Finite Group Databasebeta

← #643021

Subgroups — #643021

Order 512.

2474 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2295C211yesmaximal, characteristic, complemented
2C2×#2293C211yesmaximal, characteristic
3#17052C214yesmaximal, complemented
4#16679C212yesmaximal, complemented
5#16665C212yesmaximal, complemented
6#15202C211yesmaximal, characteristic, complemented
7#15194C211yesmaximal, characteristic, complemented
8#15157C211yesmaximal, characteristic
9#15147C211yesmaximal, characteristic
10#15114C214yesmaximal, complemented
11#15050C212yesmaximal, complemented
12#14969C212yesmaximal, complemented
13#14688C211yesmaximal, characteristic, complemented
14#14266C212yesmaximal, complemented
15#14260C212yesmaximal
16#14042C212yesmaximal, complemented
17#14018C212yesmaximal
18C22×C4×Q8C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×#117C2212yes—
21C2×#397C2212yes—
22C2×#393C2212yes—
23#2295—28no—
24C2×#391C2211yescharacteristic
25C2×#391C2211yescharacteristic
26#2293—28no—
27C2×#390C2211yescharacteristic
28C2×#390C2212yes—
29C2×#390C2212yes—
30C2×#390C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#388C2212yes—
38C2×#388C2212yes—
39C2×#387C2212yes—
40C2×#387C2212yes—
41C2×#387C2214yes—
42C2×#385C2214yes—
43C2×#383C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2212yes—
48C2×#376C2212yes—
49C2×C43C2211yescharacteristic
50#1446C2212yes—
51#1445C2212yes—
52#1445C2214yes—
53#1444C2214yes—
54#1444C2212yes—
55#1444C2212yes—
56#1443C2214yes—
57#1443C2214yes—
58#1442C2214yes—
59#1441C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1440C2214yes—
63#1440C2214yes—
64#1440C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2212yes—
68#1439C2214yes—
69#1439C2212yes—
70#1439C2214yes—
71#1439C2214yes—
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2212yes—
74C4⋊#95C2211yescharacteristic
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80#1434C2212yes—
81#1434C2214yes—
82#1434C2212yes—
83#1434C2212yes—
84C22⋊#95C2214yes—
85C23×Q8C2311yescharacteristic
86C23×D4C2311yescharacteristic
87C24×C4C2312yes—
88C2×C4×Q8—24no—
89C2×C4×Q8—28no—
90C2×C4×D4—24no—
91C2×C4×D4—24no—
92C2×C4×D4—24no—
93C2×#117—28no—
94C2×#117—48no—
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C42C2312yes—
120C22×C42C2312yes—
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129#397—44no—
130#393—44no—
131#391—24no—
132#391—24no—
133#390—28no—
134#390—24no—
135#390—24no—
136#390—28no—
137#389—28no—
138#389—24no—
139#389—24no—
140#389—24no—
141#389—24no—
142#389—28no—
143#388—28no—
144#388—28no—
145#387—28no—
146#387—28no—
147#387—48no—
148#385—48no—
149#383—44no—
150C4×C4⋊C4—24no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—28no—
153C4×C22⋊C4—28no—
154#376—28no—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C43D418yes—
184C25C2412yes—
185C22×Q8—22no—
186C22×Q8—24no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—42no—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4—24no—
193C22×D4—24no—
194C22×D4#4114yes—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—28no—
199C23×C4—44no—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2414yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C4×Q8—232no—
217C4×D4—232no—
218#117—432no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4#4114yes—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—44no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C42—24no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—28no—
297C2×C42—42no—
298C2×C42—41no—
299C2×C42—41no—
300C2×C42C2×D414yes—
301C2×C42C2×D414yes—
302C2×C42C2×D414yes—
303C2×C42C2×D414yes—
304C2×C42C2×D414yes—
305C2×C42C2×D414yes—
306C2×C42C2×D414yes—
307C2×C42#4118yes—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—28no—
319#95—28no—
320#95—24no—
321#95—24no—
322#95—28no—
323#95—42no—
324#95—44no—
325#95—42no—
