Finite Group Databasebeta

← #6529812

Subgroups — #6529812

Order 512.

1974 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
1C2×#3101C211yesmaximal, characteristic
2C2×#2984C211yesmaximal, characteristic, complemented
3#35248C211yesmaximal, characteristic, complemented
4C2×#2580C211yesmaximal, characteristic
5#28270C212yesmaximal, complemented
6#28266C212yesmaximal, complemented
7#28192C212yesmaximal, complemented
8#28189C212yesmaximal, complemented
9#25485C214yesmaximal, complemented
10#25308C214yesmaximal, complemented
11#24609C212yesmaximal, complemented
12#24601C212yesmaximal, complemented
13#24507C212yesmaximal
14#24501C212yesmaximal
15#21983C211yesmaximal, characteristic, complemented
16#21866C211yesmaximal, characteristic, complemented
17#20739C211yesmaximal, characteristic
18C2×#533C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21#3101—28no—
22C2×#483C2211yescharacteristic
23C2×#483C2211yescharacteristic
24C2×#476C2212yes—
25C2×#461C2212yes—
26C2×#450C2211yescharacteristic
27C2×#450C2211yescharacteristic
28#2984—28no—
29C2×#446C2212yescomplemented
30C2×#446C2214yescomplemented
31C2×C8×Q8C2212yescomplemented
32C2×C4×SD16C2212yes—
33C2×C4×SD16C2212yes—
34C2×#432C2211yescharacteristic
35C2×#432C2212yes—
36C2×#431C2212yes—
37C2×#431C2211yescharacteristic
38#2855C2212yes—
39#2855C2212yes—
40#2634—21no—
41#2634—21no—
42#2581C2211yescharacteristic, complemented
43#2581C2211yescharacteristic, complemented
44#2581C2211yescharacteristic, complemented
45#2581C2211yescharacteristic, complemented
46#2580—22no—
47#2580—21no—
48#2580—21no—
49#2580C2212yes—
50#2580C2212yes—
51#2580C2214yes—
52#2568C2212yescomplemented
53#2546C2212yescomplemented
54#2503—21no—
55#2484—22no—
56#2434—21no—
57C2×#399—21no—
58C2×#393—21no—
59#2327—22no—
60C2×#390C2212yes—
61C2×#389C2212yes—
62C2×#385—22no—
63#2283C2211yescharacteristic
64C2×#382C2211yescharacteristic
65#2088C2214yescomplemented
66#2088C2214yescomplemented
67#2068C2214yescomplemented
68#2067C2214yes—
69#2066C2214yescomplemented
70#2065C2214yes—
71#2055C2214yes—
72#2055C2214yes—
73#1993C2212yescomplemented
74#1973C2214yes—
75#1968C2214yes—
76#1960C2212yescomplemented
77#1949C2214yescomplemented
78#1943C2214yescomplemented
79#1941C2214yescomplemented
80#1936C2214yes—
81#1935C2214yescomplemented
82#1934C2214yes—
83#1933C2214yescomplemented
84#1872C2212yes—
85#1872C2212yes—
86#1853C2212yes—
87#1852C2212yes—
88#1846C2212yescomplemented
89#1835C2212yescomplemented
90#1823C2212yes—
91#1805C2212yescomplemented
92#1805C2212yescomplemented
93#1804C2212yes—
94#1804C2212yes—
95#1786C2212yes—
96#1762C2214yes—
97C23×Q8D411yescharacteristic
98C23×D4C2311yescharacteristic
99C24×C4—21no—
100C22×SD16C2312yes—
101C22×SD16C2312yes—
102#533—24no—
103#533—24no—
104C2×C4⋊Q8—22no—
105C2×C4⋊Q8C2311yescharacteristic
106C2×C4⋊Q8C2311yescharacteristic
107C2×#125—21no—
108C2×C22⋊Q8—21no—
109C2×C22⋊Q8C2311yescharacteristic
110C2×C22⋊Q8C2311yescharacteristic
111C2×C4×Q8—24no—
112C2×C4×Q8—21no—
113C2×C4×Q8—24no—
114C2×C4×Q8D412yes—
115C2×C4×Q8D411yescharacteristic
116C2×C4×Q8D411yescharacteristic
117C2×C4×Q8C2311yescharacteristic
118C2×C4×Q8C2311yescharacteristic
119C2×C4×Q8C2312yes—
120C2×C4×Q8C2311yescharacteristic
