Finite Group Databasebeta

← #6495909

Subgroups — #6495909

Order 512.

1494 conjugacy classes of subgroups.

The subgroup lattice has 713 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35319C211yesmaximal, characteristic
2#35291C211yesmaximal, characteristic
3#33176C211yesmaximal, characteristic
4#33172C211yesmaximal, characteristic
5#33049C211yesmaximal, characteristic
6#33043C211yesmaximal, characteristic
7#30307C211yesmaximal, characteristic, complemented
8#30262C211yesmaximal, characteristic, complemented
9#30220C211yesmaximal, characteristic, complemented
10#30206C211yesmaximal, characteristic, complemented
11#29565C214yesmaximal, complemented
12#29509C214yesmaximal, complemented
13#28588C214yesmaximal
14#25512C214yesmaximal
15#22270C211yesmaximal, characteristic, complemented
16#22270C211yesmaximal, characteristic, complemented
17#22221C211yesmaximal, characteristic, complemented
18#22218C211yesmaximal, characteristic, complemented
19#20366C211yesmaximal, characteristic
20C2×#541C2212yes—
21C2×#494C2211yescharacteristic
22C2×#494C2211yescharacteristic
23C2×#487C2211yescharacteristic
24C2×#487C2211yescharacteristic
25C2×#470C2214yes—
26C2×#467C2214yes—
27C2×C4×M16C2211yescharacteristic
28C22×C4×C8C2211yescharacteristic
29#2868C2211yescharacteristic
30#2862C2211yescharacteristic
31#2859C2211yescharacteristic
32#2859C2211yescharacteristic
33#2857C2211yescharacteristic
34#2857C2211yescharacteristic
35#2857C2211yescharacteristic
36#2857C2211yescharacteristic
37#2645C2212yes—
38#2642C2212yes—
39#2625C2212yes—
40#2621C2212yes—
41#2514—21no—
42#2509—21no—
43#2508—21no—
44#2506—21no—
45#2506—21no—
46#2506—21no—
47#2506—21no—
48#2496—21no—
49#2496—21no—
50#2496—21no—
51#2496—21no—
52#2495—21no—
53#2494—21no—
54#2489—21no—
55#2453—21no—
56#2421—21no—
57#2327C2212yes—
58#2323C2212yes—
59#2320C2212yes—
60C4×C4⋊Q8C2211yescharacteristic
61C4×C4⋊Q8C2211yescharacteristic
62C4×C4⋊Q8C2211yescharacteristic
63C4×C4⋊Q8C2211yescharacteristic
64C4×#117C2211yescharacteristic
65#2085C2214yes—
66#2074C2214yes—
67#2059C2214yes—
68#2043C2214yes—
69#2043C2214yes—
70#2026C2214yes—
71#2016C2214yes—
72#2016C2214yes—
73#1990C2211yescharacteristic
74#1990C2211yescharacteristic
75#1971C2214yes—
76#1969C2214yes—
77#1956C2211yescharacteristic
78#1956C2211yescharacteristic
79#1956C2211yescharacteristic
80#1956C2211yescharacteristic
81#1956C2211yescharacteristic
82#1956C2211yescharacteristic
83#1935C2214yes—
84#1933C2214yes—
85#1928C2214yes—
86#1927C2214yes—
87#1894C2214yes—
88#1892C2214yes—
89#1878C2214yes—
90#1878C2214yes—
91#1878C2214yes—
92#1878C2214yes—
93#1865C2214yes—
94#1863C2214yes—
95#1845C2211yescharacteristic
96#1844C2211yescharacteristic
97#1834C2211yescharacteristic
98#1834C2211yescharacteristic
99#1746C2212yes—
100#1746C2212yes—
101#1744C2211yescharacteristic
102#1744C2211yescharacteristic
103#1744C2211yescharacteristic
104#1744C2211yescharacteristic
105#1729C2212yes—
106#1729C2212yes—
107C22×Q16C2314yes—
108C22×Q16C2314yes—
109C22×M16C2312yes—
110C23×C8C2312yes—
111#541—48no—
112C2×C4⋊Q8C2311yescharacteristic
113C2×C4⋊Q8C2311yescharacteristic
114C2×C4⋊Q8C2311yescharacteristic
115C2×C4⋊Q8C2312yes—
116C2×C4⋊Q8C2312yes—
117C2×C4⋊Q8C2311yescharacteristic
