Finite Group Databasebeta

← #164201

Subgroups — #164201

Order 512.

847 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36390C211yesmaximal, characteristic
2#22590C211yesmaximal, characteristic, complemented
3#20430C211yesmaximal, characteristic, complemented
4#7713C211yesmaximal, characteristic, complemented
5#7695C211yesmaximal, characteristic, complemented
6#7689C211yesmaximal, characteristic, complemented
7#7682C211yesmaximal, characteristic, complemented
8#7681C211yesmaximal, characteristic, complemented
9#7662C211yesmaximal, characteristic, complemented
10#7635C211yesmaximal, characteristic, complemented
11#7631C211yesmaximal, characteristic
12#7614C211yesmaximal, characteristic, complemented
13#7584C211yesmaximal, characteristic, complemented
14#7578C211yesmaximal, characteristic
15#7560C211yesmaximal, characteristic, complemented
16C2×#424—21no—
17C2×#405—21no—
18#2863C2211yescharacteristic
19#2862C2211yescharacteristic
20#2855C2211yescharacteristic
21#2609—21no—
22#2576—21no—
23#2572—21no—
24#2527—21no—
25#2515—21no—
26#2514C411yescharacteristic
27#2477—21no—
28#2452C411yescharacteristic
29#2447C411yescharacteristic
30#2446C411yescharacteristic
31#2439C411yescharacteristic
32#2432C411yescharacteristic
33#2368—21no—
34#2362C411yescharacteristic
35#2343—21no—
36#2339C411yescharacteristic
37C4×#124C2211yescharacteristic
38#2297C2211yescharacteristic
39#2293C2211yescharacteristic
40C4×#117C2211yescharacteristic
41#1917—21no—
42#1916—21no—
43#1851—21no—
44#1839C2211yescharacteristic
45#1831C2211yescharacteristic
46#1817C2211yescharacteristic
47#1768—21no—
48#1768—21no—
49#1767C2211yescharacteristic
50#1767C2211yescharacteristic
51#1767C2211yescharacteristic
52C4×C4⋊C8C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54#1753—21no—
55#1744C2211yescharacteristic
56#1728C2211yescharacteristic
57#1728C2211yescharacteristic
58C2×#336C2211yescharacteristic
59#1272C2211yescharacteristic
60#1272C2211yescharacteristic
61#1272C2211yescharacteristic
62#1272C2211yescharacteristic
63#1272C2211yescharacteristic
64#1272C2211yescharacteristic
65#1272C2211yescharacteristic
66#1272C2211yescharacteristic
67#1271C2211yescharacteristic
68#1271C2211yescharacteristic
69#1271C2211yescharacteristic
70#1271C2211yescharacteristic
71#1270C2211yescharacteristic
72#1270C2211yescharacteristic
73#1270C2211yescharacteristic
74#1270C2211yescharacteristic
75C22×#41C2×C411yescharacteristic
76C22×M16—21no—
77C22×M16—21no—
78C23×C8—21no—
79C2×C4⋊Q8C2×C411yescharacteristic
80C2×C4⋊Q8C2×C411yescharacteristic
81C2×#124C2×C411yescharacteristic
82C2×C22⋊Q8—21no—
83C2×C22⋊Q8C2×C411yescharacteristic
84C2×C4×Q8C2×C411yescharacteristic
85C2×C4×D4C2×C411yescharacteristic
86C2×C4×D4C2×C411yescharacteristic
87C2×#117—21no—
88C2×#117C2×C411yescharacteristic
89C2×#117D411yescharacteristic
90C2×#117D411yescharacteristic
91C22×C42C2311yescharacteristic
92#424—28no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C22⋊C8—21no—
100#405—44no—
101C2×#97—21no—
102C2×#97—21no—
103C2×#97—21no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107#401—21no—
108#400—21no—
109#400—21no—
110#398—21no—
111#397—21no—
112#397—21no—
113#396—21no—
114#396—21no—
115#394—21no—
116#394—21no—
117#393—21no—
118#393—21no—
119#391—21no—
120#391—21no—
121#391—21no—
122#391C2×C411yescharacteristic
123#391C2×C411yescharacteristic
124#391C2×C411yescharacteristic
125#391C2×C411yescharacteristic
126#391C2×C411yescharacteristic
127#391C2×C411yescharacteristic
128#391C2311yescharacteristic
129#391C2311yescharacteristic
130#390C2×C411yescharacteristic
131#390C2×C411yescharacteristic
132#390C2311yescharacteristic
133#389C2×C411yescharacteristic
134#389C2×C411yescharacteristic
135#389C2×C411yescharacteristic
136#389C2×C411yescharacteristic
137#389C2311yescharacteristic
138#388—21no—
139#388—21no—
140#388—21no—
141#388—21no—
142#388—21no—
143#388—21no—
144#387—21no—
145#387—21no—
