Finite Group Databasebeta

← #6520268

Subgroups — #6520268

Order 512.

2522 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3216C211yesmaximal, characteristic, complemented
2C2×#3207C211yesmaximal, characteristic, complemented
3C2×#3109C211yesmaximal, characteristic, complemented
4C2×#3101C211yesmaximal, characteristic, complemented
5#37640C211yesmaximal, characteristic, complemented
6#35165C211yesmaximal, characteristic, complemented
7#28479C214yesmaximal, complemented
8#28470C212yesmaximal, complemented
9#28460C212yesmaximal, complemented
10#28430C214yesmaximal, complemented
11#28419C212yesmaximal, complemented
12#28404C212yesmaximal, complemented
13#28328C214yesmaximal, complemented
14#25736C214yesmaximal, complemented
15#20537C211yesmaximal, characteristic, complemented
16C2×#535C2212yes—
17C2×#533C2211yescharacteristic
18C2×#532C2211yescharacteristic
19C2×#532C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21#3216—28no—
22#3207—28no—
23C2×#497C2212yescomplemented
24C2×#496C2214yes—
25C2×#495C2214yes—
26C2×#493C2211yescharacteristic, complemented
27C2×#492C2211yescharacteristic, complemented
28C2×#489C2212yescomplemented
29C2×#487C2211yescharacteristic, complemented
30C2×#486C2211yescharacteristic, complemented
31#3109—28no—
32#3101—28no—
33C2×#481C2212yes—
34C2×#483C2212yes—
35C2×#476C2212yes—
36C2×#476C2212yes—
37C2×#462C2212yescomplemented
38C2×#460C2212yescomplemented
39C2×#463C2214yescomplemented
40C2×#461C2212yes—
41C2×#459C2212yes—
42C2×#452C2214yes—
43C2×#449C2214yes—
44C2×#450C2212yes—
45C2×#447C2212yes—
46C2×#442C2214yes—
47C2×C4×SD16C2212yes—
48C2×C4×SD16C2212yes—
49C2×#432C2211yescharacteristic
50C2×#431C2211yescharacteristic
51#2847—22no—
52#2820—22no—
53#2624—22no—
54#2620C2214yes—
55#2617—22no—
56#2616—22no—
57#2567C2212yescomplemented
58#2548C2212yescomplemented
59#2547C2212yes—
60#2547C2212yescomplemented
61#2544C2214yes—
62#2542C2212yescomplemented
63#2541C2212yes—
64#2538—22no—
65#2537—22no—
66#2535—22no—
67#2311C2212yes—
68#2285C2211yescharacteristic
69#2096C2214yescomplemented
70#2094C2214yescomplemented
71#2015C2214yes—
72#2015C2214yescomplemented
73#2014C2214yes—
74#2012C2214yes—
75#2012C2214yescomplemented
76#2011C2214yes—
77#1972C2214yescomplemented
78#1971C2214yescomplemented
79#1968C2214yescomplemented
80#1968C2214yescomplemented
81#1959C2214yes—
82#1957C2214yes—
83#1771C2214yescomplemented
84#1770C2212yescomplemented
85C8×C4⋊C4C2212yescomplemented
86#1750C2214yes—
87C22×#41—21no—
88C22×#41C2312yes—
89C23×D4C2311yescharacteristic
90C24×C4—21no—
91C22×Q16C2314yes—
92C22×SD16C2312yes—
93C22×SD16C2312yes—
94C22×SD16C2314yes—
95C22×SD16C2314yes—
96C22×D8C2314yes—
97C22×D8C2314yes—
98#535—48no—
99#533—24no—
100#533—24no—
101#532—24no—
102#532—24no—
103#532—24no—
104#532—24no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2311yescharacteristic
109C2×#127C2311yescharacteristic
110C2×#127C2311yescharacteristic
111C2×#127C2314yes—
112C2×#125—21no—
113C2×#124—21no—
114C2×#124C2312yes—
115C2×#124C2314yes—
116C2×#124C2314yes—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8—24no—
119C2×C22⋊Q8C2312yes—
120C2×#121—21no—
121C2×#121—21no—
122C2×#121—21no—
