Finite Group Databasebeta

← #164681

Subgroups — #164681

Order 512.

810 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#36366C211yesmaximal, characteristic
2#22589C211yesmaximal, characteristic, complemented
3#20781C211yesmaximal, characteristic, complemented
4#7718C211yesmaximal, characteristic, complemented
5#7716C211yesmaximal, characteristic, complemented
6#7689C211yesmaximal, characteristic, complemented
7#7673C211yesmaximal, characteristic, complemented
8#7656C211yesmaximal, characteristic, complemented
9#7654C211yesmaximal, characteristic, complemented
10#7645C211yesmaximal, characteristic, complemented
11#7642C211yesmaximal, characteristic, complemented
12#7621C211yesmaximal, characteristic, complemented
13#7580C211yesmaximal, characteristic, complemented
14#7578C211yesmaximal, characteristic, complemented
15#7569C211yesmaximal, characteristic, complemented
16#2863C2211yescharacteristic
17#2861C2211yescharacteristic
18#2854C2211yescharacteristic
19#2598—21no—
20#2526—21no—
21#2516—21no—
22#2511C411yescharacteristic
23#2471—21no—
24#2471—21no—
25#2457—21no—
26#2454C411yescharacteristic
27#2447C411yescharacteristic
28#2445C411yescharacteristic
29#2441C411yescharacteristic
30#2431C411yescharacteristic
31#2363C411yescharacteristic
32#2359—21no—
33#2353—21no—
34#2339C411yescharacteristic
35C4×#127C2211yescharacteristic
36#2297C2211yescharacteristic
37#2293C2211yescharacteristic
38C4×#117C2211yescharacteristic
39#1916—21no—
40#1916—21no—
41#1850—21no—
42#1839C2211yescharacteristic
43#1839C2211yescharacteristic
44#1839C2211yescharacteristic
45#1832—21no—
46#1831C2211yescharacteristic
47#1817C2211yescharacteristic
48#1817C2211yescharacteristic
49#1817C2211yescharacteristic
50#1798—21no—
51#1767C2211yescharacteristic
52C4×C4⋊C8C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54C8×C22⋊C4—21no—
55#1750—21no—
56#1746—21no—
57#1744C2211yescharacteristic
58C2×#336C2211yescharacteristic
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
75C22×#41C2×C411yescharacteristic
76C23×C8—21no—
77C2×#127C2×C411yescharacteristic
78C2×#125C2×C411yescharacteristic
79C2×#124C2×C411yescharacteristic
80C2×#123—21no—
81C2×#123—21no—
82C2×#121C2×C411yescharacteristic
83C2×C4×D4—21no—
84C2×C4×D4—21no—
85C2×C4×D4C2×C411yescharacteristic
86C2×C4×D4C2×C411yescharacteristic
87C2×C4×D4C2×C411yescharacteristic
88C2×#117—21no—
89C2×#117C2×C411yescharacteristic
90C2×#117D411yescharacteristic
91C2×#117D411yescharacteristic
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×#97—21no—
101C2×C4×C8—21no—
102C2×C4×C8—21no—
103C2×C4×C8—21no—
104#400—21no—
105#400—21no—
106#399—21no—
107#399—21no—
108#397—21no—
109#397—21no—
110#396—21no—
111#396—21no—
112#394—21no—
113#394—21no—
114#392—21no—
115#392—21no—
116#391C2×C411yescharacteristic
117#391C2×C411yescharacteristic
118#391C2×C411yescharacteristic
119#391C2×C411yescharacteristic
120#391C2311yescharacteristic
121#390—21no—
122#390—21no—
123#390—21no—
124#390C2×C411yescharacteristic
125#390C2×C411yescharacteristic
126#390C2×C411yescharacteristic
127#390C2×C411yescharacteristic
128#390C2311yescharacteristic
129#390C2311yescharacteristic
130#389—21no—
131#389C2×C411yescharacteristic
132#389C2×C411yescharacteristic
133#389C2×C411yescharacteristic
134#389C2×C411yescharacteristic
135#389C2311yescharacteristic
136#388—21no—
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#387—21no—
