Finite Group Databasebeta

← #6482038

Subgroups — #6482038

Order 512.

1868 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35240C211yesmaximal, characteristic
2C2×#2354C211yesmaximal, characteristic
3#32985C212yesmaximal, complemented
4#32985C212yesmaximal, complemented
5#29895C212yesmaximal, complemented
6#29893C212yesmaximal, complemented
7#26303C212yesmaximal, complemented
8#26301C212yesmaximal, complemented
9#26233C212yesmaximal, complemented
10#26231C212yesmaximal, complemented
11#25025C214yesmaximal, complemented
12#24980C214yesmaximal, complemented
13#21987C211yesmaximal, characteristic, complemented
14#21860C211yesmaximal, characteristic, complemented
15#21788C211yesmaximal, characteristic
16#21394C211yesmaximal, characteristic, complemented
17#21361C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×#117C2212yes—
21C2×#485C2214yescomplemented
22C2×#485C2214yescomplemented
23C2×#432C2211yescharacteristic
24C2×#431C2211yescharacteristic
25#2848C2212yes—
26#2848C2212yes—
27#2633—22no—
28#2547C2211yescharacteristic
29#2547C2211yescharacteristic
30#2542C2211yescharacteristic
31#2542C2211yescharacteristic
32#2536C2212yes—
33#2536C2212yes—
34#2502—21no—
35#2482—22no—
36#2435—21no—
37#2354—21no—
38#2354—21no—
39#2354—22no—
40#2354C2212yes—
41#2354C2212yes—
42#2354C2214yes—
43#2326—21no—
44#2326—21no—
45C2×#390—21no—
46C2×#388—22no—
47C2×#384C2211yescharacteristic
48#2283C2211yescharacteristic
49C2×#380C2212yes—
50C2×C4×C4⋊C4—21no—
51C2×C4×C4⋊C4C2211yescharacteristic
52#2077C2214yescomplemented
53#2077C2214yescomplemented
54#2076C2214yescomplemented
55#2076C2214yescomplemented
56#2036C2214yescomplemented
57#2036C2214yescomplemented
58#2035C2214yescomplemented
59#2035C2214yescomplemented
60#1990C2212yescomplemented
61#1988C2212yescomplemented
62#1981C2212yescomplemented
63#1956C2212yescomplemented
64#1955C2214yes—
65#1954C2212yes—
66#1954C2212yes—
67#1949C2212yescomplemented
68#1949C2212yescomplemented
69#1948C2212yescomplemented
70#1947C2212yescomplemented
71#1873C2214yescomplemented
72#1873C2214yescomplemented
73#1872C2212yes—
74#1872C2212yes—
75#1872C2212yes—
76#1872C2212yes—
77#1854C2214yescomplemented
78#1854C2214yescomplemented
79#1853C2211yescharacteristic
80#1853C2211yescharacteristic
81#1853C2212yes—
82#1852C2211yescharacteristic
83#1852C2211yescharacteristic
84#1852C2212yes—
85#1847C2212yescomplemented
86#1836C2212yescomplemented
87#1825C2214yescomplemented
88#1825C2214yescomplemented
89#1803C2214yescomplemented
90#1779C2212yes—
91#1779C2212yes—
92C4×#107C2212yescomplemented
93C4×#106C2212yescomplemented
94#1763C2214yescomplemented
95C23×D4D411yescharacteristic
96C24×C4C2312yes—
97#533—24no—
98#533—24no—
99C2×C4⋊Q8C2311yescharacteristic
100C2×C4⋊Q8C2311yescharacteristic
101C2×#125—21no—
102C2×#124—21no—
103C2×#124—21no—
104C2×C22⋊Q8—21no—
105C2×C22⋊Q8C2312yes—
106C2×C22⋊Q8C2312yes—
107C2×C22⋊Q8C2312yes—
108C2×C22⋊Q8C2312yes—
109C2×C22⋊Q8C2312yes—
110C2×C4×Q8—21no—
111C2×C4×Q8C2312yes—
112C2×C4×D4—24no—
113C2×C4×D4—24no—
114C2×C4×D4D411yescharacteristic
115C2×C4×D4D411yescharacteristic
116C2×C4×D4D412yes—
117C2×C4×D4C2311yescharacteristic
118C2×C4×D4C2311yescharacteristic
119C2×C4×D4C2312yes—
120C2×#117—22no—
121C2×#117—22no—
122C2×#117—21no—
123C2×#117—48no—
124C2×#117C2314yes—
