Finite Group Databasebeta

← #6501951

Subgroups — #6501951

Order 512.

1932 conjugacy classes of subgroups.

The subgroup lattice has 723 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2420C211yesmaximal, characteristic
2#34212C211yesmaximal, characteristic, complemented
3#28123C212yesmaximal, complemented
4#27686C212yesmaximal, complemented
5#27603C211yesmaximal, characteristic, complemented
6#27178C214yesmaximal, complemented
7#26676C214yesmaximal, complemented
8#26567C214yesmaximal, complemented
9#24914C214yesmaximal, complemented
10#24006C211yesmaximal, characteristic, complemented
11#24005C211yesmaximal, characteristic
12#23919C211yesmaximal, characteristic
13#22085C212yesmaximal, complemented
14#21906C211yesmaximal, characteristic, complemented
15#21453C212yesmaximal, complemented
16C22×C22⋊Q8C2211yescharacteristic
17C22×#121C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C2×#484C2212yes—
20C2×#481C2212yes—
21C2×#448C2212yes—
22C2×#450C2212yes—
23C2×#440C2214yescomplemented
24#2420—22no—
25#2420—22no—
26#2420C2214yes—
27#2420C2214yes—
28C2×#397—21no—
29C2×#394—21no—
30#2368—22no—
31#2360C2212yescomplemented
32#2360C2212yescomplemented
33#2345—22no—
34#2345C2212yescomplemented
35Q8×C22⋊C4C2212yescomplemented
36#2329—21no—
37#2329—21no—
38#2315—21no—
39C4×C22⋊Q8C2212yes—
40C4×C22⋊Q8C2212yes—
41C2×#390C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#386—21no—
44C2×#382C2211yescharacteristic
45C2×#381—21no—
46C2×#380C2211yescharacteristic
47C2×#380C2211yescharacteristic
48#2273—21no—
49C2×C4×C22⋊C4C2211yescharacteristic
50#2101C2214yescomplemented
51#2097C2214yescomplemented
52#2076C2214yescomplemented
53#2067C2214yescomplemented
54#2055C2214yescomplemented
55#2036C2214yescomplemented
56#2034C2214yes—
57#2000C2214yes—
58#1949C2214yescomplemented
59#1943C2214yescomplemented
60#1935C2214yescomplemented
61#1925C2214yescomplemented
62#1924C2214yescomplemented
63#1924C2214yescomplemented
64#1918C2214yescomplemented
65#1880C2214yescomplemented
66#1877C2214yescomplemented
67#1874C2212yes—
68#1874C2212yes—
69#1872C2212yes—
70#1872C2212yes—
71#1865C2214yescomplemented
72#1862C2214yescomplemented
73#1853C2212yes—
74#1852C2212yes—
75#1852C2212yescomplemented
76#1852C2212yescomplemented
77#1824C2212yes—
78#1824C2212yescomplemented
79#1805C2214yescomplemented
80#1797C2212yes—
81#1787C2212yescomplemented
82#1786C2212yes—
83#1779C2211yescharacteristic
84#1779C2211yescharacteristic
85#1762C2214yescomplemented
86#1749C2212yes—
87C23×Q8C2311yescharacteristic
88C23×D4—21no—
89C23×D4C2311yescharacteristic
90C24×C4—21no—
91C24×C4C2311yescharacteristic
92C22×SD16C2314yes—
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8—21no—
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8C2312yes—
97C2×C22⋊Q8C2312yes—
98C2×#121—21no—
99C2×#121—21no—
100C2×#121—44no—
101C2×#121C2312yes—
102C2×#121C2312yes—
103C2×C4×Q8C2312yes—
104C2×C4×Q8C2312yes—
105C2×C4×D4—22no—
106C2×C4×D4—44no—
107C2×C4×D4C2314yes—
108C22×C4⋊C4—21no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C22⋊C4—21no—
117C22×C22⋊C4—21no—
118C22×C22⋊C4—21no—
119C22×C22⋊C4—21no—
120C22×C22⋊C4—21no—
121C22×C22⋊C4—21no—
122C22×C22⋊C4—21no—
