Finite Group Databasebeta

← #6458652

Subgroups — #6458652

Order 512.

1786 conjugacy classes of subgroups.

The subgroup lattice has 721 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34188C211yesmaximal, characteristic
2C4×#536C211yesmaximal, characteristic
3#27639C214yesmaximal, complemented
4#27638C214yesmaximal, complemented
5#27631C212yesmaximal, complemented
6#27630C212yesmaximal, complemented
7#27629C212yesmaximal, complemented
8#27628C212yesmaximal, complemented
9#23641C212yesmaximal
10#23484C212yesmaximal
11#21171C211yesmaximal, characteristic, complemented
12#21169C211yesmaximal, characteristic, complemented
13#21066C212yesmaximal, complemented
14#21035C212yesmaximal, complemented
15#20647C211yesmaximal, characteristic, complemented
16C2×#1803C211yesmaximal, characteristic, complemented
17#20629C211yesmaximal, characteristic
18C2×#536C2211yescharacteristic
19C2×C4×#41C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C2×#428C2212yes—
22C2×#427C2212yes—
23C2×#423C2211yescharacteristic
24C22×C4⋊C8C2211yescharacteristic
25C2×#419C2212yes—
26C22×#103C2212yes—
27C2×#408C2212yes—
28#2355—21no—
29#2355—21no—
30#2354—21no—
31#2354—21no—
32#2352—22no—
33#2352—22no—
34#2327—21no—
35#2326—21no—
36#2323C2212yes—
37#2323C2212yes—
38#2322—22no—
39#2317C2211yescharacteristic
40#2314C2214yes—
41#2311C2211yescharacteristic
42#2311C2212yes—
43#2310C2211yescharacteristic
44C4×C4⋊Q8—21no—
45C4×#124—21no—
46C4×C22⋊Q8C2212yes—
47C4×C22⋊Q8C2212yes—
48C4×#123—22no—
49#2285C2212yes—
50#2285C2212yes—
51#2285C2211yescharacteristic
52C2×#384C2211yescharacteristic
53#2274C2212yes—
54C4×#117C2211yescharacteristic
55#2077C2214yes—
56#2077C2214yes—
57#2076C2214yes—
58#2075C2214yes—
59#2036C2214yes—
60#2036C2214yes—
61#2035C2214yes—
62#2035C2214yes—
63#1949C2212yes—
64#1949C2212yes—
65#1948C2211yescharacteristic
66#1948C2211yescharacteristic
67#1948C2212yes—
68#1948C2212yes—
69#1947C2211yescharacteristic
70#1947C2211yescharacteristic
71#1947C2212yes—
72#1947C2212yes—
73#1928C2214yes—
74#1928C2214yes—
75#1927C2214yes—
76#1927C2214yes—
77#1921C2214yes—
78#1920C2214yes—
79#1919C2214yes—
80#1918C2214yes—
81#1805C2214yes—
82#1803C2212yes—
83#1803C2212yes—
84#1803C2214yes—
85#1803C2212yes—
86#1803C2212yes—
87#1798C2214yes—
88#1797C2212yes—
89#1797C2212yes—
90#1736C2212yes—
91#1736C2212yes—
92#1735C2212yes—
93C2×#340C2212yes—
94#1729C2212yes—
95C22×#41C2311yescharacteristic
96C23×Q8C2311yescharacteristic
97C22×M16C2312yes—
98C23×C8C2312yes—
99#536—24no—
100#536—24no—
101C2×C4⋊Q8—21no—
102C2×#124—21no—
103C2×#123—22no—
104C2×C22⋊Q8C2312yes—
105C2×C22⋊Q8C2312yes—
106C4×#41—24no—
107C4×#41—24no—
108C2×C4×Q8—21no—
109C2×C4×Q8—21no—
110C2×C4×Q8—24no—
111C2×C4×Q8—24no—
112C2×C4×Q8D411yescharacteristic
113C2×C4×Q8D411yescharacteristic
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2312yes—
116C2×C4×D4D412yes—
117C2×C4×D4D411yescharacteristic
118C2×C4×D4D412yes—
119C2×C4×D4D411yescharacteristic
120C2×#117—22no—
121C2×#117—22no—
122C2×#117C2311yescharacteristic
123C2×#117C2312yes—
124C2×#117C2311yescharacteristic
125C2×#117C2312yes—
126C2×#117C2311yescharacteristic
