemented
7#81893C213yesmaximal, complemented
8#81089C213yesmaximal, complemented
9#81060C213yesmaximal, complemented
10#81040C213yesmaximal, complemented
11#81012C213yesmaximal, complemented
12Q8×D24C211yesmaximal, characteristic, complemented
13Q8×ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
14C4×#914C213yesmaximal
15#79635C213yesmaximal, complemented
16Q8×AGL(1,ℤ/8)—31nomaximal, Sylow
17C22×D3×Q8—21no—
18C3×Q82C2211yescharacteristic, complemented
19C3×D4×Q8C2211yescharacteristic
20Q8×#67—21no—
21C2×#914C2211yescharacteristic
22#5730—23no—
23#5730C2213yescomplemented
24#5728—23no—
25#5728C2213yes—
26#5727C2213yes—
27D3×C4⋊Q8C2213yes—
28#5632—23no—
29D3×C22⋊Q8—23no—
30Q8×D12—21no—
31Q8×D12C2211yescharacteristic
32Q8×D12C2211yescharacteristic, complemented
33C4×#239C2213yes—
34C4×D3×Q8—21no—
35C4×D3×Q8C2211yescharacteristic
36#5570C2213yescomplemented
37C4×D3×D4C2213yes—
38D4×Dic6C2213yes—
39C3×#498C2213yescomplemented
40C3×#476C2213yes—
41C3×#475C2213yescomplemented
42C3×C8×Q8C2211yescharacteristic, complemented
43C3×C4×SD16C2213yes—
44#5164C2213yes—
45#5087C2213yescomplemented
46#5084C2213yes—
47#5080C2213yescomplemented
48#5078C2213yescomplemented
49C4×#215C2213yes—
50Q8×C3⋊C8C2211yescharacteristic, complemented
51#5020C2213yescomplemented
52C4×#213C2213yes—
53#5007C2213yes—
54#4871C2216yes—
55#4870C2216yescomplemented
56#4869C2213yes—
57#4861C2213yes—
58#4861C2216yes—
59#4856C2216yescomplemented
60#4856C2216yescomplemented
61#4846C2213yes—
62#4846C2213yescomplemented
63#4837C2213yes—
64#4831C2213yescomplemented
65#4817C2213yes—
66#4803C2213yes—
67#4793C2216yescomplemented
68#4785C2213yes—
69#4770C2213yes—
70#4766C2216yes—
71C4×D24C2213yes—
72C4×ΓL(1,𝔽25)C2213yes—
73#4682C2213yescomplemented
74#4679C2213yescomplemented
75Q8×#41—31no—
76C2×D4×Q8—31no—
77#3386—33no—
78#3386—33no—
79#3384—33no—
80#3384—33no—
81Q8×SD16—31no—
82Q8×SD16D311yescharacteristic, complemented
83Q8×D8—31no—
84Q8×D8—31no—
85#3165—33no—
86#3079—33no—
87#3077—33no—
88Q8×M16—31no—
89C4×AGL(1,ℤ/8)—33no—
90C22×C3×Q8—21no—
91C2×#239C2311yescharacteristic
92C2×D3×Q8—21no—
93C2×D3×Q8—21no—
94C2×D3×Q8—23no—
95C2×D3×Q8—23no—
96C2×D3×Q8—48no—
97C2×D3×Q8D411yescharacteristic
98C2×D3×D4C2311yescharacteristic
99C22×C4×D3—23no—
100C22×Dic6—23no—
101C2×C3×SD16C2311yescharacteristic
102C3×C4⋊Q8—23no—
103C3×C4⋊Q8C2313yes—
104C3×C4⋊Q8C2313yes—
105C3×C22⋊Q8—23no—
