Finite Group Databasebeta

← #6432376

Subgroups — #6432376

Order 512.

2326 conjugacy classes of subgroups.

The subgroup lattice has 713 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2947C212yesmaximal, complemented
2#34254C211yesmaximal, characteristic, complemented
3#34207C211yesmaximal, characteristic
4#23695C214yesmaximal, complemented
5#23694C214yesmaximal, complemented
6#23685C212yesmaximal, complemented
7#23684C212yesmaximal, complemented
8#23679C212yesmaximal, complemented
9#23678C212yesmaximal, complemented
10#23658C211yesmaximal, characteristic, complemented
11C2×#2072C211yesmaximal, characteristic, complemented
12#23616C212yesmaximal, complemented
13#23545C212yesmaximal, complemented
14#23504C211yesmaximal, characteristic
15#23429C211yesmaximal, characteristic
16#20430C211yesmaximal, characteristic, complemented
17#20415C212yesmaximal, complemented
18C22×#137C2212yes—
19C2×#532C2211yescharacteristic
20C2×#520C2212yes—
21C2×#519C2211yescharacteristic
22#2947—216no—
23C2×#441C2214yes—
24C2×#440C2214yes—
25C2×C4×Q16C2214yes—
26C2×C4×SD16C2214yes—
27C2×#428C2212yes—
28C2×#424C2212yes—
29C2×#417C2212yes—
30C2×#416C2212yes—
31C2×#416C2212yes—
32C22×#103C2212yes—
33C2×#405C2212yes—
34#2361—22no—
35#2361—22no—
36#2356—21no—
37#2356—21no—
38#2355—21no—
39#2355—21no—
40#2354—21no—
41#2354—21no—
42D4×C4⋊C4—21no—
43D4×C4⋊C4—21no—
44#2343—22no—
45#2343—22no—
46#2326C2214yes—
47#2320C2212yes—
48#2320C2212yes—
49#2317C2211yescharacteristic
50#2311C2212yes—
51#2310C2211yescharacteristic
52C4×C4⋊Q8C2212yes—
53C4×#127C2212yes—
54#2296C2211yescharacteristic
55#2295C2212yes—
56#2295C2212yes—
57C2×#391C2211yescharacteristic
58C2×#384C2211yescharacteristic
59#2274C2211yescharacteristic
60C4×#117C2211yescharacteristic
61#2084C2212yes—
62#2084C2212yes—
63#2083C2212yes—
64#2083C2212yes—
65#2072C2212yes—
66#2072C2212yes—
67#2072C2214yes—
68#2068C2214yes—
69#2067C2214yes—
70#2066C2214yes—
71#2065C2214yes—
72#2056C2214yes—
73#2056C2214yes—
74#2055C2214yes—
75#2055C2214yes—
76#2043C2211yescharacteristic
77#2043C2211yescharacteristic
78#2042C2212yes—
79#2042C2212yes—
80#2041C2211yescharacteristic
81#2041C2211yescharacteristic
82#2016C2212yes—
83#2015C2214yes—
84#2014C2212yes—
85#1768C2214yes—
86#1767C2212yes—
87C4×C4⋊C8C2212yes—
88#1763C2214yes—
89#1762C2214yes—
90C4×#103C2214yes—
91C4×#102C2214yes—
92#1736C2212yes—
93C2×#340C2212yes—
94#1728C2211yescharacteristic
95C2×#336C2211yescharacteristic
96C22×#41C2312yes—
97C23×Q8C2311yescharacteristic
98C2×#137—28no—
99C2×#137—28no—
100C2×#137—28no—
101C22×Q16C2314yes—
102C22×SD16C2314yes—
103C22×M16C2312yes—
104C23×C8C2312yes—
105#532—28no—
106C2×C4⋊Q8C2311yescharacteristic
107C2×#127C2311yescharacteristic
108C2×#124C2312yes—
109C2×#121—21no—
110C2×#121—21no—
111#520—28no—
112#520—28no—
113#519—24no—
114#519D418yescomplemented
115C2×C4×Q8—21no—
116C2×C4×Q8—21no—
117C2×C4×Q8C2314yes—
118C2×C4×Q8C2312yes—
119C2×C4×Q8C2312yes—
120C2×C4×D4—24no—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4—22no—
124C2×C4×D4—22no—
125C2×C4×D4C2314yes—
126C2×#117—22no—
127C2×#117—22no—
128C2×#117C2311yescharacteristic
129C2×#117C2311yescharacteristic
130C2×#117C2311yescharacteristic
131C2×#117C2311yescharacteristic
132C22×C4⋊C4—21no—
133C22×C4⋊C4—21no—
134C22×C4⋊C4—22no—
135C22×C4⋊C4C2312yes—
