Finite Group Databasebeta

← #6489115

Subgroups — #6489115

Order 512.

2020 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36645C211yesmaximal, characteristic, complemented
2C4×#532C211yesmaximal, characteristic
3#32907C214yesmaximal, complemented
4#30046C214yesmaximal, complemented
5#29141C211yesmaximal, characteristic, complemented
6#29139C211yesmaximal, characteristic, complemented
7#28440C211yesmaximal, characteristic, complemented
8#28438C211yesmaximal, characteristic, complemented
9#28159C214yesmaximal, complemented
10#27975C212yesmaximal, complemented
11#25719C212yesmaximal
12#23935C214yesmaximal, complemented
13#22270C212yesmaximal, complemented
14#22268C212yesmaximal, complemented
15#20777C211yesmaximal, characteristic, complemented
16C2×#533C2212yes—
17C2×#532C2211yescharacteristic
18C2×C4×#41C2212yes—
19C2×#494C2212yes—
20C2×#492C2212yes—
21C2×#462C2214yes—
22C2×#461C2214yes—
23C2×#452C2218yes—
24#2868C2211yescharacteristic
25#2868C2211yescharacteristic
26#2862C2212yes—
27#2859—21no—
28#2859—21no—
29#2857—21no—
30#2857—21no—
31#2857C2212yes—
32#2857C2212yes—
33C4×C4⋊Q8—21no—
34C4×C4⋊Q8—21no—
35C4×C4⋊Q8C2211yescharacteristic
36C4×C4⋊Q8C2212yes—
37C4×C4⋊Q8C2212yes—
38C4×C4⋊Q8C2211yescharacteristic
39C4×#127C2211yescharacteristic
40C4×#127C2211yescharacteristic
41C4×#124C2212yes—
42C4×#124C2212yes—
43C4×#121—21no—
44C4×#121—21no—
45C4×#121—21no—
46C4×#121—21no—
47#2297C2212yes—
48#2293—21no—
49#2293—21no—
50#2285—21no—
51#2285—21no—
52#2285C2211yescharacteristic
53C42×D4—21no—
54C42×D4—21no—
55C2×C43C2211yescharacteristic
56#2074C2218yes—
57#2024C2214yes—
58#2023C2214yes—
59#2016C2214yes—
60#2015C2218yes—
61#2013C2214yes—
62#1957C2212yes—
63#1956C2212yes—
64#1956C2212yes—
65#1956C2212yes—
66#1880C2218yes—
67#1878C2218yes—
68#1878C2218yes—
69#1877C2218yes—
70#1873C2218yes—
71#1839C2212yes—
72#1834C2212yes—
73#1834C2212yes—
74#1831C2212yes—
75#1817C2212yes—
76#1803C2214yes—
77#1802C2214yes—
78#1797C2212yes—
79C4×C4⋊C8C2211yescharacteristic
80C4×C4⋊C8C2211yescharacteristic
81C4×C22⋊C8C2212yes—
82#1744C2212yes—
83C22×#41C2312yes—
84C22×Q16C2318yes—
85C22×SD16C2318yes—
86#533—48no—
87#532—24no—
88#532—24no—
89C2×C4⋊Q8—21no—
90C2×C4⋊Q8—21no—
91C2×C4⋊Q8C2312yes—
92C2×C4⋊Q8C2312yes—
93C2×C4⋊Q8C2311yescharacteristic
94C2×C4⋊Q8C2312yes—
95C2×C4⋊Q8C2312yes—
96C2×#127C2311yescharacteristic
97C2×#125—22no—
98C2×#124C2312yes—
99C2×C22⋊Q8—22no—
100C2×C22⋊Q8C2314yes—
101C2×#121—21no—
102C2×#121—21no—
103C4×#41—48no—
104C2×C4×Q8—22no—
105C2×C4×Q8C2312yes—
106C2×C4×Q8C2314yes—
107C2×C4×Q8C2312yes—
108C2×C4×D4—22no—
109C2×C4×D4—22no—
110C2×C4×D4—21no—
111C2×C4×D4—21no—
112C2×C4×D4—21no—
113C2×C4×D4—21no—
114C2×C4×D4—21no—
115C2×C4×D4—21no—
116C2×C4×D4C2312yes—
117C2×C4×D4C2312yes—
118C2×#117—22no—
119C2×#117—22no—
120C2×#117C2312yes—
121C22×C42—22no—
122C22×C42D411yescharacteristic
123C22×C42D411yescharacteristic
124C22×C42D411yescharacteristic
125C22×C42D411yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129#494—28no—
130#492—28no—
131#462—48no—
132#461—48no—
133#452—416no—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2312yes—
136C2×#103C2318yes—
137C2×#103C2318yes—
138C2×#103C2318yes—
139C2×#102C2318yes—
140C2×C22⋊C8C2314yes—
141C2×C4×C8C2312yes—
142C2×C4×C8C2312yes—
143#391—22no—
144#391—22no—
145#391C2314yes—
146#391C2314yes—
147#391C2312yes—
148#391C2312yes—
149#390—21no—
150#390—21no—
151#390—21no—
152#390—21no—
153#390C2312yes—
154#390C2312yes—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389—22no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—21no—
167#388—22no—
168#388—22no—
169#388—22no—
170#388—22no—
171#386—22no—
172#386—22no—
173#384—21no—
174#384—21no—
175#384—21no—
176#384—21no—
177#384—21no—
178#384—21no—
179#384C2311yescharacteristic
180#384C2311yescharacteristic
181#384C2312yes—
182#384C2312yes—
