Finite Group Databasebeta

← #164141

Subgroups — #164141

Order 512.

844 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36502C211yesmaximal, characteristic
2#22589C211yesmaximal, characteristic, complemented
3#20468C211yesmaximal, characteristic, complemented
4#7692C211yesmaximal, characteristic, complemented
5#7690C211yesmaximal, characteristic, complemented
6#7689C211yesmaximal, characteristic, complemented
7#7673C211yesmaximal, characteristic, complemented
8#7670C211yesmaximal, characteristic, complemented
9#7666C211yesmaximal, characteristic, complemented
10#7621C211yesmaximal, characteristic, complemented
11#7620C211yesmaximal, characteristic, complemented
12#7618C211yesmaximal, characteristic, complemented
13#7569C211yesmaximal, characteristic, complemented
14#7567C211yesmaximal, characteristic, complemented
15#7565C211yesmaximal, characteristic, complemented
16C2×#433—21no—
17#2863C2211yescharacteristic
18#2860C2211yescharacteristic
19#2850C2211yescharacteristic
20#2598—21no—
21#2595—21no—
22#2570C411yescharacteristic
23#2544C411yescharacteristic
24#2531—21no—
25#2476—21no—
26#2468—21no—
27#2467—21no—
28#2428C411yescharacteristic
29#2427C411yescharacteristic
30#2423C411yescharacteristic
31#2420C411yescharacteristic
32#2367C411yescharacteristic
33#2364—21no—
34#2353—21no—
35#2349C411yescharacteristic
36C4×#127C2211yescharacteristic
37C4×#124C2211yescharacteristic
38C4×#121C2211yescharacteristic
39C4×#117C2211yescharacteristic
40#1916—21no—
41#1916—21no—
42#1850—21no—
43#1839C2211yescharacteristic
44#1831C2211yescharacteristic
45#1817C2211yescharacteristic
46#1768—21no—
47#1767C2211yescharacteristic
48C4×C4⋊C8C2211yescharacteristic
49C4×C4⋊C8C2211yescharacteristic
50C8×C22⋊C4—21no—
51#1750—21no—
52#1749C2211yescharacteristic
53C4×C22⋊C8C2211yescharacteristic
54#1745C2211yescharacteristic
55#1744C2211yescharacteristic
56#1744C2211yescharacteristic
57C2×#336C2211yescharacteristic
58#1726—21no—
59#1272C2211yescharacteristic
60#1272C2211yescharacteristic
61#1272C2211yescharacteristic
62#1272C2211yescharacteristic
63#1271C2211yescharacteristic
64#1271C2211yescharacteristic
65#1271C2211yescharacteristic
66#1271C2211yescharacteristic
67#1270C2211yescharacteristic
68#1270C2211yescharacteristic
69#1270C2211yescharacteristic
70#1270C2211yescharacteristic
71#1270C2211yescharacteristic
72#1270C2211yescharacteristic
73#1270C2211yescharacteristic
74#1270C2211yescharacteristic
75C23×C8—21no—
76C2×#127C2×C411yescharacteristic
77C2×#126—21no—
78C2×#124C2×C411yescharacteristic
79C2×#124C2×C411yescharacteristic
80C2×C22⋊Q8C2×C411yescharacteristic
81C2×#121—21no—
82C2×#121C2×C411yescharacteristic
83C2×#121C2×C411yescharacteristic
84C2×C4×Q8C2×C411yescharacteristic
85C2×C4×D4—21no—
86C2×C4×D4—21no—
87C2×C4×D4—21no—
88C2×C4×D4C2×C411yescharacteristic
89C2×C4×D4C2×C411yescharacteristic
90C2×C4×D4C2×C411yescharacteristic
91C2×C4×D4C2×C411yescharacteristic
92C2×#117D411yescharacteristic
93C2×#117D411yescharacteristic
94C22×C42C2311yescharacteristic
95#433—28no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×#97—21no—
104C2×#97—21no—
105C2×#97—21no—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108C2×C4×C8—21no—
109C2×C4×C8—21no—
110C2×C4×C8—21no—
111#400—21no—
112#399—21no—
113#398—21no—
114#398—21no—
115#397—21no—
116#397—21no—
117#394—21no—
118#394—21no—
119#394—21no—
120#394—21no—
121#394—21no—
122#392—21no—
123#391C2×C411yescharacteristic
124#391C2×C411yescharacteristic
125#391C2311yescharacteristic
126#390—21no—
127#390—21no—
128#390C2×C411yescharacteristic
129#390C2×C411yescharacteristic
130#390C2311yescharacteristic