125C2×#117C2314yes—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4D411yescharacteristic
129C22×C4⋊C4C2311yescharacteristic
130C22×C4⋊C4C2312yes—
131C22×C4⋊C4C2311yescharacteristic
132C22×C22⋊C4—22no—
133C22×C22⋊C4—22no—
134C22×C22⋊C4D412yes—
135C22×C42—22no—
136C22×C42—21no—
137C22×C42C2311yescharacteristic
138#485—48no—
139#485—48no—
140#432—24no—
141#431—24no—
142C2×#107C2312yes—
143C2×#107C2314yes—
144C2×#107C2312yes—
145C2×#106C2312yes—
146C2×#106C2314yes—
147C2×#106C2312yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2312yes—
150C2×#103C2314yes—
151C2×#103C2314yes—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×C22⋊C8C2311yescharacteristic
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2311yescharacteristic
157C2×C22⋊C8C2314yes—
158C2×C22⋊C8C2314yes—
159C2×C22⋊C8C2314yes—
160C2×C22⋊C8C2314yes—
161C2×#97C2312yes—
162C2×C4×C8C2312yes—
163#399—22no—
164#399—22no—
165#398—22no—
166#398—22no—
167#397—22no—
168#397—22no—
169#394—22no—
170#394—22no—
171#391—21no—
172#391—21no—
173#390—21no—
174#390—21no—
175#390—22no—
176#390—22no—
177#390—24no—
178#388—22no—
179#388—22no—
180#388—22no—
181#388—22no—
182#388—24no—
183#388—24no—
184#388—24no—
185#388—24no—
186#385—22no—
187#385—22no—
188#385—22no—
189#385—22no—
190#385—22no—
191#385—22no—
192#384—21no—
193#384—21no—
194#384—21no—
195#384—21no—
196#384C2312yes—
197#384C2312yes—
198#383—21no—
199#382C2311yescharacteristic
200#382C2311yescharacteristic
201#381—21no—
202#380—22no—
203#380—22no—
204#380C2312yes—
205#380C2312yes—
206#380C2314yes—
207#380C2314yes—
208#380C2312yes—
209#380C2312yes—
210#380C2311yescharacteristic
211#380C2311yescharacteristic
212#379—22no—
213#379—22no—
214C4×C4⋊C4—22no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—24no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—21no—
219C4×C4⋊C4C2312yes—
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2311yescharacteristic
222C4×C4⋊C4C2311yescharacteristic
223C4×C22⋊C4—21no—
224#376—21no—
225C2×#95—22no—
226C2×#95—21no—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#340C2312yes—
233#340C2314yes—
234#340C2312yes—
235#340C2312yes—
236#340C2314yes—
237C25C2×D412yes—
238C22×Q8—21no—
239C22×Q8C2412yes—
240C22×D4—24no—
241C22×D4—42no—
242C22×D4—42no—
243C22×D4C2×D411yescharacteristic
244C22×D4C2×D411yescharacteristic
245C22×D4C2×D412yes—
246C23×C4—21no—
247C23×C4—24no—
248C23×C4—24no—
249C23×C4—22no—
250C23×C4—21no—
251C23×C4—44no—
252C23×C4—44no—
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D412yes—
256C23×C4C2×D412yes—
257C23×C4C2×D412yes—
258C23×C4C2412yes—
259C23×C4C2412yes—
260C23×C4C2411yescharacteristic
261C22×C8C2412yes—
262C22×C8C2412yes—
263C22×C8C2414yes—
264C4⋊Q8—22no—
265C4⋊Q8—22no—
266C4⋊Q8—22no—
267C4⋊Q8—22no—
268#125—44no—
269#124—44no—
270#124—44no—
271C22⋊Q8—24no—
272C22⋊Q8—24no—
273C22⋊Q8—44no—
274C22⋊Q8—44no—
275C22⋊Q8—44no—
276C22⋊Q8—44no—
277C22⋊Q8—44no—
278C4×Q8—24no—
279C4×Q8—24no—
280C4×Q8—44no—
281C4×D4—24no—
282C4×D4—24no—
283C4×D4—28no—
284C4×D4—22no—
285C4×D4—22no—
286C4×D4—24no—
287C4×D4—24no—
288C4×D4—22no—
289C4×D4—22no—
290#117—28no—
291#117—28no—
292#117—44no—
293#117—44no—
294#117—48no—
295#117—48no—
296#117—44no—
297C2×C4⋊C4—24no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—42no—
