Finite Group Databasebeta

← #6489001

Subgroups — #6489001

Order 512.

1872 conjugacy classes of subgroups.

The subgroup lattice has 770 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic, complemented
2C4×#532C211yesmaximal, characteristic
3#32958C212yesmaximal, complemented
4#32907C212yesmaximal, complemented
5#30132C212yesmaximal, complemented
6#30128C212yesmaximal, complemented
7#29123C211yesmaximal, characteristic, complemented
8#29121C211yesmaximal, characteristic, complemented
9#28373C211yesmaximal, characteristic, complemented
10#28370C211yesmaximal, characteristic, complemented
11#28282C212yesmaximal, complemented
12#28254C212yesmaximal, complemented
13#27685C212yesmaximal, complemented
14#24505C214yesmaximal, complemented
15#24216C212yesmaximal
16#22150C214yesmaximal, complemented
17#20808C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C2×C4×#41C2212yes—
20C2×#477C2212yes—
21C2×#476C2212yes—
22C2×#464C2214yes—
23C2×#462C2212yes—
24C2×#461C2212yes—
25C2×#452C2214yes—
26C2×#431C2212yes—
27#2864—21no—
28#2863—21no—
29#2863C2212yes—
30#2862C2211yescharacteristic
31#2859C2211yescharacteristic
32#2857C2211yescharacteristic
33#2857C2211yescharacteristic
34#2855—21no—
35#2855C2212yes—
36#2851—21no—
37C4×C4⋊Q8C2211yescharacteristic
38C4×C4⋊Q8C2211yescharacteristic
39C4×C4⋊Q8C2211yescharacteristic
40C4×C4⋊Q8C2211yescharacteristic
41C4×#127C2211yescharacteristic
42C4×#127C2211yescharacteristic
43C4×#125—21no—
44C4×#124C2212yes—
45C4×#124C2212yes—
46C4×C22⋊Q8—21no—
47C4×C22⋊Q8C2212yes—
48C4×#121—21no—
49C4×#121—21no—
50C4×#121—21no—
51C4×#121—21no—
52#2297C2212yes—
53#2295—21no—
54#2295C2212yes—
55#2293—21no—
56#2293—21no—
57#2285C2211yescharacteristic
58#2283—21no—
59C42×D4—21no—
60C42×D4—21no—
61C2×C43C2211yescharacteristic
62#2100C2212yes—
63#2099C2212yes—
64#2096C2212yes—
65#2095C2212yes—
66#2067C2214yes—
67#2066C2214yes—
68#2056C2214yes—
69#2055C2214yes—
70#2043C2212yes—
71#2041C2212yes—
72#2016C2212yes—
73#2014C2212yes—
74#1983C2214yes—
75#1969C2214yes—
76#1968C2214yes—
77#1949C2214yes—
78#1934C2214yes—
79#1933C2214yes—
80#1878C2214yes—
81#1877C2214yes—
82#1873C2214yes—
83#1865C2214yes—
84#1862C2214yes—
85#1854C2214yes—
86#1842C2212yes—
87#1831C2212yes—
88#1805C2214yes—
89#1803C2212yes—
90#1803C2212yes—
91#1802C2212yes—
92#1802C2212yes—
93#1797C2212yes—
94#1797C2212yes—
95#1788C2214yes—
96C4×C4⋊C8C2212yes—
97C4×C4⋊C8C2211yescharacteristic
98C4×C4⋊C8C2211yescharacteristic
99C22×#41C2312yes—
100C22×Q16C2314yes—
101C22×SD16C2314yes—
102#532—28no—
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2311yescharacteristic
106C2×#127C2311yescharacteristic
107C2×#125—21no—
108C2×#124C2312yes—
109C2×C22⋊Q8—21no—
110C2×C22⋊Q8C2312yes—
111C2×#121—21no—
112C2×#121—21no—
113C4×#41—44no—
114C4×#41—44no—
115C2×C4×Q8—21no—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2312yes—
118C2×C4×Q8C2312yes—
119C2×C4×D4—22no—
120C2×C4×D4—22no—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4—21no—
125C2×C4×D4—21no—
126C2×C4×D4—21no—
127C2×C4×D4C2312yes—
128C2×C4×D4C2312yes—
129C2×#117—22no—
130C2×#117C2312yes—
131C22×C42—22no—
132C22×C42D411yescharacteristic
133C22×C42D411yescharacteristic
134C22×C42D411yescharacteristic
