Finite Group Databasebeta

← #6434740

Subgroups — #6434740

Order 512.

1784 conjugacy classes of subgroups.

The subgroup lattice has 731 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36639C211yesmaximal, characteristic
2C4×#533C211yesmaximal, characteristic, complemented
3#30067C212yesmaximal, complemented
4#30066C212yesmaximal, complemented
5#30065C214yesmaximal, complemented
6#30064C212yesmaximal, complemented
7#30063C211yesmaximal, characteristic, complemented
8#30062C211yesmaximal, characteristic, complemented
9#30061C212yesmaximal, complemented
10#30060C211yesmaximal, characteristic, complemented
11#30059C211yesmaximal, characteristic, complemented
12#23730C212yesmaximal, complemented
13#23551C212yesmaximal, complemented
14#23504C212yesmaximal
15#23451C212yesmaximal, complemented
16#21058C214yesmaximal, complemented
17C2×#1839C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C2×#533C2211yescharacteristic
20C2×#532C2211yescharacteristic
21C2×#427C2214yes—
22C2×#416C2214yes—
23C2×#416C2214yes—
24#2862C2211yescharacteristic
25#2862C2211yescharacteristic
26#2861C2212yes—
27#2861C2212yes—
28#2858C2211yescharacteristic
29#2858C2211yescharacteristic
30#2850—21no—
31#2850—21no—
32#2850—21no—
33#2850—21no—
34C4×C4⋊Q8C2211yescharacteristic
35C4×C4⋊Q8C2211yescharacteristic
36C4×C4⋊Q8C2211yescharacteristic
37C4×C4⋊Q8C2211yescharacteristic
38C4×#125C2212yes—
39C4×#125C2212yes—
40C4×#124—21no—
41C4×#124—21no—
42C4×#124—21no—
43C4×#124—21no—
44C4×C22⋊Q8—21no—
45C4×C22⋊Q8—21no—
46C4×C22⋊Q8—21no—
47C4×C22⋊Q8—21no—
48#2297C2212yes—
49#2297C2212yes—
50#2296C2212yes—
51#2295—21no—
52#2295—21no—
53#2285C2211yescharacteristic
54C42×Q8—21no—
55C42×Q8—21no—
56C4×#117C2212yes—
57C2×C43C2211yescharacteristic
58#2097C2214yes—
59#2096C2212yes—
60#2095C2212yes—
61#2094C2212yes—
62#2093C2212yes—
63#2092C2214yes—
64#2059C2214yes—
65#2059C2214yes—
66#2058C2212yes—
67#2058C2212yes—
68#2057C2212yes—
69#2057C2212yes—
70#2043C2212yes—
71#2043C2212yes—
72#2042C2214yes—
73#2042C2214yes—
74#2041C2212yes—
75#2041C2212yes—
76#2026C2214yes—
77#2025C2214yes—
78#2024C2212yes—
79#2024C2212yes—
80#2023C2212yes—
81#2023C2212yes—
82#1928C2214yes—
83#1928C2214yes—
84#1927C2214yes—
85#1927C2214yes—
86#1926C2214yes—
87#1926C2214yes—
88#1925C2214yes—
89#1925C2214yes—
90#1839C2211yescharacteristic
91#1839C2211yescharacteristic
92#1839C2212yes—
93#1839C2212yes—
94#1839C2212yes—
95#1735C2214yes—
96C2×#336C2212yes—
97C2×#336C2212yes—
98C2×#336C2212yes—
99C22×#41C2312yes—
100C23×C8C2314yes—
101#533—28no—
102#533—44no—
103#532—44no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2311yescharacteristic
108C2×C4⋊Q8C2311yescharacteristic
109C2×#127C2311yescharacteristic
110C2×#125C2312yes—
111C2×#125C2312yes—
112C2×#124—21no—
113C2×#124—21no—
114C2×#124—21no—
115C2×#124—21no—
116C2×#124C2312yes—
117C2×C22⋊Q8—21no—
118C2×C22⋊Q8—21no—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8—21no—
121C2×#121—21no—
122C2×#121—21no—
123C2×C4×Q8—21no—
124C2×C4×Q8—21no—
125C2×C4×Q8—21no—
126C2×C4×Q8—21no—
127C2×C4×Q8—21no—
128C2×C4×Q8—21no—
129C2×C4×Q8—21no—
130C2×C4×Q8—21no—
131C2×C4×D4—21no—
132C2×C4×D4—21no—
133C2×#117C2312yes—
134C2×#117C2312yes—
135C2×#117C2312yes—
136C22×C42D411yescharacteristic
137C22×C42D411yescharacteristic
138C22×C42D411yescharacteristic
139C22×C42D411yescharacteristic
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142C22×C42C2311yescharacteristic
143#427—48no—
144C2×#107C2314yes—
145C2×#107C2314yes—
146C2×#106C2314yes—
147C2×#106C2314yes—
148#416—216no—
149#416—48no—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×#102C2314yes—
157C2×#102C2314yes—
158#391—21no—
159#391—21no—
160#391—21no—
161#391—21no—
162#391—21no—
163#391—21no—
164#391—21no—
