Finite Group Databasebeta

← Q8×#4057 #12979870

Subgroups — Q8×#4057 #12979870

Order 1280.

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
1D4×Q8×D5C211yesmaximal, characteristic, complemented
2Q8×#711C211yesmaximal, characteristic, complemented
3C5×Q8×D8C211yesmaximal, characteristic, complemented
4Q8×#687C211yesmaximal, characteristic, complemented
5Q8×#686C211yesmaximal, characteristic, complemented
6#10613686C213yesmaximal, complemented
7#10613671C213yesmaximal, complemented
8#10612697C213yesmaximal, complemented
9#10612696C213yesmaximal, complemented
10#10612692C213yesmaximal, complemented
11#10612690C213yesmaximal, complemented
12Q8×#677C211yesmaximal, characteristic, complemented
13Q8×#676C211yesmaximal, characteristic, complemented
14C4×#4057C213yesmaximal
15#10611307C213yesmaximal, complemented
16C22×Q8×D5—21no—
17C5×D4×Q8C2211yescharacteristic
18C5×D4×Q8C2211yescharacteristic, complemented
19Q8×C22⋊D5—21no—
20Q8×C22⋊D5—21no—
21C2×Q8×Dic5—21no—
22C2×#4057C2211yescharacteristic
23#66062C2213yes—
24#66061C2213yes—
25D5×C4⋊Q8C2213yes—
26#66055C2213yescomplemented
27#65969—23no—
28#65966—23no—
29D5×C22⋊Q8—23no—
30#65956—23no—
31Q8×D20C2211yescharacteristic
32C4×Q8×D5C2211yescharacteristic
33Q8×Dic10C2211yescharacteristic, complemented
34C4×D4×D5C2213yes—
35#65877C2213yescomplemented
36D4×Dic10C2213yes—
37C4×#711C2213yes—
38C5×#500C2213yescomplemented
39C5×#474C2213yes—
40C5×#474C2213yescomplemented
41C5×C8×Q8C2211yescharacteristic, complemented
42C4×C5×D8C2213yes—
43#65441C2213yes—
44#65383C2213yescomplemented
45#65376C2213yes—
46#65368C2213yescomplemented
47#65367C2213yescomplemented
48Q8×#168C2211yescharacteristic, complemented
49C4×#687C2213yes—
50C4×#686C2213yes—
51#65297C2213yes—
52#65296C2213yescomplemented
53#65180C2216yes—
54#65178C2213yes—
55#65170C2213yes—
56#65170C2216yes—
57#65165C2216yescomplemented
58#65165C2216yescomplemented
59#65164C2216yescomplemented
60#65140C2213yescomplemented
61#65131C2213yes—
62#65126C2213yes—
63#65120C2213yescomplemented
64#65074C2213yes—
65#65060C2213yes—
66#65059C2213yes—
67#65052C2216yes—
68#65050C2216yescomplemented
69#65046C2213yes—
70C4×#677C2213yes—
71C4×#676C2213yes—
72#64969C2213yescomplemented
73#64968C2213yescomplemented
74Q8×AGL(1,ℤ/8)—51nomaximal, Sylow
75C22×C5×Q8—21no—
76C22×C5×Q8—21no—
77C2×Q8×D5—21no—
78C2×Q8×D5—23no—
79C2×Q8×D5—23no—
80C2×Q8×D5—48no—
81C2×Q8×D5D411yescharacteristic
82C2×#711C2311yescharacteristic
83C2×D4×D5C2311yescharacteristic
84C22×C4×D5—23no—
