Finite Group Databasebeta

← #4929042

Subgroups — #4929042

Order 512.

3030 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2579C211yesmaximal, characteristic
2C2×#2486C211yesmaximal, characteristic
3C22×#400C211yesmaximal, characteristic, complemented
4C2×#2275C211yesmaximal, characteristic
5#14908C212yesmaximal, complemented
6#14869C212yesmaximal, complemented
7#14846C214yesmaximal, complemented
8#14841C212yesmaximal, complemented
9#14833C214yesmaximal, complemented
10#14826C212yesmaximal, complemented
11#14347C211yesmaximal, characteristic, complemented
12#13992C212yesmaximal, complemented
13#13992C212yesmaximal, complemented
14#13968C212yesmaximal
15#13968C212yesmaximal
16#13923C211yesmaximal, characteristic, complemented
17#13904C211yesmaximal, characteristic, complemented
18C22×#123C2212yes—
19C22×C4×D4C2212yes—
20C22×C4×D4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2211yescharacteristic
23#2579—28no—
24#2486—28no—
25C2×#400—28no—
26C2×#400C2212yes—
27C2×#400C2212yes—
28C2×#400C2214yes—
29C2×#398C2212yes—
30C2×#398C2212yes—
31C2×#397C2212yes—
32C2×#396C2212yes—
33C2×#389C2212yes—
34C2×#387C2212yes—
35C2×#386C2211yescharacteristic
36C2×#386C2211yescharacteristic
37C2×#385C2211yescharacteristic
38C2×#385C2211yescharacteristic
39C2×#385C2212yes—
40C2×#381C2212yes—
41C2×#381C2212yes—
42C2×#381C2212yes—
43C2×#380C2211yescharacteristic
44C2×#380C2211yescharacteristic
45C2×#380C2212yes—
46#2275—28no—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C22⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2212yes—
52C2×C4×C22⋊C4C2211yescharacteristic
53C2×#376C2211yescharacteristic
54C22×#95C2212yes—
55C22×#95C2212yes—
56C22×#95C2211yescharacteristic
57C22×#95C2211yescharacteristic
58#1445C2212yes—
59#1445C2212yes—
60#1443C2214yes—
61#1443C2214yes—
62#1442C2212yes—
63#1442C2212yes—
64#1442C2214yes—
65#1442C2214yes—
66#1442C2214yes—
67#1441C2212yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2212yes—
71#1440C2214yes—
72#1440C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78C4⋊#95C2214yes—
79#1434C2214yes—
80#1434C2214yes—
81C22⋊#95C2212yes—
82C22⋊#95C2212yes—
83C22⋊#95C2212yes—
84C22⋊#95C2212yes—
85C22⋊#95C2214yes—
86C22⋊#95C2214yes—
87C23×D4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2312yes—
92C24×C4C2312yes—
93C24×C4C2311yescharacteristic
94C24×C4C2311yescharacteristic
95C2×#123—28no—
96C2×#123—28no—
97C2×#123—28no—
98C2×C4×D4—28no—
99C2×C4×D4—28no—
100C2×C4×D4—24no—
101C2×C4×D4—24no—
102C2×C4×D4—28no—
103C2×C4×D4D418yes—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4—24no—
107C22×C4⋊C4—24no—
108C22×C4⋊C4—24no—
109C22×C4⋊C4—24no—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2311yescharacteristic
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133C22×C42C2312yes—
134C22×C42C2312yes—
135C22×C42C2312yes—
136#400—28no—
137#400—28no—
138#400—216no—
139#398—28no—
140#398—28no—
141#397—28no—
142#396—28no—
143#389—28no—
144#387—28no—
145#386—24no—
146#386—24no—
147#385—28no—
148#385—24no—
149#385—24no—
150#381—28no—
151#381—28no—
152#381—28no—
153#380—24no—
154#380—28no—
155#380—24no—
156C4×C4⋊C4—24no—
157C4×C4⋊C4—28no—
158C4×C22⋊C4—28no—
159C4×C22⋊C4—24no—
160C4×C22⋊C4—28no—
161C4×C22⋊C4—24no—
162#376—24no—
163C2×#95—28no—
164C2×#95—28no—
165C2×#95—24no—
166C2×#95D418yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2311yescharacteristic
184C2×#95C2311yescharacteristic
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2311yescharacteristic
196C2×#95C2311yescharacteristic
197C2×#95C2312yes—
198C25C2411yescharacteristic
199C25C2411yescharacteristic
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—24no—
203C22×D4C2×D414yes—
204C22×D4#4118yes—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—22no—
214C23×C4—24no—
215C23×C4—22no—
216C23×C4—24no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—44no—
227C23×C4—42no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4—44no—
234C23×C4C2×D418yes—
235C23×C4C2×D414yes—
236C23×C4C2×D414yes—
237C23×C4C2×D414yes—
238C23×C4#4114yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2414yes—
242C23×C4C2414yes—
243C23×C4C2412yes—
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C23×C4C2412yes—
247C23×C4C2412yes—
248C23×C4C2412yes—
249C23×C4C2412yes—
250C23×C4C2412yes—
251C23×C4C2411yescharacteristic
252C23×C4C2411yescharacteristic
253C23×C4C2411yescharacteristic
254#123—432no—
255C4×D4—264no—
256C4×D4—232no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—28no—
278C2×C4⋊C4—28no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—24no—
282C2×C4⋊C4—42no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—416no—