106C3×C4×Q8—21no—
107C3×C4×Q8C2313yes—
108C3×C4×Q8C2311yescharacteristic
109C3×C4×D4C2313yes—
110#946—23no—
111#946—23no—
112#946—26no—
113Q8×Dic3—23no—
114Q8×Dic3D411yescharacteristic
115Q8×Dic3C2313yes—
116#944—23no—
117C2×#215C2311yescharacteristic
118D4×Dic3C2313yes—
119C2×#213C2311yescharacteristic
120C4×#67—23no—
121#924—23no—
122#914Q818yesdirect product factor
123C2×D24C2311yescharacteristic
124C2×ΓL(1,𝔽25)C2313yes—
125C2×ΓL(1,𝔽25)C2311yescharacteristic
126#897—23no—
127#897—23no—
128#897—23no—
129#897—23no—
130#896—26no—
131#893—23no—
132#893C2313yes—
133#893C2313yes—
134#892—23no—
135#892C2313yes—
136D3×C4⋊C4—23no—
137D3×C4⋊C4—23no—
138D3×C4⋊C4D413yes—
139D3×C4⋊C4C2313yes—
140#890—26no—
141#890C2316yes—
142#888—23no—
143#888C2316yes—
144#887—23no—
145#887D413yes—
146#67⋊C4—23no—
147D3×C22⋊C4—23no—
148#878—26no—
149C4×D12—23no—
150C4×D12C2313yes—
151C4×D12C2313yes—
152C42×D3—23no—
153C42×D3C2313yes—
154#869—23no—
155#869C2313yes—
156#869C2313yes—
157C4×Dic6—23no—
158C4×Dic6C2313yes—
159C3×#106C2313yes—
160C3×#106C2316yes—
161C3×C4⋊C8C2313yes—
162C3×#103C2313yes—
163C3×#102C2313yes—
164C3×C4×C8C2313yes—
165#835C2313yes—
166#832C2313yes—
167#821C2313yes—
168#821C2313yes—
169#818C2313yes—
170#818C2316yes—
171#815C2313yes—
172#815C2313yes—
173Dic3⋊C8C2316yes—
174#809C2313yes—
175#809C2313yes—
176#808C2316yes—
177#808C2313yes—
178#807C2316yes—
179#807C2313yes—
180C4⋊(C3⋊C8)C2313yes—
181C4×C3⋊C8C2313yes—
182C23×Q8—61no—
183C2×AGL(1,ℤ/8)—31no—
184Q82C2×D311yescharacteristic, complemented
185#557—33no—
186#554—33no—
187#554—63no—
188D4×Q8—33no—
189D4×Q8—31no—
190D4×Q8—31no—
191D4×Q8—33no—
192D4×Q8—61no—
193D4×Q8—63no—
194D4×Q8—61no—
195D4×Q8C2×D311yescharacteristic
196#533—33no—
197#533—63no—
198C2×C4⋊Q8—33no—
199C2×C22⋊Q8—63no—
200C4×#41—33no—
201C2×C4×Q8—31no—
202C2×C4×Q8—61no—
203C2×C4×D4—33no—
204#501—36no—
205#501—36no—
206#500—33no—
207#500—33no—
208#498—33no—
209#498C2×D313yescomplemented
210#478—36no—
211#478—36no—
212#477—36no—
213#476—33no—
214#476—36no—
215#476C2×D313yes—
216#475—33no—
217#475C2×D313yescomplemented
218#474—33no—
219#474—33no—
220#474—33no—
221#474—33no—
222#446—33no—
223#446—33no—
224C8×Q8—31no—
225C8×Q8C2×D311yescharacteristic, complemented
226#442—33no—
227#442—33no—
228#440—33no—
229#440—33no—
230C4×SD16—33no—
231C4×SD16C2×D313yes—
232C4×D8—33no—
233C4×D8—33no—
234#428—36no—
235#428—33no—
236#423—33no—
