Finite Group Databasebeta

← #6432496

Subgroups — #6432496

Order 512.

1888 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34267C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic, complemented
3#23768C214yesmaximal, complemented
4#23766C214yesmaximal, complemented
5#23763C212yesmaximal, complemented
6#23761C212yesmaximal, complemented
7#23756C212yesmaximal, complemented
8#23754C212yesmaximal, complemented
9C2×#2099C211yesmaximal, characteristic, complemented
10#23734C211yesmaximal, characteristic, complemented
11#23657C211yesmaximal, characteristic, complemented
12#23643C212yesmaximal, complemented
13#23627C211yesmaximal, characteristic, complemented
14#23485C212yesmaximal
15#20453C212yesmaximal, complemented
16#20415C212yesmaximal, complemented
17#20378C211yesmaximal, characteristic, complemented
18C2×#537C2211yescharacteristic
19C2×#520C2211yescharacteristic
20C2×C4×#41C2211yescharacteristic
21C2×#428C2212yes—
22C2×#428C2212yes—
23C2×#417C2212yes—
24C2×#416C2212yes—
25C2×#408C2212yes—
26C2×#405C2212yes—
27#2364—22no—
28#2363—21no—
29#2362—21no—
30#2361—22no—
31#2361—21no—
32#2360—21no—
33#2355—21no—
34#2355—21no—
35#2354—21no—
36#2354—21no—
37#2352—22no—
38#2352—22no—
39#2324C2212yes—
40#2323C2212yes—
41#2314C2214yes—
42#2311C2211yescharacteristic
43#2311C2212yes—
44C4×C22⋊Q8C2212yes—
45C4×#121C2212yes—
46#2297C2211yescharacteristic
47C2×#389C2211yescharacteristic
48#2285C2212yes—
49#2285C2212yes—
50#2285C2211yescharacteristic
51C2×#384C2211yescharacteristic
52#2274C2212yes—
53#2274C2211yescharacteristic
54#2099C2212yes—
55#2099C2212yes—
56#2099C2214yes—
57#2096C2212yes—
58#2095C2212yes—
59#2094C2212yes—
60#2093C2212yes—
61#2085C2214yes—
62#2084C2212yes—
63#2083C2212yes—
64#2077C2214yes—
65#2077C2214yes—
66#2076C2214yes—
67#2075C2214yes—
68#2074C2212yes—
69#2073C2212yes—
70#2072C2211yescharacteristic
71#2072C2211yescharacteristic
72#2071C2211yescharacteristic
73#2071C2211yescharacteristic
74#2036C2214yes—
75#2035C2214yes—
76#2034C2214yes—
77#2033C2214yes—
78#1776C2214yes—
79#1776C2214yes—
80C4×#107C2214yes—
81C4×#106C2214yes—
82#1763C2214yes—
83#1762C2214yes—
84C4×#103C2214yes—
85C4×#102C2214yes—
86#1750C2214yes—
87#1749C2212yes—
88C4×C22⋊C8C2212yes—
89#1736C2212yes—
90#1735C2212yes—
91C2×#340C2212yes—
92C2×#340C2212yes—
93#1728C2211yescharacteristic
94C2×#336C2211yescharacteristic
95C22×#41C2311yescharacteristic
96C22×#41C2311yescharacteristic
97C22×M16C2312yes—
98C23×C8C2312yes—
99#537—24no—
100#537—24no—
101C2×C22⋊Q8—21no—
102C2×C22⋊Q8C2312yes—
103C2×#121—21no—
104C2×#121—22no—
105C2×#121C2312yes—
106#520—28no—
107C4×#41—28no—
108C2×C4×Q8D411yescharacteristic
109C2×C4×Q8D411yescharacteristic
110C2×C4×Q8C2312yes—
111C2×C4×D4—22no—
112C2×C4×D4—22no—
113C2×C4×D4D412yes—
114C2×C4×D4D412yes—
115C2×C4×D4D411yescharacteristic
116C2×C4×D4D411yescharacteristic
117C2×C4×D4C2312yes—
118C2×#117—22no—
119C2×#117—22no—
120C2×#117—21no—
121C2×#117C2312yes—
122C2×#117C2311yescharacteristic
123C2×#117C2312yes—
124C2×#117C2311yescharacteristic
125C22×C4⋊C4—21no—
126C22×C4⋊C4—21no—
127C22×C4⋊C4—21no—