121C2×C4×Q8C2311yescharacteristic
122C2×C4×D4—21no—
123C2×C4×D4—21no—
124C2×C4×D4—24no—
125C2×C4×D4—24no—
126C2×C4×D4C2312yes—
127C2×C4×D4C2312yes—
128C2×#117—21no—
129C22×C4⋊C4—22no—
130C22×C4⋊C4D411yescharacteristic
131C22×C4⋊C4D412yes—
132C22×C4⋊C4C2311yescharacteristic
133C22×C4⋊C4C2312yes—
134C22×C22⋊C4—21no—
135C22×C22⋊C4—22no—
136C22×C22⋊C4—22no—
137C22×C42C2312yes—
138C22×C42C2311yescharacteristic
139#483—24no—
140#483—24no—
141#476—28no—
142#461—28no—
143#450—24no—
144#450—24no—
145#446—216no—
146#446—28no—
147C8×Q8—28no—
148C4×SD16—28no—
149C4×SD16—28no—
150#432—24no—
151#432—28no—
152#431—28no—
153#431D418yes—
154C2×#107C2314yes—
155C2×#106C2312yes—
156C2×#106C2312yes—
157C2×#106C2314yes—
158C2×C4⋊C8C2314yes—
159C2×C4⋊C8C2314yes—
160C2×C4⋊C8C2312yes—
161C2×C4⋊C8C2312yes—
162C2×#103C2312yes—
163C2×#103C2312yes—
164C2×#103C2314yes—
165C2×#103C2314yes—
166C2×#102C2314yes—
167C2×#102C2312yes—
168C2×#102C2312yes—
169C2×#102C2314yes—
170C2×C22⋊C8C2312yes—
171C2×C22⋊C8C2311yescharacteristic
172C2×C22⋊C8C2311yescharacteristic
173C2×#97C2314yes—
174C2×#97C2312yes—
175C2×C4×C8C2314yes—
176C2×C4×C8C2312yes—
177#400—22no—
178#400—22no—
179#400—21no—
180#400—21no—
181#400—21no—
182#400—21no—
183#399—24no—
184#399—22no—
185#399—22no—
186#398—22no—
187#398—22no—
188#395—21no—
189#395—21no—
190#395—21no—
191#395—21no—
192#393—22no—
193#393—24no—
194#393—22no—
195#391—22no—
196#391—22no—
197#391C2312yes—
198#391C2312yes—
199#391C2312yes—
200#391C2312yes—
201#390—22no—
202#390—22no—
203#390C2314yes—
204#390C2314yes—
205#389—22no—
206#389—22no—
207#389C2312yes—
208#389C2314yes—
209#389C2312yes—
210#389C2314yes—
211#389C2312yes—
212#389C2312yes—
213#387—22no—
214#387—22no—
215#387—22no—
216#387—22no—
217#387—22no—
218#387—22no—
219#387—22no—
220#387—22no—
221#385—22no—
222#385—22no—
223#385—24no—
224#385—24no—
225#385—24no—
226#385—24no—
227#383—21no—
228#382—22no—
229#382—21no—
230#382—21no—
231#382C2311yescharacteristic
232#382C2311yescharacteristic
233#382C2312yes—
234#382C2312yes—
235#381—22no—
236#381—22no—
237#381—21no—
238#380C2312yes—
239C4×C4⋊C4C2312yes—
240C4×C22⋊C4—21no—
241#376—21no—
242C2×#95—21no—
243C2×#95—22no—
244C2×#95—21no—
245#340C2314yes—
246#340C2314yes—
247#340C2314yes—
248#340C2314yes—
249#340C2314yes—
250#340C2314yes—
251C25—21no—
252C22×Q8—24no—
253C22×Q8—21no—
254C22×Q8—44no—
255C22×Q8C2×D411yescharacteristic
256C22×Q8C2×D412yes—
257C22×Q8C2×D411yescharacteristic
258C22×Q8C2411yescharacteristic
259C22×Q8C2411yescharacteristic
260C22×D4—24no—
261C22×D4—21no—
262C22×D4—22no—
263C22×D4—22no—
264C22×D4—42no—
265C22×D4C2412yes—
266C23×C4—22no—
267C23×C4—21no—
268C23×C4—22no—
269C23×C4—22no—
270C23×C4—44no—
271C23×C4—84no—
272C23×C4C2×D412yes—
273C23×C4C2×D412yes—
274C23×C4C2×D411yescharacteristic
275C23×C4C2×D411yescharacteristic
276C23×C4C2411yescharacteristic
277C2×SD16—24no—
278C2×SD16—24no—
279C2×SD16—24no—
280C2×SD16—24no—
281C2×SD16—42no—
282C2×SD16—42no—
283C22×C8C2414yes—
284C22×C8C2412yes—
285C22×C8C2412yes—
286C4⋊Q8—22no—
287C4⋊Q8—22no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—48no—
291#125—44no—
292C22⋊Q8—22no—
293C22⋊Q8—22no—
294C22⋊Q8—22no—
295C22⋊Q8—22no—
296C22⋊Q8—44no—
297C4×Q8—28no—
298C4×Q8—24no—
299C4×Q8—24no—