165#391—21no—
166#391C2312yes—
167#391C2312yes—
168#391C2312yes—
169#391C2312yes—
170#390—21no—
171#390—21no—
172#390—21no—
173#390—21no—
174#390C2312yes—
175#390C2312yes—
176#390C2312yes—
177#390C2312yes—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389C2312yes—
183#389C2312yes—
184#389C2312yes—
185#389C2312yes—
186#388—22no—
187#388—22no—
188#388—22no—
189#388—22no—
190#388—22no—
191#388—22no—
192#388—22no—
193#388—22no—
194#387—22no—
195#387—22no—
196#387—22no—
197#387—22no—
198#386—22no—
199#386—22no—
200#386—22no—
201#386—22no—
202#385—22no—
203#385—22no—
204#384C2311yescharacteristic
205#384C2311yescharacteristic
206#382—21no—
207#382—21no—
208#382—21no—
209#382—21no—
210#382C2312yes—
211#380—21no—
212#380—21no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
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⋊C4C2312yes—
226C4×C4⋊C4C2312yes—
227C4×C4⋊C4C2312yes—
228C4×C4⋊C4C2312yes—
229C4×C22⋊C4—22no—
230C4×C22⋊C4—22no—
231C4×C22⋊C4—22no—
232C4×C22⋊C4—22no—
233C4×C22⋊C4—22no—
234C4×C22⋊C4—22no—
235#376—22no—
236C43—22no—
237C43D411yescharacteristic
238C43D411yescharacteristic
239C43D411yescharacteristic
240C43D411yescharacteristic
241C43C2311yescharacteristic
242C43C2311yescharacteristic
243C43C2311yescharacteristic
244C43C2311yescharacteristic
245#340C2314yes—
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#336C2312yes—
250#336C2312yes—
251#336C2314yes—
252#336C2314yes—
253#336C2312yes—
254#336C2312yes—
255#336C2312yes—
256#336C2314yes—
257#336C2312yes—
258C2×#41—28no—
259C2×#41—28no—
260C2×#41—28no—
261C22×Q8—21no—
262C22×Q8—21no—
263C22×Q8—21no—
264C22×Q8—21no—
265C22×Q8C2412yes—
266C22×D4—22no—
267C22×D4—21no—
268C22×D4—21no—
269C22×D4C2412yes—
270C23×C4—22no—
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D411yescharacteristic
276C23×C4C2×D411yescharacteristic
277C23×C4C2411yescharacteristic
278C22×C8—28no—
279C22×C8—28no—
280C22×C8—28no—
281C22×C8C2414yes—
282C22×C8C2414yes—
283C4⋊Q8—24no—
284C4⋊Q8—24no—
285C4⋊Q8—42no—
286C4⋊Q8—42no—
287C4⋊Q8—42no—
288#127—42no—
289#125—24no—
290#125—24no—
291#125—44no—
292#124—44no—
293#124—44no—
294#124—44no—
295#124—44no—
296#124—44no—
297C22⋊Q8—28no—
298C22⋊Q8—28no—
299C22⋊Q8—44no—
300C22⋊Q8—44no—
301#121—44no—
302#121—44no—
303C4×Q8—28no—
304C4×Q8—28no—
305C4×Q8—44no—
306C4×Q8—44no—
307C4×Q8—44no—
308C4×Q8—44no—
309C4×Q8—44no—
310C4×Q8—44no—
311C4×D4—44no—
312C4×D4—44no—
313#117—24no—
314#117—28no—
315#117—28no—
316#117#4118yes—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C42—22no—
355C2×C42—24no—
356C2×C42—24no—
357C2×C42—24no—
358C2×C42—24no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—42no—
366C2×C42—42no—
367C2×C42—42no—
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
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×C42C2411yescharacteristic
391C2×C42C2411yescharacteristic
392C2×C42C2411yescharacteristic
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395C2×C42C2411yescharacteristic
396#107—44no—
397#107—44no—
398#106—44no—
399#106—44no—
400#103—28no—
401#103—28no—
402#103—44no—
403#103—44no—
404#102—28no—
405#102—28no—
406#102—44no—
407#102—44no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
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#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426C24—22no—
427C24C22×D411yescharacteristic
428#41—432no—
429C2×Q8—24no—
430C2×Q8—24no—
431C2×Q8—24no—
432C2×Q8—24no—
433C2×Q8—24no—
434C2×Q8—42no—
435C2×Q8—42no—
436C2×Q8—42no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×Q8—42no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—42no—
443C2×Q8—42no—
444C2×D4—24no—
445C2×D4—24no—
446C2×D4—24no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—42no—
