Finite Group Databasebeta

← #6480221

Subgroups — #6480221

Order 512.

1774 conjugacy classes of subgroups.

The subgroup lattice has 762 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36639C211yesmaximal, characteristic
2C4×#533C211yesmaximal, characteristic, complemented
3#33033C212yesmaximal, complemented
4#33031C212yesmaximal, complemented
5#30061C212yesmaximal, complemented
6#30043C212yesmaximal, complemented
7#29557C214yesmaximal, complemented
8#29347C211yesmaximal, characteristic, complemented
9#29346C211yesmaximal, characteristic, complemented
10#28374C211yesmaximal, characteristic, complemented
11#28373C211yesmaximal, characteristic, complemented
12#27584C212yesmaximal, complemented
13#24303C212yesmaximal, complemented
14#24213C212yesmaximal
15#23935C212yesmaximal, complemented
16#22042C214yesmaximal, complemented
17#20777C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C2×#532C2211yescharacteristic
21C2×#477C2212yes—
22C2×#475C2212yes—
23C2×#462C2212yes—
24C2×#460C2212yes—
25C2×#452C2214yes—
26#2862C2211yescharacteristic
27#2862C2211yescharacteristic
28#2861C2212yes—
29#2861C2212yes—
30#2858C2211yescharacteristic
31#2858C2211yescharacteristic
32#2850—21no—
33#2850—21no—
34#2850—21no—
35#2850—21no—
36C4×C4⋊Q8C2211yescharacteristic
37C4×C4⋊Q8C2211yescharacteristic
38C4×C4⋊Q8C2211yescharacteristic
39C4×C4⋊Q8C2211yescharacteristic
40C4×#125—21no—
41C4×#125—21no—
42C4×#124—21no—
43C4×#124—21no—
44C4×#124—21no—
45C4×#124—21no—
46C4×C22⋊Q8—21no—
47C4×C22⋊Q8—21no—
48C4×C22⋊Q8C2212yes—
49C4×C22⋊Q8C2212yes—
50#2297C2212yes—
51#2297C2212yes—
52#2296—21no—
53#2295—21no—
54#2295C2212yes—
55#2285C2211yescharacteristic
56C42×Q8—21no—
57C42×Q8C2212yes—
58C4×#117—21no—
59C2×C43C2211yescharacteristic
60#2095C2212yes—
61#2094C2212yes—
62#2072C2212yes—
63#2071C2212yes—
64#2059C2214yes—
65#2042C2214yes—
66#2026C2214yes—
67#2024C2212yes—
68#2023C2212yes—
69#2016C2212yes—
70#2015C2214yes—
71#2013C2212yes—
72#1988C2212yes—
73#1987C2212yes—
74#1982C2212yes—
75#1981C2212yes—
76#1969C2214yes—
77#1968C2214yes—
78#1934C2214yes—
79#1933C2214yes—
80#1927C2214yes—
81#1927C2214yes—
82#1925C2214yes—
83#1925C2214yes—
84#1880C2214yes—
85#1880C2214yes—
86#1878C2214yes—
87#1878C2214yes—
88#1873C2214yes—
89#1854C2214yes—
90#1839C2212yes—
91#1831C2212yes—
92#1817C2212yes—
93#1797C2212yes—
94#1788C2214yes—
95C4×C4⋊C8C2211yescharacteristic
96C4×C4⋊C8C2211yescharacteristic
97C4×C22⋊C8C2212yes—
98#1744C2212yes—
99C22×#41C2312yes—
100C22×Q16C2314yes—
101C22×SD16C2314yes—
102#533—28no—
103#533—44no—
104#532—44no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2311yescharacteristic
109C2×C4⋊Q8C2311yescharacteristic
110C2×#127C2311yescharacteristic
111C2×#125—21no—
112C2×#125C2312yes—
113C2×#124—21no—
114C2×#124—21no—
115C2×#124—21no—
116C2×#124—21no—
117C2×#124C2312yes—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8—21no—
121C2×C22⋊Q8C2312yes—
122C2×#121—21no—
123C2×#121—21no—
124C2×C4×Q8—21no—
125C2×C4×Q8—21no—
126C2×C4×Q8—21no—
127C2×C4×Q8—21no—
128C2×C4×Q8—21no—
129C2×C4×Q8C2312yes—
130C2×C4×Q8C2312yes—
131C2×C4×Q8C2312yes—
