Finite Group Databasebeta

← #4868746

Subgroups — #4868746

Order 512.

2756 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic, complemented
2C2×#2534C211yesmaximal, characteristic
3#17107C214yesmaximal, complemented
4#17057C212yesmaximal, complemented
5#16685C214yesmaximal, complemented
6#16666C212yesmaximal, complemented
7#16506C212yesmaximal, complemented
8#16401C212yesmaximal, complemented
9#16395C212yesmaximal, complemented
10#16297C212yesmaximal, complemented
11#15983C212yesmaximal, complemented
12#15281C212yesmaximal
13#15079C212yesmaximal, complemented
14#14820C212yesmaximal, complemented
15#14344C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic, complemented
17C22×#123C2211yescharacteristic
18C22×#120C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21#2563—28no—
22#2534—28no—
23C2×#397C2212yes—
24C2×#397C2212yes—
25C2×#396C2212yes—
26C2×#394C2212yes—
27C2×#394C2214yes—
28C2×#393C2212yes—
29C2×#393C2214yes—
30C2×#393C2212yescomplemented
31C2×#390C2212yes—
32C2×#388C2212yes—
33C2×#388C2214yes—
34C2×#387C2214yescomplemented
35C2×#386C2212yes—
36C2×#382C2212yescomplemented
37C2×#381C2211yescharacteristic
38C2×#381C2211yescharacteristic, complemented
39C2×#380C2211yescharacteristic
40C2×#380C2212yes—
41C2×#380C2212yescomplemented
42C2×#380C2211yescharacteristic, complemented
43C2×#379C2211yescharacteristic
44C2×#379C2214yes—
45C2×C4×C4⋊C4C2211yescharacteristic, complemented
46#1446C2214yescomplemented
47#1445C2212yescomplemented
48#1445C2212yescomplemented
49#1445C2214yescomplemented
50#1445C2214yescomplemented
51#1444C2212yescomplemented
52#1444C2212yescomplemented
53#1444C2214yescomplemented
54#1443C2214yescomplemented
55#1441C2212yescomplemented
56#1441C2214yescomplemented
57#1441C2214yescomplemented
58#1440C2214yes—
59#1440C2214yes—
60#1440C2212yescomplemented
61#1440C2214yescomplemented
62#1439C2212yes—
63#1439C2214yes—
64#1439C2212yes—
65#1439C2214yes—
66#1439C2212yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2214yescomplemented
72#1438C2212yes—
73#1438C2214yes—
74#1434C2212yescomplemented
75C22⋊#95C2212yes—
76C22⋊#95C2214yes—
77C26C2311yescharacteristic
78C23×Q8C2312yes—
79C23×D4C2311yescharacteristic
80C23×D4C2312yes—
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C2×C4⋊Q8—24no—
85C2×C4⋊Q8—42no—
86C2×C4⋊Q8—42no—
87C2×#123—24no—
88C2×#123—44no—
89C2×#120—24no—
90C2×#120—44no—
91C2×C4×D4—42no—
92C2×C4×D4—42no—
93C2×C4×D4D418yes—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C22⋊C4—24no—
104C22×C22⋊C4—42no—
105C22×C22⋊C4—42no—
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2312yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127#397—44no—
128#397—44no—
129#396—28no—
130#394—28no—
131#394—48no—
132#393—28no—
133#393—28no—
134#393—48no—
135#390—28no—
136#388—28no—
137#388—48no—
138#387—48no—
139#386—28no—
140#382—28no—
141#381—24no—
142#381—42no—
143#380—28no—
144#380—28no—
145#380—24no—
146#380—42no—
147#379—24no—
148#379—48no—
149C4×C4⋊C4—24no—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C25—22no—
173C25—24no—
174C25—24no—
175C25—41no—
176C25—41no—
177C25—42no—
178C25—42no—
179C25—44no—
180C25C2411yescharacteristic
181C25C2412yes—
182C25C2411yescharacteristic
183C22×Q8—24no—
184C22×Q8—42no—
185C22×Q8—42no—
186C22×Q8—44no—
187C22×Q8—44no—
188C22×D4—24no—
189C22×D4—24no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—41no—
193C22×D4—41no—
194C22×D4—44no—
195C22×D4—42no—
196C22×D4—44no—
197C22×D4C2×D414yes—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—41no—
205C23×C4—41no—
206C23×C4—42no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4C2×D414yes—
210C23×C4C2×D414yes—
211C23×C4C2412yes—
212C23×C4C2414yes—
213C23×C4C2414yes—
214C23×C4C2414yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C4⋊Q8—416no—
224#123—416no—
225#120—416no—
226C4×D4—416no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4—41no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—42no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4C2×D414yes—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—28no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—28no—
262C2×C22⋊C4—28no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—41no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—41no—
275C2×C22⋊C4—41no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—416no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4C2×D414yes—
300C2×C22⋊C4C2×D414yes—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—24no—
304C2×C42—28no—
305C2×C42C2×D414yes—
306#95—24no—
307#95—28no—
308#95—28no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—28no—
315#95—44no—
316#95—44no—
317#95—42no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328C24—22no—
329C24—22no—
330C24—21no—
