Finite Group Databasebeta

← #165035

Subgroups — #165035

Order 512.

771 conjugacy classes of subgroups.

The subgroup lattice has 691 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36402C211yesmaximal, characteristic
2#22839C211yesmaximal, characteristic, complemented
3#22785C211yesmaximal, characteristic, complemented
4#7730C211yesmaximal, characteristic, complemented
5#7721C211yesmaximal, characteristic, complemented
6#7690C211yesmaximal, characteristic, complemented
7#7675C211yesmaximal, characteristic, complemented
8#7672C211yesmaximal, characteristic, complemented
9#7666C211yesmaximal, characteristic, complemented
10#7644C211yesmaximal, characteristic, complemented
11#7620C211yesmaximal, characteristic, complemented
12#7613C211yesmaximal, characteristic, complemented
13#7582C211yesmaximal, characteristic, complemented
14#7579C211yesmaximal, characteristic, complemented
15#7562C211yesmaximal, characteristic, complemented
16#2860C2211yescharacteristic
17#2855C2211yescharacteristic
18#2850C2211yescharacteristic
19#2603—21no—
20#2600—21no—
21#2597—21no—
22#2595—21no—
23#2594—21no—
24#2576—21no—
25#2508C411yescharacteristic
26#2494C411yescharacteristic
27#2447C411yescharacteristic
28#2426C411yescharacteristic
29#2415C411yescharacteristic
30#2414C411yescharacteristic
31#2373—21no—
32#2363C411yescharacteristic
33#2362C411yescharacteristic
34#2357—21no—
35C4×#121C2211yescharacteristic
36#2296C2211yescharacteristic
37#2295C2211yescharacteristic
38C42×D4C2211yescharacteristic
39#1954—21no—
40#1924—21no—
41#1924—21no—
42#1923—21no—
43#1916—21no—
44#1869—21no—
45#1839C2211yescharacteristic
46#1831C2211yescharacteristic
47#1817C2211yescharacteristic
48#1817C2211yescharacteristic
49#1817C2211yescharacteristic
50#1797C2211yescharacteristic
51#1770—21no—
52C4×C4⋊C8C2211yescharacteristic
53C8×C22⋊C4—21no—
54#1749C2211yescharacteristic
55#1749C2211yescharacteristic
56C4×C22⋊C8C2211yescharacteristic
57#1744C2211yescharacteristic
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
75C23×C8—21no—
76C2×#126—21no—
77C2×#124C2×C411yescharacteristic
78C2×C22⋊Q8C2×C411yescharacteristic
79C2×C22⋊Q8C2×C411yescharacteristic
80C2×#121C2×C411yescharacteristic
81C2×#121C2×C411yescharacteristic
82C2×C4×Q8C2×C411yescharacteristic
83C2×C4×D4—21no—
84C2×C4×D4C2×C411yescharacteristic
85C2×C4×D4C2×C411yescharacteristic
86C2×C4×D4D411yescharacteristic
87C2×C4×D4D411yescharacteristic
88C2×#117C2×C411yescharacteristic
89C2×#117C2×C411yescharacteristic
90C22×C42C2311yescharacteristic
91C22×C42C2311yescharacteristic
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×#97—21no—
103C2×#97—21no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107#401—21no—
108#400—21no—
109#398—21no—
110#397—21no—
111#397—21no—
112#397—21no—
113#397—21no—
114#396—21no—
115#394—21no—
116#394—21no—
117#394—21no—
118#392—21no—
119#391C2×C411yescharacteristic
120#391C2×C411yescharacteristic
121#391C2311yescharacteristic
122#390—21no—
123#390—21no—
124#390—21no—
125#390C2×C411yescharacteristic
126#390C2×C411yescharacteristic
127#390C2311yescharacteristic
128#389—21no—
129#389C2×C411yescharacteristic
130#389C2×C411yescharacteristic
131#389C2×C411yescharacteristic
132#389C2×C411yescharacteristic
133#389C2×C411yescharacteristic
