Finite Group Databasebeta

← C2×#18766 #6348681

Subgroups — C2×#18766 #6348681

Order 512.

4454 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#550C211yesmaximal, characteristic, complemented
2C2×#3205C211yesmaximal, characteristic, complemented
3C2×#3100C211yesmaximal, characteristic, complemented
4C2×#3037C211yesmaximal, characteristic, complemented
5#18766C2116yesmaximal, direct product factor
6C2×#1641C214yesmaximal, complemented
7C2×#1621C214yesmaximal, complemented
8C2×#1545C211yesmaximal, characteristic, complemented
9C2×#1486C211yesmaximal, characteristic, complemented
10C2×#1479C211yesmaximal, characteristic, complemented
11C23×#41—21no—
12C22×#137—21no—
13C2×#550—22no—
14C2×#550—24no—
15C2×#550—22no—
16C2×#550—24no—
17C2×#550—22no—
18C2×#550—24no—
19C2×#550C2214yes—
20C2×#550C2218yescomplemented
21C2×#532C2211yescharacteristic
22C22×#127—21no—
23C22×#127C2211yescharacteristic
24C22×#121—22no—
25C22×#121—21no—
26C22×C4×Q8C2211yescharacteristic
27C22×C4×D4—21no—
28C22×C4×D4C2211yescharacteristic
29#3205—24no—
30#3205C2218yescomplemented
31C2×#493C2212yescomplemented
32C2×#486C2212yescomplemented
33#3100—24no—
34#3100C2218yescomplemented
35C2×#480—21no—
36C2×#474C2212yescomplemented
37#3037—24no—
38#3037C2218yescomplemented
39C2×#464—22no—
40C2×#462C2212yescomplemented
41C2×#460C2212yescomplemented
42C2×#459—22no—
43C2×#459C2212yescomplemented
44C2×#447—22no—
45C2×#447—21no—
46C2×C4×D8—22no—
47C2×#431C2211yescharacteristic
48C2×#431C2211yescharacteristic
49C2×#423C2211yescharacteristic
50C2×#417—21no—
51#1641C22132yescomplemented
52#1621C22132yescomplemented
53#1545C2218yescomplemented
54C2×#331C2214yescomplemented
55#1486C2218yescomplemented
56#1479C2218yescomplemented
57C2×Q8⋊C8C2214yescomplemented
58C2×D4⋊C8C2214yescomplemented
59C22×#41—28no—
60C22×#41—24no—
61C22×#41—24no—
62C22×#41—28no—
63C22×#41—21no—
64C22×#41—48no—
65C22×#41—48no—
66C23×Q8D411yescharacteristic
67C23×D4—22no—
68C23×D4—22no—
69C23×D4—21no—
70C23×D4—21no—
71C23×D4D411yescharacteristic
72C23×D4D412yes—
73C24×C4—22no—
74C24×C4—21no—
75C2×#137—28no—
76C2×#137—48no—
77C22×Q16—22no—
78C22×SD16—22no—
79C22×D8—22no—
80C22×D8—24no—
81C22×M16—21no—
82#550—216no—
83#550—216no—
84#550—216no—
85#550—28no—
86#550—216no—
87#550—28no—
88#550—216no—
89#550—28no—
90#550—216no—
91#550C23116yes—
92#532—22no—
93#532—22no—
94#532—22no—
95#532C2314yes—
96C2×C4⋊Q8C2311yescharacteristic
97C2×#127—28no—
98C2×#127—24no—
99C2×#127—24no—
100C2×#127—28no—
101C2×#127—21no—
102C2×#127—21no—
103C2×#127—22no—
104C2×#127—22no—
105C2×#127D414yes—
106C2×#127D412yes—
107C2×#127C2311yescharacteristic
108C2×#127C2312yes—
109C2×#127C2314yes—
110C2×#124—21no—
111C2×#121—28no—
112C2×#121—24no—
113C2×#121—216no—
114C2×#121—28no—
115C2×#121—28no—