123C2×#121—24no—
124C2×#121—24no—
125C2×#121C2312yes—
126C2×#120—24no—
127C2×C4×Q8—21no—
128C2×C4×Q8C2312yes—
129C2×C4×D4—21no—
130C2×C4×D4—21no—
131C2×C4×D4—21no—
132C2×C4×D4—21no—
133C2×C4×D4—21no—
134C2×C4×D4—24no—
135C2×C4×D4—24no—
136C2×C4×D4C2312yes—
137C2×C4×D4C2312yes—
138C2×C4×D4C2312yes—
139C2×#117—21no—
140C2×#117C2312yes—
141C22×C4⋊C4C2311yescharacteristic
142C22×C22⋊C4—21no—
143C22×C42C2311yescharacteristic
144#497—28no—
145#496—48no—
146#495—48no—
147#493—24no—
148#492—24no—
149#489—28no—
150#487—24no—
151#486—24no—
152#483—28no—
153#481—28no—
154#476—28no—
155#476—28no—
156#463—216no—
157#462—28no—
158#461—28no—
159#460—28no—
160#459—28no—
161#452—216no—
162#450—28no—
163#449—216no—
164#447—28no—
165#442—216no—
166C4×SD16—28no—
167C4×SD16—28no—
168#432D418yescomplemented
169#431D418yescomplemented
170C2×#106C2312yes—
171C2×#106C2312yes—
172C2×#106C2314yes—
173C2×C4⋊C8C2312yes—
174C2×C4⋊C8C2312yes—
175C2×#103C2312yes—
176C2×#103C2312yes—
177C2×#103C2314yes—
178C2×#103C2314yes—
179C2×#102C2314yes—
180C2×#102C2314yes—
181C2×#102C2314yes—
182C2×#102C2312yes—
183C2×#102C2312yes—
184C2×#102C2314yes—
185C2×C22⋊C8C2312yes—
186C2×C22⋊C8C2312yes—
187C2×#97C2314yes—
188C2×#97C2312yes—
189C2×C4×C8C2314yes—
190C2×C4×C8C2311yescharacteristic
191C2×C4×C8C2311yescharacteristic
192#399—24no—
193#397—24no—
194#397—24no—
195#396—28no—
196#395—24no—
197#394—28no—
198#394—28no—
199#394—24no—
200#392—24no—
201#391C2314yes—
202#391C2314yes—
203#390—24no—
204#390C2314yes—
205#390C2314yes—
206#389—24no—
207#386—24no—
208#386—28no—
209#384D412yes—
210#384C2314yes—
211#382D412yes—
212#380—24no—
213#380—24no—
214#380—24no—
215#380D412yes—
216#380D412yes—
217C4×C4⋊C4C2312yes—
218C4×C4⋊C4C2312yes—
219#336C2314yes—
220#336C2314yes—
221C25—21no—
222C2×#41—24no—
223C2×#41—24no—
224C2×#41—44no—
225C2×#41—44no—
226C2×#41—44no—
227C2×#41—84no—
228C22×Q8—21no—
229C22×Q8—21no—
230C22×Q8C2412yes—
231C22×Q8C2412yes—
232C22×D4—24no—
233C22×D4—22no—
234C22×D4—22no—
235C22×D4—21no—
236C22×D4—22no—
237C22×D4—22no—
238C22×D4—21no—
239C22×D4—21no—
240C22×D4—21no—
241C22×D4—28no—
242C22×D4—21no—
243C22×D4—42no—
244C22×D4C2412yes—
245C22×D4C2412yes—
246C22×D4C2412yes—
247C23×C4—21no—
248C23×C4—22no—
249C23×C4—22no—
250C23×C4—22no—
251C23×C4—22no—
252C23×C4—44no—
253C23×C4—84no—
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2411yescharacteristic
257C2×Q16—28no—
258C2×Q16—44no—
259C2×Q16—44no—
260C2×SD16—24no—
261C2×SD16—24no—
262C2×SD16—24no—
263C2×SD16—24no—
264C2×SD16—28no—
265C2×SD16—28no—
266C2×SD16—42no—
267C2×SD16—42no—
268C2×SD16—44no—
269C2×SD16—44no—
270C2×SD16—44no—
271C2×SD16—44no—
272C2×D8—28no—
273C2×D8—28no—
274C2×D8—28no—
275C2×D8—44no—
276C2×D8—44no—
277C2×D8—44no—
278C22×C8C2412yes—
279C22×C8C2414yes—
280C22×C8C2412yes—
281C4⋊Q8—22no—
282C4⋊Q8—22no—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289#127—28no—
290#127—22no—
291#127—22no—
292#127—22no—
293#127—22no—
294#127—44no—
295#125—44no—
296#124—28no—
297#124—24no—
298#124—24no—
299#124—44no—
300#124—44no—
301#124—44no—
302#124—44no—
303C22⋊Q8—24no—
304C22⋊Q8—24no—
305C22⋊Q8—44no—
306C22⋊Q8—88no—
307#121—24no—
308#121—24no—
309#121—416no—