134#389C2×C411yescharacteristic
135#389C2311yescharacteristic
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#388—21no—
149#388—21no—
150#387—21no—
151#387—21no—
152#387—21no—
153#387—21no—
154#387—21no—
155#387—21no—
156#387—21no—
157#386—21no—
158#386—21no—
159#386—21no—
160#386—21no—
161#385—21no—
162#385—21no—
163#385—21no—
164#385—21no—
165#382—21no—
166#382C2×C411yescharacteristic
167#381—21no—
168#381—21no—
169#380C2×C411yescharacteristic
170#380C2×C411yescharacteristic
171C4×C4⋊C4—21no—
172C4×C4⋊C4—21no—
173C4×C4⋊C4—21no—
174C4×C4⋊C4—21no—
175C4×C4⋊C4C2×C411yescharacteristic
176C4×C4⋊C4C2×C411yescharacteristic
177C4×C4⋊C4C2×C411yescharacteristic
178C4×C4⋊C4D411yescharacteristic
179C4×C4⋊C4D411yescharacteristic
180C4×C22⋊C4—21no—
181C4×C22⋊C4—21no—
182C4×C22⋊C4C2×C411yescharacteristic
183C4×C22⋊C4C2×C411yescharacteristic
184C4×C22⋊C4D411yescharacteristic
185C4×C22⋊C4D411yescharacteristic
186C4×C22⋊C4D411yescharacteristic
187C4×C22⋊C4D411yescharacteristic
188#376—21no—
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
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
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—21no—
226C22×C8—42no—
227C22×C8—42no—
228C22×C8—81no—
229#126—42no—
230#126—42no—
231#124—81no—
232C22⋊Q8—42no—
233C22⋊Q8—81no—
234#121—22no—
235#121—41no—
236#121—42no—
237C4×Q8—22no—
238C4×Q8—41no—
239C4×D4—22no—
240C4×D4—22no—
241C4×D4—41no—
242C4×D4—41no—
243C4×D4—42no—
244C4×D4—42no—
245C4×D4—42no—
246C4×D4—81no—
247#117—22no—
248#117—41no—
249#117—81no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
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⋊C4C22⋊C411yescharacteristic
262C2×C4⋊C4C22⋊C411yescharacteristic
263C2×C4⋊C4C22⋊C411yescharacteristic
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×C22⋊C4—21no—
271C2×C22⋊C4—21no—
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⋊C4C22⋊C411yescharacteristic
288C2×C22⋊C4C22⋊C411yescharacteristic
289C2×C22⋊C4C22⋊C411yescharacteristic
290C2×C22⋊C4C22⋊C411yescharacteristic
291C2×C22⋊C4C22⋊C411yescharacteristic
292C2×C22⋊C4C22⋊C411yescharacteristic
293C2×C42—21no—
294C2×C42—21no—
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—
316C22⋊C8—42no—
317C22⋊C8—42no—
318C22⋊C8—42no—
319C22⋊C8—42no—
320C22⋊C8—81no—
321C22⋊C8—81no—
322C22⋊C8—81no—
323#97—81no—
324#97—81no—
325C4×C8—42no—
326C4×C8—42no—
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#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
357C2×Q8—21no—
358C2×Q8—21no—
359C2×Q8—41no—
360C2×Q8—41no—
361C2×D4—22no—
362C2×D4—21no—
363C2×D4—21no—
364C2×D4—22no—
365C2×D4—41no—
366C2×D4—42no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—42no—
370C2×D4—42no—
371C2×D4—81no—
372C2×D4—81no—
373C2×D4—81no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—21no—
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—81no—
407C22×C4—81no—
408C22×C4C2×C22⋊C411yescharacteristic
409C22×C4C2×C22⋊C411yescharacteristic
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×C4C23×C411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C2×C8—22no—
422C2×C8—22no—
423C2×C8—41no—
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—84no—
446C4⋊C4—21no—
447C4⋊C4—22no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—42no—