183#382—22no—
184#382—22no—
185#382C2312yes—
186#380—22no—
187#380—22no—
188#380—22no—
189#380—22no—
190#380—21no—
191#380—21no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—21no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C22⋊C4—22no—
203C4×C22⋊C4—22no—
204C4×C22⋊C4—22no—
205C4×C22⋊C4—22no—
206C4×C22⋊C4—22no—
207C4×C22⋊C4—22no—
208C43D411yescharacteristic
209C43D411yescharacteristic
210C43D411yescharacteristic
211C43D411yescharacteristic
212C43C2311yescharacteristic
213C43C2311yescharacteristic
214C43C2311yescharacteristic
215C43C2311yescharacteristic
216#340C2318yes—
217#340C2318yes—
218#336C2314yes—
219#336C2314yes—
220#336C2312yes—
221#336C2312yes—
222#336C2314yes—
223C2×#41—24no—
224C2×#41—24no—
225C2×#41—28no—
226C2×#41—44no—
227C22×Q8—22no—
228C22×Q8C2414yes—
229C22×Q8C2412yes—
230C22×D4—22no—
231C22×D4—21no—
232C22×D4—21no—
233C22×D4C2412yes—
234C23×C4—22no—
235C23×C4—22no—
236C23×C4—22no—
237C23×C4C2×D411yescharacteristic
238C23×C4C2×D411yescharacteristic
239C23×C4C2×D411yescharacteristic
240C23×C4C2×D411yescharacteristic
241C23×C4C2×D411yescharacteristic
242C23×C4C2×D411yescharacteristic
243C23×C4C2411yescharacteristic
244C2×Q16—216no—
245C2×Q16—48no—
246C2×Q16—48no—
247C2×SD16—216no—
248C2×SD16—48no—
249C2×SD16—48no—
250C22×C8C2414yes—
251C22×C8C2414yes—
252C4⋊Q8—22no—
253C4⋊Q8—24no—
254C4⋊Q8—24no—
255C4⋊Q8—28no—
256C4⋊Q8—44no—
257C4⋊Q8—44no—
258C4⋊Q8—44no—
259C4⋊Q8—44no—
260C4⋊Q8#4114yes—
261#127—22no—
262#127#4114yes—
263#125—48no—
264#124—24no—
265#124—24no—
266C22⋊Q8—216no—
267C22⋊Q8—48no—
268#121—28no—
269#121—28no—
270C4×Q8—28no—
271C4×Q8—44no—
272C4×Q8—48no—
273C4×Q8—48no—
274C4×D4—28no—
275C4×D4—28no—
276C4×D4—28no—
277C4×D4—44no—
278C4×D4—44no—
279C4×D4—48no—
280C4×D4—48no—
281C4×D4—44no—
282C4×D4—44no—
283C4×D4—44no—
284#117—48no—
285#117—44no—
286#117—48no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—22no—
298C2×C4⋊C4—24no—
299C2×C4⋊C4—24no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2412yes—
310C2×C4⋊C4C2414yes—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—22no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—22no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—24no—
328C2×C42—22no—
329C2×C42—24no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—42no—
333C2×C42—48no—
334C2×C42C2×D411yescharacteristic
335C2×C42C2×D411yescharacteristic
336C2×C42C2×D411yescharacteristic
337C2×C42C2×D411yescharacteristic
338C2×C42C2×D411yescharacteristic
339C2×C42C2×D411yescharacteristic
340C2×C42C2×D411yescharacteristic
341C2×C42C2×D411yescharacteristic
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D411yescharacteristic
345C2×C42C2×D411yescharacteristic
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2×D411yescharacteristic
349C2×C42C2×D411yescharacteristic
350C2×C42C2×D411yescharacteristic
351C2×C42C2×D411yescharacteristic
352C2×C42C2×D411yescharacteristic
353C2×C42C2×D411yescharacteristic
354C2×C42C2×D411yescharacteristic
355C2×C42C2×D411yescharacteristic
356C2×C42C2411yescharacteristic
357C2×C42C2411yescharacteristic
358C2×C42C2411yescharacteristic
359C2×C42C2411yescharacteristic
360C2×C42C2411yescharacteristic
361C2×C42C2411yescharacteristic
362C4⋊C8—24no—
363C4⋊C8—24no—
364#103—48no—
365#103—48no—
366#103—48no—
367#102—48no—
368C22⋊C8—28no—
369C4×C8—24no—
370C4×C8—24no—
371#95—22no—
372#95—22no—
373#95—22no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—24no—
378#95—24no—
379#95—24no—
380#95—24no—
381C24—22no—
382C24C22×D411yescharacteristic
383#41—416no—
384#41—416no—
385C2×Q8—22no—
386C2×Q8—24no—
387C2×Q8—24no—
388C2×Q8—24no—
389C2×Q8—24no—
390C2×Q8—28no—
391C2×Q8—28no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8C2×#4114yes—
396C2×D4—28no—
397C2×D4—24no—
398C2×D4—24no—
399C2×D4—22no—
400C2×D4—24no—
401C2×D4—24no—
402C2×D4—44no—
