Finite Group Databasebeta

← #6481773

Subgroups — #6481773

Order 512.

1857 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3117C211yesmaximal, characteristic
2C2×#3104C211yesmaximal, characteristic, complemented
3#35241C211yesmaximal, characteristic
4C2×#2356C211yesmaximal, characteristic, complemented
5#33110C212yesmaximal
6#33109C212yesmaximal, complemented
7#29719C212yesmaximal
8#29715C212yesmaximal, complemented
9#24699C212yesmaximal, complemented
10#24694C212yesmaximal, complemented
11#24687C214yesmaximal, complemented
12#24648C212yesmaximal, complemented
13#24645C212yesmaximal, complemented
14#24637C214yesmaximal, complemented
15#21953C211yesmaximal, characteristic, complemented
16#21866C211yesmaximal, characteristic, complemented
17#21861C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C2×#529C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21#3117—28no—
22#3104—28no—
23C2×#484C2211yescharacteristic
24C2×#484C2212yes—
25C2×#481C2211yescharacteristic
26C2×#479C2212yes—
27C2×#477C2212yes—
28C2×#464C2212yes—
29C2×#462C2212yes—
30C2×#451C2211yescharacteristic
31C2×#451C2212yes—
32C2×#448C2211yescharacteristic
33C2×#441C2212yes—
34C2×#440C2212yes—
35C2×C4×Q16C2214yes—
36C2×#432C2211yescharacteristic
37C2×#431C2211yescharacteristic
38#2857C2212yes—
39#2850C2212yes—
40#2664—21no—
41#2664—21no—
42#2568C2211yescharacteristic
43#2565C2212yes—
44#2547C2211yescharacteristic
45#2546C2211yescharacteristic
46#2545C2212yes—
47#2542C2211yescharacteristic
48#2498—21no—
49#2498—21no—
50#2417—22no—
51#2356—22no—
52#2356—21no—
53#2356—21no—
54#2356C2214yes—
55#2356C2212yes—
56#2356C2212yes—
57#2323—22no—
58C2×#391C2212yes—
59C2×#387—22no—
60C2×#383—21no—
61C2×#382C2211yescharacteristic
62C2×C4×C4⋊C4C2212yes—
63C2×C4×C4⋊C4C2211yescharacteristic
64C2×#376—21no—
65#2074C2214yes—
66#2072C2214yes—
67#2056C2214yes—
68#2056C2214yes—
69#2056C2214yes—
70#2055C2214yes—
71#2016C2214yes—
72#2015C2214yes—
73#1993C2212yes—
74#1991C2212yes—
75#1983C2214yes—
76#1981C2214yes—
77#1960C2212yes—
78#1958C2212yes—
79#1949C2214yes—
80#1947C2214yes—
81#1940C2212yes—
82#1940C2212yes—
83#1939C2214yes—
84#1928C2214yes—
85#1928C2214yes—
86#1927C2214yes—
87#1925C2214yes—
88#1874C2212yes—
89#1872C2212yes—
90#1872C2212yes—
91#1872C2212yes—
92#1805C2214yes—
93#1805C2212yes—
94#1804C2212yes—
95#1797C2212yes—
96#1797C2212yes—
97C23×Q8C2311yescharacteristic
98C22×Q16C2314yes—
99C22×Q16C2312yes—
100C22×SD16C2312yes—
101#533—24no—
102#533—24no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2312yes—
105C2×C4⋊Q8C2311yescharacteristic
106#529—24no—
107#529—24no—
108C2×#126—21no—
109C2×#125—21no—
110C2×#125C2311yescharacteristic
111C2×#124C2311yescharacteristic
112C2×#124C2312yes—
113C2×#123—21no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—22no—
116C2×C22⋊Q8C2312yes—
117C2×C22⋊Q8C2311yescharacteristic
118C2×#121C2311yescharacteristic
119C2×C4×Q8—21no—
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8—24no—