128C22×C4⋊C4—21no—
129C22×C4⋊C4C2312yes—
130C22×C4⋊C4C2311yescharacteristic
131C22×C4⋊C4C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134#428—28no—
135#428—28no—
136C2×#107C2314yes—
137C2×#107C2314yes—
138C2×#106C2314yes—
139C2×#106C2314yes—
140#417—28no—
141#416—28no—
142C2×#103C2312yes—
143C2×#103C2312yes—
144C2×#103C2314yes—
145C2×#102C2312yes—
146C2×#102C2312yes—
147C2×#102C2314yes—
148#408—28no—
149C2×C22⋊C8C2314yes—
150C2×C22⋊C8C2314yes—
151#405—28no—
152C2×#97C2314yes—
153C2×C4×C8C2314yes—
154#391—22no—
155#391—21no—
156#391C2311yescharacteristic
157#391C2311yescharacteristic
158#390—22no—
159#390—21no—
160#390—22no—
161#390C2311yescharacteristic
162#390C2311yescharacteristic
163#389—22no—
164#389—21no—
165#389—21no—
166#389C2312yes—
167#389C2312yes—
168#388—22no—
169#388—24no—
170#388—24no—
171#388—22no—
172#388—22no—
173#388—22no—
174#387—24no—
175#387—24no—
176#387—22no—
177#387—22no—
178#386—24no—
179#386—22no—
180#386—21no—
181#386—22no—
182#386—21no—
183#386—22no—
184#385—22no—
185#385—22no—
186#384—21no—
187#384—21no—
188#384C2312yes—
189#384C2311yescharacteristic
190#384C2311yescharacteristic
191#384C2312yes—
192#382—22no—
193#382—22no—
194#382C2312yes—
195#380—21no—
196#380—21no—
197#380—22no—
198#380—22no—
199#380C2314yes—
200#380C2314yes—
201#380C2312yes—
202#380C2312yes—
203C4×C4⋊C4—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—21no—
206C4×C4⋊C4—21no—
207C4×C4⋊C4—22no—
208C4×C4⋊C4—22no—
209C4×C4⋊C4—21no—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—22no—
212C4×C4⋊C4C2314yes—
213C4×C4⋊C4C2314yes—
214C4×C4⋊C4C2312yes—
215C4×C4⋊C4C2312yes—
216C4×C22⋊C4—22no—
217C4×C22⋊C4—24no—
218C4×C22⋊C4—22no—
219C4×C22⋊C4—21no—
220C4×C22⋊C4—22no—
221C4×C22⋊C4—21no—
222C2×#95—21no—
223#340C2312yes—
224#340C2312yes—
225#340C2312yes—
226#340C2312yes—
227#340C2314yes—
228#340C2314yes—
229#340C2314yes—
230#340C2314yes—
231#340C2314yes—
232#340C2314yes—
233#336C2311yescharacteristic
234#336C2311yescharacteristic
235#336C2312yes—
236#336C2312yes—
237#336C2312yes—
238C2×#41—24no—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—24no—
243C2×#41—24no—
244C2×#41—24no—
245C2×#41#4114yes—
246C22×Q8C2×D411yescharacteristic
247C22×Q8C2411yescharacteristic
248C22×D4—21no—
249C22×D4—22no—
250C22×D4C2×D412yes—
251C22×D4C2×D411yescharacteristic
252C22×D4C2411yescharacteristic
253C23×C4—21no—
254C23×C4—21no—
255C23×C4—22no—
256C23×C4—21no—
257C23×C4—22no—
258C23×C4C2×D411yescharacteristic
259C23×C4C2×D411yescharacteristic
260C23×C4C2×D412yes—
261C23×C4C2×D412yes—
262C23×C4C2412yes—
263C23×C4C2411yescharacteristic
264C2×M16—24no—
265C2×M16—24no—
266C2×M16#4118yes—
267C22×C8—24no—
268C22×C8—24no—
269C22×C8—24no—
270C22×C8C2412yes—
271C22×C8C2412yes—
272C22×C8C2414yes—
273C22⋊Q8—28no—
274C22⋊Q8—24no—
275C22⋊Q8—24no—
276#121—216no—
277#121—28no—
278#121—24no—
279#121—24no—
280C4×Q8—24no—
281C4×Q8—24no—
282C4×Q8—28no—
283C4×D4—24no—
284C4×D4—24no—
285C4×D4—28no—
286C4×D4—28no—
287C4×D4—28no—
288C4×D4—48no—
289C4×D4—84no—
290#117—28no—
291#117—24no—
292#117—24no—
293#117—24no—
294#117—44no—
295#117—84no—