123C22×C22⋊C4D411yescharacteristic
124C22×C22⋊C4D411yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127#484—28no—
128#481—44no—
129#450—28no—
130#448—44no—
131#440—48no—
132C2×#107C2314yes—
133C2×#107C2314yes—
134C2×#106C2314yes—
135C2×C4⋊C8C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#102C2314yes—
140C2×#102C2314yes—
141C2×#102C2314yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145C2×C22⋊C8C2312yes—
146C2×C22⋊C8C2312yes—
147C2×C22⋊C8C2312yes—
148C2×#97C2314yes—
149#397—24no—
150#397—24no—
151#394—24no—
152#394—24no—
153#391—22no—
154#391—22no—
155#391C2312yes—
156#391C2312yes—
157#390—22no—
158#390C2314yes—
159#389—22no—
160#389—22no—
161#389—22no—
162#389C2312yes—
163#389C2312yes—
164#389C2314yes—
165#387—22no—
166#387—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#386—24no—
172#386—24no—
173#385—22no—
174#385—22no—
175#385—22no—
176#385—22no—
177#382—22no—
178#382C2314yes—
179#381—24no—
180#381—24no—
181#380—21no—
182#380—21no—
183#380—21no—
184#380—21no—
185#380—22no—
186#380—22no—
187#380C2312yes—
188#380C2312yes—
189#380C2312yes—
190#380C2312yes—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4C2312yes—
194C4×C4⋊C4C2312yes—
195C4×C22⋊C4—22no—
196C4×C22⋊C4—22no—
197C4×C22⋊C4—22no—
198C4×C22⋊C4—22no—
199C4×C22⋊C4—22no—
200C4×C22⋊C4—22no—
201C4×C22⋊C4D412yes—
202C4×C22⋊C4D412yes—
203C4×C22⋊C4C2312yes—
204C4×C22⋊C4C2312yes—
205C2×#95—21no—
206C2×#95—21no—
207C2×#95—21no—
208C2×#95—21no—
209C2×#95—21no—
210C2×#95D411yescharacteristic
211C2×#95D411yescharacteristic
212#340C2314yes—
213#340C2314yes—
214#340C2314yes—
215#340C2314yes—
216#340C2314yes—
217#340C2314yes—
218#336C2314yes—
219#336C2314yes—
220C25—21no—
221C25C2×D411yescharacteristic
222C22×Q8—24no—
223C22×Q8—21no—
224C22×Q8—42no—
225C22×Q8—42no—
226C22×Q8C2412yes—
227C22×D4—24no—
228C22×D4—22no—
229C22×D4—22no—
230C22×D4—21no—
231C22×D4—24no—
232C22×D4—42no—
233C22×D4—44no—
234C22×D4—44no—
235C22×D4—82no—
236C22×D4C2412yes—
237C23×C4—21no—
238C23×C4—21no—
239C23×C4—24no—
240C23×C4—21no—
241C23×C4—21no—
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4—24no—
246C23×C4—42no—
247C23×C4—42no—
248C23×C4—44no—
249C23×C4—44no—
250C23×C4—82no—
251C23×C4C2×D411yescharacteristic
252C23×C4C2×D411yescharacteristic
253C23×C4C2×D411yescharacteristic
254C23×C4C2×D411yescharacteristic
255C23×C4C2×D411yescharacteristic
256C23×C4C2×D411yescharacteristic
257C23×C4C2×D411yescharacteristic
258C23×C4C2411yescharacteristic
259C23×C4C2411yescharacteristic
260C23×C4C2411yescharacteristic
261C2×SD16—28no—
262C2×SD16—28no—
263C2×SD16—44no—
264C22×C8C2414yes—
265C22×C8C2414yes—
266C22⋊Q8—28no—
267C22⋊Q8—28no—
268C22⋊Q8—24no—
269C22⋊Q8—24no—
270C22⋊Q8—24no—
271C22⋊Q8#4118yes—
272#121—24no—
273#121—24no—
274#121—42no—
275#121—42no—
276#121—44no—
277#121—44no—
278C4×Q8—44no—
279C4×Q8—44no—
280C4×D4—28no—
281C4×D4—44no—
282C4×D4—48no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—24no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—22no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—42no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4—42no—
303C2×C4⋊C4—82no—
304C2×C4⋊C4C2412yes—
305C2×C4⋊C4C2412yes—
