Finite Group Databasebeta

← #6992941

Subgroups — #6992941

Order 512.

1841 conjugacy classes of subgroups.

The subgroup lattice has 717 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36481C211yesmaximal, characteristic, complemented
2#35285C211yesmaximal, characteristic
3#32799C212yesmaximal, complemented
4#31012C214yesmaximal, complemented
5#30464C214yesmaximal, complemented
6#30343C212yesmaximal, complemented
7#30041C212yesmaximal, complemented
8#29664C211yesmaximal, characteristic, complemented
9#28473C212yesmaximal, complemented
10#28447C212yesmaximal, complemented
11#27758C212yesmaximal, complemented
12#25717C211yesmaximal, characteristic
13#25369C212yesmaximal, complemented
14#24733C212yesmaximal, complemented
15#24211C211yesmaximal, characteristic
16#23571C211yesmaximal, characteristic, complemented
17#22643C211yesmaximal, characteristic, complemented
18C2×#533C2211yescharacteristic
19C22×#127C2211yescharacteristic
20C2×#497C2214yescomplemented
21C2×#492C2212yes—
22C2×#464C2214yescomplemented
23C2×#464C2214yescomplemented
24C2×#462C2212yescomplemented
25C2×#460C2212yes—
26C2×#450C2212yes—
27C2×#450C2214yes—
28C2×C4×Q16C2212yescomplemented
29#2863C2212yescomplemented
30#2858C2212yes—
31#2858C2212yes—
32#2857C2211yescharacteristic
33#2641—22no—
34#2606—22no—
35#2552C2211yescharacteristic, complemented
36#2548—21no—
37#2548—21no—
38#2548C2212yes—
39#2548C2212yes—
40#2546C2211yescharacteristic, complemented
41#2522—22no—
42#2507—21no—
43#2441—21no—
44#2435—22no—
45#2435C2214yes—
46#2429C2212yes—
47#2423C2212yes—
48#2370—21no—
49#2365C2211yescharacteristic, complemented
50Q8×C4⋊C4C2211yescharacteristic, complemented
51#2347—21no—
52#2309—21no—
53#2309—21no—
54C4×C4⋊Q8C2212yescomplemented
55C4×C22⋊Q8C2211yescharacteristic
56C4×#117C2211yescharacteristic
57#2071C2214yescomplemented
58#2067C2214yescomplemented
59#2057C2211yescharacteristic, complemented
60#2055C2214yescomplemented
61#2043C2211yescharacteristic, complemented
62#2042C2214yescomplemented
63#2042C2214yescomplemented
64#2041C2212yes—
65#2036C2212yes—
66#2024C2211yescharacteristic, complemented
67#2024C2212yescomplemented
68#2016C2211yescharacteristic, complemented
69#2016C2212yescomplemented
70#2015C2214yescomplemented
71#2015C2214yescomplemented
72#2014C2212yes—
73#2014C2212yes—
74#2014C2212yes—
75#2013C2214yescomplemented
76#2001C2212yes—
77#2001C2214yes—
78#1982C2212yes—
79#1973C2214yescomplemented
80#1959C2214yescomplemented
81#1957C2212yes—
82#1939C2212yescomplemented
83#1936C2214yescomplemented
84#1924C2214yes—
85#1880C2214yescomplemented
86#1880C2214yescomplemented
87#1839C2212yescomplemented
88#1803C2212yescomplemented
89#1797C2211yescharacteristic
90#1786C2212yes—
91#1770C2212yescomplemented
92#1767C2212yescomplemented
93C4×#103C2212yescomplemented
94C4×C22⋊C8C2211yescharacteristic
95C23×D4—21no—
96C23×D4—21no—
