Finite Group Databasebeta

← #6489269

Subgroups — #6489269

Order 512.

1720 conjugacy classes of subgroups.

The subgroup lattice has 747 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic, complemented
2#36631C211yesmaximal, characteristic
3#33184C212yesmaximal, complemented
4#33051C212yesmaximal, complemented
5#30347C214yesmaximal, complemented
6#29212C211yesmaximal, characteristic, complemented
7#29197C212yesmaximal, complemented
8#29150C211yesmaximal, characteristic, complemented
9#28754C211yesmaximal, characteristic, complemented
10#28620C212yesmaximal, complemented
11#28451C211yesmaximal, characteristic, complemented
12#25499C214yesmaximal, complemented
13#24753C212yesmaximal
14#24752C212yesmaximal, complemented
15#22154C212yesmaximal, complemented
16#22153C212yesmaximal, complemented
17C2×#1767C211yesmaximal, characteristic, complemented
18C2×#529C2212yes—
19C2×#497C2212yes—
20C2×#489C2212yes—
21C2×#467C2214yes—
22C2×#465C2214yes—
23C2×#462C2212yes—
24C2×#460C2212yes—
25C22×C4⋊C8C2212yes—
26C22×#97C2212yes—
27#2864—21no—
28#2863—21no—
29#2863C2212yes—
30#2862C2211yescharacteristic
31#2861C2212yes—
32#2860—21no—
33#2860—21no—
34#2859C2211yescharacteristic
35#2859C2211yescharacteristic
36#2858C2211yescharacteristic
37#2857C2211yescharacteristic
38#2857C2211yescharacteristic
39#2857C2211yescharacteristic
40#2856C2211yescharacteristic
41#2855—21no—
42#2855C2212yes—
43#2854—21no—
44#2854—21no—
45#2851—21no—
46#2850C2212yes—
47C4×C4⋊Q8C2211yescharacteristic
48C4×C4⋊Q8C2211yescharacteristic
49C4×C4⋊Q8C2211yescharacteristic
50C4×#127C2211yescharacteristic
51C4×#125—21no—
52C4×#124C2212yes—
53C4×C22⋊Q8—21no—
54C4×C22⋊Q8C2212yes—
55C4×#121—21no—
56C4×#121—21no—
57#2295—21no—
58#2295C2212yes—
59#2293—21no—
60#2293—21no—
61#2283—21no—
62C2×C43C2211yescharacteristic
63#2068C2214yes—
64#2065C2214yes—
65#2059C2214yes—
66#2056C2214yes—
67#2055C2214yes—
68#2042C2214yes—
69#2026C2214yes—
70#2015C2214yes—
71#1993C2212yes—
72#1982C2212yes—
73#1981C2212yes—
74#1973C2214yes—
75#1971C2214yes—
76#1960C2212yes—
77#1959C2212yes—
78#1959C2212yes—
79#1948C2212yes—
80#1947C2212yes—
81#1936C2214yes—
82#1935C2214yes—
83#1893C2214yes—
84#1892C2214yes—
85#1878C2214yes—
86#1877C2214yes—
87#1865C2214yes—
88#1862C2214yes—
89#1847C2212yes—
90#1836C2212yes—
91#1803C2212yes—
92#1803C2212yes—
93#1767C2212yes—
94#1767C2211yescharacteristic
95#1767C2211yescharacteristic
96#1767C2212yes—
97#1767C2212yes—
98C2×#336C2212yes—
99C22×Q16C2314yes—
100C22×SD16C2314yes—
101C23×C8C2314yes—
102C2×C4⋊Q8C2311yescharacteristic
103C2×C4⋊Q8C2311yescharacteristic
104C2×C4⋊Q8C2311yescharacteristic
105C2×#127C2311yescharacteristic
106#529—48no—
107C2×#126—22no—
108C2×#125—21no—
109C2×#125C2312yes—
110C2×#124C2312yes—
111C2×#124C2312yes—
112C2×#123—22no—
113C2×C22⋊Q8—21no—
114C2×C22⋊Q8C2312yes—
115C2×C22⋊Q8C2312yes—
116C2×#121—21no—
117C2×#121—21no—
118C2×#121C2312yes—
119C2×C4×Q8—21no—
120C2×C4×Q8C2312yes—
121C2×C4×Q8C2312yes—
122C2×C4×D4—22no—
123C2×C4×D4—21no—
124C2×C4×D4—21no—
125C2×C4×D4C2312yes—
126C2×#117—22no—
127C22×C42D411yescharacteristic
128C22×C42D411yescharacteristic
129C22×C42D411yescharacteristic
130C22×C42D411yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134#497—44no—
135#489—44no—
136#467—48no—
137#465—48no—
138#462—28no—
139#460—28no—
140C2×#107C2314yes—
141C2×#106C2314yes—
142C2×C4⋊C8—28no—
143C2×C4⋊C8C2314yes—
144C2×C4⋊C8C2312yes—