146#387—21no—
147#387—21no—
148#387—21no—
149#387—21no—
150#387—21no—
151#386—21no—
152#386—21no—
153#386—21no—
154#386—21no—
155#386—21no—
156#386—21no—
157#386—21no—
158#386—21no—
159#386—21no—
160#385—21no—
161#385—21no—
162#384C2×C411yescharacteristic
163#383—21no—
164#381—21no—
165#381—21no—
166#380—21no—
167#380C2×C411yescharacteristic
168C4×C4⋊C4C2×C411yescharacteristic
169C4×C4⋊C4C2×C411yescharacteristic
170C4×C4⋊C4C2×C411yescharacteristic
171C4×C4⋊C4D411yescharacteristic
172C4×C4⋊C4D411yescharacteristic
173C4×C22⋊C4—21no—
174C4×C22⋊C4—21no—
175C4×C22⋊C4—21no—
176C4×C22⋊C4—21no—
177C4×C22⋊C4C2×C411yescharacteristic
178C4×C22⋊C4C2×C411yescharacteristic
179C4×C22⋊C4D411yescharacteristic
180C4×C22⋊C4D411yescharacteristic
181#376D411yescharacteristic
182#376D411yescharacteristic
183C2×#95—21no—
184C43C2311yescharacteristic
185C43C2311yescharacteristic
186#336—21no—
187#336—21no—
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×D4—21no—
214C22×D4—21no—
215C22×D4C22×C411yescharacteristic
216C22×D4C22×C411yescharacteristic
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4—21no—
221C23×C4C22×C411yescharacteristic
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—81no—
235#127—22no—
236#127—41no—
237#125—81no—
238#124—81no—
239#123—44no—
240#123—81no—
241#123—81no—
242#121—42no—
243C4×D4—22no—
244C4×D4—41no—
245C4×D4—42no—
246C4×D4—42no—
247C4×D4—81no—
248C4×D4—81no—
249C4×D4—81no—
250C4×D4—81no—
251#117—22no—
252#117—22no—
253#117—42no—
254#117—81no—
255#117—81no—
256#117—81no—
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⋊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⋊C4C22⋊C411yescharacteristic
296C2×C22⋊C4C22⋊C411yescharacteristic
297C2×C22⋊C4C22⋊C411yescharacteristic
298C2×C22⋊C4C22⋊C411yescharacteristic
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—42no—
302C2×C42C22×C411yescharacteristic
303C2×C42C22×C411yescharacteristic
304C2×C42C22×C411yescharacteristic
305C2×C42C22×C411yescharacteristic
306C2×C42C22×C411yescharacteristic
307C2×C42C2×D411yescharacteristic
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×C42C2411yescharacteristic
318C4⋊C8—42no—
319C4⋊C8—42no—
320C4⋊C8—42no—
321C4⋊C8—81no—
322C22⋊C8—42no—
323C22⋊C8—42no—
324C22⋊C8—42no—
325#97—42no—
326C4×C8—42no—
327C4×C8—42no—
328C4×C8—81no—
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#95C22⋊C411yescharacteristic
347#95C22⋊C411yescharacteristic
348#95C22⋊C411yescharacteristic
349#95C22⋊C411yescharacteristic
350#95C22⋊C411yescharacteristic
351#95C22⋊C411yescharacteristic
352#95C22⋊C411yescharacteristic
353#95C22⋊C411yescharacteristic
354C24—21no—
355C24—21no—
356C24C2×C22⋊C411yescharacteristic
357#41—84no—
358#41—84no—
359C2×Q8—21no—
360C2×Q8—21no—
361C2×Q8—41no—
362C2×Q8—41no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—21no—
366C2×D4—41no—
367C2×D4—41no—
368C2×D4—42no—
369C2×D4—42no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—42no—
373C2×D4—42no—
374C2×D4—42no—
375C2×D4—42no—
376C2×D4—42no—
377C2×D4—81no—
378C2×D4—81no—
379C2×D4—81no—
380C2×D4—81no—
381C2×D4—81no—
382C2×D4#13112yes—
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—21no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—81no—
408C22×C4—81no—
409C22×C4—81no—
410C22×C4—81no—
411C22×C4—81no—
412C22×C4—81no—
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×C4C2×C22⋊C411yescharacteristic
421C22×C4C2×C22⋊C411yescharacteristic
