Finite Group Databasebeta

← #763553

Subgroups — #763553

Order 512.

3646 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2482C211yesmaximal, characteristic
2C22×#393C211yesmaximal, characteristic
3C2×#2310C211yesmaximal, characteristic, complemented
4C2×#2310C211yesmaximal, characteristic, complemented
5#15883C214yesmaximal, complemented
6#15851C214yesmaximal, complemented
7#15504C214yesmaximal, complemented
8#15499C214yesmaximal, complemented
9#14919C212yesmaximal, complemented
10#14918C212yesmaximal
11#14128C211yesmaximal, characteristic, complemented
12#13962C212yesmaximal, complemented
13#13958C211yesmaximal, characteristic, complemented
14#13868C212yesmaximal
15#13868C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C22×#117C2212yes—
19C22×#117C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23#2482—28no—
24C2×#399C2212yes—
25C2×#397C2212yes—
26C2×#393—28no—
27C2×#393C2214yes—
28C2×#393C2214yes—
29#2310—28no—
30#2310—28no—
31C2×#391C2212yescomplemented
32C2×#391C2214yescomplemented
33C2×#391C2212yescomplemented
34C2×#391C2214yescomplemented
35C2×#391C2212yescomplemented
36C2×#391C2212yescomplemented
37C2×#388C2214yes—
38C2×#385C2212yes—
39C2×#385C2212yes—
40C2×#381C2211yescharacteristic
41C2×#381C2212yes—
42C2×#381C2212yes—
43C2×#381C2212yes—
44C2×#381C2211yescharacteristic
45C2×#379C2212yes—
46C2×#379C2212yes—
47C22×#95C2211yescharacteristic
48C22×#95C2211yescharacteristic
49C22×#95C2211yescharacteristic
50#1445C2214yescomplemented
51#1445C2212yescomplemented
52#1445C2212yescomplemented
53#1444C2212yescomplemented
54#1443C2214yescomplemented
55#1443C2214yescomplemented
56#1442C2212yescomplemented
57#1442C2214yescomplemented
58#1441C2214yescomplemented
59#1441C2214yescomplemented
60#1440C2214yes—
61#1440C2214yes—
62#1439C2212yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2212yes—
66#1439C2214yes—
67#1439C2214yes—
68#1437C2214yes—
69#1437C2214yes—
70#1437C2214yes—
71#1436C2214yescomplemented
72C4⋊#95C2214yescomplemented
73#1434C2214yescomplemented
74#1434C2214yescomplemented
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C26C2311yescharacteristic
80C23×Q8—24no—
81C23×Q8—24no—
82C23×Q8—28no—
83C23×Q8—44no—
84C23×Q8D418yes—
85C23×Q8C2312yes—
86C23×Q8C2312yes—
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C2×#117—24no—
94C2×#117—24no—
95C2×#117—24no—
96C2×#117—28no—
97C2×#117—24no—
98C2×#117—28no—
99C2×#117—44no—
100C2×#117—44no—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C22⋊C4—24no—
109C22×C22⋊C4—24no—
110C22×C22⋊C4—24no—
111C22×C22⋊C4—24no—
112C22×C22⋊C4—24no—
113C22×C22⋊C4—28no—
114C22×C22⋊C4—44no—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C42C2312yes—
128C22×C42C2314yes—
129C22×C42C2312yes—
130#399—28no—
131#397—28no—
132#393—216no—
133#393—216no—
134#391—28no—
135#391—28no—
136#391—28no—
137#391—28no—
138#391—216no—
139#391—216no—
140#388—216no—
141#385—28no—
142#385—44no—
143#381—28no—
144#381—24no—
145#381—28no—
146#381—28no—
147#381D418yes—
148#379—28no—
149#379—44no—
150C2×#95—24no—
151C2×#95—24no—
152C2×#95—24no—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C25—22no—
182C25—24no—
183C25—22no—
184C25—22no—
185C25—24no—
186C25—24no—
187C25—42no—
188C25—42no—
189C25—42no—
190C25C2412yes—
191C25C2411yescharacteristic
192C22×Q8—232no—
193C22×Q8—232no—
194C22×Q8—24no—