145C2×C4⋊C8C2312yes—
146C2×#103C2314yes—
147C2×#103C2314yes—
148C2×#103C2314yes—
149C2×#103C2314yes—
150C2×#103C2314yes—
151C2×#102C2314yes—
152C2×#102C2314yes—
153C2×#102C2314yes—
154C2×#97—44no—
155C2×#97C2312yes—
156C2×#97C2312yes—
157C2×#97C2314yes—
158#391—21no—
159#391—21no—
160#391—21no—
161#391—21no—
162#391C2312yes—
163#391C2312yes—
164#391C2312yes—
165#391C2312yes—
166#391C2312yes—
167#391C2312yes—
168#390—21no—
169#390—21no—
170#390—21no—
171#390—21no—
172#390—21no—
173#390—21no—
174#390—21no—
175#390—21no—
176#390C2312yes—
177#390C2312yes—
178#389—21no—
179#389—21no—
180#389—21no—
181#389—21no—
182#389—21no—
183#389—21no—
184#389—21no—
185#389—21no—
186#389—21no—
187#389—21no—
188#389—21no—
189#389—21no—
190#389—21no—
191#389—21no—
192#389—21no—
193#389—21no—
194#389C2312yes—
195#389C2312yes—
196#389C2312yes—
197#389C2312yes—
198#388—22no—
199#388—22no—
200#388—22no—
201#388—22no—
202#388—22no—
203#388—22no—
204#388—22no—
205#388—22no—
206#387—22no—
207#387—22no—
208#387—22no—
209#387—22no—
210#387—22no—
211#387—22no—
212#387—22no—
213#387—22no—
214#386—22no—
215#386—22no—
216#385—22no—
217#385—22no—
218#384C2311yescharacteristic
219#384C2311yescharacteristic
220#384C2311yescharacteristic
221#384C2311yescharacteristic
222#383—22no—
223#382—21no—
224#382—21no—
225#382—21no—
226#382C2312yes—
227#381—22no—
228#380—21no—
229#380—21no—
230#380—21no—
231#380—21no—
232#380—21no—
233#380C2312yes—
234C4×C4⋊C4—21no—
235C4×C4⋊C4—21no—
236C4×C4⋊C4—22no—
237C4×C4⋊C4—22no—
238C4×C4⋊C4—21no—
239C4×C4⋊C4—21no—
240C4×C4⋊C4—21no—
241C4×C4⋊C4C2312yes—
242C4×C22⋊C4—22no—
243C4×C22⋊C4—22no—
244C4×C22⋊C4—22no—
245#376—22no—
246C43D411yescharacteristic
247C43D411yescharacteristic
248C43D411yescharacteristic
249C43D411yescharacteristic
250C43C2311yescharacteristic
251C43C2311yescharacteristic
252C43C2311yescharacteristic
253C43C2311yescharacteristic
254#340C2314yes—
255#340C2314yes—
256#340C2314yes—
257#340C2314yes—
258#336C2312yes—
259#336C2314yes—
260#336C2312yes—
261C22×Q8—21no—
262C22×Q8C2412yes—
263C22×Q8C2412yes—
264C22×D4—22no—
265C22×D4—21no—
266C22×D4—21no—
267C22×D4C2412yes—
268C23×C4—22no—
269C23×C4C2×D411yescharacteristic
270C23×C4C2×D411yescharacteristic
271C23×C4C2×D411yescharacteristic
272C23×C4C2×D411yescharacteristic
273C23×C4C2×D411yescharacteristic
274C23×C4C2×D411yescharacteristic
275C23×C4C2411yescharacteristic
276C2×Q16—28no—
277C2×Q16—44no—
278C2×Q16—44no—
279C2×SD16—28no—
280C2×SD16—44no—
281C2×SD16—44no—
282C22×C8—28no—
283C22×C8—28no—
284C22×C8—44no—
285C22×C8C2414yes—
286C22×C8C2414yes—
287C4⋊Q8—24no—
288C4⋊Q8—22no—
289C4⋊Q8—22no—
290C4⋊Q8—22no—
291C4⋊Q8—22no—
292#127—24no—
293#126—48no—
294#125—44no—
295#125—44no—
296#124—24no—
297#124—24no—
298#124—44no—
299#123—48no—
300C22⋊Q8—28no—
301C22⋊Q8—24no—
302C22⋊Q8—24no—
303C22⋊Q8—44no—
304#121—28no—
305#121—44no—
306#121—44no—
307C4×Q8—24no—
308C4×Q8—24no—
309C4×Q8—44no—
310C4×Q8—44no—
311C4×D4—44no—
312C4×D4—44no—
313C4×D4—44no—
314C4×D4—48no—
315#117—48no—
316C2×C4⋊C4—22no—
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—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—22no—
329C2×C4⋊C4—22no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—21no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—22no—
334C2×C4⋊C4—21no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—21no—
339C2×C4⋊C4C2412yes—
340C2×C4⋊C4C2412yes—