97C23×D4C2312yes—
98C22×Q16C2312yes—
99C22×Q16C2314yes—
100C22×SD16C2314yes—
101C22×SD16C2314yes—
102C22×SD16C2314yes—
103#533—44no—
104C2×C4⋊Q8C2311yescharacteristic
105C2×C4⋊Q8C2312yes—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2311yescharacteristic
108C2×#127—21no—
109C2×#127—21no—
110C2×#127—44no—
111C2×#127C2312yes—
112C2×#127C2312yes—
113C2×#125—21no—
114C2×#125—21no—
115C2×#124—22no—
116C2×#124—22no—
117C2×#124C2314yes—
118C2×#124C2314yes—
119C2×C22⋊Q8—21no—
120C2×C22⋊Q8C2311yescharacteristic
121C2×C22⋊Q8C2312yes—
122C2×C22⋊Q8C2312yes—
123C2×C4×Q8—21no—
124C2×C4×Q8C2312yes—
125C2×C4×Q8C2311yescharacteristic
126C2×C4×Q8C2312yes—
127C2×C4×Q8C2312yes—
128C2×#117—21no—
129C2×#117D411yescharacteristic
130C2×#117D411yescharacteristic
131C22×C42C2311yescharacteristic
132#497—48no—
133#492—44no—
134#464—48no—
135#464—48no—
136#462—44no—
137#460—44no—
138#450—48no—
139#450—44no—
140C4×Q16—44no—
141C2×#107C2312yes—
142C2×C4⋊C8C2312yes—
143C2×C4⋊C8C2314yes—
144C2×#103C2312yes—
145C2×#103C2314yes—
146C2×#103C2314yes—
147C2×#103C2312yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#102C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×C22⋊C8C2312yes—
156C2×C22⋊C8C2312yes—
157C2×#97C2314yes—
158C2×C4×C8C2312yes—
159#399—22no—
160#398—22no—
161#398—22no—
162#398—22no—
163#398—22no—
164#396—22no—
165#394—24no—
166#394—24no—
167#394—24no—
168#394—24no—
169#391C2311yescharacteristic
170#391C2312yes—
171#391C2312yes—
172#391C2312yes—
173#390—22no—
174#390—22no—
175#390—22no—
176#390—22no—
177#390—22no—
178#390—22no—
179#390—22no—
180#390—22no—
181#390—22no—
182#390—22no—
183#390C2314yes—
184#390C2314yes—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—22no—
190#389—22no—
191#389—21no—
192#389—21no—
193#389—22no—
194#389—22no—
195#389C2311yescharacteristic
196#389C2312yes—
197#389C2312yes—
198#388—22no—
199#388—22no—
200#387—21no—
201#387—22no—
202#387—22no—
203#387—22no—
204#387—22no—
205#387—22no—
206#385—21no—
207#384—21no—
208#384C2312yes—
209#384C2311yescharacteristic
210#383—22no—
211#380—21no—
212#380—21no—
213#380C2311yescharacteristic
214#379—24no—
215#379—24no—
216C4×C4⋊C4—21no—
217C4×C4⋊C4—21no—
218C4×C4⋊C4—22no—
219C4×C4⋊C4—22no—
220C4×C4⋊C4—21no—
221C4×C4⋊C4D411yescharacteristic
222C4×C4⋊C4D411yescharacteristic
223C4×C4⋊C4C2311yescharacteristic
224C4×C4⋊C4C2311yescharacteristic
225C4×C4⋊C4C2311yescharacteristic
226C4×C22⋊C4—21no—
227C4×C22⋊C4D412yes—
228C4×C22⋊C4C2312yes—
229#376D412yes—
230C43C2312yes—
231#340C2312yes—
232#340C2314yes—
233#340C2312yes—
234#336C2312yes—
235#336C2311yescharacteristic
236#336C2314yes—
237#336C2311yescharacteristic
238#336C2314yes—
239C25—22no—
240C22×Q8—21no—
241C22×Q8C2412yes—
242C22×Q8C2412yes—
243C22×Q8C2411yescharacteristic
244C22×D4—22no—
245C22×D4—22no—
246C22×D4—22no—
247C22×D4—22no—
248C22×D4—22no—
249C22×D4—44no—
250C22×D4—42no—
