Finite Group Databasebeta

← #162976

Subgroups — #162976

Order 512.

787 conjugacy classes of subgroups.

The subgroup lattice has 663 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35177C211yesmaximal, characteristic
2#22788C211yesmaximal, characteristic
3#20556C211yesmaximal, characteristic
4#7705C211yesmaximal, characteristic, complemented
5#7695C211yesmaximal, characteristic, complemented
6#7688C211yesmaximal, characteristic, complemented
7#7685C211yesmaximal, characteristic, complemented
8#7681C211yesmaximal, characteristic, complemented
9#7675C211yesmaximal, characteristic, complemented
10#7668C211yesmaximal, characteristic, complemented
11#7661C211yesmaximal, characteristic, complemented
12#7638C211yesmaximal, characteristic, complemented
13#7630C211yesmaximal, characteristic, complemented
14#7625C211yesmaximal, characteristic, complemented
15#7611C211yesmaximal, characteristic, complemented
16C2×#435—21no—
17#2857C2211yescharacteristic
18#2857C2211yescharacteristic
19#2855C2211yescharacteristic
20#2855C2211yescharacteristic
21#2665—21no—
22#2665—21no—
23#2567C411yescharacteristic
24#2563C411yescharacteristic
25#2551C411yescharacteristic
26#2538C411yescharacteristic
27#2509—21no—
28#2497C411yescharacteristic
29#2497C411yescharacteristic
30#2492C411yescharacteristic
31#2492C411yescharacteristic
32#2465—21no—
33#2465—21no—
34#2452—21no—
35#2331—21no—
36#2319—21no—
37C4×C4⋊Q8C2211yescharacteristic
38#2295C2211yescharacteristic
39C42×D4C2211yescharacteristic
40#1924—21no—
41#1916—21no—
42#1870—21no—
43#1839C2211yescharacteristic
44#1817C2211yescharacteristic
45#1797C2211yescharacteristic
46#1771—21no—
47#1767C2211yescharacteristic
48#1753—21no—
49#1753—21no—
50#1752—21no—
51#1749C2211yescharacteristic
52C4×C22⋊C8C2211yescharacteristic
53C4×C22⋊C8C2211yescharacteristic
54#1745C2211yescharacteristic
55#1745C2211yescharacteristic
56#1728C2211yescharacteristic
57#1728C2211yescharacteristic
58C2×#336C2211yescharacteristic
59#1272C2211yescharacteristic
60#1272C2211yescharacteristic
61#1272C2211yescharacteristic
62#1272C2211yescharacteristic
63#1271C2211yescharacteristic
64#1271C2211yescharacteristic
65#1271C2211yescharacteristic
66#1271C2211yescharacteristic
67#1271C2211yescharacteristic
68#1271C2211yescharacteristic
69#1271C2211yescharacteristic
70#1271C2211yescharacteristic
71#1271C2211yescharacteristic
72#1271C2211yescharacteristic
73#1271C2211yescharacteristic
74#1271C2211yescharacteristic
75C22×M16—21no—
76C22×M16—21no—
77C23×C8—21no—
78C2×C4⋊Q8—21no—
79C2×C4⋊Q8C2×C411yescharacteristic
80C2×C4⋊Q8C2×C411yescharacteristic
81C2×C4⋊Q8C2×C411yescharacteristic
82C2×C22⋊Q8—21no—
83C2×C22⋊Q8—21no—
84C2×C22⋊Q8C2×C411yescharacteristic
85C2×C22⋊Q8C2×C411yescharacteristic
86C2×C4×Q8—21no—
87C2×C4×Q8C2×C411yescharacteristic
88C2×C4×D4C2×C411yescharacteristic
89C2×C4×D4D411yescharacteristic
90C2×C4×D4D411yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93#435—28no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×#97—21no—
102C2×#97—21no—
103C2×#97—21no—
104C2×#97—21no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107#400—21no—
108#400—21no—
109#400—21no—
110#400—21no—
111#396—21no—
112#396—21no—
113#396—21no—
114#396—21no—
115#395—21no—
116#395—21no—
117#395—21no—
118#395—21no—
119#393—21no—
120#393—21no—
121#393—21no—
122#393—21no—
123#391—21no—
124#391—21no—
125#391—21no—
126#391—21no—
127#391—21no—
128#391C2×C411yescharacteristic
129#391C2×C411yescharacteristic
130#391C2×C411yescharacteristic
131#391C2311yescharacteristic
132#390—21no—
133#390—21no—
134#389—21no—
135#389—21no—
136#389—21no—
137#389—21no—
138#389—21no—
139#389—21no—
140#389—21no—
141#389C2×C411yescharacteristic
142#389C2×C411yescharacteristic
143#389C2×C411yescharacteristic
144#389C2311yescharacteristic
145#389C2311yescharacteristic
