Finite Group Databasebeta

← #164877

Subgroups — #164877

Order 512.

779 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36464C211yesmaximal, characteristic
2#22000C211yesmaximal, characteristic, complemented
3#21998C211yesmaximal, characteristic, complemented
4#7723C211yesmaximal, characteristic, complemented
5#7720C211yesmaximal, characteristic, complemented
6#7718C211yesmaximal, characteristic, complemented
7#7714C211yesmaximal, characteristic, complemented
8#7678C211yesmaximal, characteristic, complemented
9#7674C211yesmaximal, characteristic, complemented
10#7649C211yesmaximal, characteristic, complemented
11#7648C211yesmaximal, characteristic, complemented
12#7634C211yesmaximal, characteristic, complemented
13#7601C211yesmaximal, characteristic, complemented
14#7600C211yesmaximal, characteristic, complemented
15#7566C211yesmaximal, characteristic, complemented
16#2859C2211yescharacteristic
17#2854C2211yescharacteristic
18#2854C2211yescharacteristic
19#2558—21no—
20#2558—21no—
21#2538C411yescharacteristic
22#2537C411yescharacteristic
23#2518—21no—
24#2518—21no—
25#2497—21no—
26#2495C411yescharacteristic
27#2495C411yescharacteristic
28#2426C411yescharacteristic
29#2426C411yescharacteristic
30#2422—21no—
31#2363—21no—
32#2354C411yescharacteristic
33Q8×C4⋊C4C411yescharacteristic
34#2347—21no—
35C4×#124C2211yescharacteristic
36C4×C22⋊Q8C2211yescharacteristic
37C4×C22⋊Q8C2211yescharacteristic
38C42×D4C2211yescharacteristic
39#1924—21no—
40#1923—21no—
41#1917—21no—
42#1916—21no—
43#1870—21no—
44#1869—21no—
45#1851—21no—
46#1850—21no—
47#1839C2211yescharacteristic
48#1839C2211yescharacteristic
49#1831C2211yescharacteristic
50#1831C2211yescharacteristic
51#1817C2211yescharacteristic
52#1817C2211yescharacteristic
53#1797C2211yescharacteristic
54#1797C2211yescharacteristic
55#1749C2211yescharacteristic
56C4×C22⋊C8C2211yescharacteristic
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#1270C2211yescharacteristic
72#1270C2211yescharacteristic
73#1270C2211yescharacteristic
74#1270C2211yescharacteristic
75C22×M16—21no—
76C23×C8—21no—
77C2×#125—21no—
78C2×#124C2×C411yescharacteristic
79C2×#123—21no—
80C2×#123—21no—
81C2×C22⋊Q8C2×C411yescharacteristic
82C2×C22⋊Q8C2×C411yescharacteristic
83C2×#121—21no—
84C2×#121C2×C411yescharacteristic
85C2×#121C2×C411yescharacteristic
86C2×C4×Q8C2×C411yescharacteristic
87C2×C4×Q8C2×C411yescharacteristic
88C2×C4×Q8C2×C411yescharacteristic
89C2×C4×D4C2×C411yescharacteristic
90C2×C4×D4C2×C411yescharacteristic
91C2×C4×D4D411yescharacteristic
92C2×C4×D4D411yescharacteristic
93C2×#117—21no—
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
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×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×#97—21no—
107C2×C4×C8—21no—
108#400—21no—
109#400—21no—
110#397—21no—
111#397—21no—
112#396—21no—
113#396—21no—
114#396—21no—
115#396—21no—
116#394—21no—
117#394—21no—
118#394—21no—
119#394—21no—
120#391C2311yescharacteristic
121#390—21no—
122#390—21no—
123#390—21no—
124#390C2×C411yescharacteristic
125#390C2×C411yescharacteristic
126#390C2311yescharacteristic
127#389—21no—
128#389—21no—
129#389—21no—
130#389—21no—
131#389—21no—
132#389C2×C411yescharacteristic
133#389C2×C411yescharacteristic
134#389C2×C411yescharacteristic
135#389C2×C411yescharacteristic
136#389C2311yescharacteristic
137#389C2311yescharacteristic
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#388—21no—
149#388—21no—
150#387—21no—
151#387—21no—
152#386—21no—
153#386—21no—
154#385—21no—
155#385—21no—