251C22×D4—44no—
252C22×D4—42no—
253C22×D4—44no—
254C22×D4—44no—
255C22×D4—44no—
256C22×D4—44no—
257C22×D4—82no—
258C22×D4—82no—
259C22×D4C2414yes—
260C23×C4C2×D411yescharacteristic
261C23×C4C2×D411yescharacteristic
262C23×C4C2411yescharacteristic
263C2×Q16—24no—
264C2×Q16—24no—
265C2×Q16—42no—
266C2×Q16—44no—
267C2×Q16—44no—
268C2×Q16—44no—
269C2×SD16—28no—
270C2×SD16—28no—
271C2×SD16—44no—
272C2×SD16—44no—
273C2×SD16—44no—
274C2×SD16—44no—
275C2×SD16—44no—
276C2×SD16—44no—
277C2×SD16—44no—
278C22×C8C2412yes—
279C22×C8C2414yes—
280C22×C8C2412yes—
281C4⋊Q8—24no—
282C4⋊Q8—24no—
283C4⋊Q8—42no—
284C4⋊Q8—42no—
285C4⋊Q8—42no—
286#127—24no—
287#127—24no—
288#127—44no—
289#127—44no—
290#127—44no—
291#125—44no—
292#125—44no—
293#124—44no—
294#124—44no—
295#124—44no—
296#124—44no—
297#124—84no—
298#124—84no—
299C22⋊Q8—22no—
300C22⋊Q8—41no—
301C22⋊Q8—42no—
302C22⋊Q8—42no—
303C22⋊Q8—44no—
304C22⋊Q8—44no—
305C4×Q8—24no—
306C4×Q8—24no—
307C4×Q8—24no—
308C4×Q8—44no—
309C4×Q8—44no—
310C4×Q8—44no—
311#117—24no—
312#117—42no—
313#117—44no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4—22no—
323C2×C4⋊C4—22no—
324C2×C4⋊C4—21no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—22no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4C2×D411yescharacteristic
336C2×C4⋊C4C2×D411yescharacteristic
337C2×C4⋊C4C2×D411yescharacteristic
338C2×C4⋊C4C2×D411yescharacteristic
339C2×C4⋊C4C2411yescharacteristic
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2411yescharacteristic
343C2×C4⋊C4C2411yescharacteristic
344C2×C22⋊C4—22no—
345C2×C22⋊C4—21no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—24no—
349C2×C22⋊C4—24no—
350C2×C22⋊C4—24no—
351C2×C22⋊C4—24no—
352C2×C22⋊C4—24no—
353C2×C22⋊C4—24no—
354C2×C22⋊C4—24no—
355C2×C22⋊C4—24no—
356C2×C22⋊C4C2×D412yes—
357C2×C22⋊C4C2×D412yes—
358C2×C42—24no—
359C2×C42—22no—
360C2×C42—21no—
361C2×C42—21no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42C2×D412yes—
365C2×C42C2×D411yescharacteristic
366C2×C42C2×D412yes—
367C2×C42C2×D411yescharacteristic
368C2×C42C2×D412yes—
369C2×C42C2×D412yes—
370C2×C42C2412yes—
371C2×C42C2412yes—
372C2×C42C2411yescharacteristic
373C2×C42C2411yescharacteristic
374#107—42no—
375C4⋊C8—28no—
376C4⋊C8—24no—
377#103—24no—
378#103—42no—
379#103—42no—
380#103—42no—
381#103—44no—
382#103—44no—
383#102—28no—
384#102—44no—
385#102—44no—
386#102—44no—
387#102—44no—
388C22⋊C8—24no—
389C22⋊C8—42no—
390#97—44no—
391C4×C8—24no—
392#95—21no—
393#95—21no—
394#95—21no—
395#95—22no—
396#95—22no—
397#95—21no—
398#95—22no—
399#95—22no—
400#95—21no—
401#95—22no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95C2×D412yes—
406#95C2×D412yes—
407#95C2×D412yes—
408#95C2×D412yes—
409C24—24no—
410C24—44no—
411C24—44no—
412C24—44no—
413C24—82no—
414C24—82no—
415C24—82no—
416C24—82no—
417C24C22×D411yescharacteristic
418C2×Q8—21no—
419C2×Q8—22no—
420C2×Q8—22no—
