Finite Group Databasebeta

← #4308754

Subgroups — #4308754

Order 512.

4450 conjugacy classes of subgroups.

The subgroup lattice has 786 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2626C211yesmaximal, characteristic, complemented
2C22×#386C211yesmaximal, characteristic
3#16368C214yesmaximal, complemented
4#16348C214yesmaximal, complemented
5#16285C214yesmaximal, complemented
6#16269C214yesmaximal, complemented
7#15014C211yesmaximal, characteristic, complemented
8#14914C211yesmaximal, characteristic, complemented
9#14820C211yesmaximal, characteristic, complemented
10#14727C212yesmaximal, complemented
11#14724C212yesmaximal
12#14717C211yesmaximal, characteristic, complemented
13#14693C212yesmaximal, complemented
14#14341C211yesmaximal, characteristic, complemented
15#14265C212yesmaximal, complemented
16C24×D4C2211yescharacteristic
17C25×C4C2211yescharacteristic
18C22×C22⋊Q8C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22#2626—28no—
23C2×#399C2212yescomplemented
24C2×#397C2214yescomplemented
25C2×#397C2212yescomplemented
26C2×#396C2212yescomplemented
27C2×#396C2212yescomplemented
28C2×#394C2214yes—
29C2×#394C2214yes—
30C2×#393C2212yescomplemented
31C2×#393C2212yescomplemented
32C2×#386—28no—
33C2×#386C2214yes—
34C2×#386C2214yes—
35C2×#385C2212yes—
36C2×#385C2212yes—
37C2×#385C2212yes—
38C2×#385C2214yescomplemented
39C2×#385C2214yescomplemented
40C2×#380C2211yescharacteristic
41C2×#379C2212yes—
42C2×#379C2212yes—
43C2×#379C2212yes—
44C2×#379C2212yes—
45C22×#95C2211yescharacteristic
46C22×#95C2211yescharacteristic
47#1446C2212yescomplemented
48#1445C2214yescomplemented
49#1445C2212yescomplemented
50#1444C2212yescomplemented
51#1444C2212yescomplemented
52#1443C2214yescomplemented
53#1442C2214yescomplemented
54#1442C2214yescomplemented
55#1441C2214yescomplemented
56#1441C2214yescomplemented
57#1440C2214yescomplemented
58#1440C2214yescomplemented
59#1440C2214yescomplemented
60#1440C2214yescomplemented
61#1440C2214yescomplemented
62#1439C2212yescomplemented
63#1439C2214yescomplemented
64#1439C2214yescomplemented
65#1439C2212yescomplemented
66#1438C2214yes—
67#1438C2214yes—
68C22⋊#95C2211yescharacteristic
69C22⋊#95C2212yes—
70C22⋊#95C2211yescharacteristic
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2214yescomplemented
76C22⋊#95C2214yescomplemented
77C26C2311yescharacteristic
78C26C2311yescharacteristic
79C23×Q8C2311yescharacteristic
80C23×D4—24no—
81C23×D4—24no—
82C23×D4—24no—
83C23×D4—44no—
84C23×D4—44no—
85C23×D4C2314yes—
86C24×C4—24no—
87C24×C4—44no—
88C24×C4—44no—
89C24×C4—44no—
90C24×C4C2311yescharacteristic
91C24×C4C2314yes—
92C24×C4C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C24×C4C2311yescharacteristic
95C24×C4C2311yescharacteristic
96C2×C22⋊Q8—24no—
97C2×C22⋊Q8—44no—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C22⋊C4—24no—
107C22×C22⋊C4—24no—
108C22×C22⋊C4—44no—
109C22×C22⋊C4—44no—
110C22×C22⋊C4—44no—
111C22×C22⋊C4D418yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131#399—28no—
132#397—48no—
133#397—44no—
134#396—28no—
135#396—28no—
136#394—48no—
137#394—48no—
138#393—28no—