403C2×D4—44no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4C2×#4114yes—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—28no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4C22×D411yescharacteristic
445C22×C4C22×D411yescharacteristic
446C22×C4C22×D411yescharacteristic
447C22×C4C22×D411yescharacteristic
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C22×D411yescharacteristic
456C22×C4C22×D411yescharacteristic
457C22×C4H(5,𝔽2)14yes—
458C22×C4C2511yescharacteristic
459Q16—816no—
460SD16—816no—
461C2×C8—24no—
462C2×C8—24no—
463C2×C8—24no—
464C2×C8—24no—
465C2×C8—24no—
466C2×C8—24no—
467C4⋊C4—24no—
468C4⋊C4—24no—
469C4⋊C4—28no—
470C4⋊C4—24no—
471C4⋊C4—24no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—48no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—48no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—44no—
491C22⋊C4—28no—
492C22⋊C4—28no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—48no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—44no—
501C42—22no—
502C42—22no—
503C42—22no—
504C42—22no—
505C42—22no—
506C42—22no—
507C42—22no—
508C42—22no—
509C42—22no—
510C42—22no—
511C42—22no—
512C42—22no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—22no—
518C42—21no—
519C42—21no—
520C42—22no—
521C42—22no—
522C42—22no—
523C42—41no—
524C42—41no—
525C42—44no—
526C42—44no—
527C42—48no—
528C42—44no—
529C42#13514yes—
530C42#13714yes—
531C42#13714yes—
532C42C2×#4112yes—
533C42C2×#4112yes—
534C42H(5,𝔽2)14yes—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—24no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23#53512yes—
549C23C23×D411yescharacteristic
550Q8—28no—
551Q8—44no—
552Q8—48no—
553Q8—48no—
554Q8—88no—
555D4—28no—
556D4—416no—
557D4—48no—
558D4—48no—
559D4—44no—
560C2×C4—28no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—24no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—416no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—88no—
609C2×C4—88no—
610C2×C4#52512yes—
611C2×C4#52512yes—
612C2×C4#53512yes—
613C2×C4#53512yes—
614C2×C4#53512yes—
615C2×C4C2×#13512yes—
616C2×C4C2×#13512yes—
617C2×C4C2×#13512yes—
618C2×C4C2×#13712yes—
619C2×C4C2×#13712yes—
620C2×C4C2×#13712yes—
621C2×C4C2×#13712yes—
622C2×C4C2×#13712yes—
623C2×C4C2×#13712yes—
624C2×C4#57512yes—
625C2×C4C22×#4111yescharacteristic
626C2×C4C22×#4111yescharacteristic
627C2×C4C2×H(5,𝔽2)12yes—
628C2×C4C2×H(5,𝔽2)12yes—
629C8—44no—
630C8—44no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—44no—
639C22—44no—
640C22—44no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22C2×#52511yescharacteristic
646C22C2×#53511yescharacteristic
647C22C2×#53511yescharacteristic
648C22C22×#13511yescharacteristic
649C22C22×#13711yescharacteristic
650C22C22×#13711yescharacteristic
651C22C2×#57511yescharacteristic
652C4—22no—
653C4—22no—
654C4—22no—
655C4—22no—
656C4—22no—
657C4—22no—
658C4—41no—
659C4—41no—
660C4—41no—
661C4—41no—
662C4—44no—
663C4—82no—
664C4—84no—
665C4—82no—
666C4—84no—
667C4#310714yes—
668C4#311614yes—
669C4#321014yes—
670C4#351912yes—
671C4#351912yes—
672C2—42no—
673C2—84no—
674C2#3523711yescharacteristic
675C2C2×#310711yescharacteristic
676C2C2×#311611yescharacteristic
677C2C2×#321011yescharacteristic
678C2#6078211yescharacteristic
679C2#6080011yescharacteristic
680C2#6080411yescharacteristic
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. #36645C2semidirect
3. #32907C2semidirect
4. #30046C2semidirect
5. #29141C2semidirect
6. #29139C2semidirect
7. #28440C2semidirect
8. #28438C2semidirect
9. #28159C2semidirect
10. #27975C2semidirect
12. #23935C2semidirect
13. #22270C2semidirect
14. #22268C2semidirect
15. #20777C2semidirect

Maximal subgroups

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

subgroupindex
#366452
C4×#5322
#329072
#300462
#291412
#291392
#284402
#284382
#281592
#279752
#257192
#239352
#222702
#222682
#207772