Finite Group Databasebeta

← #6446025

Subgroups — #6446025

Order 512.

1748 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34952C211yesmaximal, characteristic
2C2×#2355C211yesmaximal, characteristic, complemented
3#25071C214yesmaximal, complemented
4#25069C214yesmaximal, complemented
5#25063C212yesmaximal, complemented
6#25061C212yesmaximal, complemented
7#25059C212yesmaximal, complemented
8#25057C212yesmaximal, complemented
9#23733C212yesmaximal
10#23659C212yesmaximal, complemented
11#21303C211yesmaximal, characteristic, complemented
12#21170C211yesmaximal, characteristic, complemented
13#21059C212yesmaximal
14#21033C212yesmaximal, complemented
15#20800C211yesmaximal, characteristic, complemented
16#20648C211yesmaximal, characteristic
17C4×#417C211yesmaximal, characteristic
18C2×C4×#41C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C2×#428C2212yes—
21C2×#428C2212yes—
22C2×#428C2212yes—
23C2×#423C2211yescharacteristic
24C2×#417C2211yescharacteristic
25C2×#417C2212yes—
26C2×#417C2211yescharacteristic
27C2×C4×M16C2211yescharacteristic
28#2582C2211yescharacteristic
29#2582C2211yescharacteristic
30#2581C2212yes—
31#2580C2212yes—
32#2579—22no—
33#2538C2211yescharacteristic
34#2538C2211yescharacteristic
35#2487—21no—
36#2409—21no—
37#2396—22no—
38#2355—22no—
39#2355—21no—
40#2355—21no—
41#2355C2212yes—
42#2355C2212yes—
43#2355C2214yes—
44C4×C22⋊Q8—21no—
45C4×C22⋊Q8—21no—
46#2297C2212yes—
47#2296C2212yes—
48C2×#391—21no—
49C2×#387—22no—
50#2285C2211yescharacteristic
51C2×#384C2211yescharacteristic
52C2×#382C2212yes—
53C2×C4×C4⋊C4—21no—
54C2×C4×C4⋊C4C2212yes—
55C2×C4×C4⋊C4C2211yescharacteristic
56#2097C2214yes—
57#2097C2214yes—
58#2092C2214yes—
59#2092C2214yes—
60#2085C2214yes—
61#2085C2214yes—
62#2085C2214yes—
63#2085C2214yes—
64#1985C2214yes—
65#1984C2212yes—
66#1984C2212yes—
67#1949C2214yes—
68#1948C2212yes—
69#1947C2212yes—
70#1928C2214yes—
71#1927C2214yes—
72#1926C2214yes—
73#1925C2214yes—
74#1921C2214yes—
75#1920C2214yes—
76#1919C2214yes—
77#1918C2214yes—
78#1847C2214yes—
79#1845C2212yes—
80#1844C2212yes—
81#1803C2212yes—
82#1803C2212yes—
83#1802C2212yes—
84#1802C2212yes—
85#1763C2212yes—
86#1762C2212yes—
87C4×#103C2212yes—
88C4×#102C2212yes—
89#1736C2211yescharacteristic
90#1736C2212yes—
91#1736C2212yes—
92#1736C2211yescharacteristic
93#1728C2212yes—
94#1728C2212yes—
95C22×#41C2311yescharacteristic
96C23×Q8D411yescharacteristic
97C22×M16C2312yes—
98C22×M16C2311yescharacteristic
99C22×M16C2311yescharacteristic
100C2×C22⋊Q8—21no—
101C2×C22⋊Q8—21no—
102C4×#41—24no—
103C4×#41—24no—
104C2×C4×Q8—24no—
105C2×C4×Q8—24no—
106C2×C4×Q8D412yes—
107C2×C4×Q8D411yescharacteristic
108C2×C4×Q8D411yescharacteristic
109C2×C4×Q8C2311yescharacteristic
110C2×C4×Q8C2311yescharacteristic
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×Q8C2311yescharacteristic
114C2×C4×D4—21no—
115C2×C4×D4—21no—
116C2×C4×D4C2311yescharacteristic
117C2×C4×D4C2311yescharacteristic
118C2×#117—21no—
119C2×#117C2312yes—
120C2×#117C2311yescharacteristic
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4D412yes—
124C22×C4⋊C4D411yescharacteristic
125C22×C4⋊C4C2311yescharacteristic
126C22×C4⋊C4C2312yes—
127C22×C4⋊C4C2311yescharacteristic
128C22×C42—21no—
129C22×C42—22no—