237#418—33no—
238#417—33no—
239C4×M16—33no—
240C23×D3—21no—
241C2×C3×Q8—23no—
242C2×C3×Q8—23no—
243C2×C3×Q8—44no—
244C2×C3×Q8C2×D411yescharacteristic
245C2×C3×Q8C2411yescharacteristic
246C2×C3×D4C2411yescharacteristic
247C22×C3×C4—23no—
248C2×#67—21no—
249C22×Dic3—23no—
250#239—22no—
251#239C2×Q814yescomplemented
252D3×Q8—24no—
253D3×Q8—412no—
254D3×Q8—412no—
255D3×Q8—44no—
256D3×Q8—44no—
257D3×D4—22no—
258D3×D4C2×Q814yescomplemented
259C2×D12—21no—
260C2×D12C2411yescharacteristic
261C2×D12C2411yescharacteristic
262C2×C4×D3—21no—
263C2×C4×D3—23no—
264C2×C4×D3—23no—
265C2×C4×D3—23no—
266C2×C4×D3—23no—
267C2×C4×D3—23no—
268C2×C4×D3—412no—
269C2×C4×D3C2×D413yes—
270C2×C4×D3C2×D413yes—
271C2×C4×D3C2411yescharacteristic
272C2×Dic6—23no—
273C2×Dic6—23no—
274C2×Dic6—26no—
275C2×Dic6—412no—
276C2×Dic6C2×D413yes—
277C3×SD16C2×Q814yescomplemented
278C2×C3×C8C2413yes—
279C2×C3×C8C2411yescharacteristic
280C3×C4⋊C4—23no—
281C3×C4⋊C4—23no—
282C3×C4⋊C4—23no—
283C3×C4⋊C4C2×D413yes—
284C3×C4⋊C4C2413yes—
285C3×C4⋊C4C2413yes—
286C3×C22⋊C4—23no—
287C3×C42—23no—
288C3×C42C2413yes—
289#217—23no—
290#215C2×Q814yescomplemented
291#213C2×Q814yescomplemented
292(C2×D3)⋊C4—23no—
293(C2×D3)⋊C4—23no—
294(C2×D3)⋊C4—23no—
295(C2×D3)⋊C4—23no—
296(C2×D3)⋊C4—23no—
297C4⋊Dic3—23no—
298C4⋊Dic3—23no—
299C4⋊Dic3—23no—
300C4⋊Dic3—26no—
301C4⋊Dic3C2×D413yes—
302C4⋊Dic3C2413yes—
303C4⋊Dic3C2413yes—
304Dic3⋊C4—26no—
305Dic3⋊C4—26no—
306Dic3⋊C4—23no—
307Dic3⋊C4C2×D416yes—
308C4×Dic3—23no—
309C4×Dic3—23no—
310C4×Dic3C2×D413yes—
311C4×Dic3C2×D413yes—
312C4×Dic3C2413yes—
313C2×C3⋊C8C2413yes—
314C2×C3⋊C8C2411yescharacteristic
315D24C2×Q814yescomplemented
316ΓL(1,𝔽25)—26no—
317ΓL(1,𝔽25)C2×Q814yescomplemented
318C2×#41—31no—
319C22×Q8—21no—
320C22×Q8—31no—
321C22×Q8—63no—
322C22×Q8—63no—
323C22×Q8—63no—
324C22×Q8—61no—
325C22×Q8—61no—
326C22×Q8—128no—
327C22×D4—31no—
328C23×C4—63no—
329AGL(1,ℤ/8)—38no—
330C2×SD16—31no—
331C2×SD16C22×D311yescharacteristic
332C2×D8—31no—
333C2×D8—31no—
334C2×M16—31no—
335C2×M16—33no—
336C4⋊Q8—23no—
337C4⋊Q8—36no—
338C4⋊Q8—33no—
339C4⋊Q8—33no—
340C4⋊Q8—63no—
341C4⋊Q8—63no—
342C4⋊Q8—63no—
343C4⋊Q8C22×D313yes—
344C4⋊Q8C22×D313yes—
345#125—36no—
346#125—66no—
347C22⋊Q8—23no—
348C22⋊Q8—66no—
349C22⋊Q8—66no—
350C22⋊Q8—63no—
351C22⋊Q8—63no—
352C22⋊Q8—63no—
353C22⋊Q8—63no—
354C22⋊Q8—66no—
355C22⋊Q8—63no—
356C22⋊Q8—63no—
357C22⋊Q8—63no—
358C4×Q8—21no—
359C4×Q8—31no—
