Finite Group Databasebeta

← #6484723

Subgroups — #6484723

Order 512.

1874 conjugacy classes of subgroups.

The subgroup lattice has 761 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3233C212yesmaximal, complemented
2#36631C211yesmaximal, characteristic
3C4×#533C211yesmaximal, characteristic, complemented
4#33112C214yesmaximal, complemented
5#29759C212yesmaximal, complemented
6#29758C212yesmaximal
7#28751C212yesmaximal, complemented
8#28690C212yesmaximal, complemented
9#28657C214yesmaximal, complemented
10#28282C212yesmaximal, complemented
11#28281C212yesmaximal, complemented
12#28206C211yesmaximal, characteristic, complemented
13#28205C211yesmaximal, characteristic, complemented
14#28202C211yesmaximal, characteristic, complemented
15#28201C211yesmaximal, characteristic, complemented
16#21862C212yesmaximal, complemented
17#20808C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#529C2212yes—
20#3233—216no—
21C2×#499C2214yes—
22C2×#488C2214yes—
23C2×#485C2214yes—
24C2×#482C2214yes—
25C2×#477C2212yes—
26C2×#475C2212yes—
27C2×#476C2212yes—
28C2×#474C2212yes—
29C2×#431C2212yes—
30#2861C2212yes—
31#2860—21no—
32#2860—21no—
33#2859C2211yescharacteristic
34#2858C2211yescharacteristic
35#2857C2211yescharacteristic
36#2856C2211yescharacteristic
37#2854—21no—
38#2854—21no—
39#2850C2212yes—
40C4×C4⋊Q8C2211yescharacteristic
41C4×C4⋊Q8C2211yescharacteristic
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×#127C2211yescharacteristic
46C4×#125C2212yes—
47C4×#125C2212yes—
48C4×#124C2212yes—
49C4×C22⋊Q8—21no—
50C4×C22⋊Q8—21no—
51C4×C22⋊Q8—21no—
52C4×C22⋊Q8—21no—
53C4×#121—21no—
54C4×#121—21no—
55#2296C2212yes—
56#2295—21no—
57#2295—21no—
58#2293—21no—
59#2293—21no—
60#2285C2211yescharacteristic
61C42×Q8—21no—
62C42×Q8—21no—
63C4×#117C2212yes—
64C2×C43C2211yescharacteristic
65#2084C2212yes—
66#2084C2212yes—
67#2083C2212yes—
68#2083C2212yes—
69#2068C2214yes—
70#2067C2214yes—
71#2066C2214yes—
72#2065C2214yes—
73#2043C2212yes—
74#2043C2212yes—
75#2041C2212yes—
76#2041C2212yes—
77#1991C2214yes—
78#1983C2214yes—
79#1983C2214yes—
80#1958C2214yes—
81#1949C2214yes—
82#1949C2214yes—
83#1928C2214yes—
84#1927C2214yes—
85#1926C2214yes—
86#1925C2214yes—
87#1875C2214yes—
88#1873C2214yes—
89#1842C2212yes—
90#1831C2211yescharacteristic
91#1831C2211yescharacteristic
92#1803C2212yes—
93#1803C2212yes—
94#1802C2212yes—
95#1802C2212yes—
96#1797C2212yes—
97#1797C2212yes—
98C4×C4⋊C8C2212yes—
99C4×C4⋊C8C2212yes—
100#533—24no—
101#533D418yescomplemented
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×#127C2311yescharacteristic
106#529—28no—
107#529—28no—
108C2×#126—22no—
109C2×#125C2312yes—
110C2×#125C2312yes—
111C2×#124C2312yes—
112C2×#124C2312yes—
113C2×#123—22no—
114C2×C22⋊Q8—21no—
115C2×C22⋊Q8—21no—
116C2×C22⋊Q8C2312yes—
117C2×#121—21no—
118C2×#121—21no—
119C2×#121C2312yes—
120C2×C4×Q8—21no—
121C2×C4×Q8—21no—
122C2×C4×Q8—21no—
123C2×C4×Q8—21no—
124C2×C4×Q8—21no—
125C2×C4×Q8—21no—
126C2×C4×Q8C2312yes—
127C2×C4×D4—22no—
128C2×C4×D4—21no—
129C2×C4×D4—21no—
130C2×C4×D4C2312yes—
131C2×#117—22no—
132C2×#117C2312yes—
133C2×#117C2312yes—
134C22×C42D411yescharacteristic