310#121—44no—
311#121—44no—
312#121—44no—
313#121—88no—
314#120—416no—
315C4×Q8—24no—
316C4×Q8—24no—
317C4×Q8—44no—
318C4×D4—24no—
319C4×D4—24no—
320C4×D4—24no—
321C4×D4—24no—
322C4×D4—24no—
323C4×D4—24no—
324C4×D4—44no—
325C4×D4—44no—
326C4×D4—44no—
327C4×D4—44no—
328C4×D4—44no—
329#117—44no—
330#117—44no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—28no—
343C2×C4⋊C4C2×D414yes—
344C2×C4⋊C4C2×D414yes—
345C2×C4⋊C4C2×D414yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—28no—
357C2×C22⋊C4—28no—
358C2×C22⋊C4—28no—
359C2×C22⋊C4—28no—
360C2×C22⋊C4—44no—
361C2×C22⋊C4C2×D414yes—
362C2×C42—22no—
363C2×C42C2×D411yescharacteristic
364C2×C42C2×D411yescharacteristic
365C2×C42C2×D414yes—
366C2×C42C2×D414yes—
367C2×C42C2414yes—
368C2×C42C2411yescharacteristic
369C2×C42C2411yescharacteristic
370#106—24no—
371#106—24no—
372#106—28no—
373C4⋊C8C2×D418yes—
374C4⋊C8C2×D418yes—
375#103—24no—
376#103—24no—
377#103—28no—
378#103—28no—
379#102—24no—
380#102—24no—
381#102—28no—
382#102—28no—
383#102—28no—
384#102—28no—
385C22⋊C8C2×D418yes—
386C22⋊C8C2×D418yes—
387#97—44no—
388#97C2×D418yes—
389C4×C8—44no—
390C4×C8C2×D414yes—
391C4×C8C2×D414yes—
392#95—28no—
393#95—28no—
394#95C2×D414yes—
395#95C2×D414yes—
396C24—24no—
397C24—22no—
398C24—22no—
399C24—22no—
400C24—42no—
401C24—42no—
402C24—82no—
403C24—82no—
404C24C22×D411yescharacteristic
405#41—88no—
406#41—88no—
407#41—816no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—24no—
413C2×Q8—22no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—22no—
417C2×Q8—24no—
418C2×Q8—44no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×Q8—82no—
422C2×D4—24no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—22no—
430C2×D4—24no—
431C2×D4—24no—
432C2×D4—22no—
433C2×D4—22no—
434C2×D4—24no—
435C2×D4—24no—
436C2×D4—48no—
437C2×D4—48no—
438C2×D4—42no—
439C2×D4—416no—
440C2×D4—44no—
441C2×D4—44no—
442C2×D4—44no—
443C2×D4—44no—
444C2×D4—42no—
445C2×D4—42no—
446C2×D4—42no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—44no—
450C2×D4—44no—
451C2×D4—82no—
452C2×D4—82no—
453C2×D4—88no—
454C2×D4—88no—
455C2×D4—84no—
456C2×D4H(5,𝔽2)14yes—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—42no—
466C22×C4—44no—
467C22×C4—48no—
468C22×C4—44no—
469C22×C4—42no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—48no—
475C22×C4—84no—
476C22×C4—816no—
477C22×C4—82no—
478C22×C4#12714yes—
479C22×C4#12714yes—
480C22×C4C22×D418yes—
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C2511yescharacteristic
489Q16—88no—
490SD16—48no—
491SD16—48no—
492SD16—88no—
493SD16—88no—
494D8—416no—
495D8—88no—
496C2×C8—22no—
497C2×C8—24no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—24no—
501C2×C8—22no—
502C2×C8C22×D414yes—
503C2×C8C22×D414yes—
504C2×C8C22×D418yes—
505C4⋊C4—28no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—44no—
513C4⋊C4—44no—
514C4⋊C4—42no—
515C4⋊C4—42no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—88no—
524C22⋊C4—44no—
525C22⋊C4—416no—
526C22⋊C4—44no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—88no—
534C22⋊C4—88no—
535C22⋊C4—88no—
536C42—24no—
537C42—24no—
538C42—44no—
539C42#12714yes—
540C42#12714yes—
541C42C22×D412yes—