146#387—21no—
147#387—21no—
148#387—21no—
149#387—21no—
150#387—21no—
151#387—21no—
152#387—21no—
153#387—21no—
154#386—21no—
155#386—21no—
156#386—21no—
157#386—21no—
158#385—21no—
159#385—21no—
160#385—21no—
161#385—21no—
162#385—21no—
163#383—21no—
164#383—21no—
165#382—21no—
166#382—21no—
167#382C2×C411yescharacteristic
168#381—21no—
169#381—21no—
170#381—21no—
171#380C2×C411yescharacteristic
172C4×C4⋊C4—21no—
173C4×C4⋊C4C2×C411yescharacteristic
174C4×C4⋊C4C2×C411yescharacteristic
175C4×C4⋊C4C2×C411yescharacteristic
176C4×C4⋊C4D411yescharacteristic
177C4×C4⋊C4D411yescharacteristic
178C4×C22⋊C4—21no—
179C4×C22⋊C4—21no—
180C4×C22⋊C4—21no—
181C4×C22⋊C4—21no—
182C4×C22⋊C4—21no—
183C4×C22⋊C4C2×C411yescharacteristic
184C4×C22⋊C4C2×C411yescharacteristic
185C4×C22⋊C4D411yescharacteristic
186C4×C22⋊C4D411yescharacteristic
187#376D411yescharacteristic
188#376D411yescharacteristic
189C2×#95—21no—
190C43C2311yescharacteristic
191C43C2311yescharacteristic
192#336—21no—
193#336—21no—
194#336—21no—
195#336—21no—
196#336—21no—
197#336—21no—
198#336—21no—
199#336—21no—
200#336C2311yescharacteristic
201#336C2311yescharacteristic
202#336C2311yescharacteristic
203#336C2311yescharacteristic
204#336C2311yescharacteristic
205#336C2311yescharacteristic
206#336C2311yescharacteristic
207#336C2311yescharacteristic
208C2×#41—22no—
209C2×#41—41no—
210C2×#41—42no—
211C2×#41—81no—
212C22×Q8—21no—
213C22×Q8C22×C411yescharacteristic
214C22×Q8C22×C411yescharacteristic
215C22×D4—21no—
216C22×D4C22×C411yescharacteristic
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4C22×C411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C2×M16—24no—
224C2×M16—42no—
225C2×M16—42no—
226C2×M16—42no—
227C2×M16—42no—
228C2×M16—81no—
229C22×C8—24no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—21no—
233C22×C8—21no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C22×C8—21no—
238C22×C8—42no—
239C22×C8—42no—
240C4⋊Q8—81no—
241C4⋊Q8—81no—
242#124—22no—
243#124—41no—
244C22⋊Q8—44no—
245C22⋊Q8—42no—
246C4×Q8—81no—
247C4×D4—22no—
248C4×D4—41no—
249C4×D4—81no—
250#117—24no—
251#117—22no—
252#117—22no—
253#117—42no—
254#117—42no—
255#117—81no—
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⋊C4C22⋊C411yescharacteristic
267C2×C4⋊C4C22⋊C411yescharacteristic
268C2×C4⋊C4C22⋊C411yescharacteristic
269C2×C4⋊C4C22⋊C411yescharacteristic
270C2×C4⋊C4C22×C411yescharacteristic
271C2×C4⋊C4C22×C411yescharacteristic
272C2×C4⋊C4C22×C411yescharacteristic
273C2×C4⋊C4C22×C411yescharacteristic
274C2×C4⋊C4C22×C411yescharacteristic
275C2×C22⋊C4—21no—
276C2×C22⋊C4—21no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
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×C42—21no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42—42no—
296C2×C42C22×C411yescharacteristic
297C2×C42C22×C411yescharacteristic
298C2×C42C22×C411yescharacteristic
299C2×C42C22×C411yescharacteristic
300C2×C42C22×C411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2×D411yescharacteristic
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2411yescharacteristic
312C4⋊C8—24no—
313C4⋊C8—24no—
314C4⋊C8—24no—
315C4⋊C8—24no—
316C4⋊C8—42no—
317C4⋊C8—81no—
318C22⋊C8—42no—
319#97—42no—
320#97—42no—
321#97—81no—
322C4×C8—42no—
323C4×C8—42no—
324C4×C8—42no—
325#95—21no—
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#95—21no—
350#95C22⋊C411yescharacteristic
351#95C22⋊C411yescharacteristic
352#95C22⋊C411yescharacteristic
353#95C22⋊C411yescharacteristic
354#95C22⋊C411yescharacteristic
355#95C22⋊C411yescharacteristic
356#95C22⋊C411yescharacteristic
357#95C22⋊C411yescharacteristic
358C24—21no—
359C24C2×C22⋊C411yescharacteristic
360#41—84no—
361#41—84no—
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—41no—
367C2×Q8—41no—
368C2×Q8—41no—
369C2×Q8—41no—
370C2×Q8—42no—
371C2×Q8—81no—
372C2×Q8—81no—
373C2×D4—21no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—81no—