118C2×#126—22no—
119C2×#126—22no—
120C2×#125—22no—
121C2×#125—22no—
122C2×#125—21no—
123C2×#125C2312yes—
124C2×C22⋊Q8—21no—
125C2×C22⋊Q8C2312yes—
126C2×C22⋊Q8C2312yes—
127C2×C22⋊Q8C2312yes—
128C2×C4×Q8—21no—
129C2×C4×Q8C2312yes—
130C2×C4×Q8C2312yes—
131C2×C4×Q8C2312yes—
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#494—24no—
136#494—24no—
137#487—24no—
138#487—24no—
139#470—48no—
140#467—48no—
141C2×#107C2314yes—
142C2×#106C2314yes—
143C2×#103C2314yes—
144C2×#103C2314yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C4×M16—24no—
152C2×#97C2312yes—
153C2×C4×C8—24no—
154C2×C4×C8C2312yes—
155C2×C4×C8C2311yescharacteristic
156C2×C4×C8C2311yescharacteristic
157C2×C4×C8C2311yescharacteristic
158C2×C4×C8C2311yescharacteristic
159#401—22no—
160#400—22no—
161#400—22no—
162#400—22no—
163#400—22no—
164#400—22no—
165#400—22no—
166#400—22no—
167#400—22no—
168#398—22no—
169#398—22no—
170#398—22no—
171#397—22no—
172#397—22no—
173#397—22no—
174#397—22no—
175#391—21no—
176#391—21no—
177#391—21no—
178#391—21no—
179#391C2312yes—
180#391C2312yes—
181#391C2312yes—
182#391C2312yes—
183#391C2312yes—
184#391C2312yes—
185#391C2312yes—
186#391C2312yes—
187#389—22no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389—21no—
192#389—21no—
193#389—21no—
194#389—21no—
195#389—21no—
196#389—21no—
197#389—21no—
198#389—21no—
199#389—21no—
200#389—21no—
201#389—21no—
202#389—21no—
203#389—21no—
204#389—21no—
205#389—21no—
206#389—21no—
207#389—21no—
208#389C2312yes—
209#389C2312yes—
210#389C2312yes—
211#389C2312yes—
212#388—22no—
213#388—22no—
214#387—22no—
215#387—22no—
216#387—22no—
217#387—22no—
218#387—22no—
219#387—22no—
220#387—22no—
221#384—21no—
222#384—21no—
223#384—21no—
224#384—21no—
225#384C2311yescharacteristic
226#384C2311yescharacteristic
227#384C2311yescharacteristic
228#384C2311yescharacteristic
229#383—22no—
230#382—21no—
231#382—21no—
232#382—21no—
233#382—22no—
234#382C2312yes—
235#380—21no—
236#380—21no—
237#380—21no—
238#380—21no—
239#380—21no—
240#380—21no—
241#380—21no—
242#380C2312yes—
243C4×C4⋊C4—21no—
244C4×C4⋊C4—21no—
245C4×C4⋊C4—21no—
246C4×C4⋊C4—21no—
247C4×C4⋊C4—21no—
248C4×C4⋊C4—21no—
249C4×C4⋊C4—21no—
250C4×C4⋊C4D411yescharacteristic
251C4×C4⋊C4D411yescharacteristic
252C4×C4⋊C4D411yescharacteristic
253C4×C4⋊C4D411yescharacteristic
254C4×C4⋊C4C2312yes—
255C4×C22⋊C4D411yescharacteristic
256C4×C22⋊C4D411yescharacteristic
257C4×C22⋊C4D411yescharacteristic
258C4×C22⋊C4D411yescharacteristic
259#376C2311yescharacteristic
260#376C2311yescharacteristic
261C43C2311yescharacteristic
262C43C2311yescharacteristic
263#340C2314yes—
264#340C2314yes—
265#340C2314yes—
266#340C2314yes—
267#336C2312yes—
268#336C2312yes—
269#336C2312yes—
270#336C2312yes—
271#336C2312yes—
272#336C2312yes—
273#336C2312yes—
274#336C2312yes—
275C22×Q8—21no—
276C22×Q8C2412yes—
277C22×Q8C2412yes—
278C22×Q8C2412yes—
279C23×C4C2×D411yescharacteristic
280C23×C4C2×D411yescharacteristic
281C23×C4C2411yescharacteristic
282C2×Q16—28no—
283C2×Q16—28no—
284C2×Q16—44no—
285C2×Q16—44no—
286C2×Q16—44no—
287C2×Q16—44no—
288C2×M16—24no—
289C2×M16—24no—
290C2×M16—42no—
291C22×C8—24no—
292C22×C8—24no—
293C22×C8—42no—
294C22×C8C2412yes—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C4⋊Q8—22no—