341C2×C4⋊C4C2412yes—
342C2×C4⋊C4C2412yes—
343C2×C4⋊C4C2412yes—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4—22no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C22⋊C4—22no—
352C2×C22⋊C4—22no—
353C2×C22⋊C4—22no—
354C2×C22⋊C4—22no—
355C2×C22⋊C4—22no—
356C2×C22⋊C4—22no—
357C2×C42—24no—
358C2×C42—24no—
359C2×C42—24no—
360C2×C42—22no—
361C2×C42—22no—
362C2×C42—22no—
363C2×C42—22no—
364C2×C42—22no—
365C2×C42—42no—
366C2×C42—42no—
367C2×C42—42no—
368C2×C42C2×D411yescharacteristic
369C2×C42C2×D411yescharacteristic
370C2×C42C2×D411yescharacteristic
371C2×C42C2×D411yescharacteristic
372C2×C42C2×D411yescharacteristic
373C2×C42C2×D411yescharacteristic
374C2×C42C2×D411yescharacteristic
375C2×C42C2×D411yescharacteristic
376C2×C42C2×D411yescharacteristic
377C2×C42C2×D411yescharacteristic
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×C42C2×D411yescharacteristic
389C2×C42C2×D411yescharacteristic
390C2×C42C2411yescharacteristic
391C2×C42C2411yescharacteristic
392C2×C42C2411yescharacteristic
393C2×C42C2411yescharacteristic
394C2×C42C2411yescharacteristic
395C2×C42C2411yescharacteristic
396#107—44no—
397#106—44no—
398C4⋊C8—28no—
399C4⋊C8—24no—
400C4⋊C8#4118yes—
401#103—28no—
402#103—44no—
403#103—44no—
404#103—44no—
405#103—44no—
406#102—28no—
407#102—44no—
408#102—44no—
409#97—24no—
410#97—24no—
411#97—44no—
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—
432C24—22no—
433C24C22×D411yescharacteristic
434C2×Q8—24no—
435C2×Q8—22no—
436C2×Q8—22no—
437C2×Q8—22no—
438C2×Q8—22no—
439C2×Q8—22no—
440C2×Q8—24no—
441C2×Q8—22no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—42no—
445C2×Q8—42no—
446C2×Q8—42no—
447C2×D4—24no—
448C2×D4—22no—
449C2×D4—24no—
450C2×D4—22no—
451C2×D4—42no—
452C2×D4—42no—
453C2×D4—44no—
454C2×D4—44no—
455C2×D4—44no—
456C2×D4—42no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—42no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—21no—
464C22×C4—21no—
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×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—41no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—41no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4#13714yes—
491C22×C4C22×D411yescharacteristic
492C22×C4C22×D411yescharacteristic
493C22×C4C22×D411yescharacteristic
494C22×C4C22×D411yescharacteristic
495C22×C4C22×D411yescharacteristic
496C22×C4C22×D411yescharacteristic
497C22×C4C22×D411yescharacteristic
498C22×C4C22×D411yescharacteristic
499C22×C4C22×D411yescharacteristic
500C22×C4C22×D411yescharacteristic
501C22×C4C22×D411yescharacteristic
502C22×C4C22×D411yescharacteristic
503C22×C4C22×D411yescharacteristic
504C22×C4H(5,𝔽2)14yes—
505C22×C4C2511yescharacteristic
506Q16—88no—
507SD16—88no—
508C2×C8—24no—
509C2×C8—24no—
510C2×C8—24no—
511C2×C8—24no—
512C2×C8—24no—
513C2×C8—416no—
514C2×C8C2×#4118yes—
515C4⋊C4—22no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—24no—
521C4⋊C4—24no—
522C4⋊C4—22no—
523C4⋊C4—44no—
524C4⋊C4—42no—
525C4⋊C4—44no—
526C4⋊C4—42no—
527C4⋊C4—44no—
528C4⋊C4—44no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—44no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—44no—
535C4⋊C4—44no—
536C4⋊C4—42no—
537C4⋊C4—42no—
538C4⋊C4—44no—
539C4⋊C4—44no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—44no—
546C22⋊C4—44no—
547C22⋊C4—44no—
548C22⋊C4—44no—
549C22⋊C4—44no—
