Finite Group Databasebeta

← #164944

Subgroups — #164944

Order 512.

803 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36392C211yesmaximal, characteristic
2#22787C211yesmaximal, characteristic, complemented
3#22785C211yesmaximal, characteristic, complemented
4#7731C211yesmaximal, characteristic, complemented
5#7729C211yesmaximal, characteristic, complemented
6#7693C211yesmaximal, characteristic, complemented
7#7691C211yesmaximal, characteristic, complemented
8#7661C211yesmaximal, characteristic, complemented
9#7655C211yesmaximal, characteristic, complemented
10#7647C211yesmaximal, characteristic, complemented
11#7612C211yesmaximal, characteristic, complemented
12#7608C211yesmaximal, characteristic, complemented
13#7580C211yesmaximal, characteristic, complemented
14#7578C211yesmaximal, characteristic, complemented
15#7565C211yesmaximal, characteristic, complemented
16#2863C2211yescharacteristic
17#2855C2211yescharacteristic
18#2851C2211yescharacteristic
19#2600—21no—
20#2598—21no—
21#2578—21no—
22#2576—21no—
23#2576—21no—
24#2575—21no—
25#2514C411yescharacteristic
26#2489C411yescharacteristic
27#2485C411yescharacteristic
28#2454C411yescharacteristic
29#2420C411yescharacteristic
30#2407C411yescharacteristic
31#2353—21no—
32#2346C411yescharacteristic
33#2343—21no—
34#2339C411yescharacteristic
35C4×#121C2211yescharacteristic
36#2297C2211yescharacteristic
37#2293C2211yescharacteristic
38C42×D4C2211yescharacteristic
39#1924—21no—
40#1923—21no—
41#1917—21no—
42#1916—21no—
43#1869—21no—
44#1869—21no—
45#1839C2211yescharacteristic
46#1839C2211yescharacteristic
47#1817C2211yescharacteristic
48#1817C2211yescharacteristic
49#1797C2211yescharacteristic
50#1797C2211yescharacteristic
51#1752—21no—
52C8×C22⋊C4—21no—
53#1749C2211yescharacteristic
54C4×C22⋊C8C2211yescharacteristic
55#1745C2211yescharacteristic
56#1744C2211yescharacteristic
57C2×#336C2211yescharacteristic
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×#41C2×C411yescharacteristic
76C23×C8—21no—
77C23×C8—21no—
78C2×#125C2×C411yescharacteristic
79C2×C22⋊Q8C2×C411yescharacteristic
80C2×#121C2×C411yescharacteristic
81C2×C4×D4—21no—
82C2×C4×D4C2×C411yescharacteristic
83C2×C4×D4C2×C411yescharacteristic
84C2×C4×D4C2×C411yescharacteristic
85C2×C4×D4D411yescharacteristic
86C2×C4×D4D411yescharacteristic
87C2×#117—21no—
88C2×#117C2×C411yescharacteristic
89C22×C42C2311yescharacteristic
90C22×C42C2311yescharacteristic
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×#97—21no—
100C2×#97—21no—
101C2×C4×C8—21no—
102C2×C4×C8—21no—
103#401—21no—
104#400—21no—
105#400—21no—
106#399—21no—
107#397—21no—
108#397—21no—
109#397—21no—
110#396—21no—
111#394—21no—
112#394—21no—
113#394—21no—
114#392—21no—
115#391C2×C411yescharacteristic
116#391C2×C411yescharacteristic
117#391C2×C411yescharacteristic
118#391C2×C411yescharacteristic
119#391C2311yescharacteristic
120#390C2×C411yescharacteristic
121#390C2×C411yescharacteristic
122#390C2311yescharacteristic
123#389C2×C411yescharacteristic
124#389C2×C411yescharacteristic
125#389C2×C411yescharacteristic
126#389C2×C411yescharacteristic
127#389C2311yescharacteristic
128#389C2311yescharacteristic
129#388—21no—
130#388—21no—
131#388—21no—
132#388—21no—
133#388—21no—
134#388—21no—
