Finite Group Databasebeta

← #6615709

Subgroups — #6615709

Order 512.

1619 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic, complemented
2C4×#555C211yesmaximal, characteristic
3#33195C211yesmaximal, characteristic, complemented
4#33114C212yesmaximal, complemented
5#29206C211yesmaximal, characteristic, complemented
6#28284C211yesmaximal, characteristic, complemented
7#27627C212yesmaximal, complemented
8#27341C214yesmaximal, complemented
9#26799C214yesmaximal, complemented
10#25015C212yesmaximal, complemented
11#24651C212yesmaximal, complemented
12#24121C212yesmaximal
13#23063C212yesmaximal, complemented
14#22805C212yesmaximal, complemented
15#22795C211yesmaximal, characteristic, complemented
16C4×#461C211yesmaximal, characteristic, complemented
17#21362C212yesmaximal, complemented
18C2×#555C2211yescharacteristic
19C2×#485C2214yes—
20C2×#477C2212yes—
21C2×#474C2212yes—
22C2×#461C2211yescharacteristic
23C2×C4×SD16C2212yes—
24C2×#431C2212yes—
25#2868C2211yescharacteristic
26#2857C2212yes—
27#2571—21no—
28#2563—21no—
29#2538C2212yes—
30#2497—22no—
31#2492C2214yes—
32#2439—22no—
33#2372—22no—
34#2369—21no—
35#2369C2211yescharacteristic
36#2363—21no—
37#2363C2212yes—
38#2355C2212yes—
39#2354—21no—
40#2354C2211yescharacteristic
41#2354C2211yescharacteristic
42#2346—21no—
43#2346C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×#124—22no—
47C4×#121—21no—
48C4×#121C2212yes—
49#2285C2212yes—
50C42×D4C2211yescharacteristic
51C42×D4C2211yescharacteristic
52C4×#117—21no—
53C4×#117C2212yes—
54#2077C2214yes—
55#2068C2214yes—
56#2058C2214yes—
57#2035C2214yes—
58#2033C2214yes—
59#2000C2214yes—
60#1990C2212yes—
61#1988C2212yes—
62#1981C2212yes—
63#1981C2211yescharacteristic
64#1968C2214yes—
65#1954C2212yes—
66#1948C2212yes—
67#1948C2212yes—
68#1948C2211yescharacteristic
69#1947C2212yes—
70#1947C2212yes—
71#1947C2212yes—
72#1933C2214yes—
73#1928C2214yes—
74#1927C2214yes—
75#1927C2212yes—
76#1926C2214yes—
77#1923C2214yes—
78#1921C2214yes—
79#1877C2212yes—
80#1873C2214yes—
81#1854C2214yes—
82#1854C2214yes—
83#1844C2212yes—
84#1842C2211yescharacteristic
85#1803C2212yes—
86#1802C2212yes—
87#1802C2212yes—
88#1802C2211yescharacteristic
89#1797C2212yes—
90C4×#107C2214yes—
91C4×#106C2214yes—
92C8×C4⋊C4C2212yes—
93C4×C4⋊C8C2211yescharacteristic
94C4×#103C2212yes—
95C22×SD16C2312yes—
96#555—24no—
97#555—24no—
98C2×C4⋊Q8—21no—
99C2×C4⋊Q8—21no—
100C2×C4⋊Q8C2311yescharacteristic
101C2×C4⋊Q8C2311yescharacteristic
102C2×C4⋊Q8C2312yes—
103C2×#124—22no—
104C2×C22⋊Q8C2314yes—
105C2×#121—21no—
106C2×#121C2312yes—
107C2×C4×Q8—21no—
108C2×C4×Q8C2312yes—
109C2×C4×D4—22no—
110C2×C4×D4D411yescharacteristic
111C2×C4×D4D411yescharacteristic
112C2×C4×D4C2311yescharacteristic
113C2×C4×D4C2311yescharacteristic
114C2×C4×D4C2311yescharacteristic
115C2×C4×D4C2311yescharacteristic
116C2×#117—22no—
117C2×#117—21no—
118C2×#117—21no—
119C2×#117C2312yes—
120C2×#117C2312yes—
121C22×C42—21no—
122C22×C42C2312yes—
123#485—48no—
124#477—28no—
125#474—44no—
126#461—24no—
127C4×SD16—28no—
128#431—28no—
129C2×#107C2314yes—