300C4×Q8—22no—
301C4×Q8—24no—
302C4×Q8—22no—
303C4×Q8—22no—
304C4×Q8—22no—
305C4×Q8—24no—
306C4×Q8—22no—
307C4×Q8—22no—
308C4×Q8—22no—
309C4×Q8—22no—
310C4×Q8—44no—
311C4×D4—24no—
312C4×D4—24no—
313C4×D4—24no—
314C4×D4—24no—
315C4×D4—44no—
316C4×D4—44no—
317#117—44no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—28no—
320C2×C4⋊C4—22no—
321C2×C4⋊C4—22no—
322C2×C4⋊C4—21no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—24no—
336C2×C4⋊C4—24no—
337C2×C4⋊C4—44no—
338C2×C4⋊C4—84no—
339C2×C4⋊C4C2×D412yes—
340C2×C4⋊C4C2×D412yes—
341C2×C4⋊C4C2×D412yes—
342C2×C4⋊C4C2×D412yes—
343C2×C4⋊C4C2×D412yes—
344C2×C4⋊C4C2×D411yescharacteristic
345C2×C4⋊C4C2×D411yescharacteristic
346C2×C4⋊C4C2412yes—
347C2×C4⋊C4C2414yes—
348C2×C4⋊C4C2411yescharacteristic
349C2×C4⋊C4C2411yescharacteristic
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2412yes—
352C2×C22⋊C4—21no—
353C2×C22⋊C4—21no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—24no—
358C2×C22⋊C4—24no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—24no—
361C2×C22⋊C4—48no—
362C2×C22⋊C4—44no—
363C2×C22⋊C4—84no—
364C2×C42—24no—
365C2×C42—24no—
366C2×C42—21no—
367C2×C42—21no—
368C2×C42—22no—
369C2×C42C2×D412yes—
370C2×C42C2×D412yes—
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D414yes—
374C2×C42#4114yes—
375C2×C42C2412yes—
376C2×C42C2412yes—
377C2×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379#107—44no—
380#106—24no—
381#106—24no—
382#106—44no—
383C4⋊C8—28no—
384C4⋊C8—28no—
385C4⋊C8—24no—
386C4⋊C8C2×D418yes—
387#103—24no—
388#103—24no—
389#103—44no—
390#103—44no—
391#102—28no—
392#102—24no—
393#102—24no—
394#102—44no—
395C22⋊C8—24no—
396C22⋊C8C2×D414yes—
397C22⋊C8C2×D414yes—
398#97—28no—
399#97—24no—
400C4×C8—28no—
401C4×C8C2×D418yes—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—24no—
407#95—24no—
408#95—22no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426C24—24no—
427C24—22no—
428C24—44no—
429C24—42no—
430C24—42no—
431C24—44no—
432C24C22×D411yescharacteristic
433C2×Q8—22no—
434C2×Q8—24no—
435C2×Q8—22no—
436C2×Q8—24no—
437C2×Q8—24no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—42no—
447C2×Q8—42no—
448C2×Q8—44no—
449C2×Q8—416no—
450C2×Q8—42no—
451C2×D4—24no—
452C2×D4—22no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—48no—
458C2×D4—48no—
459C2×D4—42no—
460C2×D4—42no—
461C2×D42−1+414yes—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—24no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—42no—
479C22×C4—42no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—42no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—84no—
488C22×C4—84no—
489C22×C4—816no—
490C22×C4#12114yes—
491C22×C4#12114yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C2×#4112yes—
503C22×C4C2511yescharacteristic
504SD16—48no—
505SD16—48no—
506C2×C8—22no—
507C2×C8—22no—
508C2×C8—22no—
509C2×C8—22no—
510C2×C8—24no—
511C2×C8—24no—
512C2×C8—24no—
513C2×C8C22×D414yes—
514C2×C8C22×D414yes—
515C4⋊C4—24no—
516C4⋊C4—24no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—24no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—22no—
525C4⋊C4—22no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—24no—
529C4⋊C4—24no—
530C4⋊C4—24no—
531C4⋊C4—24no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—44no—
