Finite Group Databasebeta

← #7506784

Subgroups — #7506784

Order 512.

1771 conjugacy classes of subgroups.

The subgroup lattice has 719 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35673C211yesmaximal, characteristic, complemented
2#35618C211yesmaximal, characteristic
3#31951C214yesmaximal, complemented
4#31948C214yesmaximal, complemented
5#31943C214yesmaximal, complemented
6#31940C214yesmaximal, complemented
7C2×#2097C211yesmaximal, characteristic, complemented
8#23660C211yesmaximal, characteristic, complemented
9#23627C211yesmaximal, characteristic, complemented
10#23610C211yesmaximal, characteristic, complemented
11#23488C211yesmaximal, characteristic
12#23394C211yesmaximal, characteristic
13#21237C211yesmaximal, characteristic, complemented
14#21048C211yesmaximal, characteristic, complemented
15#20930C211yesmaximal, characteristic, complemented
16#20913C211yesmaximal, characteristic, complemented
17#20848C211yesmaximal, characteristic, complemented
18#20765C211yesmaximal, characteristic, complemented
19#20651C211yesmaximal, characteristic, complemented
20C22×#137C2211yescharacteristic
21C2×#537C2211yescharacteristic
22C2×#536C2211yescharacteristic
23C2×#520C2211yescharacteristic
24C2×#519C2211yescharacteristic
25C2×#428C2211yescharacteristic
26C22×#107C2211yescharacteristic
27C22×#106C2211yescharacteristic
28C2×#424C2211yescharacteristic
29C2×#417C2211yescharacteristic
30C2×#419C2211yescharacteristic
31C2×#416C2211yescharacteristic
32C2×#416C2211yescharacteristic
33C22×#103C2211yescharacteristic
34C22×#103C2211yescharacteristic
35C2×#408C2211yescharacteristic
36C2×#408C2211yescharacteristic
37C2×C4×M16C2211yescharacteristic
38#2504C2212yes—
39#2503C2212yes—
40#2500C2212yes—
41#2499C2212yes—
42#2498—24no—
43#2448—22no—
44#2446—22no—
45#2444—24no—
46#2423C2214yes—
47#2421—22no—
48#2421C2214yes—
49#2420—22no—
50#2317C2211yescharacteristic
51#2316C2211yescharacteristic
52#2311C2211yescharacteristic
53#2311C2211yescharacteristic
54#2297C2211yescharacteristic
55#2297C2211yescharacteristic
56C2×#391C2211yescharacteristic
57#2285C2211yescharacteristic
58C2×#382C2211yescharacteristic
59#2097C2214yes—
60#2097C2214yes—
61#2085C2214yes—
62#2083C2214yes—
63#2074C2212yes—
64#2073C2212yes—
65#2072C2212yes—
66#2071C2212yes—
67#2066C2214yes—
68#2065C2214yes—
69#2036C2214yes—
70#2033C2214yes—
71#2001C2214yes—
72#2000C2214yes—
73#1973C2212yes—
74#1971C2212yes—
75#1969C2212yes—
76#1968C2212yes—
77#1924C2214yes—
78#1923C2214yes—
79#1880C2214yes—
80#1878C2214yes—
81#1873C2214yes—
82#1872C2214yes—
83#1854C2214yes—
84#1853C2214yes—
85#1836C2214yes—
86#1835C2214yes—
87#1805C2214yes—
88#1803C2214yes—
89#1736C2211yescharacteristic
90#1736C2211yescharacteristic
91#1736C2211yescharacteristic
92#1736C2211yescharacteristic
93#1735C2211yescharacteristic