316C2×C4⋊C4—44no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—82no—
319C2×C4⋊C4C2×D411yescharacteristic
320C2×C4⋊C4C2×D411yescharacteristic
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2414yes—
323C2×C4⋊C4C2412yes—
324C2×C4⋊C4C2412yes—
325C2×C4⋊C4C2411yescharacteristic
326C2×C4⋊C4C2411yescharacteristic
327C2×C4⋊C4C2411yescharacteristic
328C2×C4⋊C4C2411yescharacteristic
329C2×C4⋊C4C2412yes—
330C2×C22⋊C4—28no—
331C2×C22⋊C4—24no—
332C2×C22⋊C4—24no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—22no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—48no—
347C2×C22⋊C4—84no—
348C2×C22⋊C4C2×D412yes—
349C2×C22⋊C4C2×D412yes—
350C2×C22⋊C4C2×D412yes—
351C2×C22⋊C4C2×D412yes—
352C2×C42—24no—
353C2×C42—24no—
354C2×C42—22no—
355C2×C42—21no—
356C2×C42—21no—
357C2×C42—22no—
358C2×C42—22no—
359C2×C42—48no—
360C2×C42—44no—
361C2×C42C2×D411yescharacteristic
362C2×C42C2×D411yescharacteristic
363C2×C42#4114yes—
364C2×C42#4114yes—
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367#107—24no—
368#107—42no—
369#107—44no—
370#106—24no—
371#106—42no—
372#106—44no—
373C4⋊C8—24no—
374C4⋊C8—24no—
375#103—28no—
376#103—44no—
377#103—44no—
378#103—44no—
379C22⋊C8—22no—
380C22⋊C8—22no—
381C22⋊C8—24no—
382C22⋊C8—44no—
383C22⋊C8—44no—
384C22⋊C8—44no—
385C22⋊C8—44no—
386#97—24no—
387C4×C8—24no—
388#95—24no—
389#95—24no—
390#95—22no—
391#95—22no—
392#95—21no—
393#95—21no—
394#95—21no—
395#95—21no—
396#95—21no—
397#95—21no—
398#95—21no—
399#95—21no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404C24—24no—
405C24—24no—
406C24—24no—
407C24—42no—
408C24—42no—
409C24—42no—
410C24—42no—
411C24C22×D412yes—
412C24C22×D412yes—
413C24C22×D411yescharacteristic
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—24no—
417C2×Q8—24no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—42no—
421C2×D4—24no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—22no—
428C2×D4—42no—
429C2×D4—42no—
430C2×D4—416no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—416no—
465C22×C4—416no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—44no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—44no—
472C22×C4—42no—
473C22×C4—44no—
474C22×C4#12314yes—
475C22×C4#12314yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4C2511yescharacteristic
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—24no—
489C2×C8—24no—
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—24no—
493C2×C8—22no—
494C2×C8—22no—
495C4⋊C4—24no—
496C4⋊C4—24no—
497C4⋊C4—22no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—22no—
502C4⋊C4—24no—
503C4⋊C4—24no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—42no—
511C4⋊C4—44no—
512C4⋊C4—44no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4—44no—
516C4⋊C4—44no—
517C4⋊C4—42no—
518C4⋊C4—44no—
519C4⋊C4—82no—
520C4⋊C4—82no—
521C4⋊C4—82no—
522C4⋊C4—82no—
523C4⋊C4—82no—
524C4⋊C4—82no—
525C4⋊C4H(5,𝔽2)14yes—
526C22⋊C4—24no—
527C22⋊C4—24no—
528C22⋊C4—24no—
529C22⋊C4—24no—
530C22⋊C4—48no—
531C22⋊C4—48no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—42no—
538C22⋊C4—42no—
539C22⋊C4—84no—
540C22⋊C4—84no—
541C22⋊C4—82no—
542C22⋊C4—82no—
543C22⋊C4—82no—
544C22⋊C4—82no—
545C42—48no—
546C42—48no—
547C42—44no—
548C42—44no—
549C42—44no—
550C42—42no—
551C42—42no—
552C42#12314yes—
553C42#12314yes—