116C2×#121—216no—
117C2×#121—24no—
118C2×#121—28no—
119C2×#121—21no—
120C2×#121—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8—22no—
123C2×C4×Q8—21no—
124C2×C4×Q8—22no—
125C2×C4×Q8D412yes—
126C2×C4×Q8D414yes—
127C2×C4×Q8C2312yes—
128C2×C4×Q8C2314yes—
129C2×C4×D4—28no—
130C2×C4×D4—24no—
131C2×C4×D4—24no—
132C2×C4×D4—28no—
133C2×C4×D4—21no—
134C2×C4×D4—22no—
135C2×C4×D4—21no—
136C2×C4×D4—22no—
137C2×C4×D4—21no—
138C2×C4×D4—21no—
139C2×C4×D4D414yes—
140C2×C4×D4D412yes—
141C2×C4×D4C2314yes—
142C2×C4×D4C2312yes—
143C22×C4⋊C4—21no—
144C22×C4⋊C4D411yescharacteristic
145C22×C4⋊C4D411yescharacteristic
146C22×C22⋊C4—22no—
147C22×C22⋊C4—21no—
148C22×C42—21no—
149C22×C42C2311yescharacteristic
150#493—24no—
151#493C2318yescomplemented
152#486—24no—
153#486C2318yescomplemented
154#480—28no—
155#474—24no—
156#474C2318yescomplemented
157#464—216no—
158#462—24no—
159#462C2318yescomplemented
160#460—24no—
161#460C2318yescomplemented
162#459—216no—
163#459—24no—
164#459C2318yescomplemented
165#447—28no—
166#447—216no—
167C4×D8—216no—
168#431—22no—
169#431—22no—
170#431C2314yes—
171#431C2314yes—
172C2×#107—22no—
173#423—22no—
174#423C2314yes—
175C2×C4⋊C8—21no—
176C2×C4⋊C8—21no—
177C2×C4⋊C8C2312yes—
178C2×C4⋊C8C2312yes—
179#417—28no—
180C2×#103—22no—
181C2×#102—24no—
182C2×#102—22no—
183C2×#102—22no—
184C2×#102—22no—
185C2×C22⋊C8—21no—
186C2×C22⋊C8—21no—
187C2×C4×C8—21no—
188C2×C4×C8C2312yes—
189#331C23116yescomplemented
190Q8⋊C8C23116yescomplemented
191D4⋊C8C23116yescomplemented
192C25—22no—
193C25—21no—
194C2×#41—216no—
195C2×#41—232no—
196C2×#41—216no—
197C2×#41—24no—
198C2×#41—416no—
199C2×#41—432no—
200C2×#41—432no—
201C2×#41—464no—
202C2×#41—416no—
203C2×#41—432no—
204C2×#41—42no—
205C2×#41—44no—
206C2×#41—44no—
207C22×Q8—22no—
208C22×Q8—21no—
209C22×Q8—24no—
210C22×Q8—21no—
211C22×Q8D818yescomplemented
212C22×Q8C2×D412yes—
213C22×Q8C2×D414yes—
214C22×D4—216no—
215C22×D4—28no—
216C22×D4—216no—
217C22×D4—28no—
218C22×D4—22no—
219C22×D4—21no—
220C22×D4—24no—
221C22×D4—22no—
222C22×D4—24no—
223C22×D4—22no—
224C22×D4—22no—
225C22×D4—24no—
226C22×D4—24no—
227C22×D4—22no—
228C22×D4—24no—
229C22×D4—22no—
230C22×D4—21no—
231C22×D4—21no—
232C22×D4—21no—
233C22×D4—44no—
234C22×D4—48no—
235C22×D4—416no—
236C22×D4—41no—
237C22×D4—44no—
238C22×D4—44no—
239C22×D4—84no—
240C22×D4—88no—
241C22×D4C2×D412yes—
242C22×D4C2×D414yes—
243C22×D4C2×D414yes—
244C22×D4C2×D418yes—
245C23×C4—21no—
246C23×C4—216no—
247C23×C4—28no—
248C23×C4—21no—
249C23×C4—21no—
250C23×C4—24no—
251C23×C4—22no—
252C23×C4—24no—
253C23×C4—22no—
254C23×C4—416no—
255C23×C4—44no—
256C23×C4—44no—
257C23×C4—41no—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D411yescharacteristic
261#137—432no—
262C2×Q16—28no—
263C2×Q16—48no—
264C2×SD16—28no—