296#117—84no—
297#117#4118yes—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—22no—
300C2×C4⋊C4—22no—
301C2×C4⋊C4—24no—
302C2×C4⋊C4—24no—
303C2×C4⋊C4—24no—
304C2×C4⋊C4—21no—
305C2×C4⋊C4—22no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—22no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—22no—
317C2×C4⋊C4—44no—
318C2×C4⋊C4—82no—
319C2×C4⋊C4—82no—
320C2×C4⋊C4—82no—
321C2×C4⋊C4C2×D412yes—
322C2×C4⋊C4C2×D412yes—
323C2×C4⋊C4C2×D411yescharacteristic
324C2×C4⋊C4C2×D411yescharacteristic
325C2×C4⋊C4#4114yes—
326C2×C4⋊C4C2412yes—
327C2×C4⋊C4C2414yes—
328C2×C4⋊C4C2412yes—
329C2×C4⋊C4C2412yes—
330C2×C4⋊C4C2411yescharacteristic
331C2×C4⋊C4C2411yescharacteristic
332C2×C4⋊C4C2412yes—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—22no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—21no—
340C2×C22⋊C4—24no—
341C2×C22⋊C4—24no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4C2×D412yes—
347C2×C22⋊C4C2×D412yes—
348C2×C22⋊C4C2×D412yes—
349C2×C42—24no—
350C2×C42—28no—
351C2×C42—22no—
352C2×C42—21no—
353C2×C42—22no—
354C2×C42—24no—
355C2×C42—24no—
356C2×C42—22no—
357C2×C42—22no—
358C2×C42C2×D412yes—
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D411yescharacteristic
361C2×C42C2×D412yes—
362C2×C42C2411yescharacteristic
363C2×C42C2411yescharacteristic
364C2×C42C2412yes—
365#107—28no—
366#107—28no—
367#106—28no—
368#106—28no—
369#103—24no—
370#103—24no—
371#103—28no—
372#102—24no—
373#102—24no—
374#102—28no—
375C22⋊C8—28no—
376C22⋊C8—28no—
377#97—28no—
378C4×C8—28no—
379#95—22no—
380#95—22no—
381#95—21no—
382#95—21no—
383#95—24no—
384#95—24no—
385#95—22no—
386#95—22no—
387#95—22no—
388#95—21no—
389#95—21no—
390#95—22no—
391#95—22no—
392#95—24no—
393#95—24no—
394#95—22no—
395C24—21no—
396C24C22×D412yes—
397C24C22×D411yescharacteristic
398#41—216no—
399#41—48no—
400#41—48no—
401#41—48no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8C2×#4112yes—
409C2×D4—24no—
410C2×D4—24no—
411C2×D4—24no—
412C2×D4—24no—
413C2×D4—28no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—22no—
417C2×D4—22no—
418C2×D4—22no—
419C2×D4—21no—
420C2×D4—42no—
421C2×D4—42no—
422C2×D4—82no—
423C2×D4—82no—
424C2×D4C2×#4112yes—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—82no—
457C22×C4—82no—
458C22×C4—81no—
459C22×C4—81no—
460C22×C4—82no—
461C22×C4—81no—
462C22×C4—82no—
463C22×C4—81no—
464C22×C4C22×D412yes—
465C22×C4C22×D411yescharacteristic
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D411yescharacteristic
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D411yescharacteristic
472C22×C4C2×#4114yes—
473C22×C4C2×#4112yes—
474C22×C4C2×#4112yes—
475C22×C4C2×#4114yes—
476C22×C4C2511yescharacteristic
477M16—216no—
478C2×C8—24no—
479C2×C8—24no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—22no—
486C2×C8—48no—
487C2×C8C2×#4118yes—
488C4⋊C4—28no—
489C4⋊C4—24no—
490C4⋊C4—24no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—22no—
496C4⋊C4—22no—
497C4⋊C4—22no—
498C4⋊C4—24no—
499C4⋊C4—22no—
500C4⋊C4—22no—
501C4⋊C4—24no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—42no—
510C4⋊C4—44no—