132C2×C4×D4—21no—
133C2×C4×D4—21no—
134C2×#117—21no—
135C2×#117—21no—
136C2×#117C2312yes—
137C22×C42D411yescharacteristic
138C22×C42D411yescharacteristic
139C22×C42D411yescharacteristic
140C22×C42D411yescharacteristic
141C22×C42C2311yescharacteristic
142C22×C42C2311yescharacteristic
143C22×C42C2311yescharacteristic
144#477—44no—
145#475—44no—
146#462—44no—
147#460—44no—
148#452—216no—
149C2×#107C2314yes—
150C2×#106C2314yes—
151C2×C4⋊C8C2312yes—
152C2×C4⋊C8C2312yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#103C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×#102C2314yes—
161C2×C22⋊C8C2314yes—
162C2×C4×C8C2312yes—
163C2×C4×C8C2312yes—
164#391—21no—
165#391—21no—
166#391—21no—
167#391—21no—
168#391—21no—
169#391—21no—
170#391C2312yes—
171#391C2312yes—
172#391C2312yes—
173#391C2312yes—
174#391C2312yes—
175#391C2312yes—
176#390—21no—
177#390—21no—
178#390—21no—
179#390—21no—
180#390C2312yes—
181#390C2312yes—
182#390C2312yes—
183#390C2312yes—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389C2312yes—
191#389C2312yes—
192#388—22no—
193#388—22no—
194#388—22no—
195#388—22no—
196#388—22no—
197#388—22no—
198#388—22no—
199#388—22no—
200#387—22no—
201#387—22no—
202#387—22no—
203#387—22no—
204#386—22no—
205#386—22no—
206#386—22no—
207#386—22no—
208#385—22no—
209#385—22no—
210#384C2311yescharacteristic
211#384C2311yescharacteristic
212#382—21no—
213#382—21no—
214#382—21no—
215#382—21no—
216#382—21no—
217#380—21no—
218#380C2312yes—
219C4×C4⋊C4—21no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4—21no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—22no—
224C4×C4⋊C4—22no—
225C4×C4⋊C4—22no—
226C4×C4⋊C4—22no—
227C4×C4⋊C4—21no—
228C4×C4⋊C4—21no—
229C4×C4⋊C4—21no—
230C4×C4⋊C4—22no—
231C4×C4⋊C4—22no—
232C4×C4⋊C4C2312yes—
233C4×C4⋊C4C2312yes—
234C4×C4⋊C4C2312yes—
235C4×C22⋊C4—22no—
236C4×C22⋊C4—22no—
237C4×C22⋊C4—22no—
238C4×C22⋊C4—22no—
239C4×C22⋊C4—22no—
240C4×C22⋊C4—22no—
241#376—22no—
242C43—22no—
243C43D411yescharacteristic
244C43D411yescharacteristic
245C43D411yescharacteristic
246C43D411yescharacteristic
247C43C2311yescharacteristic
248C43C2311yescharacteristic
249C43C2311yescharacteristic
250C43C2311yescharacteristic
251#340C2314yes—
252#340C2314yes—
253#340C2314yes—
254#340C2314yes—
255#336C2314yes—
256#336C2312yes—
257#336C2312yes—
258#336C2314yes—
259#336C2314yes—
260C2×#41—28no—
261C2×#41—44no—
262C2×#41—42no—
263C2×#41—42no—
264C22×Q8—21no—
265C22×Q8—21no—
266C22×Q8—21no—
267C22×Q8C2412yes—
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
281C2×Q16—28no—
282C2×Q16—44no—
283C2×Q16—44no—
284C2×SD16—28no—
285C2×SD16—44no—
286C2×SD16—44no—
287C22×C8C2414yes—
288C22×C8C2414yes—
289C4⋊Q8—24no—
290C4⋊Q8—24no—
291C4⋊Q8—24no—
292C4⋊Q8—24no—
293C4⋊Q8—24no—
294#127—24no—
295#125—28no—
296#125—44no—
297#124—44no—
298#124—44no—
299#124—44no—
300#124—44no—
301#124—44no—
302C22⋊Q8—28no—
303C22⋊Q8—24no—
304C22⋊Q8—44no—
305C22⋊Q8—44no—
306C22⋊Q8#4118yes—
307#121—44no—
308#121—44no—
309C4×Q8—28no—
310C4×Q8—24no—
311C4×Q8—24no—
312C4×Q8—24no—
313C4×Q8—24no—
314C4×Q8—42no—
315C4×Q8—42no—
316C4×Q8—44no—