360C4×Q8—33no—
361C4×Q8—33no—
362C4×Q8—33no—
363C4×Q8—63no—
364C4×Q8—63no—
365C4×Q8—63no—
366C4×Q8C22×D313yes—
367C4×Q8C22×D311yescharacteristic
368C4×D4—33no—
369C4×D4—33no—
370C4×D4—33no—
371C4×D4—33no—
372C4×D4—33no—
373C4×D4—63no—
374C4×D4—63no—
375C4×D4—63no—
376C4×D4—63no—
377C4×D4C22×D313yes—
378#117—33no—
379#117—63no—
380C2×C4⋊C4—33no—
381C2×C4⋊C4—33no—
382C2×C4⋊C4—63no—
383C2×C4⋊C4—63no—
384C2×C22⋊C4—63no—
385C2×C42—33no—
386C2×C42—63no—
387#107—33no—
388#107—36no—
389#107—33no—
390#107—36no—
391#106—33no—
392#106—36no—
393#106C22×D313yes—
394#106C22×D316yes—
395C4⋊C8—36no—
396C4⋊C8—33no—
397C4⋊C8C22×D313yes—
398#103—33no—
399#103C22×D313yes—
400#102—33no—
401#102—33no—
402#102—33no—
403#102—33no—
404#102—33no—
405#102C22×D313yes—
406#97—33no—
407#97—33no—
408C4×C8—33no—
409C4×C8C22×D313yes—
410C23×C3—21no—
411C22×D3—21no—
412C22×D3—21no—
413C22×D3—24no—
414C22×D3—42no—
415C22×D3—42no—
416C22×D3C22×D411yescharacteristic
417C3×Q8—21no—
418C3×Q8—46no—
419C3×Q8—83no—
420C3×Q8AGL(1,ℤ/8)14yescomplemented
421C3×Q8C22×Q812yescomplemented
422C3×D4—21no—
423C3×D4C22×Q812yes—
424C2×C3×C4—21no—
425C2×C3×C4—23no—
426C2×C3×C4—23no—
427C2×C3×C4—46no—
428C2×C3×C4C22×D413yes—
429C2×C3×C4C22×D413yes—
430C2×C3×C4C2511yescharacteristic
431#67—24no—
432C2×Dic3—23no—
433C2×Dic3—23no—
434C2×Dic3—46no—
435C2×Dic3C22×D413yes—
436C2×Dic3C22×D413yes—
437C2×Dic3C22×D411yescharacteristic
438D12—24no—
439D12—21no—
440D12—21no—
441D12C22×Q812yes—
442D12C22×Q812yescomplemented
443C4×D3—26no—
444C4×D3—24no—
445C4×D3—43no—
446C4×D3—46no—
447C4×D3—46no—
448C4×D3—46no—
449C4×D3—46no—
450C4×D3—83no—
451C4×D3C22×Q812yes—
452C4×D32−1+412yes—
453Dic6—26no—
454Dic6—412no—
455Dic6—46no—
456Dic6—46no—
457C3×C8—23no—
458C3×C8C22×Q812yescomplemented
459C3⋊C8—23no—
460C3⋊C8C22×Q812yescomplemented
461C24—61no—
462#41—34no—
463#41—62no—
464C2×Q8—23no—
465C2×Q8—23no—
466C2×Q8—33no—
467C2×Q8—44no—
468C2×Q8—66no—
469C2×Q8—64no—
470C2×Q8—63no—
471C2×Q8—63no—
472C2×Q8—1212no—
473C2×Q8—1212no—
474C2×Q8—1212no—
475C2×Q8—124no—
476C2×Q8—124no—
477C2×Q8D3×D411yescharacteristic
478C2×Q8C23×D311yescharacteristic
479C2×D4—34no—
480C2×D4—31no—
481C2×D4—31no—
482C2×D4—61no—
483C2×D4—62no—
484C2×D4—61no—
485C2×D4C23×D311yescharacteristic
486C22×C4—23no—
487C22×C4—31no—
488C22×C4—33no—
489C22×C4—33no—
490C22×C4—61no—
491C22×C4—63no—
492C22×C4—63no—
493C22×C4—63no—
494C22×C4—63no—