139#393—44no—
140#386—216no—
141#386—216no—
142#385—28no—
143#385—28no—
144#385—48no—
145#385—44no—
146#385—48no—
147#380—24no—
148#379—28no—
149#379—28no—
150#379—44no—
151#379—44no—
152C2×#95—24no—
153C2×#95—24no—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C25—22no—
177C25—22no—
178C25—24no—
179C25—24no—
180C25—22no—
181C25—22no—
182C25—24no—
183C25—24no—
184C25—24no—
185C25—24no—
186C25—42no—
187C25—42no—
188C25—42no—
189C25—42no—
190C25—42no—
191C25—42no—
192C25C2×D414yes—
193C25#4114yes—
194C25C2412yes—
195C25C2412yes—
196C25C2411yescharacteristic
197C22×Q8—22no—
198C22×Q8—42no—
199C22×Q8—42no—
200C22×Q8—42no—
201C22×D4—232no—
202C22×D4—28no—
203C22×D4—28no—
204C22×D4—28no—
205C22×D4—28no—
206C22×D4—28no—
207C22×D4—44no—
208C22×D4—44no—
209C22×D4—432no—
210C22×D4—416no—
211C22×D4—416no—
212C22×D4—432no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—22no—
220C23×C4—28no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—28no—
224C23×C4—28no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4—44no—
229C23×C4—44no—
230C23×C4—44no—
231C23×C4—44no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4—42no—
235C23×C4—42no—
236C23×C4—42no—
237C23×C4—42no—
238C23×C4—416no—
239C23×C4—416no—
240C23×C4—416no—
241C23×C4—464no—
242C23×C4C2×D414yes—
243C23×C4C2×D414yes—
244C23×C4#4114yes—
245C23×C4C2412yes—
246C23×C4C2412yes—
247C23×C4C2412yes—
248C23×C4C2412yes—
249C23×C4C2414yes—
250C23×C4C2414yes—
251C23×C4C2412yes—
252C23×C4C2411yescharacteristic
253C23×C4C2411yescharacteristic
254C23×C4C2412yes—
255C23×C4C2412yes—
256C23×C4C2412yes—
257C22⋊Q8—416no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—42no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—416no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—416no—
316C2×C22⋊C4—416no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C22⋊C4—44no—
331C2×C22⋊C4C2×D418yes—
332C2×C22⋊C4C2×D418yes—
333#95—24no—
334#95—24no—
335#95—28no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—28no—
340#95—28no—
341#95—42no—
342#95—42no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—42no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—42no—
352#95—44no—
353#95—44no—
354#95—44no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—28no—
360C24—28no—
361C24—216no—
362C24—28no—
363C24—216no—
364C24—22no—
365C24—22no—
366C24—24no—
367C24—24no—
368C24—22no—
369C24—22no—
370C24—24no—
371C24—24no—
372C24—28no—
373C24—216no—
374C24—22no—
375C24—22no—
376C24—24no—
377C24—22no—
378C24—22no—
379C24—24no—
380C24—24no—
381C24—22no—
382C24—22no—
383C24—24no—
384C24—28no—
385C24—28no—
386C24—48no—
387C24—44no—
388C24—44no—
389C24—44no—
390C24—44no—
391C24—416no—
392C24—416no—
393C24—416no—
394C24—416no—
395C24—44no—
396C24—44no—
397C24—48no—
398C24—48no—
399C24—48no—
400C24—48no—
401C24—48no—
402C24—44no—
403C24—44no—
404C24—48no—
405C24—48no—
406C24—48no—
407C24—88no—
408C24—88no—
409C24—88no—
410C24—88no—
411C24C22×D414yes—
412C24C22×D412yes—
413C24C2×#4114yes—
414C24C2×#4112yes—
415C24C2×#4112yes—
416C24H(5,𝔽2)14yes—
417C24H(5,𝔽2)14yes—
418C24H(5,𝔽2)14yes—
419C242−1+414yes—