135C22×C42D411yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139#477—44no—
140#476—44no—
141#464—216no—
142#462—44no—
143#461—44no—
144#452—48no—
145#431—44no—
146C2×#107C2314yes—
147C2×#106C2314yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2312yes—
150C2×C4⋊C8C2314yes—
151C2×C4⋊C8C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156C2×#103C2314yes—
157C2×#102C2314yes—
158C2×#102C2314yes—
159C2×#102C2314yes—
160C2×C22⋊C8C2314yes—
161C2×C4×C8C2312yes—
162C2×C4×C8C2312yes—
163#391—21no—
164#391—21no—
165#391—21no—
166#391—21no—
167#391C2312yes—
168#391C2312yes—
169#391C2312yes—
170#391C2312yes—
171#391C2312yes—
172#391C2312yes—
173#390—21no—
174#390—21no—
175#390—21no—
176#390—21no—
177#390C2312yes—
178#390C2312yes—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389C2312yes—
192#389C2312yes—
193#389C2312yes—
194#389C2312yes—
195#388—22no—
196#388—22no—
197#388—22no—
198#388—22no—
199#387—22no—
200#387—22no—
201#387—22no—
202#387—22no—
203#387—22no—
204#387—22no—
205#387—22no—
206#387—22no—
207#386—22no—
208#386—22no—
209#385—22no—
210#385—22no—
211#384C2311yescharacteristic
212#384C2311yescharacteristic
213#384C2311yescharacteristic
214#384C2311yescharacteristic
215#383—22no—
216#382—21no—
217#382—21no—
218#382—21no—
219#382C2312yes—
220#381—22no—
221#380—21no—
222#380—21no—
223#380—21no—
224#380—21no—
225#380—21no—
226#380C2312yes—
227C4×C4⋊C4—21no—
228C4×C4⋊C4—21no—
229C4×C4⋊C4—22no—
230C4×C4⋊C4—22no—
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⋊C4C2312yes—
239C4×C22⋊C4—22no—
240C4×C22⋊C4—22no—
241C4×C22⋊C4—22no—
242C4×C22⋊C4—22no—
243C4×C22⋊C4—22no—
244C4×C22⋊C4—22no—
245C4×C22⋊C4—22no—
246#376—22no—
247C43D411yescharacteristic
248C43D411yescharacteristic
249C43D411yescharacteristic
250C43D411yescharacteristic
251C43C2311yescharacteristic
252C43C2311yescharacteristic
253C43C2311yescharacteristic
254C43C2311yescharacteristic
255#340C2314yes—
256#340C2314yes—
257#340C2314yes—
258#340C2314yes—
259#336C2312yes—
260#336C2312yes—
261#336C2314yes—
262C2×#41—24no—
263C2×#41—24no—
264C2×#41—24no—
265C2×#41—24no—
266C2×#41—42no—
267C2×#41—42no—
268C22×Q8—21no—
269C22×Q8C2412yes—
270C22×Q8C2412yes—
271C22×D4—22no—
272C22×D4—21no—
273C22×D4—21no—
274C22×D4C2412yes—
275C23×C4—22no—
276C23×C4—22no—
277C23×C4—22no—
278C23×C4C2×D411yescharacteristic
279C23×C4C2×D411yescharacteristic
280C23×C4C2×D411yescharacteristic
281C23×C4C2×D411yescharacteristic
282C23×C4C2×D411yescharacteristic
283C23×C4C2×D411yescharacteristic
284C23×C4C2411yescharacteristic
285C2×Q16—28no—
286C2×Q16—44no—
287C2×Q16—44no—
288C2×SD16—28no—
289C2×SD16—44no—
290C2×SD16—44no—
291C22×C8C2414yes—
292C22×C8C2414yes—
293C4⋊Q8—24no—
294C4⋊Q8—24no—
295C4⋊Q8—24no—
296#127—24no—
297#125—44no—
298#124—24no—
299#124#4118yes—
300C22⋊Q8—24no—
301C22⋊Q8—24no—
302C22⋊Q8—44no—
303#121—28no—
304#121—28no—
305C4×Q8—28no—
306C4×Q8—24no—
307C4×Q8—24no—
308C4×Q8—42no—
309C4×Q8—42no—
310C4×Q8—44no—
311C4×D4—28no—
312C4×D4—28no—
313C4×D4—24no—
314C4×D4—24no—
315C4×D4—42no—
316C4×D4—42no—
317C4×D4—44no—
318C4×D4—44no—
319C4×D4—48no—
320C4×D4—48no—
321C4×D4—44no—
322C4×D4—44no—
323#117—48no—
324#117—42no—
325#117—42no—