135#388—21no—
136#387—21no—
137#387—21no—
138#387—21no—
139#387—21no—
140#387—21no—
141#387—21no—
142#387—21no—
143#387—21no—
144#387—21no—
145#386—21no—
146#386—21no—
147#386—21no—
148#386—21no—
149#386—21no—
150#386—21no—
151#386—21no—
152#385—21no—
153#385—21no—
154#385—21no—
155#385—21no—
156#385—21no—
157#383—21no—
158#383—21no—
159#382—21no—
160#382—21no—
161#382C2×C411yescharacteristic
162#381—21no—
163#381—21no—
164#381—21no—
165#380—21no—
166#380C2×C411yescharacteristic
167#380C2×C411yescharacteristic
168#380C2×C411yescharacteristic
169#380C2×C411yescharacteristic
170C4×C4⋊C4C2×C411yescharacteristic
171C4×C4⋊C4C2×C411yescharacteristic
172C4×C4⋊C4C2×C411yescharacteristic
173C4×C4⋊C4D411yescharacteristic
174C4×C4⋊C4D411yescharacteristic
175C4×C22⋊C4—21no—
176C4×C22⋊C4—21no—
177C4×C22⋊C4—21no—
178C4×C22⋊C4—21no—
179C4×C22⋊C4C2×C411yescharacteristic
180C4×C22⋊C4C2×C411yescharacteristic
181C4×C22⋊C4D411yescharacteristic
182C4×C22⋊C4D411yescharacteristic
183C4×C22⋊C4D411yescharacteristic
184C4×C22⋊C4D411yescharacteristic
185#376—21no—
186C2×#95—21no—
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
206C2×#41—22no—
207C2×#41—41no—
208C2×#41—42no—
209C2×#41—81no—
210C22×Q8C22×C411yescharacteristic
211C22×D4—21no—
212C22×D4—21no—
213C22×D4C22⋊C411yescharacteristic
214C22×D4C22×C411yescharacteristic
215C23×C4—21no—
216C23×C4—21no—
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—42no—
234C22×C8—42no—
235C22×C8—42no—
236C22×C8—81no—
237C22×C8—81no—
238#125—81no—
239C22⋊Q8—81no—
240#121—22no—
241#121—41no—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—41no—
246C4×D4—41no—
247C4×D4—41no—
248C4×D4—42no—
249C4×D4—42no—
250C4×D4—42no—
251C4×D4—81no—
252#117—42no—
253#117—42no—
254#117—42no—
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×C4⋊C4C22×C411yescharacteristic
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⋊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—81no—
298C2×C42—81no—
299C2×C42C22×C411yescharacteristic
300C2×C42C22×C411yescharacteristic
301C2×C42C22×C411yescharacteristic
302C2×C42C22×C411yescharacteristic
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—81no—
313C4⋊C8—81no—
314C22⋊C8—42no—
315C22⋊C8—42no—
316C22⋊C8—42no—
317C22⋊C8—42no—
318C22⋊C8—81no—
319C22⋊C8—81no—
320#97—42no—
321#97—81no—
322C4×C8—42no—
323C4×C8—81no—
324#95—21no—
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#95C22⋊C411yescharacteristic
349#95C22⋊C411yescharacteristic
350#95C22⋊C411yescharacteristic
351#95C22⋊C411yescharacteristic
352#95C22⋊C411yescharacteristic
353#95C22⋊C411yescharacteristic
354C24—21no—
355C24C2×C22⋊C411yescharacteristic
356C24C2×C22⋊C411yescharacteristic
357#41—84no—
358#41—84no—
359C2×Q8—21no—
360C2×Q8—21no—
361C2×Q8—41no—
362C2×Q8—41no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—21no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—81no—
374C2×D4—81no—
375C2×D4#13112yes—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
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—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
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—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—81no—
411C22×C4—81no—
412C22×C4—81no—
413C22×C4—81no—