420C24C2511yescharacteristic
421C2×Q8—44no—
422C2×Q8—82no—
423C2×Q8—82no—
424C2×Q8—88no—
425C2×D4—232no—
426C2×D4—416no—
427C2×D4—416no—
428C2×D4—416no—
429C2×D4—416no—
430C2×D4—4128no—
431C2×D4—416no—
432C2×D4—416no—
433C22×C4—28no—
434C22×C4—28no—
435C22×C4—28no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—28no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—22no—
486C22×C4—22no—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—48no—
497C22×C4—416no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—416no—
501C22×C4—416no—
502C22×C4—416no—
503C22×C4—416no—
504C22×C4—416no—
505C22×C4—44no—
506C22×C4—416no—
507C22×C4—416no—
508C22×C4—48no—
509C22×C4—44no—
510C22×C4—44no—
511C22×C4—4128no—
512C22×C4—416no—
513C22×C4—44no—
514C22×C4—44no—
515C22×C4—44no—
516C22×C4—44no—
517C22×C4—88no—
518C22×C4—88no—
519C22×C4C22×D412yes—
520C22×C4C22×D414yes—
521C22×C4C22×D412yes—
522C22×C4C2×#4114yes—
523C22×C4H(5,𝔽2)14yes—
524C4⋊C4—48no—
525C4⋊C4—44no—
526C4⋊C4—82no—
527C4⋊C4—84no—
528C4⋊C4—84no—
529C4⋊C4—82no—
530C4⋊C4—88no—
531C4⋊C4—88no—
532C22⋊C4—48no—
533C22⋊C4—48no—
534C22⋊C4—48no—
535C22⋊C4—416no—
536C22⋊C4—48no—
537C22⋊C4—416no—
538C22⋊C4—48no—
539C22⋊C4—48no—
540C22⋊C4—416no—
541C22⋊C4—48no—
542C22⋊C4—48no—
543C22⋊C4—84no—
544C22⋊C4—84no—
545C22⋊C4—84no—
546C22⋊C4—84no—
547C22⋊C4—88no—
548C22⋊C4—88no—
549C22⋊C4—88no—
550C22⋊C4—88no—
551C23—24no—
552C23—24no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—24no—
558C23—24no—
559C23—28no—
560C23—28no—
561C23—28no—
562C23—28no—
563C23—232no—
564C23—24no—
565C23—24no—
566C23—28no—
567C23—28no—
568C23—24no—
569C23—24no—
570C23—24no—
571C23—24no—
572C23—28no—
573C23—28no—
574C23—28no—
575C23—28no—
576C23—24no—
577C23—24no—
578C23—24no—
579C23—24no—
580C23—24no—
581C23—24no—
582C23—24no—
583C23—464no—
584C23—416no—
585C23—416no—
586C23—416no—
587C23—416no—
588C23—464no—
589C23—416no—
590C23—416no—
591C23—416no—
592C23—416no—
593C23—42no—
594C23—42no—
595C23—44no—
596C23—42no—
597C23—42no—
598C23—44no—
599C23—44no—
600C23—44no—
601C23—44no—
602C23—44no—
603C23—416no—
604C23—44no—
605C23—42no—
606C23—42no—
607C23—42no—
608C23—42no—
609C23—44no—
610C23—44no—
611C23—44no—
612C23—44no—
613C23—44no—
614C23—44no—
615C23—44no—
616C23—44no—
617C23—44no—
618C23—44no—
619C23—44no—
620C23—88no—
621C23—88no—
622C23—88no—
623C23—88no—
624C23—832no—
625C23—832no—
626C23#52514yes—
627C23#53414yes—
628C23#53614yes—
629C23#53818yes—
630C23#53918yes—
631C23#54218yes—
632C23C23×D411yescharacteristic
633C23C22×#4111yescharacteristic
634C23C22×#4111yescharacteristic
635C23C2×H(5,𝔽2)12yes—
636C23C2×H(5,𝔽2)12yes—
637C23C2×H(5,𝔽2)12yes—
638C23C2×H(5,𝔽2)12yes—
639C23C2×2−1+412yes—
640C23#58512yes—
641Q8—88no—
642D4—464no—
643C2×C4—24no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—24no—
647C2×C4—22no—
648C2×C4—24no—
649C2×C4—22no—
650C2×C4—24no—
651C2×C4—24no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—28no—
655C2×C4—28no—
656C2×C4—416no—
657C2×C4—416no—
658C2×C4—416no—
659C2×C4—464no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—42no—