85C22×Dic10—23no—
86C2×C5×D8C2311yescharacteristic
87C5×C4⋊Q8C2313yes—
88C5×C4⋊Q8C2313yes—
89C5×C22⋊Q8—23no—
90C5×C22⋊Q8—23no—
91C4×C5×Q8C2311yescharacteristic
92C4×C5×D4C2313yes—
93C4×C5×D4C2313yes—
94#4092—23no—
95#4092—23no—
96Q8×Dic5—21no—
97Q8×Dic5—23no—
98Q8×Dic5—23no—
99Q8×Dic5D411yescharacteristic
100#4090—23no—
101#4090—23no—
102D4×Dic5C2313yes—
103D4×Dic5C2313yes—
104C2×#687C2311yescharacteristic
105C2×#686C2311yescharacteristic
106C4×C22⋊D5—23no—
107C4×C22⋊D5—23no—
108C2×C4⋊Dic5—23no—
109#4070—23no—
110#4070—23no—
111C2×C4×Dic5—23no—
112#4057Q818yesdirect product factor
113C2×#677C2311yescharacteristic
114C2×#676C2313yes—
115C2×#676C2311yescharacteristic
116#4043—23no—
117#4043—23no—
118#4042—26no—
119#4039C2313yes—
120#4038C2313yes—
121D5×C4⋊C4—23no—
122D5×C4⋊C4D413yes—
123D5×C4⋊C4C2313yes—
124#4036C2316yes—
125#4035—26no—
126#4034—23no—
127#4034—23no—
128#4034C2316yes—
129#4033—23no—
130#4033D413yes—
131#4033C2313yes—
132#4027—23no—
133#4027—23no—
134D5×C22⋊C4—23no—
135#4024—26no—
136#4024—26no—
137#4023—23no—
138C4×D20C2313yes—
139C42×D5C2313yes—
140#4015C2313yes—
141#4015C2313yes—
142C4×Dic10C2313yes—
143C4×Dic10C2313yes—
144C5×#107C2313yes—
145C5×#107C2316yes—
146C5×C4⋊C8C2313yes—
147C5×#102C2313yes—
148C5×#102C2313yes—
149C4×C5×C8C2313yes—
150#3964C2313yes—
151#3964C2313yes—
152#3953C2313yes—
153#3949C2316yes—
154#3949C2313yes—
155#3948C2313yes—
156#3947C2313yes—
157#3947C2313yes—
158#3946C2316yes—
159#3942C2313yes—
160#3941C2313yes—
161#3940C2316yes—
162#3940C2313yes—
163#3939C2316yes—
164#3939C2313yes—
165C4⋊#168C2313yes—
166C4×#168C2313yes—
167Q8×#41—51no—
168C2×D4×Q8—51no—
169#3386—53no—
170#3386—53no—
171#3384—53no—
172#3384—53no—
173Q8×SD16—51no—
174Q8×SD16—51no—
175Q8×D8—51no—
176Q8×D8D511yescharacteristic, complemented
177#3165—53no—
178#3079—53no—
179#3077—53no—
180Q8×M16—51no—
181C4×AGL(1,ℤ/8)—53no—
182C23×D5—21no—
183C2×C5×Q8—23no—
184C2×C5×Q8—23no—
185C2×C5×Q8—44no—
186C2×C5×Q8—44no—
187C2×C5×Q8C2×D411yescharacteristic
188C2×C5×D4C2411yescharacteristic
189C2×C5×D4C2411yescharacteristic
190C22×C4×C5—23no—
191C22×C4×C5—23no—
192C2×C22⋊D5—21no—
193C2×C22⋊D5—21no—
194C22×Dic5—21no—
195C22×Dic5—23no—
196C22×Dic5—23no—
197Q8×D5—24no—
198Q8×D5—412no—
199Q8×D5—412no—
200Q8×D5—44no—
201#711—22no—
202#711C2×Q814yescomplemented
203D4×D5—22no—
204D4×D5C2×Q814yescomplemented
205C2×D20C2411yescharacteristic
206C2×C4×D5—23no—
207C2×C4×D5—23no—
208C2×C4×D5—23no—
209C2×C4×D5—412no—
210C2×C4×D5C2×D413yes—
