Finite Group Databasebeta

← #4308804

Subgroups — #4308804

Order 512.

2906 conjugacy classes of subgroups.

The subgroup lattice has 737 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2626C211yesmaximal, characteristic
2C22×#388C211yesmaximal, characteristic
3#16439C214yesmaximal, complemented
4#16428C214yesmaximal, complemented
5#16314C214yesmaximal, complemented
6#16278C214yesmaximal, complemented
7#15026C212yesmaximal, complemented
8#14786C212yesmaximal, complemented
9#14727C212yesmaximal
10#14727C212yesmaximal
11#14702C211yesmaximal, characteristic, complemented
12#14697C211yesmaximal, characteristic, complemented
13#14338C211yesmaximal, characteristic
14#14281C211yesmaximal, characteristic, complemented
15#14275C211yesmaximal, characteristic, complemented
16C22×C22⋊Q8C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2626—28no—
23C2×#399C2212yes—
24C2×#398C2214yescomplemented
25C2×#397C2214yes—
26C2×#397C2214yes—
27C2×#396C2212yes—
28C2×#396C2212yes—
29C2×#393C2212yes—
30C2×#391C2212yescomplemented
31C2×#389C2212yescomplemented
32C2×#388—28no—
33C2×#388C2214yes—
34C2×#388C2214yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#385C2212yes—
38C2×#380C2211yescharacteristic
39C2×#380C2211yescharacteristic
40C2×#380C2211yescharacteristic
41C2×#379C2212yes—
42C2×#379C2212yes—
43C22×#95C2211yescharacteristic
44C22×#95C2211yescharacteristic
45#1446C2212yescomplemented
46#1445C2212yescomplemented
47#1445C2214yescomplemented
48#1444C2214yescomplemented
49#1444C2212yescomplemented
50#1443C2214yescomplemented
51#1443C2214yescomplemented
52#1442C2214yescomplemented
53#1442C2214yescomplemented
54#1442C2212yescomplemented
55#1441C2214yescomplemented
56#1441C2214yescomplemented
57#1441C2214yescomplemented
58#1440C2214yes—
59#1440C2214yes—
60#1440C2214yes—
61#1440C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1436C2214yescomplemented
67#1436C2214yescomplemented
68C4⋊#95C2212yescomplemented
69C4⋊#95C2212yescomplemented
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C26C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C2×C22⋊Q8—24no—
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8—24no—
88C22×C4⋊C4—24no—
89C22×C4⋊C4—44no—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2311yescharacteristic
92C22×C4⋊C4C2314yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C22⋊C4—24no—
102C22×C22⋊C4—24no—
103C22×C22⋊C4—24no—
104C22×C22⋊C4—24no—
105C22×C22⋊C4—44no—
106C22×C22⋊C4—44no—
107C22×C22⋊C4D418yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C42—24no—
123C22×C42—44no—
124C22×C42C2312yes—
125C22×C42C2312yes—
126#399—28no—
127#398—48no—
128#397—48no—
129#397—48no—
130#396—28no—
131#396—28no—
132#393—28no—
133#391—44no—
134#389—44no—
135#388—216no—
136#388—216no—
137#385—28no—
138#385—44no—
139#385—44no—
140#380—24no—
141#380—24no—
142#380—24no—
143#379—28no—
144#379—44no—
145C2×#95—24no—
146C2×#95—24no—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2312yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C25—22no—
172C25—22no—
173C25—22no—
174C25—22no—
175C25—22no—
176C25—22no—
177C25—42no—
178C25—42no—
179C25—42no—
180C25—42no—
181C25C2×D414yes—
182C25C2412yes—
183C25C2411yescharacteristic
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—22no—
187C22×Q8—42no—
188C22×Q8—42no—
189C23×C4—22no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4C2×D414yes—
216C23×C4C2×D414yes—
217C23×C4#4114yes—
218C23×C4#4114yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2414yes—
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C22⋊Q8—416no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—416no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C22⋊C4—232no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—416no—
291C2×C22⋊C4—416no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4C2×D418yes—
303C2×C22⋊C4C2×D418yes—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—42no—
309C2×C42—42no—
310C2×C42—416no—
311#95—28no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—28no—
320#95—28no—
321#95—28no—
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—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—28no—
340C24—22no—
341C24—22no—
342C24—24no—
343C24—24no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—24no—
349C24—24no—
350C24—28no—
351C24—28no—
352C24—44no—
353C24—44no—
354C24—44no—
355C24—44no—
356C24—48no—
357C24—48no—
358C24—48no—