195C22×Q8—24no—
196C22×Q8—24no—
197C22×Q8—24no—
198C22×Q8—24no—
199C22×Q8—24no—
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8—416no—
203C22×Q8—42no—
204C22×Q8—42no—
205C22×Q8—42no—
206C22×Q8—42no—
207C22×Q8—432no—
208C22×Q8—432no—
209C22×Q8C2×D418yes—
210C22×Q8C2×D418yes—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—24no—
226C23×C4—24no—
227C23×C4—42no—
228C23×C4—42no—
229C23×C4—42no—
230C23×C4—42no—
231C23×C4—42no—
232C23×C4—42no—
233C23×C4C2×D414yes—
234C23×C4C2×D414yes—
235C23×C4C2×D414yes—
236C23×C4C2×D414yes—
237C23×C4#4114yes—
238C23×C4#4118yes—
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C23×C4C2414yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2412yes—
247C23×C4C2414yes—
248C23×C4C2412yes—
249C23×C4C2411yescharacteristic
250C23×C4C2411yescharacteristic
251C23×C4C2412yes—
252#117—432no—
253#117—416no—
254#117—416no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
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—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—416no—
303C2×C22⋊C4—416no—
304C2×C22⋊C4—416no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4C2×D418yes—
313C2×C42—24no—
314C2×C42—28no—
315C2×C42—24no—
316C2×C42—24no—
317C2×C42—28no—
318C2×C42—24no—
319C2×C42—44no—
320C2×C42—42no—
321C2×C42—42no—
322#95—28no—
323#95—24no—
324#95—24no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—28no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95C2×D418yes—
349#95C2×D418yes—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—28no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—24no—
359C24—24no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—24no—
364C24—24no—
365C24—48no—
366C24—48no—
367C24—416no—
368C24—416no—
369C24—44no—
370C24—44no—
371C24—44no—
372C24—44no—
373C24—44no—
374C24—44no—
375C24—48no—
376C24—48no—
377C24—84no—
378C24—84no—
379C24—88no—
380C24—88no—
381C24C22×D412yes—
382C24C22×D412yes—
383C24C2×#4112yes—
384C24C2×#4114yes—
385C24H(5,𝔽2)14yes—
386C242−1+414yes—
387C24C2511yescharacteristic
388C2×Q8—216no—
389C2×Q8—216no—
390C2×Q8—216no—
391C2×Q8—216no—
392C2×Q8—48no—
393C2×Q8—48no—
394C2×Q8—4128no—
395C2×Q8—48no—
396C2×Q8—48no—
397C2×Q8—48no—
398C2×Q8—48no—
399C2×Q8—48no—
400C2×Q8—48no—
401C2×Q8—48no—
402C2×Q8—48no—
403C22×C4—28no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—216no—
408C22×C4—216no—
409C22×C4—216no—
410C22×C4—216no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—28no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—44no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—84no—
488C22×C4—84no—
489C22×C4—88no—
490C22×C4—88no—
491C22×C4#121116yes—
492C22×C4#123116yes—
493C22×C4C22×D414yes—
494C22×C4C22×D418yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C22×D412yes—
498C22×C4C22×D412yes—
499C22×C4C22×D414yes—
500C22×C4C2×#4112yes—
501C22×C4C2×#4112yes—
502C22×C4C2×#4114yes—
503C22×C4C2×#4114yes—
504C4⋊C4—416no—
505C4⋊C4—44no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—44no—
509C4⋊C4—88no—
510C4⋊C4—88no—
511C22⋊C4—216no—
512C22⋊C4—416no—
513C22⋊C4—416no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—416no—
520C22⋊C4—88no—
521C22⋊C4—88no—
522C22⋊C4—88no—
523C42—416no—
524C42—48no—
525C42—48no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—24no—
531C23—24no—
532C23—28no—
533C23—24no—
534C23—24no—