495C22×C4—63no—
496C22×C4—63no—
497C22×C4—1212no—
498SD16—34no—
499SD16D3×Q814yescomplemented
500D8—34no—
501D8—34no—
502M16—34no—
503M16—66no—
504C2×C8—31no—
505C2×C8—33no—
506C2×C8C23×D313yes—
507C2×C8C23×D311yescharacteristic
508C4⋊C4—23no—
509C4⋊C4—23no—
510C4⋊C4—23no—
511C4⋊C4—33no—
512C4⋊C4—36no—
513C4⋊C4—33no—
514C4⋊C4—33no—
515C4⋊C4—63no—
516C4⋊C4—66no—
517C4⋊C4—66no—
518C4⋊C4—63no—
519C4⋊C4—66no—
520C4⋊C4—63no—
521C4⋊C4—63no—
522C4⋊C4D3×D413yes—
523C4⋊C4C23×D313yes—
524C4⋊C4C23×D313yes—
525C22⋊C4—23no—
526C22⋊C4—63no—
527C22⋊C4—63no—
528C22⋊C4—63no—
529C22⋊C4—63no—
530C22⋊C4—63no—
531C22⋊C4—63no—
532C42—23no—
533C42—33no—
534C42—33no—
535C42—33no—
536C42—63no—
537C42—63no—
538C42C23×D313yes—
539C22×C3—22no—
540C22×C3—41no—
541C22×C3—41no—
542C22×C3C23×D411yescharacteristic
543C2×D3—21no—
544C2×D3—21no—
545C2×D3—22no—
546C2×D3—22no—
547C2×D3—48no—
548C2×D3—41no—
549C2×D3—41no—
550C2×D3—41no—
551C2×D3—41no—
552C2×D3D4×Q812yes—
553C3×C4—23no—
554C3×C4—22no—
555C3×C4—43no—
556C3×C4—43no—
557C3×C4C2×AGL(1,ℤ/8)16yes—
558C3×C4C23×Q811yescharacteristic
559C3×C4C2×2−1+411yescharacteristic
560Dic3—23no—
561Dic3—23no—
562Dic3—43no—
563Dic3—43no—
564Dic3D4×Q812yes—
565C23—21no—
566C23—31no—
567C23—64no—
568C23—61no—
569C23—61no—
570C23—122no—
571C23—122no—
572Q8—21no—
573Q8—46no—
574Q8—66no—
575Q8—83no—
576Q8—1212no—
577Q8—126no—
578Q8—126no—
579Q8#91414yesdirect product factor
580Q8C2×D3×Q812yescomplemented
581D4—21no—
582D4—32no—
583D4—32no—
584D4—64no—
585D4—64no—
586D4—61no—
587D4—61no—
588D4C2×D3×Q812yes—
589C2×C4—21no—
590C2×C4—23no—
591C2×C4—23no—
592C2×C4—31no—
593C2×C4—32no—
594C2×C4—32no—
595C2×C4—33no—
596C2×C4—33no—
597C2×C4—46no—
598C2×C4—64no—
599C2×C4—66no—
600C2×C4—63no—
601C2×C4—63no—
602C2×C4—123no—
603C2×C4—126no—
604C2×C4—126no—
605C2×C4—126no—
606C2×C4—126no—
607C2×C4—126no—
608C2×C4—243no—
609C2×C4C2×D3×D413yes—
610C2×C4C2×D3×D413yes—
611C2×C4C24×D311yescharacteristic
612C8—23no—
613C8—32no—
614C8—63no—
615C8C2×D3×Q812yescomplemented
616C2×C3—42no—
617C2×C3C2×D4×Q811yescharacteristic
618C2×C3C22×AGL(1,ℤ/8)11yescharacteristic
619C2×C3#358611yescharacteristic
620D3—22no—
621D3—42no—
622D3—42no—
623C22—22no—
624C22—32no—
625C22—41no—
626C22—41no—
627C22—61no—
628C22—61no—
629C22—62no—
630C22—62no—
631C22—128no—
632C22—121no—
633C22—121no—
634C22—121no—
635C22—121no—
636C22C22×D3×D411yescharacteristic
637C4—23no—