299C4⋊Q8—22no—
300C4⋊Q8—22no—
301C4⋊Q8—22no—
302C4⋊Q8—22no—
303C4⋊Q8—22no—
304C4⋊Q8—22no—
305C4⋊Q8—22no—
306C4⋊Q8—44no—
307C4⋊Q8—44no—
308#126—48no—
309#126—48no—
310#125—44no—
311#125—48no—
312#125—48no—
313#125—82no—
314C22⋊Q8—28no—
315C22⋊Q8—28no—
316C22⋊Q8—28no—
317C22⋊Q8—82no—
318C4×Q8—44no—
319C4×Q8—44no—
320C4×Q8—44no—
321C4×Q8—82no—
322#117—24no—
323#117—42no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—22no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—21no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—21no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—22no—
355C2×C4⋊C4C2×D411yescharacteristic
356C2×C4⋊C4C2×D411yescharacteristic
357C2×C4⋊C4C2×D411yescharacteristic
358C2×C4⋊C4C2×D411yescharacteristic
359C2×C4⋊C4C2412yes—
360C2×C4⋊C4C2412yes—
361C2×C4⋊C4C2412yes—
362C2×C4⋊C4C2412yes—
363C2×C4⋊C4C2412yes—
364C2×C22⋊C4—22no—
365C2×C22⋊C4—22no—
366C2×C22⋊C4—22no—
367C2×C22⋊C4—22no—
368C2×C22⋊C4C2×D411yescharacteristic
369C2×C22⋊C4C2×D411yescharacteristic
370C2×C22⋊C4C2×D411yescharacteristic
371C2×C22⋊C4C2×D411yescharacteristic
372C2×C42—22no—
373C2×C42—22no—
374C2×C42—22no—
375C2×C42—22no—
376C2×C42—22no—
377C2×C42—22no—
378C2×C42C2×D411yescharacteristic
379C2×C42C2×D411yescharacteristic
380C2×C42C2×D411yescharacteristic
381C2×C42C2×D411yescharacteristic
382C2×C42C2×D411yescharacteristic
383C2×C42C2×D411yescharacteristic
384C2×C42C2×D411yescharacteristic
385C2×C42C2×D411yescharacteristic
386C2×C42C2×D411yescharacteristic
387C2×C42C2×D411yescharacteristic
388C2×C42C2411yescharacteristic
389C2×C42C2411yescharacteristic
390C2×C42C2411yescharacteristic
391C2×C42C2411yescharacteristic
392C2×C42C2411yescharacteristic
393C2×C42C2411yescharacteristic
394#107—44no—
395#106—44no—
396#103—28no—
397#103—28no—
398#103—44no—
399#103—44no—
400#103—44no—
401#103—44no—
402#103—44no—
403#103—44no—
404#97—24no—
405C4×C8—24no—
406C4×C8—22no—
407C4×C8—22no—
408C4×C8#4114yes—
409C4×C8#4114yes—
410#95—22no—
411#95—22no—
412#95—22no—
413#95—22no—
414#95—22no—
415#95—22no—
416#95—22no—
417#95—22no—
418#95—22no—
419#95—22no—
420#95—22no—
421#95—22no—
422#95—22no—
423#95—22no—
424#95—22no—
425#95—22no—
426#95—22no—
427#95—22no—
428#95—22no—
429#95—22no—
430#95—22no—
431#95—22no—
432#95—22no—
433#95—22no—
434#95C2×D411yescharacteristic
435#95C2×D411yescharacteristic
436#95C2×D411yescharacteristic
437#95C2×D411yescharacteristic
438#95C2×D411yescharacteristic
439#95C2×D411yescharacteristic
440#95C2×D411yescharacteristic
441#95C2×D411yescharacteristic
442C24C22×D411yescharacteristic
443C2×Q8—22no—
444C2×Q8—22no—
445C2×Q8—22no—
446C2×Q8—22no—
447C2×Q8—22no—
448C2×Q8—22no—
449C2×Q8—24no—
450C2×Q8—24no—
451C2×Q8—24no—
452C2×Q8—24no—
453C2×Q8—24no—
454C2×Q8—24no—
455C2×Q8—42no—
456C2×Q8—42no—
457C2×Q8—42no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4C22×D411yescharacteristic
476C22×C4C22×D411yescharacteristic
477C22×C4C22×D411yescharacteristic
478C22×C4C22×D411yescharacteristic
479C22×C4C22×D411yescharacteristic
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D411yescharacteristic
482C22×C4C22×D411yescharacteristic
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C22×D411yescharacteristic
