Finite Group Databasebeta

← #4308932

Subgroups — #4308932

Order 512.

3590 conjugacy classes of subgroups.

The subgroup lattice has 701 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2627C211yesmaximal, characteristic, complemented
2C22×#380C211yesmaximal, characteristic
3#16569C214yesmaximal, complemented
4#16568C214yesmaximal, complemented
5#16521C214yesmaximal, complemented
6#16520C214yesmaximal, complemented
7#14936C211yesmaximal, characteristic, complemented
8#14935C212yesmaximal, complemented
9#14934C211yesmaximal, characteristic, complemented
10#14734C211yesmaximal, characteristic
11#14726C212yesmaximal
12#14717C211yesmaximal, characteristic
13#14681C212yesmaximal
14#14339C211yesmaximal, characteristic, complemented
15#14263C212yesmaximal, complemented
16C25×C4C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22#2627—28no—
23C2×#400C2212yescomplemented
24C2×#399C2212yes—
25C2×#398C2212yescomplemented
26C2×#398C2212yescomplemented
27C2×#396C2214yes—
28C2×#395C2212yescomplemented
29C2×#393C2212yes—
30C2×#385C2212yes—
31C2×#385C2212yes—
32C2×#385C2212yes—
33C2×#381C2212yes—
34C2×#381C2212yes—
35C2×#381C2212yes—
36C2×#381C2212yes—
37C2×#380—28no—
38C2×#380C2211yescharacteristic
39C2×#380C2214yes—
40C2×#380C2214yes—
41C2×#380C2212yes—
42C2×#379C2212yes—
43C2×#379C2212yes—
44C22×#95C2211yescharacteristic
45C22×#95C2211yescharacteristic
46#1446C2212yescomplemented
47#1445C2214yescomplemented
48#1445C2212yescomplemented
49#1444C2212yescomplemented
50#1444C2214yescomplemented
51#1444C2214yescomplemented
52#1443C2214yescomplemented
53#1442C2214yescomplemented
54#1442C2212yescomplemented
55#1442C2214yescomplemented
56#1442C2214yescomplemented
57#1441C2214yescomplemented
58#1441C2214yescomplemented
59#1441C2214yescomplemented
60#1441C2214yescomplemented
61#1441C2214yescomplemented
62#1439C2214yes—
63#1439C2212yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2212yes—
69C22⋊#95C2211yescharacteristic
70C22⋊#95C2211yescharacteristic
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C26C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C24×C4—24no—
80C24×C4—24no—
81C24×C4—44no—
82C24×C4—44no—
83C24×C4Q818yescomplemented
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2314yes—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—44no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—44no—
95C22×C4⋊C4—44no—
96C22×C4⋊C4D418yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C22⋊C4—24no—
110C22×C22⋊C4—24no—
111C22×C22⋊C4—44no—
112C22×C22⋊C4—44no—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124#400—28no—
125#399—44no—
126#398—28no—
127#398—28no—
128#396—48no—
129#395—28no—
130#393—44no—
131#385—28no—
132#385—44no—
133#385—44no—
134#381—28no—
135#381—28no—
136#381—44no—
137#381—44no—
138#380—24no—
139#380—216no—
140#380—216no—
141#380—44no—
142#379—44no—
143#379—44no—
144C2×#95—24no—
145C2×#95—24no—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C25—22no—
174C25—22no—
175C25—24no—
176C25—24no—
177C25—24no—
178C25—24no—
179C25—42no—
180C25—42no—
181C25C2×Q814yes—
182C25C2412yes—
183C25C2411yescharacteristic
184C22×Q8—22no—
185C22×Q8—42no—
186C22×Q8—42no—
187C22×Q8—42no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—232no—
197C23×C4—28no—
198C23×C4—28no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—432no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—416no—
217C23×C4—416no—
218C23×C4—432no—
219C23×C4C2×D414yes—
220C23×C4C2×D414yes—
221C23×C4C2×D414yes—
222C23×C4C2×Q814yes—
223C23×C4C2×Q814yes—
224C23×C4C2×Q8116yes—
225C23×C4#4114yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2414yes—
231C23×C4C2414yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2412yes—
238C23×C4C2412yes—
239C22⋊Q8—416no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—416no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—416no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4C2×D418yes—
272C2×C4⋊C4C2×D418yes—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—416no—
299C2×C22⋊C4—416no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308#95—28no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—28no—
319#95—28no—
320#95—42no—
321#95—42no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—42no—
332#95—44no—
333#95—44no—
334#95—42no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—28no—
340C24—28no—
341C24—216no—
342C24—28no—
343C24—216no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—24no—
348C24—24no—
349C24—22no—
350C24—24no—
351C24—24no—
352C24—216no—
353C24—48no—
354C24—416no—
355C24—416no—
356C24—416no—
357C24—416no—
358C24—44no—
359C24—44no—
360C24—48no—
361C24—44no—
362C24—44no—
363C24—48no—
364C24C22×D412yes—