127C22×C4⋊C4—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4—21no—
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2312yes—
132C22×C42—21no—
133C22×C42C2312yes—
134C22×C42C2311yescharacteristic
135#428—28no—
136#427—44no—
137C2×#107C2314yes—
138C2×#107C2312yes—
139C2×#107C2312yes—
140C2×#106C2314yes—
141C2×#106C2312yes—
142C2×#106C2312yes—
143#423—24no—
144C2×C4⋊C8—24no—
145C2×C4⋊C8C2311yescharacteristic
146C2×C4⋊C8C2311yescharacteristic
147C2×C4⋊C8C2312yes—
148C2×C4⋊C8C2312yes—
149C2×C4⋊C8C2312yes—
150#419—44no—
151C2×#103—28no—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2314yes—
155C2×#103C2314yes—
156#408—44no—
157C2×C22⋊C8C2314yes—
158C2×C22⋊C8C2314yes—
159#391—21no—
160#391—22no—
161#391—21no—
162#391—21no—
163#391C2312yes—
164#391C2312yes—
165#390—21no—
166#390—22no—
167#389—22no—
168#389—22no—
169#389C2312yes—
170#389C2312yes—
171#388—24no—
172#388—24no—
173#388—22no—
174#388—22no—
175#388—22no—
176#388—22no—
177#387—24no—
178#387—24no—
179#387—22no—
180#387—22no—
181#387—22no—
182#387—22no—
183#386—22no—
184#386—24no—
185#385—22no—
186#385—22no—
187#385—22no—
188#384—21no—
189#384—21no—
190#384—21no—
191#384C2312yes—
192#384C2312yes—
193#384C2311yescharacteristic
194#384C2311yescharacteristic
195#382—21no—
196#382—22no—
197#382—22no—
198#382C2312yes—
199#381—22no—
200#380—21no—
201#380—21no—
202#380—22no—
203#380—22no—
204#380C2314yes—
205#380C2314yes—
206#380C2312yes—
207#380C2312yes—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—22no—
210C4×C4⋊C4—22no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—21no—
216C4×C4⋊C4C2314yes—
217C4×C4⋊C4C2314yes—
218C4×C4⋊C4C2311yescharacteristic
219C4×C4⋊C4C2311yescharacteristic
220C4×C4⋊C4C2311yescharacteristic
221C4×C4⋊C4C2311yescharacteristic
222C4×C22⋊C4—21no—
223C4×C22⋊C4—21no—
224C4×C22⋊C4—22no—
225C4×C22⋊C4—22no—
226C4×C22⋊C4—24no—
227#376—21no—
228C43—21no—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2312yes—
236#340C2312yes—
237#340C2312yes—
238#340C2312yes—
239#336C2314yes—
240#336C2314yes—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—22no—
245C2×#41—22no—
246C2×#41—22no—
247C22×Q8—21no—
248C22×Q8—22no—
249C22×Q8—22no—
250C22×Q8—22no—
251C22×Q8—22no—
252C22×Q8—22no—
253C22×Q8C2×D411yescharacteristic
254C22×Q8#4114yes—
255C22×Q8C2412yes—
256C22×D4C2×D412yes—
257C22×D4C2×D411yescharacteristic
258C23×C4—21no—
259C23×C4—22no—
260C23×C4—21no—
261C23×C4C2×D412yes—
262C23×C4C2×D411yescharacteristic
263C23×C4C2×D412yes—
264C23×C4C2×D411yescharacteristic
265C23×C4C2412yes—
266C23×C4C2411yescharacteristic
267C2×M16—24no—
268C2×M16—24no—
269C2×M16—24no—
270C22×C8—24no—
271C22×C8—24no—
272C22×C8—24no—
273C22×C8C2412yes—
274C22×C8C2412yes—
275C22×C8C2414yes—
276C4⋊Q8—28no—
277#124—28no—
278#123—216no—
279C22⋊Q8—24no—
280C22⋊Q8—24no—
281C22⋊Q8—24no—
282C22⋊Q8—24no—
283C4×Q8—24no—
284C4×Q8—24no—
285C4×Q8—24no—
286C4×Q8—24no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—44no—
290C4×Q8—44no—
291C4×D4—28no—
292C4×D4—28no—
293C4×D4—24no—
294C4×D4—24no—
295#117—22no—
296#117—22no—
297#117—22no—
298#117—24no—
299#117—24no—
300#117—24no—
301#117—22no—
302#117—22no—
303#117—24no—
304#117—84no—