638C4—22no—
639C4—32no—
640C4—43no—
641C4—43no—
642C4—63no—
643C4—63no—
644C4—123no—
645C4—123no—
646C4C2×#91416yes—
647C4C22×D3×Q811yescharacteristic
648C4D3×2−1+411yescharacteristic
649C3Q8×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
650C2—42no—
651C2—62no—
652C2—122no—
653C2—122no—
654C2D3×D4×Q811yescharacteristic
655C2C22×#91411yescharacteristic
656C2#8720811yescharacteristic
subgroupG/H
centerC22C22×D3×D4
commutator, cosocleC2×C3×C4C25
FrattiniC2×C4C24×D3
FittingC3×Q8×SD16C2
socleC22×C3C23×D4
prime radicalC2×#914C22
hypercenterQ8×SD16D3
hypocenterC3Q8×AGL(1,ℤ/8)
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×Q8 ◁ Q8×SD16

Complements

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

NH ≅ G/Ntype
1. Q8×#239C2semidirect
2. D3×D4×Q8C2semidirect
3. C3×Q8×SD16C2semidirect
4. Q8×#215C2semidirect
5. Q8×#213C2semidirect
6. #81905C2semidirect
7. #81893C2semidirect
8. #81089C2semidirect
9. #81060C2semidirect
10. #81040C2semidirect
11. #81012C2semidirect
12. Q8×D24C2semidirect
13. Q8×ΓL(1,𝔽25)C2semidirect
15. #79635C2semidirect
18. C3×Q82C22semidirect
23. #5730C22semidirect
32. Q8×D12C22semidirect
36. #5570C22semidirect
39. C3×#498C22semidirect
41. C3×#475C22semidirect
42. C3×C8×Q8C22semidirect
45. #5087C22semidirect
47. #5080C22semidirect
48. #5078C22semidirect
50. Q8×C3⋊C8C22semidirect
51. #5020C22semidirect
55. #4870C22semidirect
59. #4856C22semidirect
60. #4856C22semidirect
62. #4846C22semidirect
64. #4831C22semidirect
67. #4793C22semidirect
73. #4682C22semidirect
74. #4679C22semidirect
82. Q8×SD16D3semidirect
122. #914Q8direct (N × H)
184. Q82C2×D3semidirect
209. #498C2×D3semidirect
217. #475C2×D3semidirect
225. C8×Q8C2×D3semidirect
251. #239C2×Q8semidirect
258. D3×D4C2×Q8semidirect
277. C3×SD16C2×Q8semidirect
290. #215C2×Q8semidirect
291. #213C2×Q8semidirect
315. D24C2×Q8semidirect
317. ΓL(1,𝔽25)C2×Q8semidirect
420. C3×Q8AGL(1,ℤ/8)semidirect
421. C3×Q8C22×Q8semidirect
442. D12C22×Q8semidirect
458. C3×C8C22×Q8semidirect
460. C3⋊C8C22×Q8semidirect
499. SD16D3×Q8semidirect
579. Q8#914direct (N × H)
580. Q8C2×D3×Q8semidirect
615. C8C2×D3×Q8semidirect
649. C3Q8×AGL(1,ℤ/8)semidirect

Maximal subgroups

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

subgroupindex
Q8×#2392
D3×D4×Q82
C3×Q8×SD162
Q8×#2152
Q8×#2132
#819052
#818932
#810892
#810602
#810402
#810122
Q8×D242
Q8×ΓL(1,𝔽25)2
C4×#9142
#796352
Q8×AGL(1,ℤ/8)3