156#384—21no—
157#384C2×C411yescharacteristic
158#384C2×C411yescharacteristic
159#383—21no—
160#383—21no—
161#382—21no—
162#382C2×C411yescharacteristic
163#380—21no—
164#380—21no—
165#380C2×C411yescharacteristic
166#380C2×C411yescharacteristic
167#380C2×C411yescharacteristic
168#380C2×C411yescharacteristic
169C4×C4⋊C4—21no—
170C4×C4⋊C4—21no—
171C4×C4⋊C4—21no—
172C4×C4⋊C4C2×C411yescharacteristic
173C4×C4⋊C4C2×C411yescharacteristic
174C4×C4⋊C4C2×C411yescharacteristic
175C4×C4⋊C4D411yescharacteristic
176C4×C4⋊C4D411yescharacteristic
177C4×C22⋊C4—21no—
178C4×C22⋊C4—21no—
179C4×C22⋊C4—21no—
180C4×C22⋊C4—21no—
181C4×C22⋊C4C2×C411yescharacteristic
182C4×C22⋊C4C2×C411yescharacteristic
183C4×C22⋊C4D411yescharacteristic
184C4×C22⋊C4D411yescharacteristic
185C4×C22⋊C4D411yescharacteristic
186C4×C22⋊C4D411yescharacteristic
187C43C2311yescharacteristic
188#336—21no—
189#336—21no—
190#336—21no—
191#336—21no—
192#336—21no—
193#336—21no—
194#336—21no—
195#336—21no—
196#336—21no—
197#336—21no—
198#336C2311yescharacteristic
199#336C2311yescharacteristic
200#336C2311yescharacteristic
201#336C2311yescharacteristic
202#336C2311yescharacteristic
203#336C2311yescharacteristic
204#336C2311yescharacteristic
205#336C2311yescharacteristic
206C22×Q8C22×C411yescharacteristic
207C22×D4—21no—
208C22×D4—21no—
209C22×D4C22⋊C411yescharacteristic
210C22×D4C22×C411yescharacteristic
211C23×C4—21no—
212C23×C4C22×C411yescharacteristic
213C23×C4C22×C411yescharacteristic
214C23×C4C2×D411yescharacteristic
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2×D411yescharacteristic
218C2×M16—42no—
219C2×M16—81no—
220C2×M16—81no—
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—81no—
231C22×C8—81no—
232#125—81no—
233#125—81no—
234#124—22no—
235#124—41no—
236#123—81no—
237#123—81no—
238#123—81no—
239#123—81no—
240C22⋊Q8—22no—
241C22⋊Q8—22no—
242C22⋊Q8—41no—
243C22⋊Q8—41no—
244#121—44no—
245#121—81no—
246#121—81no—
247C4×Q8—42no—
248C4×Q8—81no—
249C4×Q8—81no—
250C4×D4—42no—
251C4×D4—42no—
252C4×D4—42no—
253C4×D4—42no—
254#117—44no—
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×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×C22⋊C4C22⋊C411yescharacteristic
301C2×C22⋊C4C22⋊C411yescharacteristic
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—81no—
308C2×C42—81no—
309C2×C42C22×C411yescharacteristic
310C2×C42C22×C411yescharacteristic
311C2×C42C22×C411yescharacteristic
312C2×C42C22×C411yescharacteristic
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2411yescharacteristic
322C4⋊C8—81no—
323C4⋊C8—81no—
324C4⋊C8—81no—
325C4⋊C8—81no—
326C22⋊C8—42no—
327C22⋊C8—42no—
328C22⋊C8—81no—
329C22⋊C8—81no—
330C22⋊C8—81no—
331C22⋊C8—81no—
332#97—81no—
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#95C22⋊C411yescharacteristic
351#95C22⋊C411yescharacteristic
352#95C22⋊C411yescharacteristic
353#95C22⋊C411yescharacteristic
354#95C22⋊C411yescharacteristic
355#95C22⋊C411yescharacteristic
356C24—21no—
357C24C2×C22⋊C411yescharacteristic
358C24C2×C22⋊C411yescharacteristic
359C2×Q8—21no—
360C2×Q8—41no—
361C2×Q8—41no—
362C2×Q8#13112yes—
363C2×D4—22no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—22no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—81no—
373C2×D4—81no—
374C2×D4—81no—
375C2×D4—81no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
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—41no—
400C22×C4—41no—
401C22×C4—41no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
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×C4C2×C22⋊C411yescharacteristic