305#117—84no—
306#117#4114yes—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—24no—
312C2×C4⋊C4—24no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—44no—
328C2×C4⋊C4—82no—
329C2×C4⋊C4—82no—
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2412yes—
335C2×C4⋊C4C2414yes—
336C2×C4⋊C4C2412yes—
337C2×C4⋊C4C2412yes—
338C2×C4⋊C4C2411yescharacteristic
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2411yescharacteristic
342C2×C22⋊C4—21no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—24no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4C2×D412yes—
352C2×C22⋊C4C2×D412yes—
353C2×C22⋊C4C2×D412yes—
354C2×C42—21no—
355C2×C42—24no—
356C2×C42—24no—
357C2×C42—21no—
358C2×C42—22no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—21no—
362C2×C42—21no—
363C2×C42—24no—
364C2×C42—24no—
365C2×C42—21no—
366C2×C42—21no—
367C2×C42—44no—
368C2×C42C2×D412yes—
369C2×C42C2×D412yes—
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42#4114yes—
373C2×C42#4114yes—
374C2×C42C2412yes—
375C2×C42C2411yescharacteristic
376C2×C42C2411yescharacteristic
377#107—28no—
378#107—42no—
379#107—42no—
380#106—28no—
381#106—42no—
382#106—42no—
383C4⋊C8—22no—
384C4⋊C8—22no—
385C4⋊C8—24no—
386C4⋊C8—24no—
387C4⋊C8—24no—
388#103—28no—
389#103—28no—
390#103—44no—
391#103—44no—
392C22⋊C8—44no—
393C22⋊C8—44no—
394#95—21no—
395#95—21no—
396#95—21no—
397#95—21no—
398#95—24no—
399#95—24no—
400#95—22no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—24no—
407#95—24no—
408#95—22no—
409#95—22no—
410#95—22no—
411#95—22no—
412C24C22×D412yes—
413C24C22×D411yescharacteristic
414#41—44no—
415#41—44no—
416#41—44no—
417#41—44no—
418C2×Q8—22no—
419C2×Q8—24no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—28no—
424C2×Q8—28no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—42no—
430C2×Q8—42no—
431C2×Q8—44no—
432C2×Q8—44no—
433C2×Q8C2×#4114yes—
434C2×D4—24no—
435C2×D4—24no—
436C2×D4—24no—
437C2×D4—22no—
438C2×D4—22no—
439C2×D4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—41no—
463C22×C4—42no—
464C22×C4—41no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—42no—
468C22×C4—42no—
469C22×C4—44no—
470C22×C4—81no—
471C22×C4—81no—
472C22×C4—82no—
473C22×C4—82no—
474C22×C4#12314yes—
475C22×C4#12314yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4H(5,𝔽2)12yes—
489C22×C42−1+412yes—
490C22×C4C2511yescharacteristic
491M16—48no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—24no—
497C2×C8—24no—
498C2×C8—24no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—48no—
502C4⋊C4—28no—
503C4⋊C4—28no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—24no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—22no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—42no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—44no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—82no—
533C4⋊C4—82no—
534C4⋊C4—82no—
535C4⋊C4—82no—
536C4⋊C4C2×#4112yes—
537C4⋊C4C2×#4112yes—
538C22⋊C4—28no—
539C22⋊C4—28no—
540C22⋊C4—24no—
541C22⋊C4—24no—
542C22⋊C4—24no—
543C22⋊C4—24no—
544C22⋊C4—44no—
545C22⋊C4—42no—
546C22⋊C4—82no—
547C22⋊C4—82no—
548C22⋊C4—84no—
549C22⋊C4—84no—
550C42—24no—