135C22×C42D411yescharacteristic
136C22×C42D411yescharacteristic
137C22×C42D411yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141#499—216no—
142#488—216no—
143#485—216no—
144#482—216no—
145#477—28no—
146#476—28no—
147#475—28no—
148#474—28no—
149#431—28no—
150C2×#107C2314yes—
151C2×#107C2314yes—
152C2×#106C2314yes—
153C2×#106C2314yes—
154C2×C4⋊C8C2312yes—
155C2×C4⋊C8C2312yes—
156C2×C4⋊C8C2314yes—
157C2×C4⋊C8C2314yes—
158C2×#103C2314yes—
159C2×#103C2314yes—
160C2×#103C2314yes—
161C2×#103C2314yes—
162C2×#102C2314yes—
163C2×#102C2314yes—
164C2×#102C2314yes—
165C2×#102C2314yes—
166C2×C22⋊C8C2314yes—
167C2×C4×C8C2314yes—
168#391—21no—
169#391—21no—
170#391—21no—
171#391—21no—
172#391C2312yes—
173#391C2312yes—
174#390—21no—
175#390—21no—
176#390—21no—
177#390—21no—
178#390—21no—
179#390—21no—
180#390—21no—
181#390—21no—
182#390C2312yes—
183#390C2312yes—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389—21no—
192#389—21no—
193#389—21no—
194#389—21no—
195#389—21no—
196#389C2312yes—
197#389C2312yes—
198#389C2312yes—
199#389C2312yes—
200#388—22no—
201#388—22no—
202#388—22no—
203#388—22no—
204#388—22no—
205#388—22no—
206#388—22no—
207#388—22no—
208#387—22no—
209#387—22no—
210#387—22no—
211#387—22no—
212#386—22no—
213#386—22no—
214#385—22no—
215#385—22no—
216#384C2311yescharacteristic
217#384C2311yescharacteristic
218#384C2311yescharacteristic
219#384C2311yescharacteristic
220#382C2312yes—
221#382C2312yes—
222#380—21no—
223#380—21no—
224#380—21no—
225#380—21no—
226C4×C4⋊C4—22no—
227C4×C4⋊C4—22no—
228C4×C4⋊C4—22no—
229C4×C4⋊C4—22no—
230C4×C4⋊C4—21no—
231C4×C4⋊C4—21no—
232C4×C4⋊C4—21no—
233C4×C4⋊C4—21no—
234C4×C4⋊C4—21no—
235C4×C4⋊C4—21no—
236C4×C4⋊C4—21no—
237C4×C4⋊C4—21no—
238C4×C4⋊C4—22no—
239C4×C4⋊C4—22no—
240C4×C4⋊C4C2312yes—
241C4×C4⋊C4C2312yes—
242C4×C4⋊C4C2312yes—
243C4×C4⋊C4C2312yes—
244C4×C22⋊C4—22no—
245C4×C22⋊C4—22no—
246C4×C22⋊C4—22no—
247C4×C22⋊C4—22no—
248#376—22no—
249C43—22no—
250C43D411yescharacteristic
251C43D411yescharacteristic
252C43D411yescharacteristic
253C43D411yescharacteristic
254C43C2311yescharacteristic
255C43C2311yescharacteristic
256C43C2311yescharacteristic
257C43C2311yescharacteristic
258#340C2314yes—
259#340C2314yes—
260#340C2314yes—
261#340C2314yes—
262#336C2312yes—
263#336C2312yes—
264#336C2312yes—
265#336C2312yes—
266C22×Q8—21no—
267C22×Q8—21no—
268C22×Q8C2412yes—
269C22×D4—22no—
270C22×D4—21no—
271C22×D4—21no—
272C22×D4C2412yes—
273C23×C4—22no—
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D411yescharacteristic
276C23×C4C2×D411yescharacteristic
277C23×C4C2×D411yescharacteristic
278C23×C4C2×D411yescharacteristic
279C23×C4C2×D411yescharacteristic
280C23×C4C2411yescharacteristic
281C22×C8C2414yes—
282C22×C8C2414yes—
283C4⋊Q8—22no—
284C4⋊Q8—22no—
285C4⋊Q8—42no—
286C4⋊Q8C2×D414yes—
287C4⋊Q8C2×D414yes—
288#127—42no—
289#126—48no—
290#125—24no—
291#125—24no—
292#125—24no—
293#125C2×D418yes—
294#124—24no—
295#124—24no—
296#124—24no—
297#124—24no—
298#123—48no—
299C22⋊Q8—28no—
300C22⋊Q8—28no—
301C22⋊Q8—24no—
302C22⋊Q8—24no—
303#121—24no—
304#121—24no—
305#121—44no—
306#121—44no—
307C4×Q8—28no—
308C4×Q8—28no—