544C4⋊C4—44no—
545C4⋊C4—84no—
546C4⋊C4—84no—
547C22⋊C4—48no—
548C22⋊C4—42no—
549C22⋊C4—42no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—48no—
554C22⋊C4—84no—
555C22⋊C4—84no—
556C42—24no—
557C42—24no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—44no—
563C42—42no—
564C42—42no—
565C42#12114yes—
566C42#12114yes—
567C42C22×D412yes—
568C42C22×D412yes—
569C42C2×#4112yes—
570C42C2×#4112yes—
571C23—21no—
572C23—21no—
573C23—22no—
574C23—24no—
575C23—48no—
576C23—416no—
577C23—42no—
578C23—42no—
579C23—44no—
580C23—44no—
581C23—84no—
582C23—88no—
583C23—84no—
584C23C2×#12112yes—
585C23#57714yes—
586C23C23×D411yescharacteristic
587Q8—28no—
588Q8—48no—
589Q8—44no—
590Q8—42no—
591Q8—42no—
592Q8—42no—
593Q8—42no—
594Q8—84no—
595D4—24no—
596D4—24no—
597D4—48no—
598D4—42no—
599D4—42no—
600C2×C4—22no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—28no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—22no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—44no—
621C2×C4—48no—
622C2×C4—48no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—88no—
645C2×C4—88no—
646C2×C4—88no—
647C2×C4—84no—
648C2×C4C2×#12112yes—
649C2×C4C2×#12112yes—
650C2×C4C2×#12112yes—
651C2×C4C2×#12112yes—
652C2×C4C2×#12112yes—
653C2×C4#57514yes—
654C2×C4C23×D411yescharacteristic
655C2×C4C22×#4111yescharacteristic
656C2×C4C2×2−1+412yes—
657C2×C4#58512yes—
658C8—24no—
659C8—24no—
660C8—44no—
661C22—22no—
662C22—22no—
663C22—24no—
664C22—24no—
665C22—44no—
666C22—48no—
667C22—816no—
668C22—82no—
669C22—82no—
670C22—84no—
671C22#305714yes—
672C22C22×#12111yescharacteristic
673C22#348611yescharacteristic
674C22#348711yescharacteristic
675C22C2×#57512yes—
676C22C2×#57712yes—
677C4—22no—
678C4—22no—
679C4—24no—
680C4—22no—
681C4—22no—
682C4—22no—
683C4—42no—
684C4—42no—
685C4—42no—
686C4—81no—
687C4—81no—
688C4—82no—
689C4—81no—
690C4—81no—
691C4—82no—
692C4—82no—
693C4—84no—
694C4#348912yes—
695C4#349512yes—
696C2—24no—
697C2—84no—
698C2#3566111yescharacteristic
699C2#6044411yescharacteristic
700C2C2×#305711yescharacteristic
701C2#6184112yes—
702C2#6190412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
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×#2984C2semidirect
3. #35248C2semidirect
5. #28270C2semidirect
6. #28266C2semidirect
7. #28192C2semidirect
8. #28189C2semidirect
9. #25485C2semidirect
10. #25308C2semidirect
11. #24609C2semidirect
12. #24601C2semidirect
15. #21983C2semidirect
16. #21866C2semidirect
29. C2×#446C22semidirect
30. C2×#446C22semidirect
31. C2×C8×Q8C22semidirect
42. #2581C22semidirect
43. #2581C22semidirect
44. #2581C22semidirect
45. #2581C22semidirect
52. #2568C22semidirect
53. #2546C22semidirect
65. #2088C22semidirect
66. #2088C22semidirect
67. #2068C22semidirect
69. #2066C22semidirect
73. #1993C22semidirect
76. #1960C22semidirect
77. #1949C22semidirect
78. #1943C22semidirect
79. #1941C22semidirect
81. #1935C22semidirect
83. #1933C22semidirect
88. #1846C22semidirect
89. #1835C22semidirect
91. #1805C22semidirect
92. #1805C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31012
C2×#29842
#352482
C2×#25802
#282702
#282662
#281922
#281892
#254852
#253082
#246092
#246012
#245072
#245012
#219832
#218662
#207392