265C2×SD16—48no—
266C2×D8—28no—
267C2×D8—216no—
268C2×D8—48no—
269C2×D8—88no—
270C2×M16—24no—
271C2×M16—44no—
272C22×C8—22no—
273C22×C8—22no—
274C22×C8—22no—
275C4⋊Q8—21no—
276C4⋊Q8—21no—
277C4⋊Q8—21no—
278C4⋊Q8C2412yes—
279#127—216no—
280#127—232no—
281#127—216no—
282#127—24no—
283#127—24no—
284#127—24no—
285#127—24no—
286#127—24no—
287#127—22no—
288#127—22no—
289#127—21no—
290#127—21no—
291#127—21no—
292#127C2×D418yes—
293#127C2×D414yes—
294#127C2412yes—
295#127C2414yes—
296#124—24no—
297#124—42no—
298#121—216no—
299#121—232no—
300#121—232no—
301#121—264no—
302#121—232no—
303#121—216no—
304#121—24no—
305#121—24no—
306#121—42no—
307#121—42no—
308C4×Q8—24no—
309C4×Q8—24no—
310C4×Q8—24no—
311C4×Q8—22no—
312C4×Q8—24no—
313C4×Q8—24no—
314C4×Q8—22no—
315C4×Q8C2×D414yes—
316C4×Q8C2×D418yes—
317C4×Q8C2414yes—
318C4×D4—216no—
319C4×D4—232no—
320C4×D4—216no—
321C4×D4—24no—
322C4×D4—24no—
323C4×D4—24no—
324C4×D4—22no—
325C4×D4—24no—
326C4×D4—24no—
327C4×D4—22no—
328C4×D4—24no—
329C4×D4—24no—
330C4×D4—42no—
331C4×D4—42no—
332C4×D4C2×D418yes—
333C4×D4C2×D414yes—
334C4×D4C2414yes—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—24no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4C2×D412yes—
346C2×C4⋊C4C2×D414yes—
347C2×C4⋊C4C2×D412yes—
348C2×C4⋊C4C2×D414yes—
349C2×C22⋊C4—216no—
350C2×C22⋊C4—28no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—41no—
356C2×C22⋊C4—41no—
357C2×C42—24no—
358C2×C42—22no—
359C2×C42—24no—
360C2×C42—22no—
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42C2×D412yes—
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D412yes—
366C2×C42C2×D412yes—
367C2×C42C2411yescharacteristic
368C2×C42C2412yes—
369#107—28no—
370C4⋊C8—24no—
371C4⋊C8—24no—
372C4⋊C8—22no—
373C4⋊C8—22no—
374C4⋊C8C2414yes—
375C4⋊C8C2414yes—
376#103—28no—
377#102—28no—
378#102—216no—
379#102—28no—
380#102—28no—
381C22⋊C8—24no—
382C22⋊C8—24no—
383C4×C8—24no—
384C4×C8—22no—
385C4×C8C2414yes—
386C24—28no—
387C24—24no—
388C24—22no—
389C24—24no—
390C24—44no—
391C24—48no—
392C24—48no—
393C24—44no—
394C24—48no—
395C24—44no—
396C24—41no—
397C24#12011yescharacteristic
398#41—464no—
399#41—4128no—
400#41—432no—
401#41—432no—
402#41—48no—
403#41—48no—
404#41—88no—
405C2×Q8—24no—
406C2×Q8—22no—
407C2×Q8—24no—
408C2×Q8—28no—
409C2×Q8—28no—
410C2×Q8—216no—
411C2×Q8—22no—
412C2×Q8—41no—
413C2×Q8—42no—
414C2×Q8—42no—
415C2×Q8C2×D818yes—
416C2×Q8C2×D8116yescomplemented
417C2×Q8C22×D414yes—
418C2×D4—232no—
419C2×D4—232no—
420C2×D4—24no—
421C2×D4—22no—
422C2×D4—28no—
423C2×D4—24no—
424C2×D4—28no—
425C2×D4—24no—
426C2×D4—28no—
427C2×D4—24no—
428C2×D4—24no—
429C2×D4—24no—
430C2×D4—24no—
431C2×D4—24no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—22no—
435C2×D4—48no—
436C2×D4—44no—
437C2×D4—48no—
438C2×D4—48no—