331C24—21no—
332C24—22no—
333C24—24no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—24no—
338C24—22no—
339C24—22no—
340C24—21no—
341C24—21no—
342C24—22no—
343C24—22no—
344C24—24no—
345C24—22no—
346C24—24no—
347C24—22no—
348C24—24no—
349C24—22no—
350C24—24no—
351C24—48no—
352C24—44no—
353C24—416no—
354C24—44no—
355C24—48no—
356C24—44no—
357C24—44no—
358C24—44no—
359C24—84no—
360C24—84no—
361C24—84no—
362C24—88no—
363C24—88no—
364C24—88no—
365C24—84no—
366C24—84no—
367C24C22×D412yes—
368C24C22×D412yes—
369C24H(5,𝔽2)14yes—
370C242−1+414yes—
371C24C2511yescharacteristic
372C2×Q8—48no—
373C2×Q8—88no—
374C2×Q8—84no—
375C2×Q8—84no—
376C2×Q8—88no—
377C2×D4—28no—
378C2×D4—28no—
379C2×D4—44no—
380C2×D4—48no—
381C2×D4—48no—
382C2×D4—48no—
383C2×D4—48no—
384C2×D4—84no—
385C2×D4—88no—
386C2×D4—88no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—84no—
452C22×C4—84no—
453C22×C4—84no—
454C22×C4—84no—
455C22×C4—84no—
456C22×C4—84no—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4H(5,𝔽2)12yes—
463C22×C4H(5,𝔽2)12yes—
464C22×C4H(5,𝔽2)14yes—
465C4⋊C4—44no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—416no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—84no—
472C4⋊C4—84no—
473C4⋊C4—88no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C22⋊C4—416no—
479C22⋊C4—44no—
480C22⋊C4—44no—
481C22⋊C4—48no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C22⋊C4—84no—
485C22⋊C4—88no—
486C22⋊C4—88no—
487C22⋊C4—88no—
488C22⋊C4—88no—
489C22⋊C4—88no—
490C22⋊C4—84no—
491C22⋊C4—88no—
492C22⋊C4—84no—
493C42—28no—
494C42—48no—
495C23—22no—
496C23—22no—
497C23—24no—
498C23—24no—
499C23—28no—
500C23—24no—
501C23—24no—
502C23—28no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—28no—
510C23—24no—
511C23—28no—
512C23—24no—
513C23—41no—
514C23—41no—
515C23—44no—
516C23—42no—
517C23—44no—
518C23—42no—
519C23—42no—
520C23—416no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—44no—
528C23—44no—
529C23—42no—
530C23—42no—
531C23—44no—
532C23—44no—
533C23—41no—
534C23—44no—
535C23—44no—
536C23—41no—
537C23—42no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—816no—
551C23—88no—
552C23—816no—
553C23—832no—
554C23—816no—
555C23—816no—
556C23—88no—
557C23#53614yes—
558C23C23×D411yescharacteristic
559C23C2×H(5,𝔽2)11yescharacteristic
560C23C2×H(5,𝔽2)11yescharacteristic
561C23C2×H(5,𝔽2)12yes—
562C23C2×H(5,𝔽2)12yes—
563C23C2×2−1+412yes—
564C23#58514yes—
565C23#58512yes—
566Q8—816no—
567D4—416no—
568D4—816no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—88no—
633C2×C4—88no—
634C2×C4—88no—
635C2×C4#53414yes—
636C2×C4#53514yes—
637C22—24no—
638C22—28no—
639C22—42no—
640C22—48no—
641C22—44no—
642C22—44no—
643C22—44no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—44no—
648C22—48no—
649C22—48no—
650C22—44no—
651C22—44no—
652C22—44no—
653C22—44no—
654C22—42no—
655C22—832no—
656C22—82no—
657C22—84no—
658C22—84no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—84no—
664C22—82no—
665C22C2×#53411yescharacteristic
666C22C2×#53511yescharacteristic
667C22C2×#53611yescharacteristic
668C22#348512yes—
669C22#348812yes—
670C22#353514yes—
671C22#353612yes—
672C22#353612yes—
673C22#353711yescharacteristic
674C22#354011yescharacteristic
675C22#354014yes—
676C22#354212yes—
677C22#354214yes—
678C22#354512yes—
679C22#354512yes—
680C22#356414yes—
681C4—24no—
682C4—24no—
683C4—42no—
684C4—42no—
685C4—42no—
686C4—82no—
687C4—84no—
688C4—84no—
689C4—84no—
690C4—84no—
691C4—82no—
692C4—82no—
693C2—44no—
694C2—82no—
695C2—84no—
696C2#3711911yescharacteristic
697C2#3726911yescharacteristic
698C2#3745111yescharacteristic
699C2#4141212yes—
700C2#4267712yes—
701C2#5392414yes—
702C2#5750912yes—
703C2#5886012yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2563C2semidirect
3. #17107C2semidirect
4. #17057C2semidirect
5. #16685C2semidirect
6. #16666C2semidirect
7. #16506C2semidirect
8. #16401C2semidirect
9. #16395C2semidirect
10. #16297C2semidirect
11. #15983C2semidirect
13. #15079C2semidirect
14. #14820C2semidirect
15. #14344C2semidirect
16. C22×C4⋊Q8C22semidirect
30. C2×#393C22semidirect
34. C2×#387C22semidirect
36. C2×#382C22semidirect
38. C2×#381C22semidirect
41. C2×#380C22semidirect
42. C2×#380C22semidirect
45. C2×C4×C4⋊C4C22semidirect
46. #1446C22semidirect
47. #1445C22semidirect
48. #1445C22semidirect
49. #1445C22semidirect
50. #1445C22semidirect
51. #1444C22semidirect
52. #1444C22semidirect
53. #1444C22semidirect
54. #1443C22semidirect
55. #1441C22semidirect
56. #1441C22semidirect
57. #1441C22semidirect
60. #1440C22semidirect
61. #1440C22semidirect
71. #1439C22semidirect
74. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
C2×#25342
#171072
#170572
#166852
#166662
#165062
#164012
#163952
#162972
#159832
#152812
#150792
#148202
#143442