317C4×Q8—44no—
318C4×Q8—44no—
319C4×Q8—44no—
320C4×D4—44no—
321C4×D4—44no—
322#117—28no—
323#117—44no—
324#117—44no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—21no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
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—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4C2412yes—
345C2×C4⋊C4C2412yes—
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—22no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—22no—
362C2×C42—22no—
363C2×C42—24no—
364C2×C42—24no—
365C2×C42—24no—
366C2×C42—24no—
367C2×C42—22no—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—42no—
374C2×C42—42no—
375C2×C42—41no—
376C2×C42—41no—
377C2×C42C2×D411yescharacteristic
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×C42C2411yescharacteristic
400C2×C42C2411yescharacteristic
401C2×C42C2411yescharacteristic
402C2×C42C2411yescharacteristic
403C2×C42C2411yescharacteristic
404C2×C42C2411yescharacteristic
405#107—44no—
406#106—44no—
407C4⋊C8—24no—
408C4⋊C8—24no—
409#103—28no—
410#103—44no—
411#103—44no—
412#103—44no—
413#103—44no—
414#102—28no—
415#102—44no—
416#102—44no—
417C22⋊C8—28no—
418C4×C8—42no—
419C4×C8—42no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
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—
438C24—22no—
439C24C22×D411yescharacteristic
440#41—88no—
441#41—88no—
442C2×Q8—24no—
443C2×Q8—22no—
444C2×Q8—24no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—24no—
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×Q8—42no—
457C2×Q8—42no—
458C2×Q8—42no—
459C2×Q8—42no—
460C2×Q8C2×#4114yes—
461C2×D4—24no—
462C2×D4—24no—
463C2×D4—22no—
464C2×D4—22no—
465C2×D4—42no—
466C2×D4—42no—
467C2×D4—44no—
468C2×D4—44no—
469C2×D4—44no—
470C2×D4—42no—
471C2×D4—42no—
472C2×D4—42no—
473C2×D4—42no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—21no—
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—22no—
489C22×C4—22no—
490C22×C4—22no—
491C22×C4—22no—
492C22×C4—22no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—41no—
497C22×C4—41no—
498C22×C4—41no—
499C22×C4—41no—
500C22×C4—41no—
501C22×C4—41no—
502C22×C4—44no—
503C22×C4—44no—
504C22×C4—44no—
505C22×C4C22×D411yescharacteristic
506C22×C4C22×D411yescharacteristic
507C22×C4C22×D411yescharacteristic
508C22×C4C22×D411yescharacteristic
509C22×C4C22×D411yescharacteristic
510C22×C4C22×D411yescharacteristic
511C22×C4C22×D411yescharacteristic
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D411yescharacteristic
517C22×C4C22×D411yescharacteristic
518C22×C4C2×#4114yes—
519C22×C4C2511yescharacteristic
520Q16—88no—
521SD16—88no—
522C2×C8—24no—
523C2×C8—24no—
524C2×C8—24no—
525C2×C8—24no—
526C2×C8—24no—
527C2×C8—24no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—28no—
531C4⋊C4—24no—
532C4⋊C4—28no—
533C4⋊C4—22no—
534C4⋊C4—22no—
535C4⋊C4—22no—
536C4⋊C4—24no—
537C4⋊C4—24no—
538C4⋊C4—22no—
539C4⋊C4—22no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—44no—
546C4⋊C4—44no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—44no—
550C4⋊C4—42no—
551C4⋊C4—42no—
552C4⋊C4—44no—
553C4⋊C4—42no—
554C4⋊C4C2×#4114yes—
