Finite Group Databasebeta

← Q8×#13063352 #25977176

Subgroups — Q8×#13063352 #25977176

Order 11008.

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×#10621902C211yesmaximal, characteristic, complemented
2D4×Q8×D43C211yesmaximal, characteristic, complemented
3#19147515C213yesmaximal, complemented
4#19147514C213yesmaximal, complemented
5#19147499C213yesmaximal, complemented
6#19147493C213yesmaximal, complemented
7Q8×#10621884C211yesmaximal, characteristic, complemented
8Q8×D344C211yesmaximal, characteristic, complemented
9Q8×#10621882C211yesmaximal, characteristic, complemented
10#19145136C213yesmaximal, complemented
11#19144405C213yesmaximal, complemented
12#19144393C213yesmaximal, complemented
13Q8×#10621872C211yesmaximal, characteristic, complemented
14C4×#13063352C213yesmaximal
15Q8×SD16×C43C211yesmaximal, characteristic, complemented
16C22×Q8×D43—21no—
17C2×#13063352C2211yescharacteristic
18#14583628C2213yescomplemented
19#14583610—23no—
20#14583610C2213yescomplemented
21Q8×D172—21no—
22Q8×D172C2211yescharacteristic
23Q8×D172C2211yescharacteristic, complemented
24D4×Dic86C2213yes—
25#14583564C2213yes—
26Q8×C22⋊D43—21no—
27#14583562—23no—
28#14583454—23no—
29#14583454C2213yes—
30C4⋊Q8×D43C2213yes—
31C22⋊Q8×D43—23no—
32C4×#10621902C2213yes—
33C4×Q8×D43—21no—
34C4×Q8×D43C2211yescharacteristic
35C4×D4×D43C2213yes—
36#14583306C2216yescomplemented
37#14583306C2216yescomplemented
38#14583304C2213yescomplemented
39#14583302C2213yescomplemented
40#14583295C2213yescomplemented
41#14583198C2216yes—
42#14583197C2216yes—
43#14583192C2216yescomplemented
44#14583188C2216yescomplemented
45#14583186C2213yes—
46#14583184C2213yescomplemented
47#14583180C2213yes—
48#14583180C2213yescomplemented
49#14583179C2213yes—
50#14583176C2213yescomplemented
51#14583029C2213yescomplemented
52#14583026C2213yescomplemented
53Q8×C43⋊C8C2211yescharacteristic, complemented
54#14583010C2213yes—
55#14583008C2213yes—
56#14583002C2213yes—
57#14583000C2213yes—
58C4×#10621884C2213yes—
59C4×D344C2213yes—
60C4×#10621882C2213yes—
61#14582938C2213yes—
62#14582938C2216yes—
63#14582921C2213yes—
64#14582902C2213yes—
65#14582899C2213yes—
66C4×#10621872C2213yes—
67Q82×C43C2211yescharacteristic, complemented
68D4×Q8×C43C2211yescharacteristic
69C43×#498C2213yescomplemented
70C43×#476C2213yes—
71C43×#475C2213yescomplemented
72C8×Q8×C43C2211yescharacteristic, complemented
73C4×SD16×C43C2213yes—
74C2×#10621902C2311yescharacteristic
75C22×Dic86—23no—
76C2×Q8×D43—21no—
77C2×Q8×D43—21no—
78C2×Q8×D43—23no—
79C2×Q8×D43—23no—
80C2×Q8×D43—48no—
81C2×Q8×D43D411yescharacteristic
82C2×D4×D43C2311yescharacteristic
83C22×C4×D43—23no—
84#13063352Q818yesdirect product factor