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—21no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—22no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—22no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—21no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
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×C42—22no—
364C2×C42—22no—
365C2×C42—24no—
366C2×C42—24no—
367C2×C42—24no—
368C2×C42—24no—
369C2×C42—22no—
370C2×C42—22no—
371C2×C42—22no—
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—42no—
375C2×C42—42no—
376C2×C42—48no—
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—28no—
408C4⋊C8—24no—
409C4⋊C8—24no—
410C4⋊C8—44no—
411#103—28no—
412#103—44no—
413#103—44no—
414#103—44no—
415#103—44no—
416#102—28no—
417#102—44no—
418#102—44no—
419C22⋊C8—28no—
420C4×C8—42no—
421C4×C8—42no—
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—
438#95—22no—
439#95—22no—
440#95—22no—
441#95—22no—
442C24—22no—
443C24C22×D411yescharacteristic
444#41—48no—
445#41—48no—
446#41—48no—
447#41—48no—
448C2×Q8—22no—
449C2×Q8—24no—
450C2×Q8—22no—
451C2×Q8—22no—
452C2×Q8—22no—
453C2×Q8—24no—
454C2×Q8—22no—
455C2×Q8—22no—
456C2×Q8—22no—
457C2×Q8—42no—
458C2×Q8—42no—
459C2×Q8—42no—
460C2×Q8C2×#4114yes—
461C2×D4—24no—
462C2×D4—24no—
463C2×D4—28no—
464C2×D4—22no—
465C2×D4—22no—
466C2×D4—22no—
467C2×D4—22no—
468C2×D4—22no—
469C2×D4—42no—
470C2×D4—42no—
471C2×D4—44no—
472C2×D4—44no—
473C2×D4—42no—
474C2×D4—42no—
475C2×D4C2×#4114yes—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—28no—
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—22no—
497C22×C4—22no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—41no—
501C22×C4—41no—
502C22×C4—41no—
503C22×C4—41no—
504C22×C4—44no—
505C22×C4—44no—
506C22×C4—44no—
507C22×C4—44no—
508C22×C4—44no—
509C22×C4—44no—
510C22×C4—44no—
511C22×C4—44no—
512C22×C4C22×D411yescharacteristic
513C22×C4C22×D411yescharacteristic
514C22×C4C22×D411yescharacteristic
515C22×C4C22×D411yescharacteristic
516C22×C4C22×D411yescharacteristic
517C22×C4C22×D411yescharacteristic
518C22×C4C22×D411yescharacteristic
519C22×C4C22×D411yescharacteristic
520C22×C4C22×D411yescharacteristic
521C22×C4C22×D411yescharacteristic
522C22×C4C22×D411yescharacteristic
523C22×C4C22×D411yescharacteristic
524C22×C4C22×D411yescharacteristic
525C22×C4H(5,𝔽2)12yes—
526C22×C42−1+412yes—
527C22×C4C2511yescharacteristic
528Q16—88no—
529SD16—88no—
530C2×C8—24no—
531C2×C8—24no—
532C2×C8—24no—
533C2×C8—24no—
534C2×C8—24no—
535C2×C8—24no—
536C4⋊C4—24no—
537C4⋊C4—24no—
538C4⋊C4—24no—
539C4⋊C4—22no—
540C4⋊C4—24no—
541C4⋊C4—22no—
542C4⋊C4—22no—
543C4⋊C4—22no—
544C4⋊C4—22no—
545C4⋊C4—22no—
546C4⋊C4—44no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—42no—
551C4⋊C4—44no—
552C4⋊C4—44no—
553C4⋊C4—42no—
554C4⋊C4—42no—
555C4⋊C4—42no—
556C4⋊C4—44no—
557C4⋊C4—42no—
558C4⋊C4—44no—
559C4⋊C4—44no—
560C4⋊C4—42no—
561C4⋊C4—42no—
562C4⋊C4—42no—
563C4⋊C4—44no—
564C4⋊C4—44no—
565C22⋊C4—28no—
566C22⋊C4—28no—
567C22⋊C4—44no—
568C22⋊C4—44no—
569C22⋊C4—44no—
570C22⋊C4—44no—
571C22⋊C4—44no—
572C22⋊C4—44no—
573C22⋊C4—44no—
574C22⋊C4—44no—
575C22⋊C4—44no—
576C42—22no—
577C42—22no—
578C42—22no—
579C42—22no—
580C42—22no—
581C42—22no—
582C42—22no—
583C42—22no—
584C42—22no—
585C42—22no—
586C42—22no—
587C42—22no—
588C42—22no—
589C42—22no—
590C42—22no—
591C42—22no—