94#1735C2211yescharacteristic
95#1729C2211yescharacteristic
96C2×#336C2211yescharacteristic
97C22×#41C2311yescharacteristic
98C22×#41C2311yescharacteristic
99C23×Q8C2311yescharacteristic
100C2×#137—24no—
101C2×#137—24no—
102C2×#137—42no—
103C22×Q16C2312yes—
104C22×SD16C2312yes—
105C22×M16C2311yescharacteristic
106C22×M16C2311yescharacteristic
107C23×C8C2311yescharacteristic
108C23×C8C2311yescharacteristic
109#537—42no—
110#537—42no—
111#536—24no—
112#536—42no—
113C2×C4⋊Q8—21no—
114C2×#124—21no—
115C2×#123—22no—
116C2×C22⋊Q8—21no—
117C2×C22⋊Q8C2314yes—
118C2×C22⋊Q8C2312yes—
119C2×#121—21no—
120C2×#121—22no—
121C2×#121C2312yes—
122#520—42no—
123#520—42no—
124#519—24no—
125#519—42no—
126C2×C4×Q8—21no—
127C2×C4×Q8—22no—
128C2×C4×Q8C2314yes—
129C2×C4×D4—21no—
130C2×C4×D4—22no—
131C2×#117—22no—
132C2×#117C2311yescharacteristic
133C2×#117C2311yescharacteristic
134C2×#117C2311yescharacteristic
135C2×#117C2312yes—
136C2×#117C2311yescharacteristic
137C22×C4⋊C4C2311yescharacteristic
138C22×C4⋊C4C2311yescharacteristic
139C22×C4⋊C4C2311yescharacteristic
140C22×C4⋊C4C2312yes—
141C22×C42C2311yescharacteristic
142#428—42no—
143C2×#107—42no—
144C2×#107C2312yes—
145C2×#107C2314yes—
146C2×#106—42no—
147C2×#106C2314yes—
148C2×#106C2312yes—
149#424—42no—
150C2×C4⋊C8C2314yes—
151#419—42no—
152#417—42no—
153#416—42no—
154#416—42no—
155C2×#103—42no—
156C2×#103—42no—
157C2×#103C2312yes—
158C2×#103C2314yes—
159C2×#103C2312yes—
160C2×#103C2312yes—
161C2×#103C2312yes—
162C2×#103C2312yes—
163C2×#103C2314yes—
164C2×#102C2312yes—
165C2×#102C2312yes—
166C2×#102C2312yes—
167#408—24no—
168#408—42no—
169C2×C22⋊C8C2314yes—
170C2×C22⋊C8C2314yes—
171C4×M16—24no—
172C2×#97C2312yes—
173C2×C4×C8C2312yes—
174#400—24no—
175#400—24no—
176#399—24no—
177#398—24no—
178#397—24no—
179#397—24no—
180#397—24no—
181#394—24no—
182#391—21no—
183#391—22no—
184#391D412yes—
185#391C2312yes—
186#391C2311yescharacteristic
187#391C2311yescharacteristic
188#391C2312yes—
189#391C2312yes—
190#390—21no—
191#390D412yes—
192#390C2312yes—
193#390C2311yescharacteristic
194#390C2311yescharacteristic
195#389—22no—
196#389—22no—
197#389C2314yes—
198#388—24no—
199#388—24no—
200#388—24no—
201#387—24no—
202#387—24no—
203#387—24no—
204#386—22no—
205#386—22no—
206#386—22no—
207#386—24no—
208#386D414yes—
209#385—22no—
210#384C2312yes—
211#384C2312yes—
212#382—22no—
213#382C2312yes—
214#382C2312yes—
215#382C2312yes—
216#382C2312yes—
217#381—24no—
218#380—22no—
219#380—22no—
220#380—22no—
221#380C2314yes—
222C4×C4⋊C4—22no—
223C4×C22⋊C4—22no—
224C4×C22⋊C4C2312yes—
225C4×C22⋊C4C2312yes—
226C2×#95—22no—
227C2×#95—22no—
228#340C2312yes—
229#340C2314yes—
230#340C2312yes—
231#340C2314yes—
232#340C2312yes—
233#340C2312yes—
234#340C2314yes—
235#340C2314yes—
236#336C2312yes—
237#336C2312yes—
238#336C2314yes—
239C2×#41—22no—
240C2×#41—22no—