130C22×C42—21no—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133#428—28no—
134#428—44no—
135#428—44no—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#107C2314yes—
139C2×#106C2314yes—
140C2×#106C2314yes—
141C2×#106C2314yes—
142#423—24no—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2312yes—
145#417—24no—
146#417—44no—
147#417Q818yescomplemented
148C2×#103C2312yes—
149C2×#103C2314yes—
150C2×#103C2312yes—
151C2×#102C2312yes—
152C2×#102C2314yes—
153C2×#102C2312yes—
154C4×M16—24no—
155C2×#97C2312yes—
156C2×C4×C8C2312yes—
157#400—22no—
158#399—22no—
159#398—22no—
160#398—22no—
161#396—22no—
162#396—22no—
163#395—22no—
164#393—22no—
165#391—21no—
166#391—21no—
167#391—22no—
168#391—22no—
169#391—24no—
170#391C2312yes—
171#391C2312yes—
172#390C2312yes—
173#390C2312yes—
174#389—21no—
175#389—21no—
176#389C2312yes—
177#389C2312yes—
178#389C2312yes—
179#389C2312yes—
180#387—22no—
181#387—22no—
182#387—24no—
183#387—24no—
184#387—24no—
185#387—24no—
186#386—22no—
187#386—22no—
188#385—22no—
189#385—22no—
190#385—22no—
191#385—22no—
192#384—21no—
193#384—21no—
194#384C2312yes—
195#384C2312yes—
196#384C2311yescharacteristic
197#384C2311yescharacteristic
198#382—22no—
199#382—22no—
200#382C2312yes—
201#382C2314yes—
202#382C2314yes—
203#380—21no—
204#380—21no—
205#380—21no—
206#380—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—24no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—22no—
213C4×C4⋊C4—22no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4—22no—
216C4×C4⋊C4—22no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4C2314yes—
219C4×C4⋊C4C2314yes—
220C4×C4⋊C4C2312yes—
221C4×C4⋊C4C2312yes—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—22no—
224C2×#95—22no—
225C2×#95—21no—
226C2×#95—22no—
227C2×#95—22no—
228#340C2314yes—
229#340C2314yes—
230#340C2312yes—
231#340C2312yes—
232#340C2312yes—
233#340C2314yes—
234#340C2314yes—
235#340C2312yes—
236#336C2314yes—
237#336C2314yes—
238#336C2314yes—
239#336C2314yes—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—22no—
245C2×#41C2×Q814yes—
246C22×Q8—24no—
247C22×Q8—42no—
248C22×Q8—42no—
249C22×Q8C2×D411yescharacteristic
250C22×Q8C2×D411yescharacteristic
251C22×Q8C2×D412yes—
252C22×Q8C2411yescharacteristic
253C22×D4—21no—
254C22×D4C2411yescharacteristic
255C23×C4—21no—
256C23×C4—21no—
257C23×C4—22no—
258C23×C4—21no—
259C23×C4—21no—
260C23×C4—21no—
261C23×C4C2×D411yescharacteristic
262C23×C4C2×D412yes—
263C23×C4C2×D412yes—
264C23×C4C2×D411yescharacteristic
265C23×C4C2411yescharacteristic
266C2×M16—22no—
267C2×M16—22no—
268C2×M16—22no—
269C2×M16—22no—
270C2×M16—24no—
271C2×M16—24no—
272C2×M16—24no—
273C2×M16C2×Q814yes—
274C2×M16#4114yes—
275C22×C8C2414yes—
276C22×C8C2412yes—
277C22×C8C2412yes—
278C22⋊Q8—44no—
279C22⋊Q8—44no—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×Q8—28no—
283C4×Q8—22no—
284C4×Q8—22no—
285C4×Q8—24no—
286C4×Q8—24no—
287C4×Q8—22no—
288C4×Q8—22no—
289C4×Q8—22no—
290C4×Q8—22no—
291C4×Q8—22no—
292C4×Q8—22no—
293C4×D4—22no—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—22no—
297C4×D4—44no—
298C4×D4—44no—
299#117—22no—
300#117—22no—
301#117—24no—
302#117—24no—
303#117—44no—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—28no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—24no—