306C2×C4⋊C4C2412yes—
307C2×C4⋊C4C2412yes—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C22⋊C4—28no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C22⋊C4—24no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—42no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—82no—
328C2×C22⋊C4—82no—
329C2×C22⋊C4—82no—
330C2×C22⋊C4C2×D412yes—
331C2×C22⋊C4C2×D412yes—
332C2×C22⋊C4C2×D412yes—
333C2×C22⋊C4C2×D412yes—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—22no—
337C2×C42—22no—
338C2×C42—42no—
339C2×C42—42no—
340C2×C42C2×D412yes—
341C2×C42C2×D412yes—
342C2×C42C2412yes—
343C2×C42C2412yes—
344#107—28no—
345#107—44no—
346#106—44no—
347C4⋊C8—44no—
348#103—28no—
349#103—44no—
350#103—44no—
351#102—28no—
352#102—28no—
353#102—44no—
354C22⋊C8—24no—
355C22⋊C8—24no—
356C22⋊C8—42no—
357C22⋊C8—42no—
358C22⋊C8—42no—
359C22⋊C8—42no—
360#97—44no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—22no—
365#95—24no—
366#95—24no—
367#95—24no—
368#95—22no—
369#95—22no—
370#95—22no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95C2×D412yes—
377#95C2×D412yes—
378#95C2×D412yes—
379#95C2×D412yes—
380C24—22no—
381C24—22no—
382C24—24no—
383C24—22no—
384C24—41no—
385C24—41no—
386C24—41no—
387C24—41no—
388C24—42no—
389C24—42no—
390C24—42no—
391C24—42no—
392C24—42no—
393C24—42no—
394C24#13714yes—
395C24C22×D411yescharacteristic
396C24C22×D411yescharacteristic
397C24C22×D411yescharacteristic
398C2×Q8—24no—
399C2×Q8—22no—
400C2×Q8—24no—
401C2×Q8—24no—
402C2×Q8—81no—
403C2×Q8—81no—
404C2×Q8—88no—
405C2×Q8C2×#4114yes—
406C2×D4—24no—
407C2×D4—24no—
408C2×D4—22no—
409C2×D4—24no—
410C2×D4—416no—
411C2×D4—42no—
412C2×D4—44no—
413C2×D4—44no—
414C2×D4—44no—
415C2×D4—44no—
416C2×D4—42no—
417C2×D4—82no—
418C2×D4—82no—
419C2×D4—816no—
420C2×D4H(5,𝔽2)14yes—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—41no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—41no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—88no—
473C22×C4—84no—
474C22×C4—81no—
475C22×C4—81no—
476C22×C4—81no—
477C22×C4—81no—
478C22×C4—81no—
479C22×C4—81no—
480C22×C4—81no—
481C22×C4—81no—
482C22×C4—816no—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D411yescharacteristic
488C22×C4C22×D412yes—
489C22×C4C22×D411yescharacteristic
490C22×C4C2×#4114yes—
491C22×C4C2511yescharacteristic
492SD16—416no—
493C2×C8—24no—
494C2×C8—24no—
495C2×C8—24no—
496C2×C8—24no—
497C2×C8—24no—
498C2×C8—24no—
499C4⋊C4—24no—
500C4⋊C4—28no—
501C4⋊C4—28no—
502C4⋊C4—22no—
503C4⋊C4—24no—
504C4⋊C4—24no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—24no—
509C4⋊C4—42no—
510C4⋊C4—42no—
511C4⋊C4—42no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4—81no—
515C4⋊C4—81no—
516C4⋊C4—84no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4C2×#4114yes—
520C22⋊C4—28no—
521C22⋊C4—28no—
522C22⋊C4—24no—
523C22⋊C4—24no—
524C22⋊C4—42no—
525C22⋊C4—42no—
526C22⋊C4—48no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—44no—
531C22⋊C4—84no—
532C22⋊C4—84no—
533C22⋊C4—82no—
534C22⋊C4—82no—
535C42—24no—
536C42—24no—
537C42—42no—
538C42—42no—
539C42—82no—
540C42—82no—
541C42—82no—
542C42—82no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23—21no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—21no—