241C2×#41—22no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—22no—
245C2×#41—22no—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41—22no—
249C2×#41—22no—
250C2×#41—22no—
251C22×Q8—21no—
252C22×Q8—22no—
253C22×Q8—22no—
254C22×Q8—22no—
255C22×Q8—22no—
256C22×Q8—22no—
257C22×Q8C2×D411yescharacteristic
258C22×Q8#4114yes—
259C22×Q8C2411yescharacteristic
260C22×Q8C2412yes—
261C22×D4—21no—
262C22×D4—22no—
263C22×D4C2×D412yes—
264C22×D4C2×D411yescharacteristic
265C22×D4C2411yescharacteristic
266C23×C4—21no—
267C23×C4—21no—
268C23×C4—22no—
269C23×C4—21no—
270C23×C4—21no—
271C23×C4—21no—
272C23×C4C2×D412yes—
273C23×C4C2412yes—
274C23×C4C2411yescharacteristic
275#137—416no—
276C2×Q16—24no—
277C2×Q16—24no—
278C2×Q16—42no—
279C2×SD16—24no—
280C2×SD16—24no—
281C2×SD16—42no—
282C2×M16—22no—
283C2×M16—22no—
284C2×M16—22no—
285C2×M16—22no—
286C2×M16—22no—
287C2×M16#4114yes—
288C22×C8—22no—
289C22×C8—22no—
290C22×C8—22no—
291C22×C8—22no—
292C22×C8—22no—
293C22×C8—22no—
294C22×C8C2412yes—
295C22×C8C2412yes—
296C22×C8C2412yes—
297C22×C8C2412yes—
298C4⋊Q8—44no—
299#124—44no—
300#123—48no—
301C22⋊Q8—28no—
302C22⋊Q8—44no—
303C22⋊Q8—42no—
304C22⋊Q8—42no—
305C22⋊Q8—44no—
306#121—42no—
307#121—42no—
308#121—48no—
309#121—44no—
310C4×Q8—28no—
311C4×Q8—44no—
312C4×Q8—48no—
313C4×Q8—44no—
314C4×D4—48no—
315C4×D4—44no—
316#117—22no—
317#117—22no—
318#117—22no—
319#117—22no—
320#117—22no—
321#117—22no—
322#117—22no—
323#117—22no—
324#117—48no—
325#117—42no—
326#117—42no—
327C2×C4⋊C4—22no—
328C2×C4⋊C4—22no—
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⋊C4—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—24no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—24no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—24no—
343C2×C4⋊C4—24no—
344C2×C4⋊C4—42no—
345C2×C4⋊C4—42no—
346C2×C4⋊C4C2×D412yes—
347C2×C4⋊C4C2412yes—
348C2×C4⋊C4C2411yescharacteristic
349C2×C4⋊C4C2412yes—
350C2×C4⋊C4C2412yes—
351C2×C4⋊C4C2411yescharacteristic
352C2×C4⋊C4C2412yes—
353C2×C4⋊C4C2412yes—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—21no—
356C2×C22⋊C4—22no—
357C2×C22⋊C4—21no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—24no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—24no—
362C2×C22⋊C4—22no—
363C2×C22⋊C4—22no—
364C2×C22⋊C4—24no—
365C2×C22⋊C4C2×D412yes—
366C2×C22⋊C4C2×D414yes—
367C2×C22⋊C4C2×D414yes—
368C2×C42—22no—
369C2×C42—22no—
370C2×C42—24no—
371C2×C42—21no—
372C2×C42—22no—
373C2×C42—21no—
374C2×C42—21no—
375C2×C42C2×D412yes—
376C2×C42C2412yes—
377C2×C42C2411yescharacteristic
378C2×C42C2411yescharacteristic
379#107—44no—
380#107—42no—
381#106—44no—
382#106—42no—
383C4⋊C8—44no—
384#103—44no—
385#103—44no—
386#103—42no—
387#103—42no—
388#103—42no—
389#103—42no—
390#103—42no—
391#102—42no—
392#102—42no—
393#102—42no—
394C22⋊C8—28no—