310C2×C4⋊C4—24no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—22no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—44no—
329C2×C4⋊C4—82no—
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2×D412yes—
333C2×C4⋊C4C2×D412yes—
334C2×C4⋊C4C2×D412yes—
335C2×C4⋊C4C2×D412yes—
336C2×C4⋊C4C2×D412yes—
337C2×C4⋊C4C2×Q814yes—
338C2×C4⋊C4C2412yes—
339C2×C4⋊C4C2414yes—
340C2×C4⋊C4C2411yescharacteristic
341C2×C4⋊C4C2411yescharacteristic
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C4⋊C4C2412yes—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C42—24no—
354C2×C42—24no—
355C2×C42—21no—
356C2×C42—22no—
357C2×C42—21no—
358C2×C42—21no—
359C2×C42—22no—
360C2×C42—22no—
361C2×C42—24no—
362C2×C42—24no—
363C2×C42—44no—
364C2×C42—44no—
365C2×C42—84no—
366C2×C42C2×D412yes—
367C2×C42C2×D412yes—
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42#4114yes—
371C2×C42#4114yes—
372C2×C42C2412yes—
373C2×C42C2412yes—
374C2×C42C2411yescharacteristic
375C2×C42C2411yescharacteristic
376#107—28no—
377#107—44no—
378#107—44no—
379#106—28no—
380#106—44no—
381#106—44no—
382C4⋊C8—24no—
383C4⋊C8—24no—
384#103—24no—
385#103—44no—
386#103C2×Q818yescomplemented
387#102—24no—
388#102—44no—
389#102C2×Q818yescomplemented
390#97—24no—
391C4×C8—24no—
392#95—24no—
393#95—24no—
394#95—22no—
395#95—22no—
396#95—24no—
397#95—24no—
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—22no—
407#95—22no—
408#95—22no—
409#95—22no—
410C24—21no—
411C24C22×D411yescharacteristic
412#41—28no—
413#41—28no—
414#41—44no—
415#41—44no—
416C2×Q8—24no—
417C2×Q8—22no—
418C2×Q8—22no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—21no—
424C2×Q8—21no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—21no—
428C2×Q8—42no—
429C2×Q8—42no—
430C2×Q8—416no—
431C2×Q8C22×Q812yes—
432C2×D4—24no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—42no—
439C2×D4—42no—
440C2×D4C22×Q812yes—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—21no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—42no—
456C22×C4—41no—
457C22×C4—41no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—42no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—44no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—42no—
474C22×C4—42no—
475C22×C4—81no—
476C22×C4—81no—
477C22×C4#12314yes—
478C22×C4#12314yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4C22×Q812yes—
489C22×C4C2×#4112yes—
490C22×C4C2×#4112yes—
491C22×C4C2×#4112yes—
492C22×C4H(5,𝔽2)12yes—
493C22×C42−1+412yes—
494C22×C4C2511yescharacteristic
495M16—28no—
496M16—28no—
497M16—48no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—24no—
503C2×C8—24no—
504C2×C8—24no—
505C2×C8C22×Q814yes—
506C2×C8C2×#4114yes—
507C4⋊C4—24no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—24no—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—24no—
519C4⋊C4—22no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—24no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—44no—
529C4⋊C4—44no—
530C4⋊C4—44no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—81no—
539C4⋊C4—81no—
540C4⋊C4—82no—
541C4⋊C4—82no—
542C4⋊C4—82no—
543C4⋊C4—82no—
544C4⋊C4C22×Q814yes—
545C22⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—42no—
548C22⋊C4—44no—
549C22⋊C4—42no—
550C22⋊C4—42no—