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—
335#95—44no—
336C24—22no—
337C24—24no—
338C24—22no—
339C24—22no—
340C24—24no—
341C24—22no—
342C24—24no—
343C24—22no—
344C24—22no—
345C24—24no—
346C24C2×#4114yes—
347C24C2511yescharacteristic
348C2×Q8—28no—
349C2×Q8—48no—
350C2×Q8—48no—
351C2×Q8—44no—
352C2×Q8—44no—
353C2×D4—28no—
354C2×D4—216no—
355C2×D4—44no—
356C2×D4—44no—
357C2×D4—48no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—216no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—416no—
437C22×C4—416no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C2×#4114yes—
448C22×C4C2×#4114yes—
449C22×C4C2×#4114yes—
450C22×C4C2×#4112yes—
451C22×C4C2×#4114yes—
452C22×C4C2×#4112yes—
453C22×C4H(5,𝔽2)14yes—
454C22×C4H(5,𝔽2)12yes—
455C22×C42−1+412yes—
456C22×C42−1+414yes—
457C4⋊C4—28no—
458C4⋊C4—216no—
459C4⋊C4—28no—
460C4⋊C4—44no—
461C4⋊C4—48no—
462C4⋊C4—416no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—44no—
466C4⋊C4—48no—
467C4⋊C4—44no—
468C4⋊C4—44no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—88no—
472C22⋊C4—216no—
473C22⋊C4—48no—
474C22⋊C4—416no—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C22⋊C4—48no—
478C22⋊C4—88no—
479C22⋊C4—88no—
480C22⋊C4—88no—
481C42—28no—
482C42—28no—
483C42—216no—
484C42—28no—
485C42—28no—
486C42—28no—
487C42—44no—
488C42—44no—
489C42—48no—
490C42—48no—
491C23—24no—
492C23—28no—
493C23—28no—
494C23—24no—
495C23—24no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—42no—
509C23—42no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23C23×D411yescharacteristic
515C23C22×#4112yes—
516C23C22×#4111yescharacteristic
517C23C22×#4111yescharacteristic
518C23C2×H(5,𝔽2)12yes—
519C23C2×H(5,𝔽2)11yescharacteristic
520C23C2×2−1+412yes—
521C23C2×2−1+411yescharacteristic
522C23#58511yescharacteristic
523C23#58511yescharacteristic
524C23#58512yes—
525Q8—416no—
526D4—416no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—28no—
555C2×C4—432no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4#52514yes—
616C2×C4#52514yes—
617C2×C4#53514yes—
618C2×C4#53614yes—
619C2×C4#53714yes—
620C2×C4#55218yes—
621C2×C4#55618yes—
622C22—44no—
623C22—48no—
624C22—48no—
625C22—44no—
626C22—44no—
627C22—82no—
628C22—82no—
629C22—84no—
630C22—84no—
631C22—82no—
632C22—82no—
633C22C2×#52511yescharacteristic
634C22C2×#52511yescharacteristic
635C22C2×#53511yescharacteristic
636C22C2×#53611yescharacteristic
637C22C2×#53711yescharacteristic
638C22#344911yescharacteristic
639C22#344911yescharacteristic
640C22#346012yes—
641C22#346012yes—
642C22C2×#55212yes—
643C22C2×#55612yes—
644C22#349412yes—
645C22#349512yes—
646C22#351312yes—
647C22#351312yes—
648C22#351411yescharacteristic
649C22#351511yescharacteristic
650C22#351711yescharacteristic
651C22#351811yescharacteristic
652C22#354812yes—
653C22#355012yes—
654C22#355712yes—
655C22#355712yes—
656C4—24no—
657C4—24no—
658C4—42no—
659C4—42no—
660C4—44no—
661C4—42no—
662C4—44no—
663C4—42no—
664C4—44no—
665C4—82no—
666C4—82no—
667C4—84no—
668C4—84no—
669C4—82no—
670C4—82no—
671C4#259918yes—
672C2—84no—
673C2#3460911yescharacteristic
674C2C2×#259911yescharacteristic
675C2#3518911yescharacteristic
676C2#3523211yescharacteristic
677C2#3530211yescharacteristic
678C2#3755111yescharacteristic
679C2#3764011yescharacteristic
680C2#3822812yes—
681C2#3823012yes—
682C2#3824012yes—
683C2#3824212yes—
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
1. C2×#2295C2semidirect
3. #17052C2semidirect
4. #16679C2semidirect
5. #16665C2semidirect
6. #15202C2semidirect
7. #15194C2semidirect
10. #15114C2semidirect
11. #15050C2semidirect
12. #14969C2semidirect
13. #14688C2semidirect
14. #14266C2semidirect
16. #14042C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22952
C2×#22932
#170522
#166792
#166652
#152022
#151942
#151572
#151472
#151142
#150502
#149692
#146882
#142662
#142602
#140422
#140182