211C2×C4×D5C2×D413yes—
212C2×C4×D5C2411yescharacteristic
213C2×Dic10—23no—
214C2×Dic10—26no—
215C2×Dic10—23no—
216C2×Dic10—412no—
217C2×Dic10C2×D413yes—
218C2×Dic10C2411yescharacteristic
219C5×D8C2×Q814yescomplemented
220C2×C5×C8C2413yes—
221C2×C5×C8C2411yescharacteristic
222C5×C4⋊C4—23no—
223C5×C4⋊C4—23no—
224C5×C4⋊C4C2×D413yes—
225C5×C4⋊C4C2413yes—
226C5×C4⋊C4C2413yes—
227C5×C22⋊C4—23no—
228C5×C22⋊C4—23no—
229C42×C5C2413yes—
230C22⋊Dic5—23no—
231C22⋊Dic5—23no—
232#687C2×Q814yescomplemented
233#686C2×Q814yescomplemented
234#685—23no—
235#685—23no—
236#685—23no—
237C4⋊Dic5—23no—
238C4⋊Dic5—23no—
239C4⋊Dic5—26no—
240C4⋊Dic5C2×D413yes—
241C4⋊Dic5C2413yes—
242C4⋊Dic5C2413yes—
243#683—23no—
244#683—23no—
245#683—26no—
246#683—26no—
247#683—23no—
248#683C2×D416yes—
249C4×Dic5—23no—
250C4×Dic5—23no—
251C4×Dic5—23no—
252C4×Dic5C2×D413yes—
253C4×Dic5C2×D413yes—
254C4×Dic5C2413yes—
255C2×#168C2413yes—
256C2×#168C2411yescharacteristic
257#677C2×Q814yescomplemented
258#676—26no—
259#676C2×Q814yescomplemented
260C23×Q8—101no—
261C2×AGL(1,ℤ/8)—51no—
262Q82—51no—
263#557—53no—
264#554—53no—
265#554—103no—
266D4×Q8—53no—
267D4×Q8—51no—
268D4×Q8—53no—
269D4×Q8—103no—
270D4×Q8—101no—
271D4×Q8—101no—
272D4×Q8C2×D511yescharacteristic
273D4×Q8C2×D511yescharacteristic, complemented
274#533—53no—
275#533—103no—
276C2×C4⋊Q8—53no—
277C2×C22⋊Q8—103no—
278C4×#41—53no—
279C2×C4×Q8—51no—
280C2×C4×Q8—101no—
281C2×C4×D4—53no—
282#501—56no—
283#501—56no—
284#500—53no—
285#500C2×D513yescomplemented
286#498—53no—
287#498—53no—
288#478—56no—
289#478—56no—
290#477—56no—
291#476—53no—
292#476—53no—
293#476—56no—
294#475—53no—
295#475—53no—
296#474—53no—
297#474—53no—
298#474C2×D513yes—
299#474C2×D513yescomplemented
300#446—53no—
301#446—53no—
302C8×Q8—51no—
303C8×Q8C2×D511yescharacteristic, complemented
304#442—53no—
305#442—53no—
306#440—53no—
307#440—53no—
308C4×SD16—53no—
309C4×SD16—53no—
310C4×D8—53no—
311C4×D8C2×D513yes—
312#428—56no—
313#428—53no—
314#423—53no—
315#418—53no—
316#417—53no—
317C4×M16—53no—
318C23×C5—21no—
319C23×C5—21no—
320C22×D5—21no—
321C22×D5—24no—
322C22×D5—42no—
323C22×D5—42no—
324C22×D5C22×D411yescharacteristic
325C5×Q8—46no—
326C5×Q8—46no—
327C5×Q8AGL(1,ℤ/8)14yescomplemented
328C5×D4—21no—
329C5×D4—21no—
330C5×D4C22×Q812yes—
331C5×D4C22×Q812yescomplemented
332C2×C4×C5—23no—
333C2×C4×C5—23no—
334C2×C4×C5—46no—
335C2×C4×C5—46no—
336C2×C4×C5C22×D413yes—
337C2×C4×C5C22×D413yes—
338C2×C4×C5C2511yescharacteristic
339C22⋊D5—24no—
340C22⋊D5—24no—
341C2×Dic5—21no—