85C2×#10621884C2311yescharacteristic
86C2×D344C2311yescharacteristic
87C2×#10621882C2311yescharacteristic
88C2×#10621872C2313yes—
89C2×#10621872C2311yescharacteristic
90#13063325—23no—
91#13063325C2313yes—
92#13063325C2313yes—
93#13063324—23no—
94#13063324C2316yes—
95#13063323—23no—
96#13063316—26no—
97#13063316C2316yes—
98#13063305—23no—
99#13063305—23no—
100#13063305—23no—
101#13063305—23no—
102#13063304—23no—
103#13063303—26no—
104#13063302—26no—
105#13063301—23no—
106#13063301—23no—
107#13063301—26no—
108Q8×Dic43—23no—
109Q8×Dic43D411yescharacteristic
110Q8×Dic43C2313yes—
111C4×Dic86—23no—
112C4×Dic86C2313yes—
113#13063290—23no—
114#13063290D413yes—
115C4×D172—23no—
116C4×D172C2313yes—
117C4×D172C2313yes—
118D4×Dic43C2313yes—
119#13063287—23no—
120#13063287C2313yes—
121#13063287C2313yes—
122C4×C22⋊D43—23no—
123(C22⋊D43)⋊C4—23no—
124#13063283—23no—
125#13063283C2313yes—
126C4⋊C4×D43—23no—
127C4⋊C4×D43—23no—
128C4⋊C4×D43D413yes—
129C4⋊C4×D43C2313yes—
130C22⋊C4×D43—23no—
131C42×D43—23no—
132C42×D43C2313yes—
133#13063259C2313yes—
134#13063259C2316yes—
135#13063258C2313yes—
136#13063258C2316yes—
137#13063256C2313yes—
138#13063256C2316yes—
139C4⋊(C43⋊C8)C2313yes—
140Dic43⋊C8C2316yes—
141#13063249C2313yes—
142#13063247C2313yes—
143#13063247C2313yes—
144#13063246C2313yes—
145#13063245C2313yes—
146#13063245C2313yes—
147#13063235C2313yes—
148#13063235C2313yes—
149C4×C43⋊C8C2313yes—
150C22×Q8×C43—21no—
151C2×SD16×C43C2311yescharacteristic
152C4⋊Q8×C43—23no—
153C4⋊Q8×C43C2313yes—
154C4⋊Q8×C43C2313yes—
155C22⋊Q8×C43—23no—
156C4×Q8×C43—21no—
157C4×Q8×C43C2313yes—
158C4×Q8×C43C2311yescharacteristic
159C4×D4×C43C2313yes—
160#106×C43C2313yes—
161#106×C43C2316yes—
162C4⋊C8×C43C2313yes—
163#103×C43C2313yes—
164#102×C43C2313yes—
165C4×C8×C43C2313yes—
166C23×D43—21no—
167C2×Q8×C43—23no—
168C2×Q8×C43—23no—
169C2×Q8×C43—44no—
170C2×Q8×C43C2×D411yescharacteristic
171C2×Q8×C43C2411yescharacteristic
172C2×D4×C43C2411yescharacteristic
173C22×C4×C43—23no—
174C2×C22⋊D43—21no—
175C22×Dic43—23no—
176#10621902—22no—
177#10621902C2×Q814yescomplemented
178Q8×D43—24no—
179Q8×D43—412no—
180Q8×D43—412no—
181Q8×D43—44no—
182Q8×D43—44no—
183D4×D43—22no—
184D4×D43C2×Q814yescomplemented
185C2×D172—21no—
186C2×D172C2411yescharacteristic
187C2×D172C2411yescharacteristic
188C2×C4×D43—21no—
189C2×C4×D43—23no—
190C2×C4×D43—23no—
191C2×C4×D43—23no—
192C2×C4×D43—23no—
193C2×C4×D43—23no—
194C2×C4×D43—412no—
195C2×C4×D43C2×D413yes—
196C2×C4×D43C2×D413yes—
197C2×C4×D43C2411yescharacteristic
198C2×Dic86—23no—
199C2×Dic86—23no—
200C2×Dic86—26no—
201C2×Dic86—412no—
202C2×Dic86C2×D413yes—
203SD16×C43C2×Q814yescomplemented