535C23—28no—
536C23—28no—
537C23—28no—
538C23—24no—
539C23—24no—
540C23—24no—
541C23—28no—
542C23—28no—
543C23—416no—
544C23—416no—
545C23—416no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—44no—
550C23—44no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—44no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—832no—
565C23—832no—
566C23—88no—
567C23—88no—
568C23—88no—
569C23—88no—
570C23C2×#12114yes—
571C23C2×#12314yes—
572C23#53414yes—
573C23#54618yes—
574C23#55118yes—
575C23C23×D411yescharacteristic
576C23C23×D411yescharacteristic
577C23C22×#4111yescharacteristic
578C23C22×#4112yes—
579C23C2×H(5,𝔽2)12yes—
580C23C2×2−1+412yes—
581C23#58512yes—
582C23#58512yes—
583C23#58512yes—
584Q8—432no—
585Q8—432no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—232no—
590C2×C4—232no—
591C2×C4—22no—
592C2×C4—22no—
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—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—28no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—416no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
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—44no—
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—88no—
660C2×C4—88no—
661C2×C4—88no—
662C2×C4C2×#12114yes—
663C2×C4C2×#12114yes—
664C2×C4C2×#12318yes—
665C22—24no—
666C22—24no—
667C22—24no—
668C22—24no—
669C22—24no—
670C22—24no—
671C22—216no—
672C22—216no—
673C22—44no—
674C22—44no—
675C22—44no—
676C22—48no—
677C22—48no—
678C22—832no—
679C22—82no—
680C22—82no—
681C22—82no—
682C22—82no—
683C22—84no—
684C22—84no—
685C22#241218yes—
686C22C22×#12111yescharacteristic
687C22C22×#12311yescharacteristic
688C22C2×#53411yescharacteristic
689C22C2×#54612yes—
690C22C2×#55112yes—
691C22#348611yescharacteristic
692C22#348711yescharacteristic
693C22#348711yescharacteristic
694C22#348711yescharacteristic
695C22#348912yes—
696C22#349512yes—
697C22#349514yes—
698C22#350012yes—
699C22#351012yes—
700C22#351014yes—
701C22#353514yes—
702C22#354514yes—
703C4—24no—
704C4—24no—
705C4—28no—
706C4—44no—
707C4—48no—
708C4—82no—
709C4—82no—
710C4—82no—
711C4—82no—
712C4—82no—
713C4—82no—
714C4—82no—
715C4—84no—
716C2—28no—
717C2—84no—
718C2C2×#241211yescharacteristic
719C2#3566111yescharacteristic
720C2#3567211yescharacteristic
721C2#3567711yescharacteristic
722C2#3567911yescharacteristic
723C2#3718611yescharacteristic
724C2#3718711yescharacteristic
725C2#4475914yes—
726C2#4478614yes—
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. C2×#2310C2semidirect
4. C2×#2310C2semidirect
5. #15883C2semidirect
6. #15851C2semidirect
7. #15504C2semidirect
8. #15499C2semidirect
9. #14919C2semidirect
11. #14128C2semidirect
12. #13962C2semidirect
13. #13958C2semidirect
31. C2×#391C22semidirect
32. C2×#391C22semidirect
33. C2×#391C22semidirect
34. C2×#391C22semidirect
35. C2×#391C22semidirect
36. C2×#391C22semidirect
50. #1445C22semidirect
51. #1445C22semidirect
52. #1445C22semidirect
53. #1444C22semidirect
54. #1443C22semidirect
55. #1443C22semidirect
56. #1442C22semidirect
57. #1442C22semidirect
58. #1441C22semidirect
59. #1441C22semidirect
71. #1436C22semidirect
72. C4⋊#95C22semidirect
73. #1434C22semidirect
74. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24822
C22×#3932
C2×#23102
#158832
#158512
#155042
#154992
#149192
#149182
#141282
#139622
#139582
#138682