146#389C2311yescharacteristic
147#388—21no—
148#388—21no—
149#388—21no—
150#388—21no—
151#387—21no—
152#387—21no—
153#387—21no—
154#387—21no—
155#387—21no—
156#387—21no—
157#387—21no—
158#387—21no—
159#385—21no—
160#385—21no—
161#384C2×C411yescharacteristic
162#384C2×C411yescharacteristic
163#384C2×C411yescharacteristic
164#382—21no—
165#381—21no—
166#381—21no—
167#380—21no—
168#380—21no—
169#380C2×C411yescharacteristic
170#380C2×C411yescharacteristic
171#380C2×C411yescharacteristic
172#380C2×C411yescharacteristic
173#380C2×C411yescharacteristic
174#380C2×C411yescharacteristic
175C4×C4⋊C4—21no—
176C4×C4⋊C4—21no—
177C4×C4⋊C4C2×C411yescharacteristic
178C4×C4⋊C4C2×C411yescharacteristic
179C4×C4⋊C4C2×C411yescharacteristic
180C4×C4⋊C4C2×C411yescharacteristic
181C4×C4⋊C4D411yescharacteristic
182C4×C4⋊C4D411yescharacteristic
183C4×C22⋊C4C2×C411yescharacteristic
184C4×C22⋊C4C2×C411yescharacteristic
185C4×C22⋊C4D411yescharacteristic
186C4×C22⋊C4D411yescharacteristic
187C4×C22⋊C4D411yescharacteristic
188C4×C22⋊C4D411yescharacteristic
189#376—21no—
190C43C2311yescharacteristic
191#336—21no—
192#336—21no—
193#336—21no—
194#336—21no—
195#336—21no—
196#336—21no—
197#336—21no—
198#336—21no—
199#336C2311yescharacteristic
200#336C2311yescharacteristic
201#336C2311yescharacteristic
202#336C2311yescharacteristic
203#336C2311yescharacteristic
204#336C2311yescharacteristic
205#336C2311yescharacteristic
206#336C2311yescharacteristic
207C22×Q8—21no—
208C22×Q8—21no—
209C22×Q8—21no—
210C22×Q8C22×C411yescharacteristic
211C22×D4C22⋊C411yescharacteristic
212C23×C4C22×C411yescharacteristic
213C23×C4C22×C411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C2×M16—24no—
219C2×M16—42no—
220C2×M16—42no—
221C2×M16—81no—
222C2×M16—81no—
223C2×M16—81no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—24no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—81no—
234C22×C8—81no—
235C4⋊Q8—42no—
236C4⋊Q8—44no—
237C4⋊Q8—81no—
238C4⋊Q8—81no—
239C22⋊Q8—81no—
240C22⋊Q8—81no—
241C22⋊Q8—81no—
242C22⋊Q8—81no—
243C22⋊Q8—81no—
244C22⋊Q8—81no—
245C4×Q8—42no—
246C4×Q8—81no—
247C4×Q8—81no—
248C4×D4—22no—
249C4×D4—22no—
250C4×D4—24no—
251C4×D4—42no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4C22⋊C411yescharacteristic
271C2×C4⋊C4C22⋊C411yescharacteristic
272C2×C4⋊C4C22⋊C411yescharacteristic
273C2×C4⋊C4C22×C411yescharacteristic
274C2×C4⋊C4C22×C411yescharacteristic
275C2×C4⋊C4C22×C411yescharacteristic
276C2×C4⋊C4C22×C411yescharacteristic
277C2×C4⋊C4C22×C411yescharacteristic
278C2×C4⋊C4C22×C411yescharacteristic
279C2×C4⋊C4C22×C411yescharacteristic
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4C22⋊C411yescharacteristic
289C2×C22⋊C4C22⋊C411yescharacteristic
290C2×C22⋊C4C22⋊C411yescharacteristic
291C2×C22⋊C4C22⋊C411yescharacteristic
292C2×C22⋊C4C22⋊C411yescharacteristic
293C2×C22⋊C4C22⋊C411yescharacteristic
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—81no—
299C2×C42—81no—
300C2×C42C22×C411yescharacteristic
301C2×C42C22×C411yescharacteristic
302C2×C42C22×C411yescharacteristic
303C2×C42C22×C411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2411yescharacteristic
313C4⋊C8—24no—
314C4⋊C8—81no—
315C22⋊C8—24no—
316C22⋊C8—24no—
317C22⋊C8—42no—
318C22⋊C8—42no—
319C22⋊C8—81no—
320#97—24no—
321#97—42no—
322#97—81no—
323#97—81no—
324C4×C8—81no—
325C4×C8—81no—
326#95—21no—
327#95—21no—
328#95—21no—
329#95—21no—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—21no—
338#95—21no—
339#95—21no—
340#95—21no—
341#95—21no—
342#95—21no—
343#95—21no—
344#95—21no—
345#95—21no—
346#95—21no—
347#95—21no—
348#95—21no—
349#95C22⋊C411yescharacteristic
350#95C22⋊C411yescharacteristic
351#95C22⋊C411yescharacteristic
352#95C22⋊C411yescharacteristic
353#95C22⋊C411yescharacteristic
354#95C22⋊C411yescharacteristic
355C24C2×C22⋊C411yescharacteristic