309C4×Q8—24no—
310C4×Q8—24no—
311C4×Q8—44no—
312C4×Q8—44no—
313C4×Q8—44no—
314C4×Q8—44no—
315C4×D4—24no—
316C4×D4—24no—
317C4×D4—44no—
318C4×D4—48no—
319C4×D4—44no—
320#117—24no—
321#117—44no—
322#117—48no—
323#117C2×D418yes—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2412yes—
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—22no—
365C2×C42—22no—
366C2×C42—24no—
367C2×C42—24no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—42no—
375C2×C42—42no—
376C2×C42—42no—
377C2×C42—42no—
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2×D411yescharacteristic
391C2×C42C2×D411yescharacteristic
392C2×C42C2×D411yescharacteristic
393C2×C42C2×D411yescharacteristic
394C2×C42C2×D411yescharacteristic
395C2×C42C2×D411yescharacteristic
396C2×C42C2×D411yescharacteristic
397C2×C42C2×D411yescharacteristic
398C2×C42C2×D411yescharacteristic
399C2×C42C2×D414yes—
400C2×C42C2×D411yescharacteristic
401C2×C42#4114yes—
402C2×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405C2×C42C2411yescharacteristic
406C2×C42C2411yescharacteristic
407C2×C42C2411yescharacteristic
408#107—28no—
409#107—28no—
410#106—28no—
411#106—28no—
412C4⋊C8—24no—
413C4⋊C8—24no—
414C4⋊C8—28no—
415C4⋊C8—44no—
416#103—28no—
417#103—28no—
418#103—44no—
419#103—44no—
420#102—28no—
421#102—28no—
422#102—44no—
423#102—44no—
424C22⋊C8—28no—
425C4×C8—28no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95—22no—
435#95—22no—
436#95—22no—
437#95—22no—
438#95—22no—
439#95—22no—
440#95—22no—
441#95—22no—
442C24—22no—
443C24C22×D411yescharacteristic
444C2×Q8—24no—
445C2×Q8—24no—
446C2×Q8—22no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—22no—
450C2×Q8—22no—
451C2×Q8—22no—
452C2×Q8—42no—
453C2×Q8—42no—
454C2×Q8—42no—
455C2×Q8—42no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—22no—
462C2×D4—42no—
463C2×D4—42no—
464C2×D4—42no—
465C2×D4—42no—
466C2×D4—44no—
467C2×D4—44no—
468C2×D4—44no—
469C2×D4—42no—
470C2×D4—42no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—22no—
488C22×C4—41no—
489C22×C4—41no—
490C22×C4—41no—
491C22×C4—41no—
492C22×C4—41no—
493C22×C4—41no—
494C22×C4—41no—
495C22×C4—41no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4#12114yes—
500C22×C4#12114yes—
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4C22×D411yescharacteristic
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D412yes—
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C2×#4112yes—
516C22×C4C2511yescharacteristic
517C2×C8—24no—
518C2×C8—24no—
519C2×C8—24no—
520C2×C8—24no—
521C2×C8—24no—
522C2×C8—24no—
523C4⋊C4—28no—
524C4⋊C4—28no—
525C4⋊C4—24no—
526C4⋊C4—24no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—22no—
533C4⋊C4—24no—
534C4⋊C4—24no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—44no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—42no—
547C4⋊C4—44no—
548C4⋊C4—42no—
549C4⋊C4—44no—
550C4⋊C4—44no—
551C4⋊C4—42no—
552C4⋊C4—42no—
553C4⋊C4C22×D414yes—
554C4⋊C4C22×D414yes—
555C22⋊C4—28no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C22⋊C4—44no—
560C22⋊C4—44no—
561C22⋊C4—44no—
562C22⋊C4—44no—
563C22⋊C4—44no—
564C22⋊C4—44no—
565C22⋊C4—44no—