342C2×Dic5—24no—
343C2×Dic5—23no—
344C2×Dic5—23no—
345C2×Dic5—46no—
346C2×Dic5—83no—
347C2×Dic5C22×D413yes—
348C2×Dic5C22×D413yes—
349C2×Dic5C22×D411yescharacteristic
350D20—21no—
351D20C22×Q812yes—
352C4×D5—26no—
353C4×D5—43no—
354C4×D5—46no—
355C4×D5—46no—
356C4×D5—46no—
357C4×D5C22×Q812yes—
358C4×D52−1+412yes—
359Dic10—26no—
360Dic10—21no—
361Dic10—412no—
362Dic10—46no—
363Dic10—83no—
364Dic10C22×Q812yescomplemented
365C5×C8—23no—
366C5×C8C22×Q812yescomplemented
367#168—23no—
368#168C22×Q812yescomplemented
369C2×#41—51no—
370C22×Q8—21no—
371C22×Q8—21no—
372C22×Q8—51no—
373C22×Q8—103no—
374C22×Q8—103no—
375C22×Q8—103no—
376C22×Q8—101no—
377C22×Q8—208no—
378C22×D4—51no—
379C23×C4—103no—
380AGL(1,ℤ/8)—58no—
381C2×SD16—51no—
382C2×SD16—51no—
383C2×D8—51no—
384C2×D8C22×D511yescharacteristic
385C2×M16—51no—
386C2×M16—53no—
387C4⋊Q8—56no—
388C4⋊Q8—53no—
389C4⋊Q8—53no—
390C4⋊Q8—103no—
391C4⋊Q8—103no—
392C4⋊Q8—103no—
393C4⋊Q8—103no—
394C4⋊Q8C22×D513yes—
395C4⋊Q8C22×D513yes—
396#125—56no—
397#125—106no—
398C22⋊Q8—23no—
399C22⋊Q8—23no—
400C22⋊Q8—106no—
401C22⋊Q8—106no—
402C22⋊Q8—106no—
403C22⋊Q8—103no—
404C22⋊Q8—103no—
405C22⋊Q8—103no—
406C22⋊Q8—103no—
407C22⋊Q8—103no—
408C22⋊Q8—103no—
409C4×Q8—51no—
410C4×Q8—53no—
411C4×Q8—53no—
412C4×Q8—53no—
413C4×Q8—53no—
414C4×Q8—103no—
415C4×Q8—103no—
416C4×Q8—103no—
417C4×Q8—101no—
418C4×Q8C22×D511yescharacteristic
419C4×D4—53no—
420C4×D4—53no—
421C4×D4—53no—
422C4×D4—53no—
423C4×D4—103no—
424C4×D4—103no—
425C4×D4—103no—
426C4×D4—103no—
427C4×D4C22×D513yes—
428C4×D4C22×D513yes—
429#117—53no—
430#117—103no—
431C2×C4⋊C4—53no—
432C2×C4⋊C4—53no—
433C2×C4⋊C4—103no—
434C2×C4⋊C4—103no—
435C2×C22⋊C4—103no—
436C2×C42—53no—
437C2×C42—103no—
438#107—53no—
439#107—56no—
440#107C22×D513yes—
441#107C22×D516yes—
442#106—53no—
443#106—56no—
444#106—53no—
445#106—56no—
446C4⋊C8—56no—
447C4⋊C8—53no—
448C4⋊C8C22×D513yes—
449#103—53no—
450#103—53no—
451#102—53no—
452#102—53no—
453#102—53no—
454#102—53no—
455#102C22×D513yes—
456#102C22×D513yes—
457#97—53no—
458#97—53no—
459C4×C8—53no—
460C4×C8C22×D513yes—
461C22×C5—22no—
462C22×C5—22no—
463C22×C5—41no—
464C22×C5—41no—
465C22×C5—41no—
466C22×C5—41no—
467C22×C5C23×D411yescharacteristic
468C2×D5—21no—
469C2×D5—21no—
470C2×D5—22no—
471C2×D5—48no—
472C2×D5—41no—
473C2×D5—41no—
474C2×D5D4×Q812yes—
475C4×C5—23no—
476C4×C5—43no—
477C4×C5—43no—
478C4×C5C2×AGL(1,ℤ/8)16yes—
479C4×C5C23×Q811yescharacteristic
480C4×C5C2×2−1+411yescharacteristic
481Dic5—23no—
482Dic5—22no—