204C2×C8×C43C2413yes—
205C2×C8×C43C2411yescharacteristic
206C4⋊C4×C43—23no—
207C4⋊C4×C43—23no—
208C4⋊C4×C43—23no—
209C4⋊C4×C43C2×D413yes—
210C4⋊C4×C43C2413yes—
211C4⋊C4×C43C2413yes—
212C22⋊C4×C43—23no—
213C42×C43—23no—
214C42×C43C2413yes—
215C22⋊Dic43—23no—
216#10621884C2×Q814yescomplemented
217#10621882C2×Q814yescomplemented
218(C2×D43)⋊C4—23no—
219(C2×D43)⋊C4—23no—
220(C2×D43)⋊C4—23no—
221(C2×D43)⋊C4—23no—
222(C2×D43)⋊C4—23no—
223C4⋊Dic43—23no—
224C4⋊Dic43—23no—
225C4⋊Dic43—23no—
226C4⋊Dic43—26no—
227C4⋊Dic43C2×D413yes—
228C4⋊Dic43C2413yes—
229C4⋊Dic43C2413yes—
230Dic43⋊C4—26no—
231Dic43⋊C4—26no—
232Dic43⋊C4—23no—
233Dic43⋊C4C2×D416yes—
234C4×Dic43—23no—
235C4×Dic43—23no—
236C4×Dic43C2×D413yes—
237C4×Dic43C2×D413yes—
238C4×Dic43C2413yes—
239C2×C43⋊C8C2413yes—
240C2×C43⋊C8C2411yescharacteristic
241D344C2×Q814yescomplemented
242#10621872—26no—
243#10621872C2×Q814yescomplemented
244C23×C43—21no—
245C22×D43—21no—
246C22×D43—21no—
247C22×D43—24no—
248C22×D43—42no—
249C22×D43—42no—
250C22×D43C22×D411yescharacteristic
251Q8×C43—21no—
252Q8×C43—46no—
253Q8×C43—83no—
254Q8×C43AGL(1,ℤ/8)14yescomplemented
255Q8×C43C22×Q812yescomplemented
256D4×C43—21no—
257D4×C43C22×Q812yes—
258C2×C4×C43—21no—
259C2×C4×C43—23no—
260C2×C4×C43—23no—
261C2×C4×C43—46no—
262C2×C4×C43C22×D413yes—
263C2×C4×C43C22×D413yes—
264C2×C4×C43C2511yescharacteristic
265C22⋊D43—24no—
266C2×Dic43—23no—
267C2×Dic43—23no—
268C2×Dic43—46no—
269C2×Dic43C22×D413yes—
270C2×Dic43C22×D413yes—
271C2×Dic43C22×D411yescharacteristic
272D172—24no—
273D172—21no—
274D172—21no—
275D172C22×Q812yes—
276D172C22×Q812yescomplemented
277C4×D43—26no—
278C4×D43—24no—
279C4×D43—43no—
280C4×D43—46no—
281C4×D43—46no—
282C4×D43—46no—
283C4×D43—46no—
284C4×D43—83no—
285C4×D43C22×Q812yes—
286C4×D432−1+412yes—
287Dic86—26no—
288Dic86—412no—
289Dic86—46no—
290Dic86—46no—
291C8×C43—23no—
292C8×C43C22×Q812yescomplemented
293C43⋊C8—23no—
294C43⋊C8C22×Q812yescomplemented
295Q8×AGL(1,ℤ/8)—431nomaximal, Sylow
296C22×C43—22no—
297C22×C43—41no—
298C22×C43—41no—
299C22×C43C23×D411yescharacteristic
300C2×D43—21no—
301C2×D43—21no—
302C2×D43—22no—
303C2×D43—22no—
304C2×D43—48no—
305C2×D43—41no—
306C2×D43—41no—
307C2×D43—41no—
308C2×D43—41no—
309C2×D43D4×Q812yes—
310C4×C43—23no—
311C4×C43—22no—
312C4×C43—43no—
313C4×C43—43no—
314C4×C43C2×AGL(1,ℤ/8)16yes—
315C4×C43C23×Q811yescharacteristic
316C4×C43C2×2−1+411yescharacteristic
317Dic43—23no—
318Dic43—23no—
319Dic43—43no—
320Dic43—43no—
321Dic43D4×Q812yes—
322Q8×#41—431no—
323C2×D4×Q8—431no—
324#3386—433no—
325#3386—433no—
326#3384—433no—
327#3384—433no—
328Q8×SD16—431no—
329Q8×SD16D4311yescharacteristic, complemented
330Q8×D8—431no—
331Q8×D8—431no—