554C42C2×#4112yes—
555C42C2×#4112yes—
556C42C2×#4112yes—
557C42C2×#4112yes—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—22no—
563C23—22no—
564C23—22no—
565C23—22no—
566C23—22no—
567C23—22no—
568C23—22no—
569C23—22no—
570C23—22no—
571C23—22no—
572C23—22no—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—48no—
577C23—48no—
578C23—48no—
579C23—84no—
580C23—84no—
581C23—84no—
582C23—84no—
583C23C2×#12312yes—
584C23#57712yes—
585C23#57812yes—
586C23C23×D411yescharacteristic
587Q8—44no—
588Q8—44no—
589Q8—84no—
590D4—48no—
591D4—44no—
592D4—44no—
593C2×C4—28no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—28no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—48no—
613C2×C4—48no—
614C2×C4—48no—
615C2×C4—48no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—41no—
645C2×C4—41no—
646C2×C4—84no—
647C2×C4—88no—
648C2×C4C2×#12312yes—
649C2×C4C2×#12312yes—
650C2×C4C2×#12312yes—
651C2×C4C2×#12312yes—
652C2×C4C2×#12312yes—
653C2×C4#57512yes—
654C2×C4#57512yes—
655C2×C4C22×#4111yescharacteristic
656C2×C4C22×#4111yescharacteristic
657C2×C4C2×H(5,𝔽2)12yes—
658C2×C4#58512yes—
659C8—42no—
660C8—42no—
661C8—44no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—42no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22—816no—
683C22#309714yes—
684C22C22×#12311yescharacteristic
685C22#348411yescharacteristic
686C22#348511yescharacteristic
687C22C2×#57511yescharacteristic
688C22C2×#57511yescharacteristic
689C22C2×#57711yescharacteristic
690C22C2×#57811yescharacteristic
691C4—22no—
692C4—22no—
693C4—22no—
694C4—22no—
695C4—42no—
696C4—42no—
697C4—44no—
698C4—81no—
699C4—81no—
700C4—82no—
701C4—82no—
702C4—82no—
703C4—81no—
704C4—81no—
705C4—84no—
706C4#309314yes—
707C4#346012yes—
708C4#350612yes—
709C2—24no—
710C2—82no—
711C2—82no—
712C2#3515211yescharacteristic
713C2C2×#309311yescharacteristic
714C2C2×#309711yescharacteristic
715C2#6159511yescharacteristic
716C2#6159611yescharacteristic
717C2#6173011yescharacteristic
718C2#6173111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32985C2semidirect
4. #32985C2semidirect
5. #29895C2semidirect
6. #29893C2semidirect
7. #26303C2semidirect
8. #26301C2semidirect
9. #26233C2semidirect
10. #26231C2semidirect
11. #25025C2semidirect
12. #24980C2semidirect
13. #21987C2semidirect
14. #21860C2semidirect
16. #21394C2semidirect
17. #21361C2semidirect
21. C2×#485C22semidirect
22. C2×#485C22semidirect
52. #2077C22semidirect
53. #2077C22semidirect
54. #2076C22semidirect
55. #2076C22semidirect
56. #2036C22semidirect
57. #2036C22semidirect
58. #2035C22semidirect
59. #2035C22semidirect
60. #1990C22semidirect
61. #1988C22semidirect
62. #1981C22semidirect
63. #1956C22semidirect
67. #1949C22semidirect
68. #1949C22semidirect
69. #1948C22semidirect
70. #1947C22semidirect
71. #1873C22semidirect
72. #1873C22semidirect
77. #1854C22semidirect
78. #1854C22semidirect
85. #1847C22semidirect
86. #1836C22semidirect
87. #1825C22semidirect
88. #1825C22semidirect
89. #1803C22semidirect
92. C4×#107C22semidirect
93. C4×#106C22semidirect
94. #1763C22semidirect

Maximal subgroups

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

subgroupindex
#352402
C2×#23542
#329852
#298952
#298932
#263032
#263012
#262332
#262312
#250252
#249802
#219872
#218602
#217882
#213942
#213612