453C4⋊C4—41no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—42no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—42no—
463C4⋊C4—42no—
464C4⋊C4—81no—
465C4⋊C4—81no—
466C4⋊C4—81no—
467C4⋊C4—81no—
468C4⋊C4—81no—
469C4⋊C4#13112yes—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—41no—
474C22⋊C4—41no—
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—42no—
486C22⋊C4—42no—
487C22⋊C4—81no—
488C22⋊C4—81no—
489C22⋊C4—81no—
490C22⋊C4—81no—
491C22⋊C4—81no—
492C22⋊C4—81no—
493C22⋊C4—81no—
494C22⋊C4—81no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—41no—
500C42—41no—
501C42—41no—
502C42—41no—
503C42—41no—
504C42—41no—
505C42—41no—
506C42—41no—
507C42—41no—
508C42—41no—
509C42—42no—
510C42—42no—
511C42—42no—
512C42—81no—
513C42—81no—
514C42#13512yes—
515C42#13512yes—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—41no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23#41312yes—
537C23C22×C22⋊C411yescharacteristic
538Q8—81no—
539Q8—81no—
540D4—44no—
541D4—81no—
542D4—81no—
543D4—84no—
544D4—162no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
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—22no—
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—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
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—81no—
605C2×C4—81no—
606C2×C4—81no—
607C2×C4—81no—
608C2×C4—81no—
609C2×C4—81no—
610C2×C4—81no—
611C2×C4—81no—
612C2×C4—162no—
613C2×C4#41612yes—
614C2×C4#41612yes—
615C2×C4C2×#13111yescharacteristic
616C2×C4#56811yescharacteristic
617C2×C4C2×#13511yescharacteristic
618C2×C4C2×#13511yescharacteristic
619C2×C4#57511yescharacteristic
620C2×C4#57511yescharacteristic
621C8—42no—
622C8—42no—
623C8—81no—
624C8—81no—
625C8—81no—
626C8—81no—
627C8—81no—
628C8—81no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—81no—
648C22—81no—
649C22—81no—
650C22#288011yescharacteristic
651C22#288311yescharacteristic
652C22C2×#41311yescharacteristic
653C22C2×#41611yescharacteristic
654C22#289611yescharacteristic
655C22#290111yescharacteristic
656C22#290111yescharacteristic
657C4—21no—
658C4—21no—
659C4—22no—
660C4—22no—
661C4—21no—
662C4—21no—
663C4—21no—
664C4—21no—
665C4—41no—
666C4—41no—
667C4—41no—
668C4—41no—
669C4—42no—
670C4—41no—
671C4—41no—
672C4—41no—
673C4—41no—
674C4—81no—
675C4—81no—
676C4—81no—
677C4—81no—
678C4—81no—
679C4—81no—
680C2—42no—
681C2—81no—
682C2—161no—
683C2#1742711yescharacteristic
684C2#1751011yescharacteristic
685C2#1751311yescharacteristic
686C2#1761711yescharacteristic
687C2#1762311yescharacteristic
688C2#1768611yescharacteristic
689C2#1771311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2414C4
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. #22839C2semidirect
3. #22785C2semidirect
4. #7730C2semidirect
5. #7721C2semidirect
6. #7690C2semidirect
7. #7675C2semidirect
8. #7672C2semidirect
9. #7666C2semidirect
10. #7644C2semidirect
11. #7620C2semidirect
12. #7613C2semidirect
13. #7582C2semidirect
14. #7579C2semidirect
15. #7562C2semidirect

Maximal subgroups

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

subgroupindex
#364022
#228392
#227852
#77302
#77212
#76902
#76752
#76722
#76662
#76442
#76202
#76132
#75822
#75792
#75622