422C22×C4C2×C22⋊C411yescharacteristic
423C22×C4C2×C22⋊C411yescharacteristic
424C22×C4C23×C411yescharacteristic
425C22×C4C22×D411yescharacteristic
426C22×C4C22×D411yescharacteristic
427C2×C8—22no—
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—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—84no—
452C4⋊C4—22no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—41no—
463C4⋊C4—41no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—81no—
467C4⋊C4—81no—
468C4⋊C4—81no—
469C4⋊C4—81no—
470C4⋊C4—81no—
471C4⋊C4—81no—
472C22⋊C4—22no—
473C22⋊C4—22no—
474C22⋊C4—22no—
475C22⋊C4—41no—
476C22⋊C4—42no—
477C22⋊C4—42no—
478C22⋊C4—42no—
479C22⋊C4—42no—
480C22⋊C4—42no—
481C22⋊C4—41no—
482C22⋊C4—41no—
483C22⋊C4—41no—
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C22⋊C4—41no—
489C22⋊C4—81no—
490C22⋊C4—81no—
491C22⋊C4—81no—
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—21no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—22no—
511C42—41no—
512C42—41no—
513C42—41no—
514C42—41no—
515C42—41no—
516C42—41no—
517C42—41no—
518C42—41no—
519C42—41no—
520C42—81no—
521C42—81no—
522C42#13512yes—
523C42#13512yes—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—22no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23#41012yes—
545C23C22×C22⋊C411yescharacteristic
546Q8—81no—
547Q8—81no—
548D4—42no—
549D4—42no—
550D4—84no—
551D4—84no—
552D4—81no—
553D4—81no—
554D4—84no—
555D4—162no—
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—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
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—41no—
616C2×C4—84no—
617C2×C4—84no—
618C2×C4—162no—
619C2×C4—162no—
620C2×C4#41612yes—
621C2×C4#41612yes—
622C2×C4#42112yes—
623C2×C4C2×#13111yescharacteristic
624C2×C4#56811yescharacteristic
625C2×C4C2×#13511yescharacteristic
626C2×C4C2×#13511yescharacteristic
627C2×C4#57611yescharacteristic
628C2×C4#57711yescharacteristic
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—42no—
645C22—42no—
646C22—42no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22—81no—
658C22#288011yescharacteristic
659C22C2×#41011yescharacteristic
660C22#288811yescharacteristic
661C22C2×#41611yescharacteristic
662C22#289711yescharacteristic
663C22C2×#42111yescharacteristic
664C22#290111yescharacteristic
665C4—21no—
666C4—21no—
667C4—22no—
668C4—22no—
669C4—22no—
670C4—21no—
671C4—21no—
672C4—21no—
673C4—21no—
674C4—41no—
675C4—41no—
676C4—41no—
677C4—41no—
678C4—41no—
679C4—41no—
680C4—41no—
681C4—41no—
682C4—81no—
683C4—81no—
684C4—81no—
685C4—81no—
686C4—81no—
687C4—81no—
688C2—42no—
689C2—82no—
690C2—161no—
691C2#1741211yescharacteristic
692C2#1750811yescharacteristic
693C2#1752111yescharacteristic
694C2#1757611yescharacteristic
695C2#1760411yescharacteristic
696C2#1768411yescharacteristic
697C2#1772411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2431C4
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. #20781C2semidirect
4. #7718C2semidirect
5. #7716C2semidirect
6. #7689C2semidirect
7. #7673C2semidirect
8. #7656C2semidirect
9. #7654C2semidirect
10. #7645C2semidirect
11. #7642C2semidirect
12. #7621C2semidirect
13. #7580C2semidirect
14. #7578C2semidirect
15. #7569C2semidirect

Maximal subgroups

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

subgroupindex
#363662
#225892
#207812
#77182
#77162
#76892
#76732
#76562
#76542
#76452
#76422
#76212
#75802
#75782
#75692