439C2×D4—416no—
440C2×D4—416no—
441C2×D4—432no—
442C2×D4—464no—
443C2×D4—48no—
444C2×D4—44no—
445C2×D4—44no—
446C2×D4—41no—
447C2×D4—41no—
448C2×D4—41no—
449C2×D4—44no—
450C2×D4—48no—
451C2×D4—42no—
452C2×D4—44no—
453C2×D4—48no—
454C2×D4—42no—
455C2×D4—48no—
456C2×D4—44no—
457C2×D4—48no—
458C2×D4—416no—
459C2×D4—84no—
460C2×D4—88no—
461C2×D4—88no—
462C2×D4—832no—
463C2×D4—816no—
464C2×D4—84no—
465C2×D4C22×D414yes—
466C2×D4C22×D418yes—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—28no—
470C22×C4—24no—
471C22×C4—232no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—28no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—42no—
480C22×C4—432no—
481C22×C4—464no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—48no—
486C22×C4—44no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—416no—
490C22×C4—41no—
491C22×C4—44no—
492C22×C4—84no—
493C22×C4#12012yes—
494C22×C4#12011yescharacteristic
495C22×C4#12011yescharacteristic
496C22×C4#12012yes—
497C22×C4#12012yes—
498C22×C4#12011yescharacteristic
499C22×C4C2×D814yes—
500C22×C4AGL(1,ℤ/8)14yes—
501C22×C4C22×D412yes—
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D412yes—
504C22×C4C22×D412yes—
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507Q16—416no—
508SD16—416no—
509D8—416no—
510D8—816no—
511M16—48no—
512C2×C8—24no—
513C2×C8—24no—
514C2×C8—24no—
515C2×C8—44no—
516C2×C8—44no—
517C2×C8—44no—
518C4⋊C4—28no—
519C4⋊C4—24no—
520C4⋊C4—24no—
521C4⋊C4—22no—
522C4⋊C4—24no—
523C4⋊C4—22no—
524C4⋊C4—24no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—41no—
528C4⋊C4—41no—
529C4⋊C4C22×D414yes—
530C4⋊C4C22×D414yes—
531C22⋊C4—232no—
532C22⋊C4—28no—
533C22⋊C4—24no—
534C22⋊C4—24no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C42—28no—
538C42—24no—
539C42—24no—
540C42#12012yes—
541C42#12012yes—
542C42#12012yes—
543C42C22×D412yes—
544C42C22×D411yescharacteristic
545C42C22×D412yes—
546C42C22×D412yes—
547C42C22×D411yescharacteristic
548C42C22×D411yescharacteristic
549C42C2511yescharacteristic
550C23—22no—
551C23—22no—
552C23—24no—
553C23—28no—
554C23—48no—
555C23—416no—
556C23—416no—
557C23—48no—
558C23—416no—
559C23—48no—
560C23—48no—
561C23—44no—
562C23—44no—
563C23—48no—
564C23—48no—
565C23—44no—
566C23—42no—
567C23—816no—
568C23—832no—
569C23—82no—
570C23—82no—
571C23—82no—
572C23#44714yes—
573C23C2×#12012yes—
574C23C2×#12011yescharacteristic
575Q8—28no—
576Q8—216no—
577Q8—24no—
578Q8—42no—
579Q8—42no—
580Q8—82no—
581Q8C22×D818yes—
582D4—24no—
583D4—24no—
584D4—48no—
585D4—48no—
586D4—416no—
587D4—464no—
588D4—44no—
589D4—416no—
590D4—48no—
591D4—44no—
592D4—48no—
593D4—44no—
594D4—42no—
595D4—44no—
596D4—42no—
597D4—48no—
598D4—44no—
599D4—84no—
600D4—88no—
601D4—82no—
602D4—832no—
603D4—88no—
604D4—82no—
605D4—82no—
606C2×C4—24no—