450C2×D4—42no—
451C2×D4—44no—
452C2×D4—44no—
453C2×D4—44no—
454C2×D4—42no—
455C2×D4—42no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—41no—
476C22×C4—41no—
477C22×C4—41no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D411yescharacteristic
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C2×#4114yes—
498C22×C4H(5,𝔽2)14yes—
499C22×C42−1+414yes—
500C22×C4C2511yescharacteristic
501C2×C8—24no—
502C2×C8—24no—
503C2×C8—24no—
504C2×C8—24no—
505C2×C8—24no—
506C2×C8—24no—
507C2×C8—416no—
508C4⋊C4—28no—
509C4⋊C4—28no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—24no—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—24no—
518C4⋊C4—42no—
519C4⋊C4—42no—
520C4⋊C4—42no—
521C4⋊C4—42no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—44no—
529C4⋊C4—42no—
530C4⋊C4—44no—
531C4⋊C4—42no—
532C4⋊C4C2×#4114yes—
533C4⋊C4C2×#4114yes—
534C22⋊C4—28no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—44no—
539C22⋊C4—44no—
540C22⋊C4—44no—
541C22⋊C4—44no—
542C22⋊C4—44no—
543C22⋊C4—44no—
544C22⋊C4—44no—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C42—22no—
548C42—22no—
549C42—28no—
550C42—22no—
551C42—22no—
552C42—22no—
553C42—22no—
554C42—22no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—41no—
570C42—41no—
571C42—41no—
572C42—41no—
573C42—41no—
574C42—41no—
575C42—44no—
576C42—44no—
577C42—44no—
578C42—44no—
579C42—44no—
580C42—44no—
581C42#13514yes—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—42no—
591C23—42no—
592C23—42no—
593C23#52512yes—
594C23#53512yes—
595C23#53612yes—
596C23C23×D411yescharacteristic
597Q8—48no—
598Q8—48no—
599Q8—48no—
600Q8—84no—
601Q8—84no—
602D4—48no—
603D4—84no—
604D4—84no—
605D4—88no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—24no—
621C2×C4—24no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—21no—
632C2×C4—21no—
633C2×C4—21no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—88no—
664C2×C4#52512yes—
665C2×C4#53512yes—
666C2×C4#53612yes—
667C2×C4#542116yes—
668C2×C4C2×#13512yes—
669C2×C4C2×#13512yes—
670C2×C4C2×#13512yes—
671C2×C4#57512yes—
672C2×C4#57512yes—
673C2×C4#57512yes—
674C2×C4C22×#4112yes—
675C2×C4C2×H(5,𝔽2)12yes—
676C2×C4C2×2−1+412yes—
677C8—44no—
678C8—44no—
679C22—22no—
680C22—22no—
681C22—22no—
682C22—22no—
683C22—22no—
684C22—22no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22#263214yes—
693C22C2×#52511yescharacteristic
694C22C2×#53511yescharacteristic
695C22C2×#53611yescharacteristic
696C22C22×#13511yescharacteristic
697C22C2×#57511yescharacteristic
698C22C2×#57511yescharacteristic
699C22C2×#57511yescharacteristic
700C4—22no—
701C4—22no—
702C4—22no—
703C4—22no—
704C4—22no—
705C4—22no—
706C4—41no—
707C4—41no—
708C4—41no—
709C4—44no—
710C4—41no—
711C4—44no—
712C4—41no—
713C4—41no—
714C4—82no—
715C4—82no—
716C4—82no—
717C4—82no—
718C4—82no—
719C4#263214yes—
720C4C2×#54214yes—
721C2—24no—
722C2—162no—
723C2C2×#263211yescharacteristic
724C2#6053611yescharacteristic
725C2#6054611yescharacteristic
726C2#6078211yescharacteristic
727C2#6079011yescharacteristic
728C2#6084611yescharacteristic
729C2#6085411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ 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. #30067C2semidirect
4. #30066C2semidirect
5. #30065C2semidirect
6. #30064C2semidirect
7. #30063C2semidirect
8. #30062C2semidirect
9. #30061C2semidirect
10. #30060C2semidirect
11. #30059C2semidirect
12. #23730C2semidirect
13. #23551C2semidirect
15. #23451C2semidirect
16. #21058C2semidirect
17. C2×#1839C2semidirect

Maximal subgroups

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

subgroupindex
#366392
C4×#5332
#300672
#300662
#300652
#300642
#300632
#300622
#300612
#300602
#300592
#237302
#235512
#235042
#234512
#210582
C2×#18392