395C22⋊C8—44no—
396#97—24no—
397C4×C8—24no—
398#95—24no—
399#95—24no—
400#95—24no—
401#95—24no—
402#95—22no—
403#95—22no—
404#95—22no—
405#95—22no—
406#95—24no—
407#95—24no—
408#95—24no—
409#95—24no—
410#95C2×D414yes—
411#95C2×D414yes—
412C24—21no—
413C24C22×D412yes—
414C24C22×D411yescharacteristic
415#41—44no—
416#41—44no—
417#41—44no—
418#41—44no—
419#41—44no—
420#41—44no—
421#41—44no—
422#41—44no—
423C2×Q8—22no—
424C2×Q8—24no—
425C2×Q8—22no—
426C2×Q8—22no—
427C2×Q8—22no—
428C2×Q8—28no—
429C2×Q8—28no—
430C2×Q8—22no—
431C2×Q8—21no—
432C2×Q8—21no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—22no—
439C2×Q8—21no—
440C2×Q8—42no—
441C2×Q8—44no—
442C2×Q8—42no—
443C2×Q8—44no—
444C2×Q8C2×#4114yes—
445C2×D4—24no—
446C2×D4—24no—
447C2×D4—24no—
448C2×D4—22no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—21no—
452C2×D4—21no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—21no—
456C2×D4—21no—
457C2×D4—42no—
458C2×D4—42no—
459C2×D4—42no—
460C2×D4—44no—
461C2×D4—84no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—22no—
474C22×C4—21no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—42no—
480C22×C4—42no—
481C22×C4—42no—
482C22×C4—42no—
483C22×C4—44no—
484C22×C4—42no—
485C22×C4—42no—
486C22×C4—42no—
487C22×C4—42no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—82no—
491C22×C4—82no—
492C22×C4—82no—
493C22×C4—84no—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D414yes—
497C22×C4C22×D412yes—
498C22×C4C22×D411yescharacteristic
499C22×C4C2×#4112yes—
500C22×C4C2×#4112yes—
501C22×C4H(5,𝔽2)12yes—
502C22×C4H(5,𝔽2)12yes—
503C22×C4H(5,𝔽2)12yes—
504C22×C42−1+412yes—
505C22×C4C2511yescharacteristic
506Q16—48no—
507SD16—48no—
508M16—28no—
509M16—44no—
510C2×C8—22no—
511C2×C8—22no—
512C2×C8—22no—
513C2×C8—22no—
514C2×C8—22no—
515C2×C8—22no—
516C2×C8—22no—
517C2×C8—22no—
518C2×C8—22no—
519C2×C8—22no—
520C2×C8—22no—
521C2×C8—44no—
522C2×C8—44no—
523C2×C8C2×#4114yes—
524C4⋊C4—24no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—22no—
528C4⋊C4—22no—
529C4⋊C4—22no—
530C4⋊C4—22no—
531C4⋊C4—22no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—22no—
535C4⋊C4—21no—
536C4⋊C4—22no—
537C4⋊C4—22no—
538C4⋊C4—21no—
539C4⋊C4—48no—
540C4⋊C4—48no—
541C4⋊C4—44no—
542C4⋊C4—44no—
543C4⋊C4—44no—
544C4⋊C4—42no—
545C4⋊C4—42no—
546C4⋊C4—42no—
547C4⋊C4—84no—
548C4⋊C4—84no—
549C4⋊C4—82no—
550C22⋊C4—24no—
551C22⋊C4—44no—
552C22⋊C4—44no—
553C22⋊C4—48no—
554C22⋊C4—44no—
555C22⋊C4—44no—
556C22⋊C4—44no—
557C22⋊C4—42no—
558C22⋊C4—42no—
559C22⋊C4—42no—
560C22⋊C4—82no—
561C22⋊C4—84no—
562C22⋊C4—82no—
563C22⋊C4—84no—
564C42—24no—
565C42—21no—
566C42—21no—
567C42—22no—
568C42—22no—
569C42—21no—
570C42—21no—
571C42—48no—
572C42—44no—
573C42—42no—
574C42—42no—
575C42—42no—
576C23—22no—
577C23—22no—
578C23—24no—
579C23—24no—
580C23—22no—
581C23—22no—
582C23—22no—