566C22⋊C4—44no—
567C22⋊C4—44no—
568C42—28no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—22no—
575C42—22no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—41no—
584C42—41no—
585C42—41no—
586C42—41no—
587C42—41no—
588C42—41no—
589C42—41no—
590C42—41no—
591C42—44no—
592C42—44no—
593C42—44no—
594C42—44no—
595C42—44no—
596C42—44no—
597C42#12114yes—
598C42#12114yes—
599C42#13514yes—
600C42C22×D412yes—
601C42C22×D412yes—
602C42C2×#4112yes—
603C42C2×#4112yes—
604C422−1+414yes—
605C23—21no—
606C23—21no—
607C23—21no—
608C23—21no—
609C23—21no—
610C23—21no—
611C23—42no—
612C23—42no—
613C23—42no—
614C23—42no—
615C23—42no—
616C23—42no—
617C23—42no—
618C23C2×#12112yes—
619C23C23×D411yescharacteristic
620Q8—48no—
621Q8—48no—
622Q8—42no—
623Q8—42no—
624Q8—42no—
625Q8—42no—
626D4—42no—
627D4—42no—
628D4—42no—
629D4—42no—
630D4—84no—
631D4—84no—
632D4—88no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—21no—
653C2×C4—24no—
654C2×C4—24no—
655C2×C4—28no—
656C2×C4—24no—
657C2×C4—24no—
658C2×C4—21no—
659C2×C4—21no—
660C2×C4—21no—
661C2×C4—21no—
662C2×C4—21no—
663C2×C4—21no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—88no—
689C2×C4C2×#12112yes—
690C2×C4C2×#12112yes—
691C2×C4C2×#12112yes—
692C2×C4C2×#12112yes—
693C2×C4C2×#12112yes—
694C2×C4#53612yes—
695C2×C4#53612yes—
696C2×C4C2×#13512yes—
697C2×C4C2×#13512yes—
698C2×C4C2×#13512yes—
699C2×C4#57512yes—
700C2×C4#57512yes—
701C2×C4#57512yes—
702C2×C4C23×D411yescharacteristic
703C2×C4C22×#4111yescharacteristic
704C2×C4C2×2−1+412yes—
705C2×C4#58512yes—
706C8—44no—
707C8—44no—
708C22—22no—
709C22—22no—
710C22—22no—
711C22—41no—
712C22—41no—
713C22—41no—
714C22—41no—
715C22—82no—
716C22—82no—
717C22—82no—
718C22—82no—
719C22—82no—
720C22—82no—
721C22—82no—
722C22C22×#12111yescharacteristic
723C22C2×#53611yescharacteristic
724C22#344911yescharacteristic
725C22C22×#13511yescharacteristic
726C22C2×#57511yescharacteristic
727C22C2×#57511yescharacteristic
728C22C2×#57511yescharacteristic
729C4—22no—
730C4—22no—
731C4—22no—
732C4—22no—
733C4—22no—
734C4—22no—
735C4—22no—
736C4—22no—
737C4—22no—
738C4—22no—
739C4—41no—
740C4—41no—
741C4—44no—
742C4—44no—
743C4—82no—
744C4—82no—
745C4—82no—
746C4—82no—
747C4—82no—
748C4#303514yes—
749C4#349712yes—
750C4#350712yes—
751C2—42no—
752C2—162no—
753C2#3517811yescharacteristic
754C2C2×#303511yescharacteristic
755C2#6041511yescharacteristic
756C2#6084611yescharacteristic
757C2#6086311yescharacteristic
758C2#6094211yescharacteristic
759C2#6097011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3233C2semidirect
3. C4×#533C2semidirect
4. #33112C2semidirect
5. #29759C2semidirect
7. #28751C2semidirect
8. #28690C2semidirect
9. #28657C2semidirect
10. #28282C2semidirect
11. #28281C2semidirect
12. #28206C2semidirect
13. #28205C2semidirect
14. #28202C2semidirect
15. #28201C2semidirect
16. #21862C2semidirect
17. #20808C2semidirect
101. #533D4semidirect

Maximal subgroups

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

subgroupindex
C2×#32332
#366312
C4×#5332
#331122
#297592
#297582
#287512
#286902
#286572
#282822
#282812
#282062
#282052
#282022
#282012
#218622
#208082