332#3165—433no—
333#3079—433no—
334#3077—433no—
335Q8×M16—431no—
336C4×AGL(1,ℤ/8)—433no—
337C2×C43—42no—
338C2×C43C2×D4×Q811yescharacteristic
339C2×C43C22×AGL(1,ℤ/8)11yescharacteristic
340C2×C43#358611yescharacteristic
341D43—22no—
342D43—42no—
343D43—42no—
344C23×Q8—861no—
345C2×AGL(1,ℤ/8)—431no—
346Q82C2×D4311yescharacteristic, complemented
347#557—433no—
348#554—433no—
349#554—863no—
350D4×Q8—433no—
351D4×Q8—431no—
352D4×Q8—431no—
353D4×Q8—433no—
354D4×Q8—861no—
355D4×Q8—863no—
356D4×Q8—861no—
357D4×Q8C2×D4311yescharacteristic
358#533—433no—
359#533—863no—
360C2×C4⋊Q8—433no—
361C2×C22⋊Q8—863no—
362C4×#41—433no—
363C2×C4×Q8—431no—
364C2×C4×Q8—861no—
365C2×C4×D4—433no—
366#501—436no—
367#501—436no—
368#500—433no—
369#500—433no—
370#498—433no—
371#498C2×D4313yescomplemented
372#478—436no—
373#478—436no—
374#477—436no—
375#476—433no—
376#476—436no—
377#476C2×D4313yes—
378#475—433no—
379#475C2×D4313yescomplemented
380#474—433no—
381#474—433no—
382#474—433no—
383#474—433no—
384#446—433no—
385#446—433no—
386C8×Q8—431no—
387C8×Q8C2×D4311yescharacteristic, complemented
388#442—433no—
389#442—433no—
390#440—433no—
391#440—433no—
392C4×SD16—433no—
393C4×SD16C2×D4313yes—
394C4×D8—433no—
395C4×D8—433no—
396#428—436no—
397#428—433no—
398#423—433no—
399#418—433no—
400#417—433no—
401C4×M16—433no—
402C43Q8×AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
403C2×#41—431no—
404C22×Q8—21no—
405C22×Q8—431no—
406C22×Q8—863no—
407C22×Q8—863no—
408C22×Q8—863no—
409C22×Q8—861no—
410C22×Q8—861no—
411C22×Q8—1728no—
412C22×D4—431no—
413C23×C4—863no—
414AGL(1,ℤ/8)—438no—
415C2×SD16—431no—
416C2×SD16C22×D4311yescharacteristic
417C2×D8—431no—
418C2×D8—431no—
419C2×M16—431no—
420C2×M16—433no—
421C4⋊Q8—23no—
422C4⋊Q8—436no—
423C4⋊Q8—433no—
424C4⋊Q8—433no—
425C4⋊Q8—863no—
426C4⋊Q8—863no—
427C4⋊Q8—863no—
428C4⋊Q8C22×D4313yes—
429C4⋊Q8C22×D4313yes—
430#125—436no—
431#125—866no—
432C22⋊Q8—23no—
433C22⋊Q8—866no—
434C22⋊Q8—866no—
435C22⋊Q8—863no—
436C22⋊Q8—863no—
437C22⋊Q8—863no—
438C22⋊Q8—863no—
439C22⋊Q8—866no—
440C22⋊Q8—863no—
441C22⋊Q8—863no—
442C22⋊Q8—863no—
443C4×Q8—21no—
444C4×Q8—431no—
445C4×Q8—433no—
446C4×Q8—433no—
447C4×Q8—433no—
448C4×Q8—863no—
449C4×Q8—863no—
450C4×Q8—863no—
451C4×Q8C22×D4313yes—
452C4×Q8C22×D4311yescharacteristic
453C4×D4—433no—
454C4×D4—433no—
455C4×D4—433no—
456C4×D4—433no—
457C4×D4—433no—
458C4×D4—863no—
459C4×D4—863no—
460C4×D4—863no—
461C4×D4—863no—
462C4×D4C22×D4313yes—
463#117—433no—
464#117—863no—
465C2×C4⋊C4—433no—
466C2×C4⋊C4—433no—
467C2×C4⋊C4—863no—
468C2×C4⋊C4—863no—
469C2×C22⋊C4—863no—
470C2×C42—433no—
471C2×C42—863no—
472#107—433no—
473#107—436no—
474#107—433no—
475#107—436no—
476#106—433no—