379C2×D4#13112yes—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—81no—
405C22×C4—81no—
406C22×C4—81no—
407C22×C4—81no—
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
410C22×C4C2×C22⋊C411yescharacteristic
411C22×C4C2×C22⋊C411yescharacteristic
412C22×C4C2×C22⋊C411yescharacteristic
413C22×C4C2×C22⋊C411yescharacteristic
414C22×C4C2×C22⋊C411yescharacteristic
415C22×C4C2×C22⋊C411yescharacteristic
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C23×C411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422M16—48no—
423M16—84no—
424C2×C8—22no—
425C2×C8—22no—
426C2×C8—22no—
427C2×C8—22no—
428C2×C8—22no—
429C2×C8—48no—
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—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—22no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—42no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—42no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—81no—
465C4⋊C4—81no—
466C4⋊C4—81no—
467C4⋊C4—81no—
468C4⋊C4—81no—
469C4⋊C4#10414yes—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—41no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—42no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—81no—
486C22⋊C4—81no—
487C22⋊C4—81no—
488C22⋊C4—81no—
489C42—21no—
490C42—21no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—22no—
500C42—22no—
501C42—41no—
502C42—41no—
503C42—41no—
504C42—41no—
505C42—41no—
506C42—41no—
507C42—41no—
508C42—41no—
509C42—42no—
510C42—81no—
511C42—81no—
512C42AGL(1,ℤ/8)12yes—
513C42#13712yes—
514C23—21no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—22no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23#41012yes—
528C23C22×C22⋊C411yescharacteristic
529Q8—81no—
530Q8—81no—
531Q8—81no—
532Q8—81no—
533Q8—162no—
534D4—42no—
535D4—42no—
536D4—84no—
537C2×C4—24no—
538C2×C4—24no—
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—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
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—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—84no—
599C2×C4—84no—
600C2×C4—162no—
601C2×C4#41712yes—
602C2×C4#41712yes—
603C2×C4C2×#10412yes—
604C2×C4C2×#10412yes—
605C2×C4C2×#10412yes—
606C2×C4C2×#13111yescharacteristic
607C2×C4#56811yescharacteristic
608C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
609C2×C4C2×#13711yescharacteristic
610C2×C4#57711yescharacteristic
611C2×C4#57811yescharacteristic
612C8—42no—
613C8—42no—
614C8—42no—
615C8—42no—
616C8—42no—
617C8—81no—
618C8—81no—
619C8—81no—
620C22—22no—
621C22—22no—
622C22—22no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—42no—
628C22—42no—
629C22—42no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22—81no—
634C22#288011yescharacteristic
635C22C2×#41011yescharacteristic
636C22#288811yescharacteristic
637C22C2×#41711yescharacteristic
638C22#289811yescharacteristic
639C22C22×#10411yescharacteristic
640C22#290111yescharacteristic
641C4—21no—
642C4—21no—
643C4—22no—
644C4—22no—
645C4—21no—
646C4—21no—
647C4—22no—
648C4—21no—
649C4—21no—
650C4—22no—
651C4—22no—
652C4—41no—
653C4—41no—
654C4—41no—
655C4—41no—
656C4—41no—
657C4—41no—
658C4—81no—
659C4—81no—
660C4—81no—
661C4—81no—
662C4—81no—
663C4—81no—
664C4—81no—
665C2—42no—
666C2—82no—
667C2#1741211yescharacteristic
668C2#1751211yescharacteristic
669C2#1751811yescharacteristic
670C2#1754611yescharacteristic
671C2#1760911yescharacteristic
672C2#1765211yescharacteristic
673C2#1776411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22590C2semidirect
3. #20430C2semidirect
4. #7713C2semidirect
5. #7695C2semidirect
6. #7689C2semidirect
7. #7682C2semidirect
8. #7681C2semidirect
9. #7662C2semidirect
10. #7635C2semidirect
12. #7614C2semidirect
13. #7584C2semidirect
15. #7560C2semidirect

Maximal subgroups

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

subgroupindex
#363902
#225902
#204302
#77132
#76952
#76892
#76822
#76812
#76622
#76352
#76312
#76142
#75842
#75782
#75602