421C2×Q8—22no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—24no—
425C2×Q8—22no—
426C2×Q8—24no—
427C2×Q8—22no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8—22no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×Q8—42no—
434C2×Q8H(5,𝔽2)12yes—
435C2×D4—24no—
436C2×D4—24no—
437C2×D4—24no—
438C2×D4—24no—
439C2×D4—24no—
440C2×D4—24no—
441C2×D4—44no—
442C2×D4—44no—
443C2×D4—44no—
444C2×D4—44no—
445C2×D4—44no—
446C2×D4—44no—
447C2×D4—44no—
448C2×D4—44no—
449C2×D4—44no—
450C2×D4—44no—
451C2×D4—44no—
452C2×D4—82no—
453C2×D4—82no—
454C2×D4—82no—
455C2×D4—82no—
456C2×D4—816no—
457C2×D4—88no—
458C2×D4—88no—
459C2×D4—816no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—22no—
475C22×C4—22no—
476C22×C4—41no—
477C22×C4#13714yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D411yescharacteristic
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D411yescharacteristic
484C22×C4C22×D411yescharacteristic
485C22×C4C22×D411yescharacteristic
486C22×C4C22×D411yescharacteristic
487C22×C4C2511yescharacteristic
488Q16—48no—
489Q16—88no—
490SD16—88no—
491SD16—88no—
492SD16—88no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—24no—
497C2×C8—22no—
498C2×C8—24no—
499C2×C8—24no—
500C2×C8—22no—
501C2×C8H(5,𝔽2)14yes—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—24no—
507C4⋊C4—21no—
508C4⋊C4—22no—
509C4⋊C4—22no—
510C4⋊C4—22no—
511C4⋊C4—22no—
512C4⋊C4—21no—
513C4⋊C4—42no—
514C4⋊C4—44no—
515C4⋊C4—44no—
516C4⋊C4—42no—
517C4⋊C4—42no—
518C4⋊C4—44no—
519C4⋊C4—42no—
520C4⋊C4—44no—
521C4⋊C4—44no—
522C4⋊C4—42no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—81no—
529C4⋊C4—82no—
530C4⋊C4—82no—
531C4⋊C4—82no—
532C4⋊C4—82no—
533C4⋊C4—82no—
534C22⋊C4—24no—
535C22⋊C4—42no—
536C22⋊C4—42no—
537C22⋊C4—44no—
538C22⋊C4—84no—
539C22⋊C4—84no—
540C22⋊C4—84no—
541C22⋊C4—84no—
542C22⋊C4—84no—
543C22⋊C4—84no—
544C22⋊C4—84no—
545C22⋊C4—84no—
546C22⋊C4—81no—
547C22⋊C4—82no—
548C42—22no—
549C42—22no—
550C42—24no—
551C42—24no—
552C42—22no—
553C42—24no—
554C42—22no—
555C42—22no—
556C42—42no—
557C42—42no—
558C42—44no—
559C42—42no—
560C42—42no—
561C42—82no—
562C42—82no—
563C42H(5,𝔽2)14yes—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—88no—
578C23—88no—
579C23—88no—
580C23—164no—
581C23—164no—
582C23—164no—
583C23—164no—
584C23C2×#13712yes—
585C23C23×D411yescharacteristic
586Q8—24no—
587Q8—42no—
588Q8—44no—
589Q8—42no—
590Q8—42no—
591Q8—44no—
592Q8—42no—
593Q8—42no—
594Q8—84no—
595D4—44no—
596D4—44no—
597D4—44no—
598D4—44no—
599D4—88no—
600D4—88no—
601D4—88no—
602D4—88no—
603D4—88no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—28no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—21no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—21no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—21no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—44no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—41no—
