Finite Group Databasebeta

← #162440

Subgroups — #162440

Order 512.

788 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35046C211yesmaximal, characteristic
2#22835C211yesmaximal, characteristic, complemented
3#21996C211yesmaximal, characteristic, complemented
4#7732C211yesmaximal, characteristic, complemented
5#7730C211yesmaximal, characteristic, complemented
6#7722C211yesmaximal, characteristic, complemented
7#7721C211yesmaximal, characteristic, complemented
8#7672C211yesmaximal, characteristic, complemented
9#7670C211yesmaximal, characteristic, complemented
10#7662C211yesmaximal, characteristic, complemented
11#7661C211yesmaximal, characteristic, complemented
12#7583C211yesmaximal, characteristic, complemented
13#7578C211yesmaximal, characteristic, complemented
14#7566C211yesmaximal, characteristic, complemented
15#7563C211yesmaximal, characteristic, complemented
16#2862C2211yescharacteristic
17#2860C2211yescharacteristic
18#2855C2211yescharacteristic
19#2850C2211yescharacteristic
20#2653—21no—
21#2636—21no—
22#2589—21no—
23#2576—21no—
24#2518—21no—
25#2514C411yescharacteristic
26#2491C411yescharacteristic
27#2459—21no—
28#2448C411yescharacteristic
29#2439C411yescharacteristic
30#2432C411yescharacteristic
31#2429C411yescharacteristic
32#2415C411yescharacteristic
33#2407C411yescharacteristic
34#2337—21no—
35#2334—21no—
36C4×C22⋊Q8C2211yescharacteristic
37#2297C2211yescharacteristic
38C42×D4C2211yescharacteristic
39#1954—21no—
40#1923—21no—
41#1923—21no—
42#1923—21no—
43#1916—21no—
44#1869—21no—
45#1850—21no—
46#1839C2211yescharacteristic
47#1831C2211yescharacteristic
48#1831C2211yescharacteristic
49#1817C2211yescharacteristic
50#1817C2211yescharacteristic
51#1817C2211yescharacteristic
52#1797C2211yescharacteristic
53C8×C4⋊C4—21no—
54C4×C4⋊C8C2211yescharacteristic
55C4×C22⋊C8C2211yescharacteristic
56C4×C22⋊C8C2211yescharacteristic
57C4×C22⋊C8C2211yescharacteristic
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
76C23×C8—21no—
77C2×C4⋊Q8C2×C411yescharacteristic
78C2×C4⋊Q8C2×C411yescharacteristic
79C2×#126—21no—
80C2×#124—21no—
81C2×#124C2×C411yescharacteristic
82C2×C22⋊Q8C2×C411yescharacteristic
83C2×C22⋊Q8C2×C411yescharacteristic
84C2×C22⋊Q8C2×C411yescharacteristic
85C2×#121C2×C411yescharacteristic
86C2×C4×Q8C2×C411yescharacteristic
87C2×C4×D4C2×C411yescharacteristic
88C2×C4×D4D411yescharacteristic
89C2×C4×D4D411yescharacteristic
90C2×#117C2×C411yescharacteristic
91C22×C42C2311yescharacteristic
92C22×C42C2311yescharacteristic
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108#401—21no—
109#400—21no—
110#399—21no—
111#398—21no—
112#397—21no—
113#397—21no—
114#396—21no—
115#396—21no—
116#394—21no—
117#394—21no—
118#394—21no—
119#394—21no—
120#394—21no—
121#393—21no—
122#393—21no—
123#392—21no—
124#391—21no—
125#391—21no—
126#391—21no—
127#391C2×C411yescharacteristic
128#391C2×C411yescharacteristic
129#391C2×C411yescharacteristic
130#391C2×C411yescharacteristic
131#391C2311yescharacteristic
132#391C2311yescharacteristic
133#390C2×C411yescharacteristic
134#390C2×C411yescharacteristic
135#390C2×C411yescharacteristic
136#390C2311yescharacteristic
137#389—21no—
138#389—21no—
139#389C2×C411yescharacteristic
140#389C2×C411yescharacteristic
141#389C2×C411yescharacteristic
142#389C2311yescharacteristic
143#388—21no—