592C42—22no—
593C42—21no—
594C42—21no—
595C42—22no—
596C42—21no—
597C42—21no—
598C42—41no—
599C42—41no—
600C42—41no—
601C42—41no—
602C42—44no—
603C42—44no—
604C42—44no—
605C42—44no—
606C42—44no—
607C42#13514yes—
608C42#13714yes—
609C42C2×#4114yes—
610C42H(5,𝔽2)14yes—
611C23—21no—
612C23—21no—
613C23—21no—
614C23—21no—
615C23—21no—
616C23—21no—
617C23—24no—
618C23—42no—
619C23—42no—
620C23—42no—
621C23—42no—
622C23—42no—
623C23—42no—
624C23#53712yes—
625C23C23×D411yescharacteristic
626Q8—24no—
627Q8—24no—
628Q8—42no—
629Q8—42no—
630Q8—44no—
631Q8—44no—
632Q8—84no—
633D4—24no—
634D4—24no—
635D4—48no—
636D4—48no—
637D4—416no—
638D4—42no—
639D4—42no—
640C2×C4—28no—
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—21no—
654C2×C4—21no—
655C2×C4—21no—
656C2×C4—21no—
657C2×C4—21no—
658C2×C4—21no—
659C2×C4—21no—
660C2×C4—21no—
661C2×C4—24no—
662C2×C4—24no—
663C2×C4—24no—
664C2×C4—21no—
665C2×C4—21no—
666C2×C4—21no—
667C2×C4—21no—
668C2×C4—21no—
669C2×C4—21no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—416no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—44no—
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—42no—
692C2×C4—42no—
693C2×C4—42no—
694C2×C4—42no—
695C2×C4—42no—
696C2×C4—88no—
697C2×C4—88no—
698C2×C4#52512yes—
699C2×C4#52512yes—
700C2×C4#53512yes—
701C2×C4#53512yes—
702C2×C4#53712yes—
703C2×C4C2×#13512yes—
704C2×C4C2×#13512yes—
705C2×C4C2×#13512yes—
706C2×C4C2×#13712yes—
707C2×C4C2×#13712yes—
708C2×C4C2×#13712yes—
709C2×C4#57512yes—
710C2×C4#57512yes—
711C2×C4C22×#4112yes—
712C2×C4C2×H(5,𝔽2)11yescharacteristic
713C2×C4C2×H(5,𝔽2)12yes—
714C2×C4C2×2−1+411yescharacteristic
715C8—44no—
716C8—44no—
717C22—22no—
718C22—22no—
719C22—22no—
720C22—41no—
721C22—41no—
722C22—41no—
723C22—41no—
724C22—44no—
725C22—44no—
726C22—44no—
727C22—82no—
728C22—82no—
729C22—82no—
730C22—82no—
731C22C2×#52511yescharacteristic
732C22C2×#53511yescharacteristic
733C22C2×#53711yescharacteristic
734C22C22×#13511yescharacteristic
735C22C22×#13711yescharacteristic
736C22C2×#57511yescharacteristic
737C22C2×#57511yescharacteristic
738C4—22no—
739C4—22no—
740C4—22no—
741C4—22no—
742C4—22no—
743C4—22no—
744C4—22no—
745C4—22no—
746C4—22no—
747C4—22no—
748C4—41no—
749C4—44no—
750C4—41no—
751C4—82no—
752C4—82no—
753C4—82no—
754C4—82no—
755C4—82no—
756C4—82no—
757C4#321014yes—
758C4#351912yes—
759C4#352012yes—
760C2—42no—
761C2—84no—
762C2#3523611yescharacteristic
763C2#6053611yescharacteristic
764C2#6056211yescharacteristic
765C2C2×#321011yescharacteristic
766C2#6080511yescharacteristic
767C2#6090011yescharacteristic
768C2#6091811yescharacteristic
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
1. #36635C2semidirect
3. #32958C2semidirect
4. #32907C2semidirect
5. #30132C2semidirect
6. #30128C2semidirect
7. #29123C2semidirect
8. #29121C2semidirect
9. #28373C2semidirect
10. #28370C2semidirect
11. #28282C2semidirect
12. #28254C2semidirect
13. #27685C2semidirect
14. #24505C2semidirect
16. #22150C2semidirect
17. #20808C2semidirect

Maximal subgroups

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

subgroupindex
#366352
C4×#5322
#329582
#329072
#301322
#301282
#291232
#291212
#283732
#283702
#282822
#282542
#276852
#245052
#242162
#221502
#208082