130C2×#107C2314yes—
131C2×#106C2312yes—
132C2×#106C2314yes—
133C2×#106C2314yes—
134C2×C4⋊C8C2312yes—
135C2×C4⋊C8C2311yescharacteristic
136C2×C4⋊C8C2312yes—
137C2×C4⋊C8C2312yes—
138C2×C4⋊C8C2311yescharacteristic
139C2×#103C2314yes—
140C2×#103C2312yes—
141C2×#103C2312yes—
142C2×#103C2312yes—
143C2×#103C2314yes—
144C2×#102C2312yes—
145C2×#102C2314yes—
146C2×C22⋊C8C2314yes—
147C2×C22⋊C8C2314yes—
148C2×C4×C8C2312yes—
149C2×C4×C8C2314yes—
150#400—24no—
151#396—24no—
152#393—24no—
153#391—21no—
154#391—24no—
155#391—22no—
156#391C2312yes—
157#391C2314yes—
158#390—22no—
159#390—21no—
160#390—22no—
161#390C2312yes—
162#389—22no—
163#389—21no—
164#389—21no—
165#389—21no—
166#389—21no—
167#389—22no—
168#389—22no—
169#389C2312yes—
170#389C2312yes—
171#388—22no—
172#388—22no—
173#388—24no—
174#388—22no—
175#388—22no—
176#388—22no—
177#388—22no—
178#387—24no—
179#386—22no—
180#386—22no—
181#384—21no—
182#384—21no—
183#384C2311yescharacteristic
184#384C2311yescharacteristic
185#384C2312yes—
186#384C2312yes—
187#382—22no—
188#382—22no—
189#382—22no—
190#380—22no—
191#380—22no—
192#380—22no—
193#380—21no—
194#380C2312yes—
195#380C2314yes—
196C4×C4⋊C4—21no—
197C4×C4⋊C4—22no—
198C4×C4⋊C4—21no—
199C4×C4⋊C4—21no—
200C4×C4⋊C4—21no—
201C4×C4⋊C4—21no—
202C4×C4⋊C4—22no—
203C4×C4⋊C4D411yescharacteristic
204C4×C4⋊C4D411yescharacteristic
205C4×C4⋊C4C2312yes—
206C4×C4⋊C4C2311yescharacteristic
207C4×C4⋊C4C2312yes—
208C4×C4⋊C4C2311yescharacteristic
209C4×C4⋊C4C2314yes—
210C4×C4⋊C4C2312yes—
211C4×C4⋊C4C2311yescharacteristic
212C4×C4⋊C4C2311yescharacteristic
213C4×C22⋊C4—22no—
214C4×C22⋊C4—22no—
215C4×C22⋊C4—21no—
216C4×C22⋊C4—21no—
217C4×C22⋊C4—22no—
218C4×C22⋊C4—21no—
219C4×C22⋊C4—22no—
220C4×C22⋊C4D412yes—
221C4×C22⋊C4D412yes—
222C4×C22⋊C4C2312yes—
223#376—22no—
224#376—22no—
225C43—22no—
226C43C2311yescharacteristic
227#340C2312yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2312yes—
234#336C2314yes—
235#336C2312yes—
236C22×Q8—21no—
237C22×Q8—22no—
238C22×Q8C2412yes—
239C22×D4—22no—
240C22×D4C2×D411yescharacteristic
241C22×D4C2411yescharacteristic
242C23×C4—21no—
243C23×C4—21no—
244C23×C4—21no—
245C23×C4C2×D412yes—
246C23×C4C2×D412yes—
247C23×C4C2412yes—
248C2×SD16—24no—
249C2×SD16—24no—
250C2×SD16—42no—
251C22×C8C2412yes—
252C22×C8C2414yes—
253C22×C8C2412yes—
254C4⋊Q8—22no—
255C4⋊Q8—22no—
256C4⋊Q8—22no—
257C4⋊Q8—42no—
258C4⋊Q8—42no—
259C4⋊Q8—44no—
260C4⋊Q8—82no—
261C4⋊Q8#4114yes—
262#124—216no—
263C22⋊Q8—44no—
264C22⋊Q8—44no—
265#121—28no—
266#121—24no—
267#121—24no—
268C4×Q8—24no—
269C4×Q8—24no—
270C4×Q8—82no—
271C4×D4—24no—
272C4×D4—22no—
273C4×D4—24no—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—22no—
277C4×D4—22no—
278C4×D4—22no—
279C4×D4—41no—
280C4×D4—84no—
281C4×D4#4114yes—
282#117—28no—
283#117—24no—
284#117—24no—
285#117—24no—
286#117—42no—
287#117—44no—
288#117—84no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—22no—
295C2×C4⋊C4—22no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—22no—
305C2×C4⋊C4C2×D411yescharacteristic
306C2×C4⋊C4C2×D411yescharacteristic