136C22×C4⋊C4C2311yescharacteristic
137C22×C42C2311yescharacteristic
138#441—216no—
139#440—216no—
140C4×Q16—216no—
141C4×SD16—216no—
142#428—28no—
143C2×#107C2314yes—
144C2×#106C2314yes—
145#424—28no—
146C2×C4⋊C8C2314yes—
147C2×C4⋊C8C2314yes—
148#417—28no—
149#416—28no—
150#416—28no—
151C2×#103—28no—
152C2×#103C2314yes—
153C2×#103C2314yes—
154C2×#103C2312yes—
155C2×#103C2312yes—
156C2×#103C2312yes—
157C2×#103C2314yes—
158C2×#103C2312yes—
159C2×#102C2312yes—
160C2×#102C2312yes—
161C2×#102C2312yes—
162C2×#102C2314yes—
163C2×#102C2312yes—
164#405—28no—
165C2×#97C2314yes—
166C2×C4×C8C2314yes—
167#391—21no—
168#391—21no—
169#391—21no—
170#391—21no—
171#391C2314yes—
172#391C2312yes—
173#391C2312yes—
174#391C2312yes—
175#391C2312yes—
176#390—22no—
177#390—22no—
178#390C2314yes—
179#389—22no—
180#389—22no—
181#389C2312yes—
182#389C2312yes—
183#389C2311yescharacteristic
184#389C2311yescharacteristic
185#389C2311yescharacteristic
186#389C2311yescharacteristic
187#388—24no—
188#388—24no—
189#387—22no—
190#387—22no—
191#387—22no—
192#387—22no—
193#387—22no—
194#387—22no—
195#387—22no—
196#387—22no—
197#386—24no—
198#385—21no—
199#385—21no—
200#385—22no—
201#385—24no—
202#385—24no—
203#384—21no—
204#384—21no—
205#384—21no—
206#384C2312yes—
207#384C2312yes—
208#383—22no—
209#383—22no—
210#382—21no—
211#382—22no—
212#382—22no—
213#381—22no—
214#380—22no—
215#380—22no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—22no—
221C4×C4⋊C4—22no—
222C4×C4⋊C4—21no—
223C4×C4⋊C4—21no—
224C4×C4⋊C4D411yescharacteristic
225C4×C4⋊C4D411yescharacteristic
226C4×C4⋊C4D411yescharacteristic
227C4×C4⋊C4D411yescharacteristic
228C4×C4⋊C4C2312yes—
229C4×C4⋊C4C2312yes—
230C4×C22⋊C4—22no—
231C4×C22⋊C4D412yes—
232C4×C22⋊C4D412yes—
233#376—22no—
234#376—22no—
235#376C2312yes—
236C2×#95—21no—
237C2×#95—21no—
238C43C2312yes—
239#340C2314yes—
240#340C2314yes—
241#340C2314yes—
242#340C2314yes—
243#336C2311yescharacteristic
244#336C2311yescharacteristic
245#336C2312yes—
246#336C2312yes—
247#336C2312yes—
248C2×#41—24no—
249C2×#41—24no—
250C2×#41—24no—
251C2×#41—24no—
252C2×#41—24no—
253C2×#41#4118yes—
254C22×Q8—21no—
255C22×Q8—22no—
256C22×Q8—24no—
257C22×Q8—24no—
258C22×Q8C2×D414yes—
259C22×Q8C2412yes—
260C22×Q8C2412yes—
261C22×D4—22no—
262C22×D4—21no—
263C22×D4—21no—
264C22×D4C2412yes—
265C23×C4—21no—
266C23×C4—22no—
267C23×C4—22no—
268C23×C4—21no—
269C23×C4—24no—
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2411yescharacteristic
273#137—264no—
274C2×Q16—28no—
275C2×Q16—28no—
276C2×Q16—28no—
277C2×SD16—28no—
278C2×SD16—28no—
279C2×SD16—28no—
280C2×M16—24no—
281C2×M16—24no—
282C2×M16#4118yes—
283C22×C8—24no—
284C22×C8—24no—
285C22×C8—24no—
286C22×C8C2412yes—
287C22×C8C2412yes—
288C22×C8C2414yes—
289C4⋊Q8—24no—
290#127—24no—
291#124—24no—
292#124—24no—
293#121—28no—
294#121—28no—
295C4×Q8—28no—
296C4×Q8—28no—
297C4×Q8—24no—
298C4×Q8—24no—
299C4×Q8—28no—
300C4×Q8—28no—
301C4×Q8C2×D418yes—
302C4×Q8C2×D418yes—
303C4×D4—28no—
304C4×D4—28no—
305C4×D4—28no—
306C4×D4—28no—
307C4×D4—416no—
308C4×D4—84no—
309C4×D4—84no—
310#117—22no—
311#117—24no—
312#117—216no—
313#117—24no—
314#117—48no—
315#117C2×D414yes—
316#117C2×D414yes—
317#117#4114yes—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—22no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—24no—