663C2×C4—42no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—42no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—44no—
678C2×C4—44no—
679C2×C4—44no—
680C2×C4—44no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—42no—
687C2×C4—42no—
688C2×C4—42no—
689C2×C4—42no—
690C2×C4—42no—
691C2×C4—44no—
692C2×C4—44no—
693C2×C4—44no—
694C2×C4—44no—
695C2×C4—44no—
696C2×C4—44no—
697C2×C4—44no—
698C2×C4—44no—
699C2×C4—44no—
700C2×C4—44no—
701C2×C4—44no—
702C2×C4—44no—
703C2×C4—44no—
704C2×C4—44no—
705C2×C4—44no—
706C2×C4—44no—
707C2×C4—44no—
708C2×C4—44no—
709C2×C4—88no—
710C2×C4—88no—
711C22—24no—
712C22—216no—
713C22—24no—
714C22—24no—
715C22—24no—
716C22—24no—
717C22—216no—
718C22—24no—
719C22—216no—
720C22—28no—
721C22—464no—
722C22—44no—
723C22—44no—
724C22—48no—
725C22—44no—
726C22—44no—
727C22—44no—
728C22—48no—
729C22—48no—
730C22—44no—
731C22—44no—
732C22—44no—
733C22—44no—
734C22—44no—
735C22—44no—
736C22—44no—
737C22—44no—
738C22—832no—
739C22—84no—
740C22—84no—
741C22—84no—
742C22—84no—
743C22#263818yes—
744C22C2×#52511yescharacteristic
745C22C2×#53411yescharacteristic
746C22C2×#53611yescharacteristic
747C22C2×#53812yes—
748C22C2×#53912yes—
749C22C2×#54212yes—
750C22#345912yes—
751C22#348411yescharacteristic
752C22#348411yescharacteristic
753C22#348411yescharacteristic
754C22#348511yescharacteristic
755C22#349012yes—
756C22#350612yes—
757C22#353214yes—
758C22#353414yes—
759C22#353514yes—
760C22#353614yes—
761C4—44no—
762C4—44no—
763C4—44no—
764C4—42no—
765C4—42no—
766C4—82no—
767C4—84no—
768C4—82no—
769C4—82no—
770C4—82no—
771C4—84no—
772C4—82no—
773C2—28no—
774C2—48no—
775C2—84no—
776C2C2×#263811yescharacteristic
777C2#3498611yescharacteristic
778C2#3498811yescharacteristic
779C2#3670211yescharacteristic
780C2#3671811yescharacteristic
781C2#3672011yescharacteristic
782C2#3679314yes—
783C2#3711711yescharacteristic
784C2#3748214yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
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×#2626C2semidirect
3. #16368C2semidirect
4. #16348C2semidirect
5. #16285C2semidirect
6. #16269C2semidirect
7. #15014C2semidirect
8. #14914C2semidirect
9. #14820C2semidirect
10. #14727C2semidirect
12. #14717C2semidirect
13. #14693C2semidirect
14. #14341C2semidirect
15. #14265C2semidirect
23. C2×#399C22semidirect
24. C2×#397C22semidirect
25. C2×#397C22semidirect
26. C2×#396C22semidirect
27. C2×#396C22semidirect
30. C2×#393C22semidirect
31. C2×#393C22semidirect
38. C2×#385C22semidirect
39. C2×#385C22semidirect
47. #1446C22semidirect
48. #1445C22semidirect
49. #1445C22semidirect
50. #1444C22semidirect
51. #1444C22semidirect
52. #1443C22semidirect
53. #1442C22semidirect
54. #1442C22semidirect
55. #1441C22semidirect
56. #1441C22semidirect
57. #1440C22semidirect
58. #1440C22semidirect
59. #1440C22semidirect
60. #1440C22semidirect
61. #1440C22semidirect
62. #1439C22semidirect
63. #1439C22semidirect
64. #1439C22semidirect
65. #1439C22semidirect
75. C22⋊#95C22semidirect
76. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#26262
C22×#3862
#163682
#163482
#162852
#162692
#150142
#149142
#148202
#147272
#147242
#147172
#146932
#143412
#142652