365C24C22×Q814yes—
366C24C22×Q812yes—
367C24C2×#4112yes—
368C24H(5,𝔽2)14yes—
369C24H(5,𝔽2)14yes—
370C24H(5,𝔽2)14yes—
371C242−1+414yes—
372C24C2511yescharacteristic
373C2×Q8—44no—
374C2×Q8—82no—
375C2×Q8—82no—
376C2×Q8—88no—
377C22×C4—28no—
378C22×C4—28no—
379C22×C4—216no—
380C22×C4—216no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—28no—
386C22×C4—216no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—28no—
402C22×C4—28no—
403C22×C4—232no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—416no—
449C22×C4—416no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—4128no—
455C22×C4—416no—
456C22×C4—416no—
457C22×C4—416no—
458C22×C4—416no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—48no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—88no—
469C22×C4—88no—
470C22×C4—88no—
471C22×C4—88no—
472C22×C4C22⋊Q8116yes—
473C22×C4C22×D414yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D414yes—
477C22×C4C22×Q814yes—
478C22×C4C22×Q814yes—
479C22×C4C2×#4114yes—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—416no—
483C4⋊C4—48no—
484C4⋊C4—44no—
485C4⋊C4—82no—
486C4⋊C4—84no—
487C4⋊C4—82no—
488C4⋊C4—84no—
489C4⋊C4—88no—
490C4⋊C4—88no—
491C4⋊C4—88no—
492C22⋊C4—416no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C22⋊C4—88no—
500C22⋊C4—88no—
501C22⋊C4—88no—
502C22⋊C4—88no—
503C23—24no—
504C23—24no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—24no—
510C23—24no—
511C23—28no—
512C23—28no—
513C23—28no—
514C23—28no—
515C23—232no—
516C23—24no—
517C23—28no—
518C23—24no—
519C23—28no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—28no—
525C23—28no—
526C23—28no—
527C23—28no—
528C23—464no—
529C23—464no—
530C23—416no—
531C23—416no—
532C23—416no—
533C23—416no—
534C23—416no—
535C23—416no—
536C23—42no—
537C23—42no—
538C23—44no—
539C23—42no—
540C23—42no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23C2×C22⋊Q814yes—
547C23#53414yes—
548C23#53514yes—
549C23#53818yes—
550C23#54318yes—
551C23#54418yes—
552C23C23×D411yescharacteristic
553C23C23×Q811yescharacteristic
554C23C22×#4111yescharacteristic
555C23C2×H(5,𝔽2)12yes—
556C23C2×H(5,𝔽2)12yes—
557C23C2×H(5,𝔽2)12yes—
558C23C2×2−1+412yes—
559C23#58512yes—
560C23#58512yes—
561Q8—88no—
562C2×C4—232no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—28no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—464no—
581C2×C4—416no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
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—44no—
633C2×C4—88no—
634C2×C4—88no—
635C2×C4—88no—
636C2×C4—88no—
637C2×C4C2×C22⋊Q818yes—
638C22—216no—
639C22—24no—
640C22—24no—
641C22—24no—
642C22—216no—
643C22—24no—
644C22—24no—
645C22—24no—
646C22—216no—
647C22—28no—
648C22—464no—
649C22—48no—
650C22—48no—
651C22—44no—
652C22—44no—
653C22—44no—
654C22—48no—
655C22—44no—
656C22—44no—
657C22—44no—
658C22#262618yes—
659C22C22×C22⋊Q811yescharacteristic
660C22C2×#53411yescharacteristic
661C22C2×#53511yescharacteristic
662C22C2×#53812yes—
663C22C2×#54312yes—
664C22C2×#54412yes—
665C22#345812yes—
666C22#348411yescharacteristic
667C22#348511yescharacteristic
668C22#348611yescharacteristic
669C22#348711yescharacteristic
670C22#349212yes—
671C22#349312yes—
672C22#353414yes—
673C22#353414yes—
674C22#353514yes—
675C22#353514yes—
676C4—28no—
677C4—44no—
678C4—42no—
679C4—44no—
680C4—42no—
681C4—84no—
682C4—84no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4—82no—
687C4—82no—
688C4—82no—
689C2—28no—
690C2—48no—
691C2C2×#262611yescharacteristic
692C2#3515111yescharacteristic
693C2#3522411yescharacteristic
694C2#3711711yescharacteristic
695C2#3711811yescharacteristic
696C2#3712314yes—
697C2#3718511yescharacteristic
698C2#3718611yescharacteristic
699C2#3747814yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
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×#2627C2semidirect
3. #16569C2semidirect
4. #16568C2semidirect
5. #16521C2semidirect
6. #16520C2semidirect
7. #14936C2semidirect
8. #14935C2semidirect
9. #14934C2semidirect
14. #14339C2semidirect
15. #14263C2semidirect
23. C2×#400C22semidirect
25. C2×#398C22semidirect
26. C2×#398C22semidirect
28. C2×#395C22semidirect
46. #1446C22semidirect
47. #1445C22semidirect
48. #1445C22semidirect
49. #1444C22semidirect
50. #1444C22semidirect
51. #1444C22semidirect
52. #1443C22semidirect
53. #1442C22semidirect
54. #1442C22semidirect
55. #1442C22semidirect
56. #1442C22semidirect
57. #1441C22semidirect
58. #1441C22semidirect
59. #1441C22semidirect
60. #1441C22semidirect
61. #1441C22semidirect
83. C24×C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26272
C22×#3802
#165692
#165682
#165212
#165202
#149362
#149352
#149342
#147342
#147262
#147172
#146812
#143392
#142632