639C2×C4—42no—
640C2×C4—41no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—41no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—41no—
652C2×C4—41no—
653C2×C4#53512yes—
654C2×C4#53512yes—
655C2×C4C2×#13712yes—
656C2×C4C2×#13712yes—
657C2×C4#57512yes—
658C2×C4#57712yes—
659C2×C4C2×H(5,𝔽2)11yescharacteristic
660C2×C4C2×H(5,𝔽2)12yes—
661C2×C4C2×H(5,𝔽2)12yes—
662C2×C4#58511yescharacteristic
663C8—24no—
664C8—44no—
665C8—42no—
666C22—22no—
667C22—22no—
668C22—22no—
669C22—41no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—84no—
674C22—84no—
675C22—84no—
676C22—84no—
677C22—84no—
678C22—84no—
679C22—84no—
680C22—1616no—
681C22C2×#53511yescharacteristic
682C22#348811yescharacteristic
683C22#349811yescharacteristic
684C22C22×#13711yescharacteristic
685C22C2×#57511yescharacteristic
686C22C2×#57711yescharacteristic
687C22#358511yescharacteristic
688C4—22no—
689C4—22no—
690C4—22no—
691C4—42no—
692C4—42no—
693C4—42no—
694C4—41no—
695C4—42no—
696C4—42no—
697C4—81no—
698C4—81no—
699C4—82no—
700C4—82no—
701C4—82no—
702C4—81no—
703C4—82no—
704C4#321514yes—
705C4#353712yes—
706C4#354612yes—
707C2—42no—
708C2—164no—
709C2#4298611yescharacteristic
710C2C2×#321511yescharacteristic
711C2#6081211yescharacteristic
712C2#6196511yescharacteristic
713C2#6199711yescharacteristic
714C2#6242211yescharacteristic
715C2#6245111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#127C22
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
1. #36481C2semidirect
3. #32799C2semidirect
4. #31012C2semidirect
5. #30464C2semidirect
6. #30343C2semidirect
7. #30041C2semidirect
8. #29664C2semidirect
9. #28473C2semidirect
10. #28447C2semidirect
11. #27758C2semidirect
13. #25369C2semidirect
14. #24733C2semidirect
16. #23571C2semidirect
17. #22643C2semidirect
20. C2×#497C22semidirect
22. C2×#464C22semidirect
23. C2×#464C22semidirect
24. C2×#462C22semidirect
28. C2×C4×Q16C22semidirect
29. #2863C22semidirect
35. #2552C22semidirect
40. #2546C22semidirect
49. #2365C22semidirect
50. Q8×C4⋊C4C22semidirect
54. C4×C4⋊Q8C22semidirect
57. #2071C22semidirect
58. #2067C22semidirect
59. #2057C22semidirect
60. #2055C22semidirect
61. #2043C22semidirect
62. #2042C22semidirect
63. #2042C22semidirect
66. #2024C22semidirect
67. #2024C22semidirect
68. #2016C22semidirect
69. #2016C22semidirect
70. #2015C22semidirect
71. #2015C22semidirect
75. #2013C22semidirect
79. #1973C22semidirect
80. #1959C22semidirect
82. #1939C22semidirect
83. #1936C22semidirect
85. #1880C22semidirect
86. #1880C22semidirect
87. #1839C22semidirect
88. #1803C22semidirect
91. #1770C22semidirect
92. #1767C22semidirect
93. C4×#103C22semidirect

Maximal subgroups

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

subgroupindex
#364812
#352852
#327992
#310122
#304642
#303432
#300412
#296642
#284732
#284472
#277582
#257172
#253692
#247332
#242112
#235712
#226432