551C42—24no—
552C42—24no—
553C42—22no—
554C42—22no—
555C42—44no—
556C42—44no—
557C42—42no—
558C42—44no—
559C42—42no—
560C42—42no—
561C42—81no—
562C42—81no—
563C42—84no—
564C42—84no—
565C42#12314yes—
566C42#12314yes—
567C42C2×#4112yes—
568C42C2×#4112yes—
569C42C2×#4112yes—
570C42C2×#4112yes—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—22no—
576C23—22no—
577C23—22no—
578C23—22no—
579C23—22no—
580C23—22no—
581C23—22no—
582C23C2×#12312yes—
583C23#52512yes—
584C23#53712yes—
585C23C23×D411yescharacteristic
586Q8—24no—
587Q8—24no—
588Q8—42no—
589Q8—42no—
590Q8—48no—
591Q8—44no—
592D4—48no—
593D4—44no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—28no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—44no—
614C2×C4—48no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—48no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—48no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—41no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—41no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—41no—
644C2×C4—42no—
645C2×C4—84no—
646C2×C4—164no—
647C2×C4C2×#12312yes—
648C2×C4C2×#12312yes—
649C2×C4C2×#12312yes—
650C2×C4C2×#12312yes—
651C2×C4C2×#12312yes—
652C2×C4#52512yes—
653C2×C4#53712yes—
654C2×C4#55214yes—
655C2×C4#55314yes—
656C2×C4#55514yes—
657C2×C4#55614yes—
658C2×C4#57712yes—
659C2×C4#57812yes—
660C2×C4C22×#4111yescharacteristic
661C2×C4C22×#4111yescharacteristic
662C2×C4C22×#4111yescharacteristic
663C2×C4C22×#4111yescharacteristic
664C2×C4C2×H(5,𝔽2)11yescharacteristic
665C2×C4C2×2−1+411yescharacteristic
666C8—42no—
667C8—42no—
668C8—44no—
669C22—22no—
670C22—22no—
671C22—22no—
672C22—22no—
673C22—22no—
674C22—22no—
675C22—42no—
676C22—42no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—42no—
682C22#248514yes—
683C22C22×#12311yescharacteristic
684C22C2×#52511yescharacteristic
685C22C2×#53711yescharacteristic
686C22C2×#57711yescharacteristic
687C22C2×#57811yescharacteristic
688C22#358511yescharacteristic
689C22#358611yescharacteristic
690C4—22no—
691C4—22no—
692C4—22no—
693C4—42no—
694C4—42no—
695C4—42no—
696C4—42no—
697C4—42no—
698C4—44no—
699C4—82no—
700C4—81no—
701C4—81no—
702C4—82no—
703C4—82no—
704C4—84no—
705C4#248514yes—
706C4#309714yes—
707C4C2×#55211yescharacteristic
708C4C2×#55311yescharacteristic
709C4C2×#55511yescharacteristic
710C4C2×#55611yescharacteristic
711C2—24no—
712C2—82no—
713C2C2×#248511yescharacteristic
714C2C2×#309711yescharacteristic
715C2#6051711yescharacteristic
716C2#6059311yescharacteristic
717C2#6059411yescharacteristic
718C2#6093011yescharacteristic
719C2#6093111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27639C2semidirect
4. #27638C2semidirect
5. #27631C2semidirect
6. #27630C2semidirect
7. #27629C2semidirect
8. #27628C2semidirect
11. #21171C2semidirect
12. #21169C2semidirect
13. #21066C2semidirect
14. #21035C2semidirect
15. #20647C2semidirect
16. C2×#1803C2semidirect

Maximal subgroups

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

subgroupindex
#341882
C4×#5362
#276392
#276382
#276312
#276302
#276292
#276282
#236412
#234842
#211712
#211692
#210662
#210352
#206472
C2×#18032
#206292