483Dic5—23no—
484Dic5—43no—
485Dic5—43no—
486Dic5D4×Q812yes—
487C24—101no—
488#41—54no—
489#41—102no—
490C2×Q8—23no—
491C2×Q8—23no—
492C2×Q8—44no—
493C2×Q8—44no—
494C2×Q8—53no—
495C2×Q8—51no—
496C2×Q8—106no—
497C2×Q8—104no—
498C2×Q8—103no—
499C2×Q8—103no—
500C2×Q8—2012no—
501C2×Q8—2012no—
502C2×Q8—2012no—
503C2×Q8—204no—
504C2×Q8D4×D511yescharacteristic
505C2×D4—54no—
506C2×D4—51no—
507C2×D4—102no—
508C2×D4—101no—
509C2×D4—101no—
510C2×D4C23×D511yescharacteristic
511C2×D4C23×D511yescharacteristic
512C22×C4—23no—
513C22×C4—23no—
514C22×C4—51no—
515C22×C4—53no—
516C22×C4—53no—
517C22×C4—103no—
518C22×C4—103no—
519C22×C4—101no—
520C22×C4—103no—
521C22×C4—103no—
522C22×C4—103no—
523C22×C4—2012no—
524SD16—54no—
525SD16—54no—
526D8—54no—
527D8Q8×D514yescomplemented
528M16—54no—
529M16—106no—
530C2×C8—51no—
531C2×C8—53no—
532C2×C8C23×D513yes—
533C2×C8C23×D511yescharacteristic
534C4⋊C4—23no—
535C4⋊C4—23no—
536C4⋊C4—53no—
537C4⋊C4—56no—
538C4⋊C4—53no—
539C4⋊C4—53no—
540C4⋊C4—103no—
541C4⋊C4—106no—
542C4⋊C4—106no—
543C4⋊C4—103no—
544C4⋊C4—106no—
545C4⋊C4—103no—
546C4⋊C4—103no—
547C4⋊C4—103no—
548C4⋊C4D4×D513yes—
549C4⋊C4C23×D513yes—
550C4⋊C4C23×D513yes—
551C22⋊C4—23no—
552C22⋊C4—23no—
553C22⋊C4—103no—
554C22⋊C4—103no—
555C22⋊C4—103no—
556C22⋊C4—103no—
557C22⋊C4—103no—
558C42—53no—
559C42—53no—
560C42—53no—
561C42—103no—
562C42—103no—
563C42—103no—
564C42C23×D513yes—
565C2×C5—42no—
566C2×C5—42no—
567C2×C5C2×D4×Q811yescharacteristic
568C2×C5C22×AGL(1,ℤ/8)11yescharacteristic
569C2×C5#358611yescharacteristic
570D5—22no—
571D5—42no—
572C23—21no—
573C23—21no—
574C23—51no—
575C23—104no—
576C23—101no—
577C23—202no—
578C23—202no—
579Q8—46no—
580Q8—46no—
581Q8—52no—
582Q8—106no—
583Q8—101no—
584Q8—2012no—
585Q8—206no—
586Q8—403no—
587Q8#405714yesdirect product factor
588D4—21no—
589D4—21no—
590D4—52no—
591D4—104no—
592D4—104no—
593D4—101no—
594D4C2×Q8×D512yes—
595D4C2×Q8×D512yescomplemented
596C2×C4—23no—
597C2×C4—23no—
598C2×C4—46no—
599C2×C4—46no—
600C2×C4—51no—
601C2×C4—52no—
602C2×C4—52no—
603C2×C4—53no—
604C2×C4—53no—
605C2×C4—104no—
606C2×C4—101no—
607C2×C4—106no—
608C2×C4—103no—
609C2×C4—103no—
610C2×C4—203no—
611C2×C4—206no—
612C2×C4—206no—
613C2×C4—206no—
614C2×C4—206no—
615C2×C4—403no—
616C2×C4C2×D4×D513yes—
617C2×C4C2×D4×D513yes—
618C2×C4C24×D511yescharacteristic
619C8—23no—
620C8—52no—
621C8—103no—
622C8C2×Q8×D512yescomplemented
623C5Q8×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
624C22—22no—
625C22—22no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—52no—