356C24C2×C22⋊C411yescharacteristic
357C2×Q8—22no—
358C2×Q8—42no—
359C2×Q8—41no—
360C2×Q8—41no—
361C2×Q8—42no—
362C2×Q8—42no—
363C2×Q8—81no—
364C2×Q8—81no—
365C2×Q8—81no—
366C2×Q8—81no—
367C2×Q8—81no—
368C2×Q8—81no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—41no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—41no—
391C22×C4—41no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4C2×C22⋊C411yescharacteristic
402C22×C4C2×C22⋊C411yescharacteristic
403C22×C4C2×C22⋊C411yescharacteristic
404C22×C4C2×C22⋊C411yescharacteristic
405C22×C4C2×C22⋊C411yescharacteristic
406C22×C4C2×C22⋊C411yescharacteristic
407C22×C4C2×C22⋊C411yescharacteristic
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4#13712yes—
411C22×C4#13712yes—
412C22×C4C23×C411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D411yescharacteristic
415M16—84no—
416M16—84no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—22no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—41no—
432C2×C8—41no—
433C2×C8—41no—
434C2×C8—41no—
435C2×C8—41no—
436C2×C8—41no—
437C2×C8—41no—
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C2×C8—84no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—42no—
462C4⋊C4—81no—
463C4⋊C4—81no—
464C4⋊C4—81no—
465C4⋊C4—81no—
466C4⋊C4—81no—
467C4⋊C4—81no—
468C4⋊C4—81no—
469C4⋊C4—81no—
470C4⋊C4—81no—
471C4⋊C4—81no—
472C4⋊C4#10414yes—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—41no—
478C22⋊C4—41no—
479C22⋊C4—81no—
480C22⋊C4—81no—
481C22⋊C4—81no—
482C22⋊C4—81no—
483C22⋊C4—81no—
484C22⋊C4—81no—
485C22⋊C4—81no—
486C22⋊C4—81no—
487C42—22no—
488C42—21no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—22no—
493C42—41no—
494C42—41no—
495C42—41no—
496C42—41no—
497C42—41no—
498C42—41no—
499C42—41no—
500C42—41no—
501C42—41no—
502C42—42no—
503C42—42no—
504C42—81no—
505C42—81no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23#41912yes—
520C23C22×C22⋊C411yescharacteristic
521Q8—82no—
522Q8—84no—
523Q8—162no—
524Q8—162no—
525D4—44no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—81no—
589C2×C4—81no—
590C2×C4—82no—
591C2×C4—82no—
592C2×C4#41912yes—
593C2×C4C2×#10412yes—
594C2×C4C2×#10412yes—
595C2×C4C2×#10412yes—
596C2×C4#56812yes—
597C2×C4C2×#13711yescharacteristic
598C2×C4C2×#13711yescharacteristic
599C2×C4#57611yescharacteristic
600C2×C4#57711yescharacteristic
601C8—42no—
602C8—42no—
603C8—42no—
604C8—42no—
605C8—42no—
606C8—81no—
607C8—81no—
608C8—81no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22—41no—
616C22—41no—
617C22—41no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22#287911yescharacteristic
624C22#288311yescharacteristic
625C22#288911yescharacteristic
626C22C2×#41911yescharacteristic
627C22#289711yescharacteristic
628C22C22×#10411yescharacteristic
629C22#290111yescharacteristic
630C4—22no—
631C4—21no—
632C4—21no—
633C4—22no—
634C4—22no—
635C4—21no—
636C4—21no—
637C4—22no—
638C4—22no—
639C4—41no—
640C4—41no—
641C4—41no—
642C4—41no—
643C4—41no—
644C4—41no—
645C4—81no—
646C4—81no—
647C4—81no—
648C4—81no—
649C4—81no—
650C4—81no—
651C4—81no—
652C4—81no—
653C2—42no—
654C2—81no—
655C2#1740711yescharacteristic
656C2#1750111yescharacteristic
657C2#1750511yescharacteristic
658C2#1754711yescharacteristic
659C2#1762111yescharacteristic
660C2#1765411yescharacteristic
661C2#1776111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7705C2semidirect
5. #7695C2semidirect
6. #7688C2semidirect
7. #7685C2semidirect
8. #7681C2semidirect
9. #7675C2semidirect
10. #7668C2semidirect
11. #7661C2semidirect
12. #7638C2semidirect
13. #7630C2semidirect
14. #7625C2semidirect
15. #7611C2semidirect

Maximal subgroups

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

subgroupindex
#351772
#227882
#205562
#77052
#76952
#76882
#76852
#76812
#76752
#76682
#76612
#76382
#76302
#76252
#76112