324C2×C4⋊C4—24no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—22no—
338C2×C4⋊C4—48no—
339C2×C4⋊C4—82no—
340C2×C4⋊C4—82no—
341C2×C4⋊C4C2×D411yescharacteristic
342C2×C4⋊C4C2×D411yescharacteristic
343C2×C4⋊C4C2×D411yescharacteristic
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2414yes—
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2411yescharacteristic
348C2×C4⋊C4C2411yescharacteristic
349C2×C4⋊C4C2411yescharacteristic
350C2×C4⋊C4C2411yescharacteristic
351C2×C22⋊C4—21no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4—24no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4C2×D412yes—
359C2×C22⋊C4C2×D412yes—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—21no—
363C2×C42—24no—
364C2×C42—24no—
365C2×C42—21no—
366C2×C42—22no—
367C2×C42—22no—
368C2×C42C2×D412yes—
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D412yes—
372C2×C42C2×D412yes—
373C2×C42C2×D412yes—
374C2×C42C2412yes—
375C2×C42C2411yescharacteristic
376C2×C42C2411yescharacteristic
377C2×C42C2412yes—
378#107—28no—
379#106—28no—
380C4⋊C8—28no—
381C4⋊C8—28no—
382#103—28no—
383#103—28no—
384#103—24no—
385#103—24no—
386#103—28no—
387#103—24no—
388#103—24no—
389#102—24no—
390#102—24no—
391#102—28no—
392#102—24no—
393#102—24no—
394#97—28no—
395C4×C8—28no—
396#95—22no—
397#95—22no—
398#95—22no—
399#95—22no—
400#95—21no—
401#95—21no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—22no—
407#95—22no—
408#95—24no—
409#95—24no—
410#95—22no—
411#95—22no—
412#95—21no—
413#95—21no—
414#95C2×D412yes—
415#95C2×D412yes—
416#95C2×D412yes—
417#95C2×D412yes—
418C24—22no—
419C24C22×D411yescharacteristic
420#41—216no—
421#41—216no—
422#41—48no—
423#41—48no—
424C2×Q8—24no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—216no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—24no—
432C2×Q8—24no—
433C2×Q8—22no—
434C2×Q8—42no—
435C2×Q8—48no—
436C2×Q8—42no—
437C2×Q8C22×D414yes—
438C2×Q8C2×#4114yes—
439C2×D4—24no—
440C2×D4—24no—
441C2×D4—28no—
442C2×D4—22no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—42no—
448C2×D4—42no—
449C2×D4—44no—
450C2×D4—44no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4C2×#4114yes—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—28no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—22no—
467C22×C4—24no—
468C22×C4—41no—
469C22×C4—41no—
470C22×C4—42no—
471C22×C4—42no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—42no—
476C22×C4—48no—
477C22×C4—84no—
478C22×C4—84no—
479C22×C4—82no—
480C22×C4—82no—
481C22×C4—81no—
482C22×C4—81no—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C22×D411yescharacteristic
490C22×C4C22×D411yescharacteristic
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D412yes—
493C22×C4C2×#4112yes—
494C22×C4C2×#4114yes—
495C22×C4C2×#4114yes—
496C22×C4C2511yescharacteristic
497Q16—232no—
498SD16—232no—
499M16—216no—
500C2×C8—24no—
501C2×C8—24no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—48no—
509C2×C8C2×#4118yes—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—28no—
513C4⋊C4—28no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—24no—
519C4⋊C4—24no—
520C4⋊C4—22no—
521C4⋊C4—22no—
522C4⋊C4—24no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—48no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—44no—
529C4⋊C4—42no—
530C4⋊C4—82no—
531C4⋊C4—81no—
