Finite Group Databasebeta

← #4308781

Subgroups — #4308781

Order 512.

3590 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2626C211yesmaximal, characteristic
2C22×#380C211yesmaximal, characteristic
3#16302C214yesmaximal, complemented
4#16301C214yesmaximal, complemented
5#16300C214yesmaximal, complemented
6#16299C214yesmaximal, complemented
7#15015C211yesmaximal, characteristic, complemented
8#14920C212yesmaximal, complemented
9#14914C211yesmaximal, characteristic, complemented
10#14822C211yesmaximal, characteristic, complemented
11#14728C212yesmaximal, complemented
12#14720C211yesmaximal, characteristic, complemented
13#14681C212yesmaximal
14#14338C211yesmaximal, characteristic
15#14262C212yesmaximal
16C25×C4C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22#2626—28no—
23C2×#400C2212yescomplemented
24C2×#399C2212yes—
25C2×#398C2214yescomplemented
26C2×#396C2212yes—
27C2×#396C2212yes—
28C2×#395C2212yescomplemented
29C2×#393C2212yes—
30C2×#385C2212yes—
31C2×#385C2212yes—
32C2×#385C2212yes—
33C2×#381C2214yes—
34C2×#381C2214yes—
35C2×#380—28no—
36C2×#380C2211yescharacteristic
37C2×#380C2214yes—
38C2×#380C2214yes—
39C2×#380C2212yes—
40C2×#379C2212yes—
41C2×#379C2212yes—
42C22×#95C2211yescharacteristic
43C22×#95C2211yescharacteristic
44#1446C2212yescomplemented
45#1445C2214yescomplemented
46#1445C2212yescomplemented
47#1444C2212yescomplemented
48#1444C2214yescomplemented
49#1444C2214yescomplemented
50#1443C2214yescomplemented
51#1442C2214yescomplemented
52#1442C2214yescomplemented
53#1442C2212yescomplemented
54#1442C2214yescomplemented
55#1441C2214yescomplemented
56#1441C2214yescomplemented
57#1441C2214yescomplemented
58#1441C2212yescomplemented
59#1441C2214yescomplemented
60#1441C2212yescomplemented
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67C22⋊#95C2211yescharacteristic
68C22⋊#95C2211yescharacteristic
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C22⋊#95C2212yes—
77C26C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C24×C4—24no—
80C24×C4—24no—
81C24×C4—44no—
82C24×C4—44no—
83C24×C4Q818yescomplemented
84C24×C4C2311yescharacteristic
85C24×C4C2314yes—
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—44no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—44no—
96C22×C4⋊C4—44no—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C22⋊C4—24no—
110C22×C22⋊C4—44no—
111C22×C22⋊C4—44no—
112C22×C22⋊C4D418yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126#400—44no—
127#399—28no—
128#398—48no—
129#396—28no—
130#396—28no—
131#395—44no—
132#393—28no—
133#385—28no—
134#385—44no—
135#385—44no—
136#381—48no—
137#381—48no—
138#380—24no—
139#380—216no—
140#380—216no—
141#380—44no—
142#379—28no—
143#379—28no—
144C2×#95—24no—
145C2×#95—24no—
146C2×#95C2312yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C25—24no—
172C25—24no—
173C25—42no—
174C25—42no—
175C25—42no—
176C25—42no—
177C25C2×D414yes—
178C25C2×Q814yes—
179C25#4114yes—
180C25C2412yes—
181C25C2411yescharacteristic
182C22×Q8—22no—
183C22×Q8—42no—
184C22×Q8—42no—
185C22×Q8—42no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—232no—
191C23×C4—22no—
192C23×C4—28no—
193C23×C4—28no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—24no—
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—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×C4—42no—
216C23×C4—416no—
217C23×C4—416no—
218C23×C4—432no—
219C23×C4C2×D414yes—
220C23×C4C2×D414yes—
221C23×C4C2×Q814yes—
222C23×C4C2×Q8116yes—
223C23×C4C2×Q814yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2414yes—
229C23×C4C2414yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2412yes—
236C23×C4C2412yes—
237C22⋊Q8—416no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—416no—
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—44no—
270C2×C4⋊C4—42no—
271C2×C22⋊C4—28no—
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—28no—
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—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—416no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—416no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4C2×D418yes—
306C2×C22⋊C4C2×D418yes—
307#95—28no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—42no—
318#95—42no—
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—
328#95—42no—
329#95—44no—
330#95—42no—
331#95—44no—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—28no—
337C24—28no—
338C24—216no—
339C24—216no—
340C24—22no—
341C24—22no—
342C24—24no—
343C24—24no—
344C24—22no—
345C24—22no—
346C24—24no—
347C24—24no—
348C24—44no—
349C24—44no—
350C24—48no—
351C24—48no—
352C24—44no—
353C24—44no—
354C24—48no—
355C24—48no—
356C24—416no—
357C24—416no—
358C24—416no—
359C24—416no—
360C24C22⋊Q8116yes—
361C24C22×D414yes—