123C2×C4×Q8D418yes—
124C2×C4×Q8C2312yes—
125C2×C4×Q8C2311yescharacteristic
126C2×C4×Q8C2312yes—
127C2×C4×Q8C2312yes—
128C2×C4×D4—21no—
129C2×C4×D4C2311yescharacteristic
130C2×#117—21no—
131C2×#117—21no—
132C22×C4⋊C4—21no—
133C22×C4⋊C4D412yes—
134C22×C4⋊C4C2311yescharacteristic
135C22×C4⋊C4C2312yes—
136C22×C42—21no—
137C22×C42—22no—
138C22×C42C2312yes—
139C22×C42C2311yescharacteristic
140#484—28no—
141#484—24no—
142#481—24no—
143#479—28no—
144#477—28no—
145#464—28no—
146#462—28no—
147#451—28no—
148#451—24no—
149#448—24no—
150#441—28no—
151#440—28no—
152C4×Q16—216no—
153#432—24no—
154#431—24no—
155C2×#107C2314yes—
156C2×#107C2312yes—
157C2×#106C2312yes—
158C2×C4⋊C8C2312yes—
159C2×C4⋊C8C2312yes—
160C2×C4⋊C8C2314yes—
161C2×C4⋊C8C2314yes—
162C2×#103C2314yes—
163C2×#103C2312yes—
164C2×#103C2312yes—
165C2×#103C2314yes—
166C2×#103C2312yes—
167C2×#103C2314yes—
168C2×#103C2314yes—
169C2×#103C2314yes—
170C2×#102C2312yes—
171C2×#102C2314yes—
172C2×C22⋊C8C2311yescharacteristic
173C2×C22⋊C8C2311yescharacteristic
174C2×C22⋊C8C2312yes—
175C2×#97C2312yes—
176C2×C4×C8C2312yes—
177#400—21no—
178#400—21no—
179#399—21no—
180#399—21no—
181#398—22no—
182#398—22no—
183#398—21no—
184#398—21no—
185#397—21no—
186#397—21no—
187#396—22no—
188#396—22no—
189#391—22no—
190#391—22no—
191#391—22no—
192#391C2312yes—
193#391C2312yes—
194#391C2314yes—
195#391C2312yes—
196#391C2314yes—
197#390C2312yes—
198#389—22no—
199#389—22no—
200#389—22no—
201#388—22no—
202#388—22no—
203#388—22no—
204#388—22no—
205#387—24no—
206#387—24no—
207#387—24no—
208#387—24no—
209#387—22no—
210#387—22no—
211#386—22no—
212#386—22no—
213#384C2312yes—
214#383—24no—
215#383—22no—
216#383—22no—
217#382—21no—
218#382—21no—
219#382C2312yes—
220#382C2312yes—
221#382C2312yes—
222#380—22no—
223#380—22no—
224#380—22no—
225C4×C4⋊C4—22no—
226C4×C4⋊C4—22no—
227C4×C4⋊C4D412yes—
228C4×C4⋊C4D411yescharacteristic
229C4×C4⋊C4D411yescharacteristic
230C4×C4⋊C4C2314yes—
231C4×C4⋊C4C2314yes—
232C4×C4⋊C4C2312yes—
233C4×C4⋊C4C2311yescharacteristic
234C4×C4⋊C4C2311yescharacteristic
235#376—24no—
236#376—22no—
237#376—22no—
238C2×#95—21no—
239C2×#95—22no—
240C2×#95—22no—
241C2×#95—21no—
242C2×#95—22no—
243C2×#95D411yescharacteristic
244C2×#95D411yescharacteristic
245#340C2314yes—
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#336C2314yes—
250#336C2314yes—
251C22×Q8—21no—
252C22×Q8—21no—
253C22×Q8—22no—
254C22×Q8—42no—
255C22×Q8—42no—
256C22×Q8C2×D414yes—
257C22×Q8C2411yescharacteristic
258C22×Q8C2412yes—
259C22×Q8C2412yes—
260C22×D4—21no—
261C22×D4C2411yescharacteristic
262C23×C4—21no—
263C23×C4—21no—
264C23×C4—22no—
265C23×C4—21no—
266C23×C4C2×D412yes—
267C23×C4C2×D411yescharacteristic
268C23×C4C2×D412yes—
269C23×C4C2×D411yescharacteristic
270C23×C4C2411yescharacteristic
271C2×Q16—28no—
272C2×Q16—24no—
273C2×Q16—44no—
274C2×Q16—42no—
275C2×Q16—42no—
276C2×Q16—44no—
277C2×SD16—24no—
278C2×SD16—42no—
279C2×SD16—42no—
280C22×C8C2412yes—
281C22×C8C2414yes—