477#106—436no—
478#106C22×D4313yes—
479#106C22×D4316yes—
480C4⋊C8—436no—
481C4⋊C8—433no—
482C4⋊C8C22×D4313yes—
483#103—433no—
484#103C22×D4313yes—
485#102—433no—
486#102—433no—
487#102—433no—
488#102—433no—
489#102—433no—
490#102C22×D4313yes—
491#97—433no—
492#97—433no—
493C4×C8—433no—
494C4×C8C22×D4313yes—
495C24—861no—
496#41—434no—
497#41—862no—
498C2×Q8—23no—
499C2×Q8—23no—
500C2×Q8—44no—
501C2×Q8—433no—
502C2×Q8—866no—
503C2×Q8—864no—
504C2×Q8—863no—
505C2×Q8—863no—
506C2×Q8—17212no—
507C2×Q8—17212no—
508C2×Q8—17212no—
509C2×Q8—1724no—
510C2×Q8—1724no—
511C2×Q8D4×D4311yescharacteristic
512C2×Q8C23×D4311yescharacteristic
513C2×D4—434no—
514C2×D4—431no—
515C2×D4—431no—
516C2×D4—861no—
517C2×D4—862no—
518C2×D4—861no—
519C2×D4C23×D4311yescharacteristic
520C22×C4—23no—
521C22×C4—431no—
522C22×C4—433no—
523C22×C4—433no—
524C22×C4—861no—
525C22×C4—863no—
526C22×C4—863no—
527C22×C4—863no—
528C22×C4—863no—
529C22×C4—863no—
530C22×C4—863no—
531C22×C4—17212no—
532SD16—434no—
533SD16Q8×D4314yescomplemented
534D8—434no—
535D8—434no—
536M16—434no—
537M16—866no—
538C2×C8—431no—
539C2×C8—433no—
540C2×C8C23×D4313yes—
541C2×C8C23×D4311yescharacteristic
542C4⋊C4—23no—
543C4⋊C4—23no—
544C4⋊C4—23no—
545C4⋊C4—433no—
546C4⋊C4—436no—
547C4⋊C4—433no—
548C4⋊C4—433no—
549C4⋊C4—863no—
550C4⋊C4—866no—
551C4⋊C4—866no—
552C4⋊C4—863no—
553C4⋊C4—866no—
554C4⋊C4—863no—
555C4⋊C4—863no—
556C4⋊C4D4×D4313yes—
557C4⋊C4C23×D4313yes—
558C4⋊C4C23×D4313yes—
559C22⋊C4—23no—
560C22⋊C4—863no—
561C22⋊C4—863no—
562C22⋊C4—863no—
563C22⋊C4—863no—
564C22⋊C4—863no—
565C22⋊C4—863no—
566C42—23no—
567C42—433no—
568C42—433no—
569C42—433no—
570C42—863no—
571C42—863no—
572C42C23×D4313yes—
573C23—21no—
574C23—431no—
575C23—864no—
576C23—861no—
577C23—861no—
578C23—1722no—
579C23—1722no—
580Q8—21no—
581Q8—46no—
582Q8—83no—
583Q8—866no—
584Q8—17212no—
585Q8—1726no—
586Q8—1726no—
587Q8#1306335214yesdirect product factor
588Q8C2×Q8×D4312yescomplemented
589D4—21no—
590D4—432no—
591D4—432no—
592D4—864no—
593D4—864no—
594D4—861no—
595D4—861no—
596D4C2×Q8×D4312yes—
597C2×C4—21no—
598C2×C4—23no—
599C2×C4—23no—
600C2×C4—46no—
601C2×C4—431no—
602C2×C4—432no—
603C2×C4—432no—
604C2×C4—433no—
605C2×C4—433no—
606C2×C4—864no—
607C2×C4—866no—
608C2×C4—863no—
609C2×C4—863no—
610C2×C4—3443no—
611C2×C4—1723no—
612C2×C4—1726no—
613C2×C4—1726no—
614C2×C4—1726no—
615C2×C4—1726no—
616C2×C4—1726no—
617C2×C4C2×D4×D4313yes—
618C2×C4C2×D4×D4313yes—
619C2×C4C24×D4311yescharacteristic
620C8—23no—
621C8—432no—
622C8—863no—
623C8C2×Q8×D4312yescomplemented
624C22—22no—
625C22—41no—
626C22—41no—
627C22—432no—
628C22—861no—
629C22—861no—
630C22—862no—
631C22—862no—
632C22—1728no—