532C4⋊C4—82no—
533C4⋊C4—81no—
534C4⋊C4—82no—
535C4⋊C4—82no—
536C4⋊C4#12114yes—
537C4⋊C4#12114yes—
538C4⋊C4C22×D412yes—
539C4⋊C4C22×D412yes—
540C22⋊C4—28no—
541C22⋊C4—28no—
542C22⋊C4—24no—
543C22⋊C4—24no—
544C22⋊C4—42no—
545C22⋊C4—48no—
546C22⋊C4—84no—
547C22⋊C4—84no—
548C22⋊C4#12118yes—
549C42—28no—
550C42—28no—
551C42—24no—
552C42—22no—
553C42—22no—
554C42—24no—
555C42—24no—
556C42—24no—
557C42—22no—
558C42—22no—
559C42—42no—
560C42—42no—
561C42—48no—
562C42—42no—
563C42—82no—
564C42—82no—
565C42C22×D414yes—
566C42C2×#4114yes—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—24no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23C2×#12112yes—
579C23#52512yes—
580C23#53212yes—
581C23C23×D411yescharacteristic
582Q8—28no—
583Q8—24no—
584Q8—24no—
585Q8—48no—
586Q8—42no—
587Q8—44no—
588Q8—42no—
589D4—24no—
590D4—24no—
591D4—48no—
592D4—48no—
593D4—416no—
594D4—42no—
595D4—42no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—22no—
607C2×C4—21no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—24no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—24no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—21no—
618C2×C4—22no—
619C2×C4—21no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—48no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—416no—
632C2×C4—41no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—41no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—168no—
645C2×C4—164no—
646C2×C4—162no—
647C2×C4C2×#12114yes—
648C2×C4C2×#12114yes—
649C2×C4C2×#12112yes—
650C2×C4#52512yes—
651C2×C4#52518yes—
652C2×C4#52918yes—
653C2×C4#53212yes—
654C2×C4#57712yes—
655C2×C4#57812yes—
656C2×C4C23×D411yescharacteristic
657C2×C4C22×#4112yes—
658C2×C4C22×#4112yes—
659C2×C4C22×#4111yescharacteristic
660C8—28no—
661C8—42no—
662C8—42no—
663C22—22no—
664C22—22no—
665C22—22no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—44no—
670C22—44no—
671C22—44no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22#239214yes—
677C22C22×#12111yescharacteristic
678C22C2×#52511yescharacteristic
679C22C2×#53211yescharacteristic
680C22C2×#57711yescharacteristic
681C22C2×#57811yescharacteristic
682C22#358511yescharacteristic
683C22#358611yescharacteristic
684C4—24no—
685C4—24no—
686C4—22no—
687C4—22no—
688C4—42no—
689C4—42no—
690C4—42no—
691C4—42no—
692C4—41no—
693C4—41no—
694C4—44no—
695C4—44no—
696C4—84no—
697C4—82no—
698C4—81no—
699C4—81no—
700C4#239214yes—
701C4C2×#52512yes—
702C4C2×#52912yes—
703C2—24no—
704C2—84no—
705C2C2×#239211yescharacteristic
706C2#6043111yescharacteristic
707C2#6043211yescharacteristic
708C2#6058511yescharacteristic
709C2#6058911yescharacteristic
710C2#6072311yescharacteristic
711C2#6072711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2947C2semidirect
2. #34254C2semidirect
4. #23695C2semidirect
5. #23694C2semidirect
6. #23685C2semidirect
7. #23684C2semidirect
8. #23679C2semidirect
9. #23678C2semidirect
10. #23658C2semidirect
11. C2×#2072C2semidirect
12. #23616C2semidirect
13. #23545C2semidirect
16. #20430C2semidirect
17. #20415C2semidirect
114. #519D4semidirect

Maximal subgroups

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

subgroupindex
C2×#29472
#342542
#342072
#236952
#236942
#236852
#236842
#236792
#236782
#236582
C2×#20722
#236162
#235452
#235042
#234292
#204302
#204152