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—22no—
584C42—22no—
585C42—22no—
586C42—21no—
587C42—21no—
588C42—21no—
589C42—21no—
590C42—41no—
591C42—41no—
592C42—41no—
593C42—41no—
594C42—41no—
595C42—41no—
596C42—44no—
597C42—44no—
598C42—44no—
599C42—44no—
600C42—44no—
601C42—44no—
602C42#13514yes—
603C42#13714yes—
604C42H(5,𝔽2)14yes—
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—
618C23#52512yes—
619C23C23×D411yescharacteristic
620Q8—24no—
621Q8—24no—
622Q8—48no—
623Q8—42no—
624Q8—42no—
625Q8—44no—
626Q8—44no—
627Q8—84no—
628Q8—84no—
629D4—44no—
630D4—44no—
631D4—84no—
632D4—88no—
633D4—84no—
634C2×C4—28no—
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—24no—
652C2×C4—24no—
653C2×C4—21no—
654C2×C4—24no—
655C2×C4—24no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—21no—
659C2×C4—21no—
660C2×C4—21no—
661C2×C4—21no—
662C2×C4—21no—
663C2×C4—21no—
664C2×C4—21no—
665C2×C4—21no—
666C2×C4—21no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—44no—
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—42no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—88no—
692C2×C4#52512yes—
693C2×C4#53512yes—
694C2×C4#53512yes—
695C2×C4C2×#13512yes—
696C2×C4C2×#13512yes—
697C2×C4C2×#13512yes—
698C2×C4C2×#13712yes—
699C2×C4C2×#13712yes—
700C2×C4C2×#13712yes—
701C2×C4#57512yes—
702C2×C4#57512yes—
703C2×C4C22×#4112yes—
704C2×C4C2×H(5,𝔽2)12yes—
705C2×C4#58511yescharacteristic
706C2×C4#58511yescharacteristic
707C8—44no—
708C8—44no—
709C22—22no—
710C22—22no—
711C22—22no—
712C22—41no—
713C22—41no—
714C22—41no—
715C22—41no—
716C22—82no—
717C22—82no—
718C22—82no—
719C22—82no—
720C22—82no—
721C22—82no—
722C22—82no—
723C22C2×#52511yescharacteristic
724C22C2×#53511yescharacteristic
725C22#344911yescharacteristic
726C22C22×#13511yescharacteristic
727C22C22×#13711yescharacteristic
728C22C2×#57511yescharacteristic
729C22C2×#57511yescharacteristic
730C4—22no—
731C4—22no—
732C4—22no—
733C4—22no—
734C4—22no—
735C4—22no—
736C4—22no—
737C4—22no—
738C4—41no—
739C4—44no—
740C4—41no—
741C4—41no—
742C4—44no—
743C4—41no—
744C4—82no—
745C4—82no—
746C4—82no—
747C4—82no—
748C4—82no—
749C4#321014yes—
750C4#351412yes—
751C4#351412yes—
752C2—42no—
753C2—162no—
754C2#3512911yescharacteristic
755C2#6053611yescharacteristic
756C2#6056011yescharacteristic
757C2C2×#321011yescharacteristic
758C2#6080211yescharacteristic
759C2#6094811yescharacteristic
760C2#6096311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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
2. C4×#533C2semidirect
3. #33033C2semidirect
4. #33031C2semidirect
5. #30061C2semidirect
6. #30043C2semidirect
7. #29557C2semidirect
8. #29347C2semidirect
9. #29346C2semidirect
10. #28374C2semidirect
11. #28373C2semidirect
12. #27584C2semidirect
13. #24303C2semidirect
15. #23935C2semidirect
16. #22042C2semidirect
17. #20777C2semidirect

Maximal subgroups

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

subgroupindex
#366392
C4×#5332
#330332
#330312
#300612
#300432
#295572
#293472
#293462
#283742
#283732
#275842
#243032
#242132
#239352
#220422
#207772