556C23—21no—
557C23—21no—
558C23—21no—
559C23—28no—
560C23—24no—
561C23—44no—
562C23—44no—
563C23—48no—
564C23—48no—
565C23—48no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—82no—
573C23—82no—
574C23—82no—
575C23—82no—
576C23—84no—
577C23—84no—
578C23—84no—
579C23—84no—
580C23#52512yes—
581C23C2×#13712yes—
582C23C2×#13712yes—
583C23C2×#13712yes—
584C23#57512yes—
585C23C23×D411yescharacteristic
586Q8—28no—
587Q8—44no—
588Q8—84no—
589D4—28no—
590D4—48no—
591D4—44no—
592D4—88no—
593D4—88no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—28no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—82no—
640C2×C4—82no—
641C2×C4—82no—
642C2×C4—82no—
643C2×C4—82no—
644C2×C4—82no—
645C2×C4—84no—
646C2×C4—84no—
647C2×C4—816no—
648C2×C4—84no—
649C2×C4—84no—
650C2×C4—84no—
651C2×C4—162no—
652C2×C4—162no—
653C2×C4#52512yes—
654C2×C4C22×#4112yes—
655C2×C4C2×H(5,𝔽2)12yes—
656C2×C4#58512yes—
657C8—44no—
658C8—44no—
659C22—22no—
660C22—22no—
661C22—22no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—22no—
671C22—41no—
672C22—41no—
673C22—41no—
674C22—41no—
675C22—41no—
676C22—41no—
677C22—41no—
678C22—41no—
679C22—44no—
680C22—44no—
681C22—44no—
682C22—88no—
683C22—816no—
684C22—82no—
685C22—82no—
686C22—82no—
687C22—82no—
688C22#311714yes—
689C22C2×#52511yescharacteristic
690C22#348411yescharacteristic
691C22#348511yescharacteristic
692C22C22×#13711yescharacteristic
693C22C2×#57511yescharacteristic
694C22#358511yescharacteristic
695C22#358511yescharacteristic
696C4—22no—
697C4—42no—
698C4—42no—
699C4—41no—
700C4—41no—
701C4—44no—
702C4—42no—
703C4—42no—
704C4—44no—
705C4—82no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4#351314yes—
711C2—24no—
712C2—42no—
713C2—42no—
714C2—84no—
715C2#3535411yescharacteristic
716C2C2×#311711yescharacteristic
717C2#6056811yescharacteristic
718C2#6157011yescharacteristic
719C2#6158111yescharacteristic
720C2#6170811yescharacteristic
721C2#6171011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#121C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34212C2semidirect
3. #28123C2semidirect
4. #27686C2semidirect
5. #27603C2semidirect
6. #27178C2semidirect
7. #26676C2semidirect
8. #26567C2semidirect
9. #24914C2semidirect
10. #24006C2semidirect
13. #22085C2semidirect
14. #21906C2semidirect
15. #21453C2semidirect
23. C2×#440C22semidirect
31. #2360C22semidirect
32. #2360C22semidirect
34. #2345C22semidirect
35. Q8×C22⋊C4C22semidirect
50. #2101C22semidirect
51. #2097C22semidirect
52. #2076C22semidirect
53. #2067C22semidirect
54. #2055C22semidirect
55. #2036C22semidirect
58. #1949C22semidirect
59. #1943C22semidirect
60. #1935C22semidirect
61. #1925C22semidirect
62. #1924C22semidirect
63. #1924C22semidirect
64. #1918C22semidirect
65. #1880C22semidirect
66. #1877C22semidirect
71. #1865C22semidirect
72. #1862C22semidirect
75. #1852C22semidirect
76. #1852C22semidirect
78. #1824C22semidirect
79. #1805C22semidirect
81. #1787C22semidirect
85. #1762C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24202
#342122
#281232
#276862
#276032
#271782
#266762
#265672
#249142
#240062
#240052
#239192
#220852
#219062
#214532