511C4⋊C4—42no—
512C4⋊C4—82no—
513C4⋊C4—82no—
514C4⋊C4—82no—
515C4⋊C4#12314yes—
516C4⋊C4#12314yes—
517C4⋊C4C2×#4114yes—
518C22⋊C4—28no—
519C22⋊C4—28no—
520C22⋊C4—24no—
521C22⋊C4—24no—
522C22⋊C4—44no—
523C22⋊C4—42no—
524C22⋊C4—44no—
525C22⋊C4—42no—
526C22⋊C4—42no—
527C22⋊C4—81no—
528C22⋊C4—81no—
529C22⋊C4—82no—
530C22⋊C4—84no—
531C22⋊C4—84no—
532C22⋊C4—82no—
533C22⋊C4#12318yes—
534C42—22no—
535C42—22no—
536C42—24no—
537C42—24no—
538C42—22no—
539C42—22no—
540C42—44no—
541C42—44no—
542C42—42no—
543C42—82no—
544C42—84no—
545C42—84no—
546C42—82no—
547C42C2×#4118yes—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23C2×#12312yes—
567C23#52512yes—
568C23#53212yes—
569C23C23×D411yescharacteristic
570Q8—24no—
571Q8—44no—
572Q8—42no—
573D4—24no—
574D4—48no—
575D4—48no—
576D4—44no—
577D4—42no—
578D4—88no—
579C2×C4—22no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—48no—
599C2×C4—48no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—48no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—44no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—41no—
627C2×C4—84no—
628C2×C4—164no—
629C2×C4—164no—
630C2×C4—162no—
631C2×C4C2×#12314yes—
632C2×C4C2×#12314yes—
633C2×C4C2×#12312yes—
634C2×C4#52512yes—
635C2×C4#52918yes—
636C2×C4#52918yes—
637C2×C4#53212yes—
638C2×C4#57612yes—
639C2×C4#57712yes—
640C2×C4C22×#4112yes—
641C2×C4C22×#4111yescharacteristic
642C2×C4C22×#4111yescharacteristic
643C2×C4C22×#4112yes—
644C8—28no—
645C8—42no—
646C8—42no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—42no—
654C22—42no—
655C22—42no—
656C22—42no—
657C22—42no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—81no—
664C22—81no—
665C22—81no—
666C22—81no—
667C22#239514yes—
668C22C22×#12311yescharacteristic
669C22C2×#52511yescharacteristic
670C22C2×#53211yescharacteristic
671C22C2×#57611yescharacteristic
672C22C2×#57711yescharacteristic
673C22#358611yescharacteristic
674C22#358611yescharacteristic
675C4—22no—
676C4—24no—
677C4—22no—
678C4—24no—
679C4—41no—
680C4—41no—
681C4—44no—
682C4—42no—
683C4—42no—
684C4—81no—
685C4—84no—
686C4—81no—
687C4—82no—
688C4—82no—
689C4—82no—
690C4#239514yes—
691C4C2×#52912yes—
692C4C2×#52912yes—
693C2—24no—
694C2—82no—
695C2—82no—
696C2C2×#239511yescharacteristic
697C2#6052111yescharacteristic
698C2#6052211yescharacteristic
699C2#6059611yescharacteristic
700C2#6059911yescharacteristic
701C2#6072211yescharacteristic
702C2#6072611yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#537C22
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
2. #34188C2semidirect
3. #23768C2semidirect
4. #23766C2semidirect
5. #23763C2semidirect
6. #23761C2semidirect
7. #23756C2semidirect
8. #23754C2semidirect
9. C2×#2099C2semidirect
10. #23734C2semidirect
11. #23657C2semidirect
12. #23643C2semidirect
13. #23627C2semidirect
15. #20453C2semidirect
16. #20415C2semidirect
17. #20378C2semidirect

Maximal subgroups

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

subgroupindex
#342672
#341882
#237682
#237662
#237632
#237612
#237562
#237542
C2×#20992
#237342
#236572
#236432
#236272
#234852
#204532
#204152
#203782