583C23—42no—
584C23—42no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23#53512yes—
589C23#54612yes—
590C23#54712yes—
591C23C23×D411yescharacteristic
592Q8—24no—
593Q8—24no—
594Q8—42no—
595Q8—48no—
596Q8—44no—
597Q8—42no—
598Q8—41no—
599Q8—41no—
600Q8—41no—
601Q8—41no—
602D4—48no—
603D4—41no—
604D4—41no—
605D4—41no—
606D4—44no—
607D4—41no—
608D4—84no—
609D4—164no—
610C2×C4—22no—
611C2×C4—24no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—24no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—24no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—22no—
622C2×C4—22no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—48no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—41no—
642C2×C4—42no—
643C2×C4—41no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—84no—
650C2×C4—84no—
651C2×C4—162no—
652C2×C4—162no—
653C2×C4—162no—
654C2×C4—164no—
655C2×C4#53512yes—
656C2×C4#53914yes—
657C2×C4#54014yes—
658C2×C4#54612yes—
659C2×C4#54712yes—
660C2×C4#55114yes—
661C2×C4#56214yes—
662C2×C4C22×#4111yescharacteristic
663C2×C4C22×#4111yescharacteristic
664C2×C4C2×H(5,𝔽2)11yescharacteristic
665C2×C4C2×H(5,𝔽2)11yescharacteristic
666C2×C4C2×H(5,𝔽2)11yescharacteristic
667C2×C4C2×2−1+411yescharacteristic
668C8—24no—
669C8—42no—
670C8—42no—
671C8—42no—
672C22—24no—
673C22—24no—
674C22—22no—
675C22—22no—
676C22—44no—
677C22—44no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22—82no—
682C22—82no—
683C22—81no—
684C22—81no—
685C22—81no—
686C22#270614yes—
687C22C2×#53511yescharacteristic
688C22C2×#54611yescharacteristic
689C22C2×#54711yescharacteristic
690C22#358412yes—
691C22#358612yes—
692C4—44no—
693C4—42no—
694C4—42no—
695C4—42no—
696C4—44no—
697C4—82no—
698C4—82no—
699C4—81no—
700C4—82no—
701C4—82no—
702C4—82no—
703C4—81no—
704C4—81no—
705C4#270614yes—
706C4C2×#53911yescharacteristic
707C4C2×#54011yescharacteristic
708C4C2×#55111yescharacteristic
709C4C2×#56211yescharacteristic
710C2—24no—
711C2—82no—
712C2—161no—
713C2C2×#270611yescharacteristic
714C2#6081911yescharacteristic
715C2#6082611yescharacteristic
716C2#6120412yes—
717C2#6127712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#537C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35673C2semidirect
3. #31951C2semidirect
4. #31948C2semidirect
5. #31943C2semidirect
6. #31940C2semidirect
7. C2×#2097C2semidirect
8. #23660C2semidirect
9. #23627C2semidirect
10. #23610C2semidirect
13. #21237C2semidirect
14. #21048C2semidirect
15. #20930C2semidirect
16. #20913C2semidirect
17. #20848C2semidirect
18. #20765C2semidirect
19. #20651C2semidirect

Maximal subgroups

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

subgroupindex
#356732
#356182
#319512
#319482
#319432
#319402
C2×#20972
#236602
#236272
#236102
#234882
#233942
#212372
#210482
#209302
#209132
#208482
#207652
#206512