359C24—48no—
360C24—44no—
361C24—44no—
362C24—44no—
363C24—44no—
364C24—84no—
365C24—84no—
366C24—84no—
367C24—84no—
368C24—84no—
369C24—84no—
370C24—84no—
371C24—84no—
372C24C22×D414yes—
373C24C22×D412yes—
374C24C2×#4112yes—
375C24C2×#4112yes—
376C24H(5,𝔽2)12yes—
377C242−1+412yes—
378C242−1+412yes—
379C242−1+412yes—
380C24C2511yescharacteristic
381C2×Q8—48no—
382C2×Q8—44no—
383C2×Q8—82no—
384C2×Q8—82no—
385C2×Q8—84no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—44no—
464C22×C4—48no—
465C22×C4—44no—
466C22×C4—48no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—84no—
476C22×C4—84no—
477C22×C4—84no—
478C22×C4—84no—
479C22×C4#123116yes—
480C22×C4C22×D414yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4C2×#4112yes—
487C22×C4H(5,𝔽2)14yes—
488C22×C4H(5,𝔽2)14yes—
489C4⋊C4—416no—
490C4⋊C4—48no—
491C4⋊C4—44no—
492C4⋊C4—48no—
493C4⋊C4—82no—
494C4⋊C4—82no—
495C4⋊C4—84no—
496C4⋊C4—84no—
497C4⋊C4—84no—
498C4⋊C4—88no—
499C4⋊C4—88no—
500C22⋊C4—216no—
501C22⋊C4—216no—
502C22⋊C4—416no—
503C22⋊C4—416no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—84no—
509C22⋊C4—84no—
510C22⋊C4—84no—
511C22⋊C4—84no—
512C22⋊C4—88no—
513C22⋊C4—88no—
514C42—48no—
515C42—48no—
516C23—24no—
517C23—24no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—416no—
540C23—416no—
541C23—416no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—88no—
559C23—88no—
560C23—88no—
561C23—88no—
562C23—816no—
563C23—816no—
564C23—816no—
565C23—816no—
566C23C2×#12314yes—
567C23#53614yes—
568C23#53714yes—
569C23#54114yes—
570C23#54114yes—
571C23#54214yes—
572C23#54214yes—
573C23C23×D411yescharacteristic
574C23C22×#4111yescharacteristic
575C23C22×#4111yescharacteristic
576C23C2×H(5,𝔽2)11yescharacteristic
577C23C2×H(5,𝔽2)12yes—
578C23C2×H(5,𝔽2)12yes—
579C23C2×2−1+411yescharacteristic
580C23C2×2−1+411yescharacteristic
581C23C2×2−1+411yescharacteristic
582C23#58512yes—
583C23#58512yes—
584Q8—88no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—416no—
603C2×C4—416no—
604C2×C4—416no—
605C2×C4—416no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—88no—
661C2×C4—88no—
662C2×C4C2×#12314yes—
663C2×C4C2×#12314yes—
664C22—24no—
665C22—28no—
666C22—28no—
667C22—24no—
668C22—24no—
669C22—24no—
670C22—24no—
671C22—28no—
672C22—28no—
673C22—24no—
674C22—44no—
675C22—44no—
676C22—44no—
677C22—44no—
678C22—44no—
679C22—44no—
680C22—44no—
681C22—832no—
682C22—84no—
683C22—84no—
684C22—84no—
685C22—84no—
686C22#262518yes—
687C22C22×#12311yescharacteristic
688C22C2×#53611yescharacteristic
689C22C2×#53711yescharacteristic
690C22C2×#54111yescharacteristic
691C22C2×#54111yescharacteristic
692C22C2×#54211yescharacteristic
693C22C2×#54211yescharacteristic
694C22#345912yes—
695C22#346012yes—
696C22#348411yescharacteristic
697C22#348411yescharacteristic
698C22#348511yescharacteristic
699C22#348511yescharacteristic
700C22#350512yes—
701C22#350612yes—
702C22#353512yes—
703C22#353512yes—
704C22#353612yes—
705C22#353712yes—
706C22#356012yes—
707C22#356112yes—
708C22#356112yes—
709C22#356412yes—
710C4—24no—
711C4—24no—
712C4—44no—
713C4—44no—
714C4—44no—
715C4—82no—
716C4—84no—
717C4—82no—
718C4—82no—
719C4—82no—
720C4—82no—
721C4—84no—
722C4—82no—
723C2—28no—
724C2—84no—
725C2C2×#262511yescharacteristic
726C2#3498311yescharacteristic
727C2#3515211yescharacteristic
728C2#3670611yescharacteristic
729C2#3672011yescharacteristic
730C2#3676012yes—
731C2#3676612yes—
732C2#3696011yescharacteristic
733C2#3696611yescharacteristic
734C2#3746012yes—
735C2#3746212yes—
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
3. #16439C2semidirect
4. #16428C2semidirect
5. #16314C2semidirect
6. #16278C2semidirect
7. #15026C2semidirect
8. #14786C2semidirect
11. #14702C2semidirect
12. #14697C2semidirect
14. #14281C2semidirect
15. #14275C2semidirect
24. C2×#398C22semidirect
30. C2×#391C22semidirect
31. C2×#389C22semidirect
45. #1446C22semidirect
46. #1445C22semidirect
47. #1445C22semidirect
48. #1444C22semidirect
49. #1444C22semidirect
50. #1443C22semidirect
51. #1443C22semidirect
52. #1442C22semidirect
53. #1442C22semidirect
54. #1442C22semidirect
55. #1441C22semidirect
56. #1441C22semidirect
57. #1441C22semidirect
66. #1436C22semidirect
67. #1436C22semidirect
68. C4⋊#95C22semidirect
69. C4⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26262
C22×#3882
#164392
#164282
#163142
#162782
#150262
#147862
#147272
#147022
#146972
#143382
#142812
#142752