420C22×C4C23×C411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D411yescharacteristic
423M16—84no—
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—41no—
442C2×C8—41no—
443C2×C8—41no—
444C2×C8—41no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—84no—
449C4⋊C4—22no—
450C4⋊C4—22no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—41no—
456C4⋊C4—42no—
457C4⋊C4—42no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—42no—
465C4⋊C4—41no—
466C4⋊C4—41no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—81no—
470C4⋊C4—81no—
471C4⋊C4—81no—
472C4⋊C4—81no—
473C4⋊C4—81no—
474C4⋊C4—81no—
475C4⋊C4—81no—
476C4⋊C4—81no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—41no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—41no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—42no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—41no—
492C22⋊C4—41no—
493C22⋊C4—41no—
494C22⋊C4—41no—
495C22⋊C4—42no—
496C22⋊C4—81no—
497C22⋊C4—81no—
498C22⋊C4—81no—
499C22⋊C4—81no—
500C22⋊C4—81no—
501C22⋊C4—81no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—22no—
509C42—41no—
510C42—41no—
511C42—41no—
512C42—41no—
513C42—41no—
514C42—41no—
515C42—41no—
516C42—41no—
517C42—41no—
518C42—42no—
519C42—42no—
520C42—42no—
521C42—81no—
522C42—81no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23#42112yes—
544C23C22×C22⋊C411yescharacteristic
545Q8—42no—
546Q8—42no—
547D4—44no—
548D4—81no—
549D4—81no—
550D4—162no—
551D4—162no—
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—22no—
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—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
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—81no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—81no—
616C2×C4—81no—
617C2×C4—81no—
618C2×C4—81no—
619C2×C4—84no—
620C2×C4#41312yes—
621C2×C4#42112yes—
622C2×C4C2×#13111yescharacteristic
623C2×C4#56811yescharacteristic
624C2×C4#57611yescharacteristic
625C2×C4#57711yescharacteristic
626C2×C4#57711yescharacteristic
627C2×C4#57811yescharacteristic
628C8—81no—
629C8—81no—
630C8—81no—
631C8—81no—
632C8—81no—
633C8—81no—
634C8—81no—
635C8—81no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22#288011yescharacteristic
658C22#288311yescharacteristic
659C22C2×#41311yescharacteristic
660C22#289711yescharacteristic
661C22#289811yescharacteristic
662C22C2×#42111yescharacteristic
663C22C2×#42111yescharacteristic
664C4—21no—
665C4—21no—
666C4—21no—
667C4—21no—
668C4—21no—
669C4—21no—
670C4—22no—
671C4—22no—
672C4—41no—
673C4—41no—
674C4—42no—
675C4—41no—
676C4—41no—
677C4—41no—
678C4—41no—
679C4—41no—
680C4—41no—
681C4—81no—
682C4—81no—
683C4—81no—
684C4—81no—
685C4—81no—
686C4—81no—
687C2—42no—
688C2—81no—
689C2—161no—
690C2#1742411yescharacteristic
691C2#1752411yescharacteristic
692C2#1752611yescharacteristic
693C2#1759711yescharacteristic
694C2#1759811yescharacteristic
695C2#1765911yescharacteristic
696C2#1778111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2537C4
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
2. #22000C2semidirect
3. #21998C2semidirect
4. #7723C2semidirect
5. #7720C2semidirect
6. #7718C2semidirect
7. #7714C2semidirect
8. #7678C2semidirect
9. #7674C2semidirect
10. #7649C2semidirect
11. #7648C2semidirect
12. #7634C2semidirect
13. #7601C2semidirect
14. #7600C2semidirect
15. #7566C2semidirect

Maximal subgroups

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

subgroupindex
#364642
#220002
#219982
#77232
#77202
#77182
#77142
#76782
#76742
#76492
#76482
#76342
#76012
#76002
#75662