362C24C22×D412yes—
363C24C22×Q814yes—
364C24C22×Q812yes—
365C24C2×#4114yes—
366C24C2×#4112yes—
367C24H(5,𝔽2)14yes—
368C242−1+414yes—
369C24C2511yescharacteristic
370C2×Q8—44no—
371C2×Q8—82no—
372C2×Q8—82no—
373C2×Q8—88no—
374C22×C4—28no—
375C22×C4—28no—
376C22×C4—216no—
377C22×C4—28no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—216no—
383C22×C4—232no—
384C22×C4—28no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—232no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
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—416no—
445C22×C4—48no—
446C22×C4—44no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—416no—
451C22×C4—416no—
452C22×C4—416no—
453C22×C4—416no—
454C22×C4—4128no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—48no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—88no—
465C22×C4—88no—
466C22×C4—88no—
467C22×C4—88no—
468C22×C4C22×D414yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×Q814yes—
472C22×C4C22×Q814yes—
473C22×C4H(5,𝔽2)14yes—
474C22×C4H(5,𝔽2)14yes—
475C4⋊C4—416no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C4⋊C4—44no—
480C4⋊C4—82no—
481C4⋊C4—84no—
482C4⋊C4—82no—
483C4⋊C4—84no—
484C4⋊C4—88no—
485C4⋊C4—88no—
486C4⋊C4—88no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—416no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—84no—
495C22⋊C4—84no—
496C22⋊C4—84no—
497C22⋊C4—84no—
498C22⋊C4—88no—
499C22⋊C4—88no—
500C23—24no—
501C23—24no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—24no—
507C23—24no—
508C23—28no—
509C23—28no—
510C23—28no—
511C23—28no—
512C23—232no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—28no—
518C23—24no—
519C23—24no—
520C23—28no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—28no—
526C23—28no—
527C23—416no—
528C23—416no—
529C23—416no—
530C23—416no—
531C23—464no—
532C23—416no—
533C23—416no—
534C23—464no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23C2×C22⋊Q818yes—
544C23C2×C22⋊Q814yes—
545C23#53414yes—
546C23#53614yes—
547C23#53918yes—
548C23#54418yes—
549C23C23×D411yescharacteristic
550C23C23×Q811yescharacteristic
551C23C22×#4111yescharacteristic
552C23C2×H(5,𝔽2)12yes—
553C23C2×H(5,𝔽2)12yes—
554C23C2×H(5,𝔽2)12yes—
555C23C2×2−1+412yes—
556C23#58512yes—
557C23#58512yes—
558Q8—88no—
559C2×C4—232no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—28no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—464no—
578C2×C4—416no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
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—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—48no—
611C2×C4—48no—
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—88no—
629C2×C4—88no—
630C2×C4—88no—
631C2×C4—88no—
632C2×C4#54318yes—
633C22—24no—
634C22—24no—
635C22—24no—
636C22—216no—
637C22—24no—
638C22—24no—
639C22—216no—
640C22—24no—
641C22—28no—
642C22—28no—
643C22—28no—
644C22—464no—
645C22—44no—
646C22—44no—
647C22—48no—
648C22—48no—
649C22—44no—
650C22—44no—
651C22—48no—
652C22—48no—
653C22#262718yes—
654C22C22×C22⋊Q811yescharacteristic
655C22C2×#53411yescharacteristic
656C22C2×#53611yescharacteristic
657C22C2×#53912yes—
658C22C2×#54312yes—
659C22C2×#54412yes—
660C22#345812yes—
661C22#348411yescharacteristic
662C22#348411yescharacteristic
663C22#348511yescharacteristic
664C22#348511yescharacteristic
665C22#349012yes—
666C22#349112yes—
667C22#354114yes—
668C22#354114yes—
669C22#354214yes—
670C22#354214yes—
671C4—28no—
672C4—44no—
673C4—42no—
674C4—42no—
675C4—44no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4—82no—
682C4—84no—
683C4—84no—
684C2—28no—
685C2—48no—
686C2C2×#262711yescharacteristic
687C2#3498211yescharacteristic
688C2#3509011yescharacteristic
689C2#3670411yescharacteristic
690C2#3670511yescharacteristic
691C2#3671511yescharacteristic
692C2#3671611yescharacteristic
693C2#3672614yes—
694C2#3747614yes—
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. #16302C2semidirect
4. #16301C2semidirect
5. #16300C2semidirect
6. #16299C2semidirect
7. #15015C2semidirect
8. #14920C2semidirect
9. #14914C2semidirect
10. #14822C2semidirect
11. #14728C2semidirect
12. #14720C2semidirect
23. C2×#400C22semidirect
25. C2×#398C22semidirect
28. C2×#395C22semidirect
44. #1446C22semidirect
45. #1445C22semidirect
46. #1445C22semidirect
47. #1444C22semidirect
48. #1444C22semidirect
49. #1444C22semidirect
50. #1443C22semidirect
51. #1442C22semidirect
52. #1442C22semidirect
53. #1442C22semidirect
54. #1442C22semidirect
55. #1441C22semidirect
56. #1441C22semidirect
57. #1441C22semidirect
58. #1441C22semidirect
59. #1441C22semidirect
60. #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×#26262
C22×#3802
#163022
#163012
#163002
#162992
#150152
#149202
#149142
#148222
#147282
#147202
#146812
#143382
#142622