607C2×C4—28no—
608C2×C4—216no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—24no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—464no—
619C2×C4—48no—
620C2×C4—48no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—416no—
624C2×C4—48no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—82no—
629C2×C4—88no—
630C2×C4#44714yes—
631C2×C4#44718yes—
632C2×C4C2×#12012yes—
633C2×C4C2×#12012yes—
634C2×C4C2×#12011yescharacteristic
635C2×C4C2×#12012yes—
636C2×C4C2×#12011yescharacteristic
637C2×C4C2×#12011yescharacteristic
638C2×C4C22×D812yes—
639C2×C4C22×D814yes—
640C2×C4C2×AGL(1,ℤ/8)12yes—
641C2×C4C2×AGL(1,ℤ/8)14yes—
642C2×C4#57514yes—
643C2×C4C23×D411yescharacteristic
644C2×C4C23×D411yescharacteristic
645C2×C4C23×D411yescharacteristic
646C8—44no—
647C8—44no—
648C8—44no—
649C22—28no—
650C22—22no—
651C22—22no—
652C22—48no—
653C22—44no—
654C22—44no—
655C22—48no—
656C22—48no—
657C22—44no—
658C22—816no—
659C22—832no—
660C22—88no—
661C22—816no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—164no—
666C22C2×#44714yes—
667C22C2×#44712yes—
668C22#300512yes—
669C22#301912yes—
670C22C22×#12011yescharacteristic
671C4—28no—
672C4—22no—
673C4—21no—
674C4—22no—
675C4—21no—
676C4—44no—
677C4—44no—
678C4—82no—
679C4C2×#44714yes—
680C4C2×#44712yes—
681C4#300512yes—
682C4C23×D811yescharacteristic
683C4C22×AGL(1,ℤ/8)11yescharacteristic
684C4C2×#57512yes—
685C2—24no—
686C2—84no—
687C2—88no—
688C2—162no—
689C2#1876614yesdirect product factor
690C2C22×#44711yescharacteristic
691C2C2×#300511yescharacteristic
692C2C2×#301911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC42C25
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#550C2semidirect
2. C2×#3205C2semidirect
3. C2×#3100C2semidirect
4. C2×#3037C2semidirect
5. #18766C2direct (N × H)
6. C2×#1641C2semidirect
7. C2×#1621C2semidirect
8. C2×#1545C2semidirect
9. C2×#1486C2semidirect
10. C2×#1479C2semidirect
20. C2×#550C22semidirect
30. #3205C22semidirect
31. C2×#493C22semidirect
32. C2×#486C22semidirect
34. #3100C22semidirect
36. C2×#474C22semidirect
38. #3037C22semidirect
40. C2×#462C22semidirect
41. C2×#460C22semidirect
43. C2×#459C22semidirect
51. #1641C22semidirect
52. #1621C22semidirect
53. #1545C22semidirect
54. C2×#331C22semidirect
55. #1486C22semidirect
56. #1479C22semidirect
57. C2×Q8⋊C8C22semidirect
58. C2×D4⋊C8C22semidirect
151. #493C23semidirect
153. #486C23semidirect
156. #474C23semidirect
159. #462C23semidirect
161. #460C23semidirect
164. #459C23semidirect
189. #331C23semidirect
190. Q8⋊C8C23semidirect
191. D4⋊C8C23semidirect
211. C22×Q8D8semidirect
416. C2×Q8C2×D8semidirect
689. C2#18766direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#5502
C2×#32052
C2×#31002
C2×#30372
#187662
C2×#16412
C2×#16212
C2×#15452
C2×#14862
C2×#14792