285C2×C4⋊C4—416no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4C2×D414yes—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4C2×D414yes—
314C2×C22⋊C4C2×D414yes—
315C2×C22⋊C4#4118yes—
316C2×C42—24no—
317C2×C42—22no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—22no—
321C2×C42—24no—
322C2×C42—24no—
323C2×C42—24no—
324C2×C42—44no—
325C2×C42—44no—
326C2×C42—44no—
327C2×C42C2×D414yes—
328#95—24no—
329#95—22no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—28no—
334#95—24no—
335#95—22no—
336#95—28no—
337#95—24no—
338#95—24no—
339#95—28no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—28no—
345#95—28no—
346#95—28no—
347#95—28no—
348#95—28no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95C2×D414yes—
357#95C2×D418yes—
358#95C2×D414yes—
359C24—22no—
360C24—24no—
361C24—22no—
362C24—21no—
363C24—21no—
364C24—22no—
365C24—22no—
366C24—24no—
367C24—22no—
368C24C22×D412yes—
369C24C22×D412yes—
370C24C22×D412yes—
371C24C2×#4114yes—
372C24C2×#4112yes—
373C24C2×#4114yes—
374C24C2×#4114yes—
375C24C2511yescharacteristic
376C2×D4—28no—
377C2×D4—216no—
378C2×D4—216no—
379C2×D4—28no—
380C2×D4#124116yes—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—28no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—216no—
417C22×C4—216no—
418C22×C4—28no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—48no—
459C22×C4—44no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—84no—
468C22×C4—84no—
469C22×C4—84no—
470C22×C4—84no—
471C22×C4—88no—
472C22×C4—88no—
473C22×C4—88no—
474C22×C4—88no—
475C22×C4—84no—
476C22×C4—84no—
477C22×C4#121116yes—
478C22×C4C22×D414yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D414yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C2×#4114yes—
488C22×C4C2×#4114yes—
489C22×C4C2×#4112yes—
490C22×C4C2×#4112yes—
491C22×C4C2×#4114yes—
492C22×C42−1+414yes—
493C4⋊C4—216no—
494C4⋊C4—28no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—48no—
501C4⋊C4—416no—
502C4⋊C4—84no—
503C4⋊C4—84no—
504C22⋊C4—216no—
505C22⋊C4—28no—
506C22⋊C4—216no—
507C22⋊C4—28no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—416no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—84no—
514C42—216no—
515C42—28no—
516C42—84no—
517C42—84no—
518C42—84no—
519C23—22no—
520C23—24no—
521C23—24no—
522C23—22no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—28no—
527C23—28no—
528C23—28no—
529C23—28no—
530C23—22no—
531C23—24no—
532C23—24no—
533C23—22no—
534C23—24no—
535C23—24no—
536C23—28no—
537C23—28no—
538C23—41no—
539C23—42no—
540C23—41no—
541C23—42no—
542C23—42no—
543C23—44no—
544C23—44no—
545C23C2×#12114yes—
546C23#52518yes—
547C23C2×#12414yes—
548C23C2×#12414yes—
549C23#52918yes—
550C23#53214yes—
551C23C23×D411yescharacteristic
552C23C23×D411yescharacteristic
553C23C22×#4111yescharacteristic
554C23C22×#4112yes—
555C23C22×#4112yes—
556C23C2×2−1+412yes—
557C23#58514yes—
558C23#58512yes—
559D4—232no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—22no—
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—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—28no—
586C2×C4—28no—
587C2×C4—416no—
588C2×C4—416no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—88no—
638C2×C4—816no—
639C2×C4—816no—
640C2×C4—88no—
641C2×C4—816no—
642C2×C4C2×#12114yes—
643C2×C4C2×#12114yes—
644C2×C4C2×#12414yes—
645C22—24no—
646C22—24no—
647C22—28no—
648C22—28no—
649C22—28no—
650C22—216no—
651C22—28no—
652C22—24no—
653C22—28no—
654C22—42no—
655C22—42no—
656C22—44no—
657C22—44no—
658C22—48no—
659C22—44no—
660C22#285018yes—
661C22C22×#12111yescharacteristic
662C22C2×#52512yes—
663C22C22×#12411yescharacteristic
664C22C2×#52912yes—
665C22C2×#53211yescharacteristic
666C22#348511yescharacteristic
667C22#348512yes—
668C22#348512yes—
669C22#348611yescharacteristic
670C22#348912yes—
671C22#349112yes—
672C22#349512yes—
673C22#351514yes—
674C22#351514yes—
675C22#352114yes—
676C22#352114yes—
677C4—24no—
678C4—24no—
679C4—24no—
680C4—42no—
681C4—44no—
682C4—42no—
683C4—44no—
684C4—44no—
685C4—44no—
686C4—44no—
687C4—84no—
688C4—82no—
689C4—84no—
690C4—82no—
691C4—82no—
692C2—28no—
693C2—44no—
694C2#3544711yescharacteristic
695C2#3545212yes—
696C2#3545312yes—
697C2#3563311yescharacteristic
698C2#3604212yes—
699C2#3604314yes—
700C2#3604512yes—
701C2C2×#285011yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C22×#400C2semidirect
5. #14908C2semidirect
6. #14869C2semidirect
7. #14846C2semidirect
8. #14841C2semidirect
9. #14833C2semidirect
10. #14826C2semidirect
11. #14347C2semidirect
12. #13992C2semidirect
13. #13992C2semidirect
16. #13923C2semidirect
17. #13904C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25792
C2×#24862
C22×#4002
C2×#22752
#149082
#148692
#148462
#148412
#148332
#148262
#143472
#139922
#139682
#139232
#139042