282C22×C8C2412yes—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—22no—
286C4⋊Q8—22no—
287C4⋊Q8—44no—
288#126—44no—
289#125—22no—
290#125—22no—
291#125—44no—
292#124—22no—
293#124—22no—
294#124—44no—
295#123—44no—
296C22⋊Q8—22no—
297C22⋊Q8—24no—
298C22⋊Q8—22no—
299C22⋊Q8—24no—
300C22⋊Q8—48no—
301C22⋊Q8—44no—
302#121—22no—
303#121—22no—
304C4×Q8—28no—
305C4×Q8—28no—
306C4×Q8—24no—
307C4×Q8—24no—
308C4×Q8—22no—
309C4×Q8—24no—
310C4×Q8—22no—
311C4×Q8—24no—
312C4×Q8—44no—
313C4×Q8C2×D418yes—
314C4×Q8C2×D418yes—
315C4×D4—22no—
316C4×D4—22no—
317C4×D4—44no—
318#117—44no—
319#117—44no—
320C2×C4⋊C4—28no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—44no—
340C2×C4⋊C4—44no—
341C2×C4⋊C4C2×D412yes—
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2×D414yes—
344C2×C4⋊C4C2×D412yes—
345C2×C4⋊C4C2×D412yes—
346C2×C4⋊C4C2414yes—
347C2×C4⋊C4C2411yescharacteristic
348C2×C4⋊C4C2411yescharacteristic
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—21no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C42—28no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—21no—
365C2×C42—22no—
366C2×C42—24no—
367C2×C42—24no—
368C2×C42—44no—
369C2×C42—48no—
370C2×C42C2×D412yes—
371C2×C42C2×D412yes—
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D414yes—
375C2×C42#4114yes—
376C2×C42C2412yes—
377C2×C42C2412yes—
378C2×C42C2411yescharacteristic
379C2×C42C2411yescharacteristic
380#107—28no—
381#107—24no—
382#106—24no—
383C4⋊C8—24no—
384C4⋊C8—24no—
385C4⋊C8—44no—
386C4⋊C8—44no—
387#103—28no—
388#103—24no—
389#103—24no—
390#103—28no—
391#103—24no—
392#103—44no—
393#103—44no—
394#103—44no—
395#102—24no—
396#102—44no—
397C22⋊C8—24no—
398C22⋊C8—22no—
399C22⋊C8—22no—
400#97—24no—
401C4×C8—24no—
402#95—22no—
403#95—22no—
404#95—24no—
405#95—24no—
406#95—24no—
407#95—24no—
408#95—22no—
409#95—22no—
410#95—24no—
411#95—24no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95C2×D412yes—
421#95C2×D411yescharacteristic
422#95C2×D411yescharacteristic
423#95C2×D412yes—
424#95C2×D411yescharacteristic
425#95C2×D411yescharacteristic
426C24—21no—
427C24C22×D411yescharacteristic
428C2×Q8—24no—
429C2×Q8—22no—
430C2×Q8—21no—
431C2×Q8—22no—
432C2×Q8—21no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—24no—
440C2×Q8—24no—
441C2×Q8—22no—
442C2×Q8—48no—
443C2×Q8—42no—
444C2×Q8—42no—
445C2×Q8—48no—
446C2×Q8—44no—
447C2×Q8—42no—
448C2×Q8C22×D414yes—
449C2×D4—21no—
450C2×D4—21no—
451C2×D4—22no—
452C2×D4—22no—
453C2×D4—42no—
454C2×D4—44no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—21no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—42no—
478C22×C4—42no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—42no—
483C22×C4#12114yes—
484C22×C4#12114yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C2×#4112yes—
496C22×C4C2511yescharacteristic
497Q16—48no—
498Q16—416no—
499SD16—48no—
500C2×C8—24no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—24no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—24no—
508C2×C8—22no—
509C4⋊C4—28no—