307C2×C4⋊C4C2×D411yescharacteristic
308C2×C4⋊C4C2412yes—
309C2×C4⋊C4C2411yescharacteristic
310C2×C4⋊C4C2412yes—
311C2×C4⋊C4C2414yes—
312C2×C4⋊C4C2411yescharacteristic
313C2×C4⋊C4C2412yes—
314C2×C4⋊C4C2411yescharacteristic
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—21no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—21no—
321C2×C22⋊C4—21no—
322C2×C22⋊C4—24no—
323C2×C22⋊C4—22no—
324C2×C22⋊C4—24no—
325C2×C22⋊C4—22no—
326C2×C22⋊C4C2×D412yes—
327C2×C22⋊C4C2×D412yes—
328C2×C22⋊C4C2×D412yes—
329C2×C42—22no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—21no—
337C2×C42—21no—
338C2×C42—21no—
339C2×C42—24no—
340C2×C42—22no—
341C2×C42—44no—
342C2×C42C2×D411yescharacteristic
343C2×C42C2×D411yescharacteristic
344C2×C42C2×D412yes—
345C2×C42C2×D412yes—
346C2×C42C2×D411yescharacteristic
347C2×C42C2×D411yescharacteristic
348C2×C42C2411yescharacteristic
349C2×C42C2412yes—
350C2×C42C2411yescharacteristic
351C2×C42C2411yescharacteristic
352#107—28no—
353#107—28no—
354#106—24no—
355#106—44no—
356#106—44no—
357C4⋊C8—24no—
358C4⋊C8—24no—
359C4⋊C8—22no—
360C4⋊C8—42no—
361C4⋊C8#4114yes—
362#103—24no—
363#103—24no—
364#103—28no—
365#103—24no—
366#103—44no—
367#102—24no—
368#102—44no—
369C22⋊C8—28no—
370C22⋊C8—44no—
371C4×C8—28no—
372C4×C8—24no—
373#95—21no—
374#95—22no—
375#95—22no—
376#95—22no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—21no—
381#95—21no—
382#95—24no—
383#95—24no—
384#95—22no—
385#95—22no—
386#95—24no—
387#95—24no—
388#95C2×D412yes—
389#95C2×D412yes—
390#95C2×D412yes—
391C24—21no—
392C24C22×D412yes—
393C2×Q8—24no—
394C2×Q8—22no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—22no—
398C2×Q8—22no—
399C2×Q8—22no—
400C2×Q8—44no—
401C2×Q8—81no—
402C2×Q8—81no—
403C2×Q8—82no—
404C2×Q8—82no—
405C2×D4—28no—
406C2×D4—22no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—82no—
415C2×D4—82no—
416C2×D4C2×#4112yes—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—42no—
438C22×C4—42no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—81no—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D411yescharacteristic
449C22×C4C22×D411yescharacteristic
450C22×C4C22×D411yescharacteristic
451C22×C4C22×D411yescharacteristic
452C22×C4C22×D412yes—
453C22×C4C22×D411yescharacteristic
454C22×C4C22×D411yescharacteristic
455C22×C4C2511yescharacteristic
456SD16—48no—
457C2×C8—22no—
458C2×C8—22no—
459C2×C8—24no—
460C2×C8—22no—
461C2×C8—24no—
462C2×C8—22no—
463C2×C8—24no—
464C2×C8—22no—
465C2×C8C2×#4114yes—
466C4⋊C4—24no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—21no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—24no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—22no—
479C4⋊C4—24no—
480C4⋊C4—42no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—44no—
486C4⋊C4—44no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—82no—
490C4⋊C4—81no—
491C4⋊C4—82no—
492C4⋊C4—84no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4#13714yes—
496C4⋊C4C2×#4112yes—
497C4⋊C4C2×#4114yes—
498C4⋊C4H(5,𝔽2)12yes—
499C22⋊C4—28no—