488C22×C4H(5,𝔽2)14yes—
489C22×C4C2511yescharacteristic
490Q16—88no—
491Q16—88no—
492M16—48no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—22no—
497C2×C8—22no—
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—48no—
504C2×C8C2×#4114yes—
505C2×C8C2×#4114yes—
506C4⋊C4—22no—
507C4⋊C4—22no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—24no—
511C4⋊C4—24no—
512C4⋊C4—24no—
513C4⋊C4—24no—
514C4⋊C4—41no—
515C4⋊C4—41no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—44no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—44no—
523C4⋊C4—44no—
524C4⋊C4—44no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—42no—
532C4⋊C4—44no—
533C4⋊C4—44no—
534C4⋊C4—44no—
535C4⋊C4—42no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—42no—
539C4⋊C4—42no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—82no—
544C4⋊C4—82no—
545C4⋊C4—82no—
546C4⋊C4—82no—
547C22⋊C4—22no—
548C22⋊C4—22no—
549C22⋊C4—41no—
550C22⋊C4—41no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—82no—
555C42—22no—
556C42—22no—
557C42—22no—
558C42—22no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—21no—
566C42—21no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—44no—
572C42—44no—
573C42—44no—
574C42—82no—
575C42C2×#4112yes—
576C42C2×#4112yes—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23#53512yes—
584C23C23×D411yescharacteristic
585Q8—44no—
586Q8—44no—
587Q8—44no—
588Q8—44no—
589Q8—44no—
590Q8—44no—
591Q8—84no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—21no—
617C2×C4—21no—
618C2×C4—21no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4#53212yes—
653C2×C4#53212yes—
654C2×C4#53512yes—
655C2×C4#57712yes—
656C2×C4#57812yes—
657C2×C4#57812yes—
658C2×C4#57812yes—
659C2×C4C22×#4111yescharacteristic
660C2×C4C22×#4111yescharacteristic
661C2×C4C2×H(5,𝔽2)12yes—
662C2×C4#58512yes—
663C8—24no—
664C8—24no—
665C8—42no—
666C8—42no—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—41no—
674C22C2×#53211yescharacteristic
675C22C2×#53511yescharacteristic
676C22#344911yescharacteristic
677C22C2×#57711yescharacteristic
678C22C2×#57811yescharacteristic
679C22C2×#57811yescharacteristic
680C22C2×#57811yescharacteristic
681C4—22no—
682C4—41no—
683C4—41no—
684C4—41no—
685C4—41no—
686C4—41no—
687C4—41no—
688C4—41no—
689C4—41no—
690C4—41no—
691C4—41no—
692C4—82no—
693C4—82no—
694C4—82no—
695C4—82no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4#317714yes—
701C4#317814yes—
702C4#349412yes—
703C4#349412yes—
704C2—42no—
705C2#3530311yescharacteristic
706C2C2×#317711yescharacteristic
707C2C2×#317811yescharacteristic
708C2#6082111yescharacteristic
709C2#6082411yescharacteristic
710C2#6100811yescharacteristic
711C2#6101311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
7. #30307C2semidirect
8. #30262C2semidirect
9. #30220C2semidirect
10. #30206C2semidirect
11. #29565C2semidirect
12. #29509C2semidirect
15. #22270C2semidirect
16. #22270C2semidirect
17. #22221C2semidirect
18. #22218C2semidirect

Maximal subgroups

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

subgroupindex
#353192
#352912
#331762
#331722
#330492
#330432
#303072
#302622
#302202
#302062
#295652
#295092
#285882
#255122
#222702
#222212
#222182
#203662