551C22⋊C4—82no—
552C22⋊C4—82no—
553C42—24no—
554C42—24no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—44no—
560C42—42no—
561C42—42no—
562C42—42no—
563C42—44no—
564C42—44no—
565C42—84no—
566C42—84no—
567C42#12314yes—
568C42#12314yes—
569C42C2×#4112yes—
570C42C2×#4112yes—
571C42C2×#4112yes—
572C42C2×#4112yes—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—22no—
578C23—41no—
579C23—41no—
580C23—41no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23C2×#12312yes—
585C23D4×Q812yes—
586C23#55012yes—
587C23C23×D411yescharacteristic
588Q8—22no—
589Q8—22no—
590Q8—48no—
591Q8—44no—
592Q8—44no—
593Q8—41no—
594Q8—41no—
595D4—22no—
596D4—22no—
597D4—48no—
598D4—41no—
599D4—41no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—28no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—24no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—44no—
622C2×C4—48no—
623C2×C4—48no—
624C2×C4—48no—
625C2×C4—42no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—42no—
629C2×C4—48no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—41no—
646C2×C4—41no—
647C2×C4—44no—
648C2×C4—84no—
649C2×C4—84no—
650C2×C4—88no—
651C2×C4—84no—
652C2×C4C2×#12312yes—
653C2×C4C2×#12312yes—
654C2×C4C2×#12312yes—
655C2×C4C2×#12312yes—
656C2×C4C2×#12312yes—
657C2×C4#53314yes—
658C2×C4#54214yes—
659C2×C4D4×Q812yes—
660C2×C4#55012yes—
661C2×C4#55214yes—
662C2×C4#55414yes—
663C2×C4#57512yes—
664C2×C4#57512yes—
665C2×C4C23×Q811yescharacteristic
666C2×C4C22×#4111yescharacteristic
667C2×C4C22×#4111yescharacteristic
668C2×C4C22×#4111yescharacteristic
669C2×C4C2×H(5,𝔽2)11yescharacteristic
670C2×C4C2×2−1+411yescharacteristic
671C8—24no—
672C8—24no—
673C8—44no—
674C22—22no—
675C22—22no—
676C22—22no—
677C22—22no—
678C22—22no—
679C22—22no—
680C22—42no—
681C22—42no—
682C22—42no—
683C22—81no—
684C22—81no—
685C22—81no—
686C22—81no—
687C22#242314yes—
688C22C22×#12311yescharacteristic
689C22C2×D4×Q811yescharacteristic
690C22C2×#55011yescharacteristic
691C22C2×#57511yescharacteristic
692C22C2×#57511yescharacteristic
693C22#358511yescharacteristic
694C22#358611yescharacteristic
695C4—22no—
696C4—22no—
697C4—22no—
698C4—42no—
699C4—42no—
700C4—42no—
701C4—42no—
702C4—42no—
703C4—44no—
704C4—81no—
705C4—81no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4—84no—
710C4#242314yes—
711C4#309314yes—
712C4C2×#53311yescharacteristic
713C4C2×#54211yescharacteristic
714C4C2×#55211yescharacteristic
715C4C2×#55411yescharacteristic
716C2—24no—
717C2—82no—
718C2C2×#242311yescharacteristic
719C2C2×#309311yescharacteristic
720C2#6051711yescharacteristic
721C2#6136711yescharacteristic
722C2#6136911yescharacteristic
723C2#6140311yescharacteristic
724C2#6140511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2355C2semidirect
3. #25071C2semidirect
4. #25069C2semidirect
5. #25063C2semidirect
6. #25061C2semidirect
7. #25059C2semidirect
8. #25057C2semidirect
10. #23659C2semidirect
11. #21303C2semidirect
12. #21170C2semidirect
14. #21033C2semidirect
15. #20800C2semidirect
147. #417Q8semidirect
386. #103C2×Q8semidirect
389. #102C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#349522
C2×#23552
#250712
#250692
#250632
#250612
#250592
#250572
#237332
#236592
#213032
#211702
#210592
#210332
#208002
#206482
C4×#4172