631C22—101no—
632C22—101no—
633C22—102no—
634C22—208no—
635C22—201no—
636C22—201no—
637C22C22×D4×D511yescharacteristic
638C4—23no—
639C4—43no—
640C4—43no—
641C4—52no—
642C4—102no—
643C4—103no—
644C4—103no—
645C4—203no—
646C4—203no—
647C4C2×#405716yes—
648C4C22×Q8×D511yescharacteristic
649C4D5×2−1+411yescharacteristic
650C2—42no—
651C2—42no—
652C2—102no—
653C2—202no—
654C2D4×Q8×D511yescharacteristic
655C2C22×#405711yescharacteristic
656C2#1061911211yescharacteristic
subgroupG/H
centerC22C22×D4×D5
commutator, cosocleC2×C4×C5C25
FrattiniC2×C4C24×D5
FittingC5×Q8×D8C2
socleC22×C5C23×D4
prime radicalC2×#4057C22
hypercenterQ8×D8D5
hypocenterC5Q8×AGL(1,ℤ/8)
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22 ◁ C4×Q8 ◁ Q8×D8

Complements

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

NH ≅ G/Ntype
1. D4×Q8×D5C2semidirect
2. Q8×#711C2semidirect
3. C5×Q8×D8C2semidirect
4. Q8×#687C2semidirect
5. Q8×#686C2semidirect
6. #10613686C2semidirect
7. #10613671C2semidirect
8. #10612697C2semidirect
9. #10612696C2semidirect
10. #10612692C2semidirect
11. #10612690C2semidirect
12. Q8×#677C2semidirect
13. Q8×#676C2semidirect
15. #10611307C2semidirect
18. C5×D4×Q8C22semidirect
26. #66055C22semidirect
33. Q8×Dic10C22semidirect
35. #65877C22semidirect
38. C5×#500C22semidirect
40. C5×#474C22semidirect
41. C5×C8×Q8C22semidirect
44. #65383C22semidirect
46. #65368C22semidirect
47. #65367C22semidirect
48. Q8×#168C22semidirect
52. #65296C22semidirect
57. #65165C22semidirect
58. #65165C22semidirect
59. #65164C22semidirect
60. #65140C22semidirect
63. #65120C22semidirect
68. #65050C22semidirect
72. #64969C22semidirect
73. #64968C22semidirect
112. #4057Q8direct (N × H)
176. Q8×D8D5semidirect
202. #711C2×Q8semidirect
204. D4×D5C2×Q8semidirect
219. C5×D8C2×Q8semidirect
232. #687C2×Q8semidirect
233. #686C2×Q8semidirect
257. #677C2×Q8semidirect
259. #676C2×Q8semidirect
273. D4×Q8C2×D5semidirect
285. #500C2×D5semidirect
299. #474C2×D5semidirect
303. C8×Q8C2×D5semidirect
327. C5×Q8AGL(1,ℤ/8)semidirect
331. C5×D4C22×Q8semidirect
364. Dic10C22×Q8semidirect
366. C5×C8C22×Q8semidirect
368. #168C22×Q8semidirect
527. D8Q8×D5semidirect
587. Q8#4057direct (N × H)
595. D4C2×Q8×D5semidirect
622. C8C2×Q8×D5semidirect
623. C5Q8×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
D4×Q8×D52
Q8×#7112
C5×Q8×D82
Q8×#6872
Q8×#6862
#106136862
#106136712
#106126972
#106126962
#106126922
#106126902
Q8×#6772
Q8×#6762
C4×#40572
#106113072
Q8×AGL(1,ℤ/8)5