144#388—21no—
145#388—21no—
146#388—21no—
147#388—21no—
148#388—21no—
149#388—21no—
150#388—21no—
151#388—21no—
152#388—21no—
153#387—21no—
154#387—21no—
155#387—21no—
156#387—21no—
157#387—21no—
158#386—21no—
159#386—21no—
160#386—21no—
161#386—21no—
162#386—21no—
163#385—21no—
164#385—21no—
165#385—21no—
166#385—21no—
167#385—21no—
168#383—21no—
169#382—21no—
170#382—21no—
171#382C2×C411yescharacteristic
172#381—21no—
173#380C2×C411yescharacteristic
174#380C2×C411yescharacteristic
175C4×C4⋊C4—21no—
176C4×C4⋊C4C2×C411yescharacteristic
177C4×C4⋊C4C2×C411yescharacteristic
178C4×C4⋊C4D411yescharacteristic
179C4×C4⋊C4D411yescharacteristic
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
187C4×C22⋊C4D411yescharacteristic
188C4×C22⋊C4D411yescharacteristic
189C43C2311yescharacteristic
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
206C2×#41—22no—
207C2×#41—22no—
208C2×#41—81no—
209C2×#41—81no—
210C22×Q8—21no—
211C22×Q8C22×C411yescharacteristic
212C22×Q8C22×C411yescharacteristic
213C22×D4—21no—
214C22×D4—21no—
215C22×D4C22⋊C411yescharacteristic
216C22×D4C22×C411yescharacteristic
217C23×C4—21no—
218C23×C4C22×C411yescharacteristic
219C23×C4C22×C411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2×D411yescharacteristic
223C23×C4C2×D411yescharacteristic
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8—21no—
232C22×C8—42no—
233C22×C8—81no—
234C22×C8—81no—
235C4⋊Q8—81no—
236C4⋊Q8—81no—
237#126—42no—
238#126—42no—
239#124—44no—
240#124—81no—
241C22⋊Q8—22no—
242C22⋊Q8—22no—
243C22⋊Q8—81no—
244C22⋊Q8—81no—
245#121—81no—
246C4×Q8—81no—
247C4×D4—22no—
248C4×D4—22no—
249C4×D4—22no—
250C4×D4—22no—
251C4×D4—42no—
252#117—81no—
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⋊C4C22⋊C411yescharacteristic
265C2×C4⋊C4C22⋊C411yescharacteristic
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×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
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⋊C4—21no—
289C2×C22⋊C4C22⋊C411yescharacteristic
290C2×C22⋊C4C22⋊C411yescharacteristic
291C2×C22⋊C4C22⋊C411yescharacteristic
292C2×C22⋊C4C22⋊C411yescharacteristic
293C2×C22⋊C4C22⋊C411yescharacteristic
294C2×C22⋊C4C22⋊C411yescharacteristic
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—42no—
314C4⋊C8—42no—
315C4⋊C8—81no—
316C4⋊C8—81no—
317C22⋊C8—42no—
318C22⋊C8—42no—
319C22⋊C8—42no—
320C22⋊C8—81no—
321C22⋊C8—81no—
322C22⋊C8—81no—
323C22⋊C8—81no—
324C4×C8—42no—
325C4×C8—81no—
326C4×C8—81no—
327C4×C8—81no—
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
355C24—21no—
356C24C2×C22⋊C411yescharacteristic
357C24C2×C22⋊C411yescharacteristic
358#41—84no—
359#41—84no—
360C2×Q8—21no—
361C2×Q8—21no—
362C2×Q8—21no—
363C2×Q8—21no—
364C2×Q8—42no—
365C2×Q8—41no—
366C2×Q8—41no—
367C2×Q8—41no—
368C2×Q8—41no—
369C2×Q8—81no—
370C2×Q8—81no—
371C2×D4—22no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4—22no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—41no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4—81no—
381C2×D4—81no—
382C2×D4—81no—
383C2×D4—81no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
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—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—41no—
403C22×C4—41no—
404C22×C4—41no—
405C22×C4—41no—
406C22×C4—41no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—42no—