131#390C2311yescharacteristic
132#389C2×C411yescharacteristic
133#389C2×C411yescharacteristic
134#389C2×C411yescharacteristic
135#389C2×C411yescharacteristic
136#389C2311yescharacteristic
137#388—21no—
138#388—21no—
139#388—21no—
140#388—21no—
141#388—21no—
142#388—21no—
143#388—21no—
144#388—21no—
145#388—21no—
146#388—21no—
147#388—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#386—21no—
159#386—21no—
160#386—21no—
161#386—21no—
162#385—21no—
163#385—21no—
164#385—21no—
165#384C2×C411yescharacteristic
166#383—21no—
167#382C2×C411yescharacteristic
168#382C2×C411yescharacteristic
169#381—21no—
170#380—21no—
171#380—21no—
172#380C2×C411yescharacteristic
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⋊C4C2×C411yescharacteristic
183C4×C22⋊C4C2×C411yescharacteristic
184C4×C22⋊C4D411yescharacteristic
185C4×C22⋊C4D411yescharacteristic
186#376D411yescharacteristic
187#376D411yescharacteristic
188C2×#95—21no—
189C43C2311yescharacteristic
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×Q8C22×C411yescharacteristic
208C22×D4—21no—
209C22×D4—21no—
210C22×D4—21no—
211C22×D4—21no—
212C22×D4C22×C411yescharacteristic
213C22×D4C22×C411yescharacteristic
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4C22×C411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—42no—
230C22×C8—42no—
231C22×C8—81no—
232#127—22no—
233#127—41no—
234#126—81no—
235#126—81no—
236#124—22no—
237#124—41no—
238#124—42no—
239C22⋊Q8—81no—
240#121—22no—
241#121—41no—
242#121—44no—
243#121—81no—
244C4×Q8—81no—
245C4×D4—42no—
246C4×D4—42no—
247C4×D4—42no—
248C4×D4—81no—
249C4×D4—81no—
250C4×D4—81no—
251C4×D4—81no—
252C4×D4—81no—
253C4×D4—81no—
254C4×D4—81no—
255#117—24no—
256#117—22no—
257#117—22no—
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⋊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×C4⋊C4C22×C411yescharacteristic
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⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4C22⋊C411yescharacteristic
297C2×C22⋊C4C22⋊C411yescharacteristic
298C2×C22⋊C4C22⋊C411yescharacteristic
299C2×C22⋊C4C22⋊C411yescharacteristic
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—42no—
303C2×C42C22×C411yescharacteristic
304C2×C42C22×C411yescharacteristic
305C2×C42C22×C411yescharacteristic
306C2×C42C22×C411yescharacteristic
307C2×C42C22×C411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2411yescharacteristic
319C4⋊C8—24no—
320C4⋊C8—24no—
321C4⋊C8—42no—
322C4⋊C8—81no—
323C22⋊C8—24no—
324C22⋊C8—24no—
325C22⋊C8—42no—
326#97—24no—
327#97—42no—
328#97—42no—
329C4×C8—24no—
330C4×C8—42no—
331C4×C8—42no—
332C4×C8—42no—
333C4×C8—81no—
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#95—21no—
351#95C22⋊C411yescharacteristic
352#95C22⋊C411yescharacteristic
353#95C22⋊C411yescharacteristic
354#95C22⋊C411yescharacteristic
355#95C22⋊C411yescharacteristic
356#95C22⋊C411yescharacteristic
357#95C22⋊C411yescharacteristic
358#95C22⋊C411yescharacteristic
359C24—21no—
360C24—21no—
361C24C2×C22⋊C411yescharacteristic
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8—41no—
365C2×Q8—41no—
366C2×D4—21no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—42no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4—81no—
381C2×D4—81no—
382C2×D4—81no—
383C2×D4—81no—
384C2×D4—81no—
385C2×D4#13112yes—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—81no—
409C22×C4—81no—
410C22×C4—81no—
411C22×C4—81no—
412C22×C4—81no—
413C22×C4—81no—
414C22×C4—81no—
415C22×C4—81no—
416C22×C4C2×C22⋊C411yescharacteristic
417C22×C4C2×C22⋊C411yescharacteristic
418C22×C4C2×C22⋊C411yescharacteristic
419C22×C4C2×C22⋊C411yescharacteristic