510C4⋊C4—28no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—22no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—22no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—22no—
521C4⋊C4—24no—
522C4⋊C4—24no—
523C4⋊C4—42no—
524C4⋊C4—42no—
525C4⋊C4—44no—
526C4⋊C4—44no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—44no—
533C4⋊C4—42no—
534C4⋊C4—44no—
535C4⋊C4—42no—
536C4⋊C4—44no—
537C4⋊C4—82no—
538C4⋊C4—82no—
539C4⋊C4C22×D414yes—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—42no—
543C22⋊C4—42no—
544C22⋊C4—42no—
545C22⋊C4—42no—
546C22⋊C4—42no—
547C22⋊C4—44no—
548C22⋊C4—82no—
549C22⋊C4—82no—
550C42—28no—
551C42—28no—
552C42—24no—
553C42—24no—
554C42—42no—
555C42—42no—
556C42—48no—
557C42—42no—
558C42—44no—
559C42—48no—
560C42#12114yes—
561C42#12114yes—
562C42C22×D412yes—
563C42C22×D412yes—
564C42C2×#4112yes—
565C42C2×#4112yes—
566C23—21no—
567C23—21no—
568C23—22no—
569C23—41no—
570C23—41no—
571C23—41no—
572C23—42no—
573C23—42no—
574C23C2×#12112yes—
575C23#57814yes—
576C23C23×D411yescharacteristic
577Q8—24no—
578Q8—24no—
579Q8—48no—
580Q8—42no—
581Q8—42no—
582Q8—42no—
583Q8—44no—
584Q8—42no—
585Q8—44no—
586Q8—84no—
587D4—42no—
588D4—42no—
589D4—84no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—28no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—44no—
613C2×C4—48no—
614C2×C4—48no—
615C2×C4—48no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—41no—
626C2×C4—44no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—84no—
636C2×C4—84no—
637C2×C4—88no—
638C2×C4C2×#12112yes—
639C2×C4C2×#12112yes—
640C2×C4C2×#12112yes—
641C2×C4C2×#12112yes—
642C2×C4C2×#12112yes—
643C2×C4#57514yes—
644C2×C4C23×D411yescharacteristic
645C2×C4C22×#4111yescharacteristic
646C2×C4#58512yes—
647C2×C4#58512yes—
648C8—42no—
649C8—44no—
650C8—42no—
651C22—24no—
652C22—22no—
653C22—24no—
654C22—22no—
655C22—81no—
656C22—82no—
657C22—81no—
658C22—82no—
659C22—81no—
660C22#304414yes—
661C22C22×#12111yescharacteristic
662C22#348511yescharacteristic
663C22#348711yescharacteristic
664C22C2×#57512yes—
665C22C2×#57812yes—
666C4—22no—
667C4—22no—
668C4—24no—
669C4—22no—
670C4—22no—
671C4—22no—
672C4—42no—
673C4—42no—
674C4—42no—
675C4—44no—
676C4—44no—
677C4—81no—
678C4—81no—
679C4—82no—
680C4—84no—
681C4—81no—
682C4—82no—
683C4#344912yes—
684C4#349312yes—
685C2—24no—
686C2—161no—
687C2#3515011yescharacteristic
688C2#6041511yescharacteristic
689C2C2×#304411yescharacteristic
690C2#6173812yes—
691C2#6190512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#3104C2semidirect
4. C2×#2356C2semidirect
6. #33109C2semidirect
8. #29715C2semidirect
9. #24699C2semidirect
10. #24694C2semidirect
11. #24687C2semidirect
12. #24648C2semidirect
13. #24645C2semidirect
14. #24637C2semidirect
15. #21953C2semidirect
16. #21866C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31172
C2×#31042
#352412
C2×#23562
#331102
#331092
#297192
#297152
#246992
#246942
#246872
#246482
#246452
#246372
#219532
#218662
#218612