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×C4#13112yes—
427C22×C4#13112yes—
428C22×C4C23×C411yescharacteristic
429C22×C4C22×D411yescharacteristic
430C22×C4C22×D411yescharacteristic
431C2×C8—22no—
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—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—84no—
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—42no—
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—22no—
480C22⋊C4—22no—
481C22⋊C4—22no—
482C22⋊C4—41no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C22⋊C4—41no—
490C22⋊C4—42no—
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—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—41no—
507C42—41no—
508C42—41no—
509C42—41no—
510C42—41no—
511C42—41no—
512C42—41no—
513C42—41no—
514C42—41no—
515C42—41no—
516C42—42no—
517C42—81no—
518C42—81no—
519C42#13512yes—
520C42#13512yes—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23#40812yes—
542C23C22×C22⋊C411yescharacteristic
543Q8—81no—
544Q8—81no—
545Q8—81no—
546Q8—81no—
547Q8—162no—
548D4—44no—
549D4—81no—
550D4—81no—
551D4—84no—
552D4—162no—
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—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—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
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—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—81no—
611C2×C4—81no—
612C2×C4—81no—
613C2×C4—81no—
614C2×C4—81no—
615C2×C4—81no—
616C2×C4—81no—
617C2×C4—81no—
618C2×C4—84no—
619C2×C4#40812yes—
620C2×C4#41612yes—
621C2×C4#41612yes—
622C2×C4#42112yes—
623C2×C4C2×#13111yescharacteristic
624C2×C4C2×#13111yescharacteristic
625C2×C4C2×#13511yescharacteristic
626C2×C4C2×#13511yescharacteristic
627C2×C4#57711yescharacteristic
628C2×C4#57811yescharacteristic
629C8—42no—
630C8—81no—
631C8—81no—
632C8—81no—
633C8—81no—
634C8—81no—
635C8—81no—
636C8—81no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22C2×#40811yescharacteristic
659C22#288311yescharacteristic
660C22#288811yescharacteristic
661C22C2×#41611yescharacteristic
662C22#289811yescharacteristic
663C22C2×#42111yescharacteristic
664C22#290111yescharacteristic
665C4—22no—
666C4—22no—
667C4—21no—
668C4—21no—
669C4—22no—
670C4—21no—
671C4—21no—
672C4—22no—
673C4—21no—
674C4—21no—
675C4—41no—
676C4—41no—
677C4—41no—
678C4—41no—
679C4—41no—
680C4—41no—
681C4—81no—
682C4—81no—
683C4—81no—
684C4—81no—
685C4—81no—
686C4—81no—
687C4—81no—
688C2—42no—
689C2—81no—
690C2—161no—
691C2#1738611yescharacteristic
692C2#1745811yescharacteristic
693C2#1748411yescharacteristic
694C2#1759811yescharacteristic
695C2#1761711yescharacteristic
696C2#1768111yescharacteristic
697C2#1772611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2407C4
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. #22835C2semidirect
3. #21996C2semidirect
4. #7732C2semidirect
5. #7730C2semidirect
6. #7722C2semidirect
7. #7721C2semidirect
8. #7672C2semidirect
9. #7670C2semidirect
10. #7662C2semidirect
11. #7661C2semidirect
12. #7583C2semidirect
13. #7578C2semidirect
14. #7566C2semidirect
15. #7563C2semidirect

Maximal subgroups

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

subgroupindex
#350462
#228352
#219962
#77322
#77302
#77222
#77212
#76722
#76702
#76622
#76612
#75832
#75782
#75662
#75632