500C22⋊C4—28no—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—24no—
504C22⋊C4—24no—
505C22⋊C4—42no—
506C22⋊C4—44no—
507C22⋊C4—42no—
508C22⋊C4—44no—
509C22⋊C4—82no—
510C22⋊C4—84no—
511C22⋊C4—82no—
512C22⋊C4—84no—
513C42—28no—
514C42—24no—
515C42—24no—
516C42—22no—
517C42—22no—
518C42—22no—
519C42—21no—
520C42—22no—
521C42—21no—
522C42—41no—
523C42—44no—
524C42—44no—
525C42—42no—
526C42—42no—
527C42—42no—
528C42—44no—
529C42—44no—
530C42—82no—
531C42—84no—
532C42#13514yes—
533C42C2×#4112yes—
534C42C2×#4112yes—
535C42C2×#4112yes—
536C422−1+412yes—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—41no—
550C23—41no—
551C23C23×D411yescharacteristic
552Q8—42no—
553Q8—42no—
554Q8—42no—
555Q8—42no—
556Q8—84no—
557Q8—164no—
558D4—22no—
559D4—22no—
560D4—44no—
561D4—41no—
562D4—41no—
563D4—88no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—48no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—48no—
592C2×C4—48no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—41no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—84no—
613C2×C4—84no—
614C2×C4#52512yes—
615C2×C4#52512yes—
616C2×C4#54712yes—
617C2×C4#54712yes—
618C2×C4C2×#13512yes—
619C2×C4C2×#13512yes—
620C2×C4C2×#13512yes—
621C2×C4C2×#13714yes—
622C2×C4C2×#13712yes—
623C2×C4#57712yes—
624C2×C4C22×#4111yescharacteristic
625C2×C4C22×#4111yescharacteristic
626C2×C4C22×#4111yescharacteristic
627C2×C4C2×H(5,𝔽2)11yescharacteristic
628C2×C4C2×2−1+411yescharacteristic
629C2×C4#58511yescharacteristic
630C8—24no—
631C8—44no—
632C8—42no—
633C22—42no—
634C22—42no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22C2×#52511yescharacteristic
648C22C2×#54711yescharacteristic
649C22#348811yescharacteristic
650C22C22×#13511yescharacteristic
651C22C22×#13711yescharacteristic
652C22C2×#57711yescharacteristic
653C22#358511yescharacteristic
654C4—22no—
655C4—24no—
656C4—22no—
657C4—22no—
658C4—42no—
659C4—41no—
660C4—42no—
661C4—42no—
662C4—44no—
663C4—42no—
664C4—81no—
665C4—82no—
666C4—82no—
667C4—82no—
668C4—84no—
669C4#310714yes—
670C4#334014yes—
671C4#347511yescharacteristic
672C4#348311yescharacteristic
673C4#351111yescharacteristic
674C4#351811yescharacteristic
675C2—82no—
676C2—82no—
677C2#3641111yescharacteristic
678C2C2×#310711yescharacteristic
679C2#6059711yescharacteristic
680C2C2×#334011yescharacteristic
681C2#6125411yescharacteristic
682C2#6194911yescharacteristic
683C2#6200211yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#555C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36194C2semidirect
3. #33195C2semidirect
4. #33114C2semidirect
5. #29206C2semidirect
6. #28284C2semidirect
7. #27627C2semidirect
8. #27341C2semidirect
9. #26799C2semidirect
10. #25015C2semidirect
11. #24651C2semidirect
13. #23063C2semidirect
14. #22805C2semidirect
15. #22795C2semidirect
16. C4×#461C2semidirect
17. #21362C2semidirect

Maximal subgroups

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

subgroupindex
#361942
C4×#5552
#331952
#331142
#292062
#282842
#276272
#273412
#267992
#250152
#246512
#241212
#230632
#228052
#227952
C4×#4612
#213622