633C22—1721no—
634C22—1721no—
635C22—1721no—
636C22—1721no—
637C22C22×D4×D4311yescharacteristic
638C4—22no—
639C4—23no—
640C4—43no—
641C4—43no—
642C4—432no—
643C4—863no—
644C4—863no—
645C4—1723no—
646C4—1723no—
647C4C2×#1306335216yes—
648C4C22×Q8×D4311yescharacteristic
649C42−1+4×D4311yescharacteristic
650C2—42no—
651C2—862no—
652C2—1722no—
653C2—1722no—
654C2D4×Q8×D4311yescharacteristic
655C2C22×#1306335211yescharacteristic
656C2#1914937711yescharacteristic
subgroupG/H
centerC22C22×D4×D43
commutator, cosocleC2×C4×C43C25
FrattiniC2×C4C24×D43
FittingQ8×SD16×C43C2
socleC22×C43C23×D4
prime radicalC2×#13063352C22
hypercenterQ8×SD16D43
hypocenterC43Q8×AGL(1,ℤ/8)
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C4×Q8 ◁ Q8×SD16

Complements

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

NH ≅ G/Ntype
1. Q8×#10621902C2semidirect
2. D4×Q8×D43C2semidirect
3. #19147515C2semidirect
4. #19147514C2semidirect
5. #19147499C2semidirect
6. #19147493C2semidirect
7. Q8×#10621884C2semidirect
8. Q8×D344C2semidirect
9. Q8×#10621882C2semidirect
10. #19145136C2semidirect
11. #19144405C2semidirect
12. #19144393C2semidirect
13. Q8×#10621872C2semidirect
15. Q8×SD16×C43C2semidirect
18. #14583628C22semidirect
20. #14583610C22semidirect
23. Q8×D172C22semidirect
36. #14583306C22semidirect
37. #14583306C22semidirect
38. #14583304C22semidirect
39. #14583302C22semidirect
40. #14583295C22semidirect
43. #14583192C22semidirect
44. #14583188C22semidirect
46. #14583184C22semidirect
48. #14583180C22semidirect
50. #14583176C22semidirect
51. #14583029C22semidirect
52. #14583026C22semidirect
53. Q8×C43⋊C8C22semidirect
67. Q82×C43C22semidirect
69. C43×#498C22semidirect
71. C43×#475C22semidirect
72. C8×Q8×C43C22semidirect
84. #13063352Q8direct (N × H)
177. #10621902C2×Q8semidirect
184. D4×D43C2×Q8semidirect
203. SD16×C43C2×Q8semidirect
216. #10621884C2×Q8semidirect
217. #10621882C2×Q8semidirect
241. D344C2×Q8semidirect
243. #10621872C2×Q8semidirect
254. Q8×C43AGL(1,ℤ/8)semidirect
255. Q8×C43C22×Q8semidirect
276. D172C22×Q8semidirect
292. C8×C43C22×Q8semidirect
294. C43⋊C8C22×Q8semidirect
329. Q8×SD16D43semidirect
346. Q82C2×D43semidirect
371. #498C2×D43semidirect
379. #475C2×D43semidirect
387. C8×Q8C2×D43semidirect
402. C43Q8×AGL(1,ℤ/8)semidirect
533. SD16Q8×D43semidirect
587. Q8#13063352direct (N × H)
588. Q8C2×Q8×D43semidirect
623. C8C2×Q8×D43semidirect

Maximal subgroups

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

subgroupindex
Q8×#106219022
D4×Q8×D432
#191475152
#191475142
#191474992
#191474932
Q8×#106218842
Q8×D3442
Q8×#106218822
#191451362
#191444052
#191443932
Q8×#106218722
C4×#130633522
Q8×SD16×C432
Q8×AGL(1,ℤ/8)43