542C42C22×D412yes—
543C42C22×D412yes—
544C42C22×D412yes—
545C23—22no—
546C23—21no—
547C23—21no—
548C23—22no—
549C23—24no—
550C23—48no—
551C23—44no—
552C23—42no—
553C23—42no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—44no—
564C23—44no—
565C23—84no—
566C23—84no—
567C23—88no—
568C23—88no—
569C23C2×#12712yes—
570C23C23×D411yescharacteristic
571Q8—42no—
572Q8—42no—
573Q8—42no—
574Q8—44no—
575Q8—42no—
576Q8—44no—
577Q8—84no—
578Q8—84no—
579D4—24no—
580D4—24no—
581D4—48no—
582D4—42no—
583D4—42no—
584D4—44no—
585D4—44no—
586D4—44no—
587D4—44no—
588D4—84no—
589D4—84no—
590D4—88no—
591D4—88no—
592D4—84no—
593D4—84no—
594D4—1616no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—28no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—28no—
604C2×C4—28no—
605C2×C4—28no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—88no—
620C2×C4—88no—
621C2×C4—88no—
622C2×C4—84no—
623C2×C4—88no—
624C2×C4C2×#12712yes—
625C2×C4C2×#12712yes—
626C2×C4C2×#12712yes—
627C2×C4C2×#12712yes—
628C2×C4C2×#12712yes—
629C2×C4#57714yes—
630C2×C4C23×D411yescharacteristic
631C2×C4C23×D411yescharacteristic
632C2×C4C2×H(5,𝔽2)12yes—
633C2×C4#58512yes—
634C8—28no—
635C8—24no—
636C8—24no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—42no—
641C22—42no—
642C22—44no—
643C22—48no—
644C22—816no—
645C22—84no—
646C22—82no—
647C22—82no—
648C22—84no—
649C22—84no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—84no—
657C22—84no—
658C22C22×#12711yescharacteristic
659C22#348411yescharacteristic
660C22#348711yescharacteristic
661C22C2×#57712yes—
662C22#358512yes—
663C4—22no—
664C4—22no—
665C4—22no—
666C4—22no—
667C4—48no—
668C4—82no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4—82no—
673C4#315414yes—
674C4#348812yes—
675C4#348812yes—
676C2—42no—
677C2—84no—
678C2—162no—
679C2—162no—
680C2#3556511yescharacteristic
681C2C2×#315411yescharacteristic
682C2#6064411yescharacteristic
683C2#6168512yes—
684C2#6191312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3216C2semidirect
2. C2×#3207C2semidirect
3. C2×#3109C2semidirect
4. C2×#3101C2semidirect
5. #37640C2semidirect
6. #35165C2semidirect
7. #28479C2semidirect
8. #28470C2semidirect
9. #28460C2semidirect
10. #28430C2semidirect
11. #28419C2semidirect
12. #28404C2semidirect
13. #28328C2semidirect
14. #25736C2semidirect
15. #20537C2semidirect
23. C2×#497C22semidirect
26. C2×#493C22semidirect
27. C2×#492C22semidirect
28. C2×#489C22semidirect
29. C2×#487C22semidirect
30. C2×#486C22semidirect
37. C2×#462C22semidirect
38. C2×#460C22semidirect
39. C2×#463C22semidirect
57. #2567C22semidirect
58. #2548C22semidirect
60. #2547C22semidirect
62. #2542C22semidirect
69. #2096C22semidirect
70. #2094C22semidirect
72. #2015C22semidirect
75. #2012C22semidirect
77. #1972C22semidirect
78. #1971C22semidirect
79. #1968C22semidirect
80. #1968C22semidirect
83. #1771C22semidirect
84. #1770C22semidirect
85. C8×C4⋊C4C22semidirect
168. #432D4semidirect
169. #431D4semidirect

Maximal subgroups

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

subgroupindex
C2×#32162
C2×#32072
C2×#31092
C2×#31012
#376402
#351652
#284792
#284702
#284602
#284302
#284192
#284042
#283282
#257362
#205372