550C22⋊C4—44no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—44no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—44no—
558C22⋊C4—44no—
559C42—22no—
560C42—22no—
561C42—22no—
562C42—22no—
563C42—22no—
564C42—22no—
565C42—22no—
566C42—22no—
567C42—22no—
568C42—22no—
569C42—22no—
570C42—22no—
571C42—22no—
572C42—22no—
573C42—22no—
574C42—21no—
575C42—22no—
576C42—22no—
577C42—21no—
578C42—41no—
579C42—41no—
580C42—41no—
581C42—41no—
582C42—41no—
583C42—41no—
584C42—41no—
585C42—41no—
586C42—44no—
587C42—44no—
588C42—44no—
589C42C2×#4114yes—
590C42H(5,𝔽2)12yes—
591C422−1+412yes—
592C23—21no—
593C23—21no—
594C23—21no—
595C23—21no—
596C23—42no—
597C23—42no—
598C23—42no—
599C23—42no—
600C23—42no—
601C23—42no—
602C23—42no—
603C23#53512yes—
604C23C2×#13712yes—
605C23#57512yes—
606C23C23×D411yescharacteristic
607Q8—42no—
608Q8—42no—
609Q8—44no—
610Q8—44no—
611Q8—42no—
612Q8—42no—
613Q8—84no—
614D4—44no—
615D4—44no—
616D4—84no—
617D4—88no—
618D4—84no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—21no—
624C2×C4—21no—
625C2×C4—21no—
626C2×C4—21no—
627C2×C4—21no—
628C2×C4—21no—
629C2×C4—21no—
630C2×C4—21no—
631C2×C4—28no—
632C2×C4—24no—
633C2×C4—24no—
634C2×C4—21no—
635C2×C4—21no—
636C2×C4—21no—
637C2×C4—21no—
638C2×C4—21no—
639C2×C4—21no—
640C2×C4—21no—
641C2×C4—21no—
642C2×C4—21no—
643C2×C4—21no—
644C2×C4—21no—
645C2×C4—21no—
646C2×C4—21no—
647C2×C4—21no—
648C2×C4—21no—
649C2×C4—21no—
650C2×C4—21no—
651C2×C4—21no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—42no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—88no—
679C2×C4#53212yes—
680C2×C4#53212yes—
681C2×C4#53512yes—
682C2×C4#53712yes—
683C2×C4#53712yes—
684C2×C4C2×#13712yes—
685C2×C4C2×#13712yes—
686C2×C4#57512yes—
687C2×C4#57512yes—
688C2×C4C22×#4112yes—
689C2×C4C2×H(5,𝔽2)11yescharacteristic
690C2×C4C2×H(5,𝔽2)12yes—
691C2×C4C2×2−1+411yescharacteristic
692C8—28no—
693C8—44no—
694C22—22no—
695C22—22no—
696C22—22no—
697C22—22no—
698C22—22no—
699C22—22no—
700C22—82no—
701C22—82no—
702C22—82no—
703C22—82no—
704C22—82no—
705C22—82no—
706C22—82no—
707C22#321614yes—
708C22C2×#53211yescharacteristic
709C22C2×#53511yescharacteristic
710C22C2×#53711yescharacteristic
711C22C22×#13711yescharacteristic
712C22C2×#57511yescharacteristic
713C22C2×#57511yescharacteristic
714C22C2×#57511yescharacteristic
715C4—22no—
716C4—22no—
717C4—22no—
718C4—22no—
719C4—22no—
720C4—22no—
721C4—41no—
722C4—41no—
723C4—41no—
724C4—41no—
725C4—41no—
726C4—41no—
727C4—82no—
728C4—82no—
729C4—82no—
730C4—82no—
731C4—82no—
732C4—82no—
733C4—82no—
734C4#318114yes—
735C4#351912yes—
736C4#352212yes—
737C2—24no—
738C2—162no—
739C2#3523811yescharacteristic
740C2C2×#318111yescharacteristic
741C2#6070211yescharacteristic
742C2C2×#321611yescharacteristic
743C2#6079011yescharacteristic
744C2#6091711yescharacteristic
745C2#6091811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36635C2semidirect
3. #33184C2semidirect
4. #33051C2semidirect
5. #30347C2semidirect
6. #29212C2semidirect
7. #29197C2semidirect
8. #29150C2semidirect
9. #28754C2semidirect
10. #28620C2semidirect
11. #28451C2semidirect
12. #25499C2semidirect
14. #24752C2semidirect
15. #22154C2semidirect
16. #22153C2semidirect
17. C2×#1767C2semidirect

Maximal subgroups

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

subgroupindex
#366352
#366312
#331842
#330512
#303472
#292122
#291972
#291502
#287542
#286202
#284512
#254992
#247532
#247522
#221542
#221532
C2×#17672