420C22×C4C2×C22⋊C411yescharacteristic
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4C2×C22⋊C411yescharacteristic
425C22×C4C2×C22⋊C411yescharacteristic
426C22×C4C2×C22⋊C411yescharacteristic
427C22×C4C23×C411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
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—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—84no—
455C4⋊C4—22no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—22no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—42no—
468C4⋊C4—42no—
469C4⋊C4—81no—
470C4⋊C4—81no—
471C4⋊C4—81no—
472C4⋊C4—81no—
473C4⋊C4—81no—
474C4⋊C4—81no—
475C22⋊C4—22no—
476C22⋊C4—22no—
477C22⋊C4—22no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—42no—
482C22⋊C4—42no—
483C22⋊C4—41no—
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C22⋊C4—41no—
490C22⋊C4—41no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—81no—
494C22⋊C4—81no—
495C22⋊C4—81no—
496C22⋊C4—81no—
497C22⋊C4—81no—
498C22⋊C4—81no—
499C22⋊C4—81no—
500C22⋊C4—81no—
501C22⋊C4—81no—
502C22⋊C4#10414yes—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—22no—
514C42—22no—
515C42—22no—
516C42—22no—
517C42—41no—
518C42—41no—
519C42—41no—
520C42—41no—
521C42—41no—
522C42—41no—
523C42—81no—
524C42—81no—
525C42#13512yes—
526C42#13512yes—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—22no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23C2×#10412yes—
548C23C22×C22⋊C411yescharacteristic
549Q8—81no—
550Q8—81no—
551D4—42no—
552D4—42no—
553D4—84no—
554D4—84no—
555D4—81no—
556D4—81no—
557D4—84no—
558D4—162no—
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—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—81no—
616C2×C4—81no—
617C2×C4—162no—
618C2×C4—162no—
619C2×C4—162no—
620C2×C4—162no—
621C2×C4#41012yes—
622C2×C4#41612yes—
623C2×C4#41612yes—
624C2×C4C2×#10412yes—
625C2×C4C2×#10412yes—
626C2×C4C2×#13111yescharacteristic
627C2×C4#56811yescharacteristic
628C2×C4C2×#13511yescharacteristic
629C2×C4C2×#13511yescharacteristic
630C2×C4#57611yescharacteristic
631C2×C4#57711yescharacteristic
632C8—42no—
633C8—42no—
634C8—42no—
635C8—42no—
636C8—42no—
637C8—81no—
638C8—81no—
639C8—81no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22—81no—
659C22—81no—
660C22—81no—
661C22#288011yescharacteristic
662C22C2×#41011yescharacteristic
663C22#288811yescharacteristic
664C22C2×#41611yescharacteristic
665C22#289711yescharacteristic
666C22C22×#10411yescharacteristic
667C22#290111yescharacteristic
668C4—21no—
669C4—21no—
670C4—22no—
671C4—22no—
672C4—22no—
673C4—21no—
674C4—21no—
675C4—21no—
676C4—21no—
677C4—22no—
678C4—22no—
679C4—41no—
680C4—41no—
681C4—41no—
682C4—41no—
683C4—41no—
684C4—41no—
685C4—81no—
686C4—81no—
687C4—81no—
688C4—81no—
689C4—81no—
690C4—81no—
691C2—42no—
692C2—82no—
693C2—161no—
694C2#1741511yescharacteristic
695C2#1750811yescharacteristic
696C2#1751711yescharacteristic
697C2#1753311yescharacteristic
698C2#1760411yescharacteristic
699C2#1765211yescharacteristic
700C2#1772411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2428C4
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. #22589C2semidirect
3. #20468C2semidirect
4. #7692C2semidirect
5. #7690C2semidirect
6. #7689C2semidirect
7. #7673C2semidirect
8. #7670C2semidirect
9. #7666C2semidirect
10. #7621C2semidirect
11. #7620C2semidirect
12. #7618C2semidirect
13. #7569C2semidirect
14. #7567C2semidirect
15. #7565C2semidirect

Maximal subgroups

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

subgroupindex
#365022
#225892
#204682
#76922
#76902
#76892
#76732
#76702
#76662
#76212
#76202
#76182
#75692
#75672
#75652