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×C4C2×C22⋊C411yescharacteristic
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4#13512yes—
425C22×C4#13512yes—
426C22×C4C23×C411yescharacteristic
427C22×C4C22×D411yescharacteristic
428C22×C4C22×D411yescharacteristic
429C2×C8—22no—
430C2×C8—22no—
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—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—84no—
454C2×C8—84no—
455C4⋊C4—22no—
456C4⋊C4—21no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—41no—
465C4⋊C4—41no—
466C4⋊C4—42no—
467C4⋊C4—41no—
468C4⋊C4—41no—
469C4⋊C4—41no—
470C4⋊C4—41no—
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—41no—
481C22⋊C4—41no—
482C22⋊C4—41no—
483C22⋊C4—42no—
484C22⋊C4—42no—
485C22⋊C4—42no—
486C22⋊C4—42no—
487C22⋊C4—42no—
488C22⋊C4—41no—
489C22⋊C4—41no—
490C22⋊C4—41no—
491C22⋊C4—41no—
492C22⋊C4—81no—
493C22⋊C4—81no—
494C22⋊C4—81no—
495C22⋊C4—81no—
496C22⋊C4—81no—
497C22⋊C4—81no—
498C22⋊C4—81no—
499C22⋊C4—81no—
500C42—21no—
501C42—21no—
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—42no—
518C42—42no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—22no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23#41312yes—
539C23#41612yes—
540C23C22×C22⋊C411yescharacteristic
541Q8—81no—
542Q8—81no—
543D4—42no—
544D4—42no—
545D4—44no—
546D4—84no—
547D4—84no—
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—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—42no—
584C2×C4—42no—
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—84no—
617C2×C4—84no—
618C2×C4—162no—
619C2×C4#41612yes—
620C2×C4C2×#13111yescharacteristic
621C2×C4#56811yescharacteristic
622C2×C4C2×#13511yescharacteristic
623C2×C4C2×#13511yescharacteristic
624C2×C4#57511yescharacteristic
625C2×C4#57511yescharacteristic
626C8—42no—
627C8—42no—
628C8—81no—
629C8—81no—
630C8—81no—
631C8—81no—
632C8—81no—
633C8—81no—
634C22—22no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22#288011yescharacteristic
656C22#288311yescharacteristic
657C22C2×#41311yescharacteristic
658C22C2×#41611yescharacteristic
659C22#289611yescharacteristic
660C22#290111yescharacteristic
661C22#290111yescharacteristic
662C4—21no—
663C4—21no—
664C4—21no—
665C4—21no—
666C4—22no—
667C4—21no—
668C4—21no—
669C4—41no—
670C4—41no—
671C4—41no—
672C4—41no—
673C4—41no—
674C4—41no—
675C4—41no—
676C4—41no—
677C4—41no—
678C4—81no—
679C4—81no—
680C4—81no—
681C4—81no—
682C4—81no—
683C4—81no—
684C4—81no—
685C2—42no—
686C2—42no—
687C2—82no—
688C2#1742211yescharacteristic
689C2#1750811yescharacteristic
690C2#1751311yescharacteristic
691C2#1761511yescharacteristic
692C2#1762911yescharacteristic
693C2#1769011yescharacteristic
694C2#1771311yescharacteristic
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. #22787C2semidirect
3. #22785C2semidirect
4. #7731C2semidirect
5. #7729C2semidirect
6. #7693C2semidirect
7. #7691C2semidirect
8. #7661C2semidirect
9. #7655C2semidirect
10. #7647C2semidirect
11. #7612C2semidirect
12. #7608C2semidirect
13. #7580C2semidirect
14. #7578C2semidirect
15. #7565C2semidirect

Maximal subgroups

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

subgroupindex
#363922
#227872
#227852
#77312
#77292
#76932
#76912
#76612
#76552
#76472
#76122
#76082
#75802
#75782
#75652