Finite Group Databasebeta

← #4414545

Subgroups — #4414545

Order 512.

2894 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2851C211yesmaximal, characteristic, complemented
2C2×#2297C211yesmaximal, characteristic
3#17011C211yesmaximal, characteristic, complemented
4#17007C211yesmaximal, characteristic, complemented
5#16993C211yesmaximal, characteristic, complemented
6#16989C211yesmaximal, characteristic, complemented
7#16884C212yesmaximal, complemented
8#16807C212yesmaximal, complemented
9#16798C214yesmaximal, complemented
10#15017C212yesmaximal, complemented
11#14925C214yesmaximal, complemented
12#14916C212yesmaximal, complemented
13#14892C212yesmaximal, complemented
14#14889C212yesmaximal
15#14683C211yesmaximal, characteristic, complemented
16#14626C212yesmaximal, complemented
17#14624C212yesmaximal, complemented
18C23×#41C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2851—28no—
23C2×#397C2214yes—
24C2×#395C2212yes—
25C2×#394C2212yes—
26C2×#394C2214yes—
27#2297—28no—
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#390C2211yescharacteristic
31C2×#390C2211yescharacteristic
32C2×#389C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#387C2212yes—
35C2×#387C2212yes—
36C2×#386C2212yes—
37C2×#386C2212yes—
38C2×#385C2212yes—
39C2×#385C2212yes—
40C2×#385C2214yes—
41C2×#383C2212yes—
42C2×#382C2211yescharacteristic
43C2×#380C2211yescharacteristic
44C2×#380C2211yescharacteristic
45C2×C4×C22⋊C4C2212yes—
46C2×C4×C22⋊C4C2212yes—
47#1446C2212yes—
48#1445C2214yes—
49#1444C2212yes—
50#1444C2212yes—
51#1444C2212yes—
52#1443C2214yes—
53#1443C2214yes—
54#1442C2212yes—
55#1442C2212yes—
56#1442C2214yes—
57#1442C2212yes—
58#1441C2212yes—
59#1441C2212yes—
60#1441C2214yes—
61#1441C2214yes—
62#1440C2214yes—
63#1440C2214yes—
64#1440C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2212yes—
69#1439C2212yes—
70#1439C2214yes—
71#1439C2212yes—
72#1439C2212yes—
73#1438C2212yes—
74#1438C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77#1434C2212yes—
78#1434C2212yes—
79#1434C2211yescharacteristic
80#1434C2211yescharacteristic
81C22⋊#95C2212yes—
82C22⋊#95C2214yes—
83C22⋊#95C2212yes—
84C22⋊#95C2212yes—
85C22×#41—24no—
86C22×#41—24no—
87C22×#41—24no—
88C22×#41—44no—
89C22×#41—44no—
90C23×Q8C2311yescharacteristic
91C23×D4C2312yes—
92C23×D4C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C24×C4C2312yes—
95C24×C4C2311yescharacteristic
96C24×C4C2311yescharacteristic
97C2×#125—24no—
98C2×#125—44no—
99C2×#117—24no—
100C2×#117—44no—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C42—44no—
126C22×C42D418yes—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132#397—48no—
133#395—28no—
134#394—28no—
135#394—48no—
136#391—24no—
137#391—24no—
138#390—24no—
139#390—24no—
140#389—24no—
141#389—24no—
142#387—28no—
143#387—28no—
144#386—28no—
145#386—28no—
146#385—28no—
147#385—28no—
148#385—48no—
149#383—44no—
150#382—24no—
151#380—24no—
152#380—24no—
153C4×C22⋊C4—28no—
154C4×C22⋊C4—28no—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C25C2412yes—
185C25C2411yescharacteristic
186C2×#41—232no—
187C2×#41—432no—
188C2×#41—432no—
189C2×#41—416no—
190C2×#41—416no—
191C22×Q8—22no—
192C22×Q8—22no—
193C22×Q8—42no—
194C22×Q8—42no—
195C22×Q8#4114yes—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—28no—
199C22×D4—22no—
200C22×D4—24no—
201C22×D4—22no—
202C22×D4—42no—
203C22×D4—42no—
204C22×D4—44no—
205C22×D4#4114yes—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—28no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—44no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4C2×D414yes—
222C23×C4C2×D414yes—
223C23×C4C2×D414yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2414yes—
230C23×C4C2414yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239#125—416no—
240#117—416no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
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×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
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—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—416no—
304C2×C42—44no—
305C2×C42—42no—
306C2×C42—42no—
307C2×C42C2×D414yes—
308C2×C42C2×D414yes—
309C2×C42C2×D414yes—
310C2×C42C2×D414yes—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—28no—
323#95—24no—
324#95—28no—
325#95—24no—
326#95—24no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—42no—
333#95—44no—
334#95—42no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340C24—21no—
341C24—21no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—24no—
350C24—24no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—24no—
356C24—24no—
357C24C22×D412yes—
358C24C2×#4114yes—
359C24H(5,𝔽2)14yes—
360C24C2511yescharacteristic
361#41—4128no—
362C2×Q8—28no—
363C2×Q8—48no—
364C2×Q8—44no—
365C2×Q8—44no—
366C2×Q8—48no—
367C2×D4—216no—
368C2×D4—28no—
369C2×D4—48no—
370C2×D4—416no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—48no—
374C2×D4—416no—
375C2×D4—48no—
376C2×D4—48no—
377C22×C4—216no—
378C22×C4—216no—
379C22×C4—28no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—28no—
388C22×C4—28no—
389C22×C4—216no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—28no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—416no—
456C22×C4—416no—
457C22×C4—416no—
458C22×C4—48no—
459C22×C4—416no—
460C22×C4—48no—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C2×#4114yes—
468C22×C4C2×#4112yes—
469C22×C4C2×#4112yes—
470C22×C4H(5,𝔽2)14yes—
471C22×C42−1+414yes—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—48no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—44no—
479C4⋊C4—48no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—88no—
483C4⋊C4—88no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—416no—
489C22⋊C4—48no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C22⋊C4—88no—
493C22⋊C4—88no—
494C22⋊C4—88no—
495C22⋊C4—88no—
496C42—28no—
497C42—28no—
498C42—44no—
499C42—44no—
500C42—48no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—22no—
507C23—24no—
508C23—24no—
509C23—28no—
510C23—28no—
511C23—24no—
512C23—28no—
513C23—24no—
514C23—24no—
515C23—28no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—44no—
521C23—44no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—44no—
529C23—42no—
530C23—42no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23#53514yes—
541C23C23×D411yescharacteristic
542C23C22×#4111yescharacteristic
543C23C22×#4111yescharacteristic
544C23C2×H(5,𝔽2)12yes—
545C23C2×H(5,𝔽2)12yes—
546C23C2×2−1+412yes—
547C23#58512yes—
548C23#58512yes—
549C23#58512yes—
550Q8—416no—
551D4—416no—
552D4—432no—
553C2×C4—232no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—28no—
576C2×C4—416no—
577C2×C4—432no—
578C2×C4—416no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
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—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
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—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4#52514yes—
632C2×C4#53414yes—
633C2×C4#53614yes—
634C22—24no—
635C22—24no—
636C22—24no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—48no—
642C22—48no—
643C22—44no—
644C22—44no—
645C22—48no—
646C22—48no—
647C22—44no—
648C22—82no—
649C22—82no—
650C22—82no—
651C22—84no—
652C22—84no—
653C22—82no—
654C22C2×#52511yescharacteristic
655C22C2×#53411yescharacteristic
656C22C2×#53511yescharacteristic
657C22C2×#53611yescharacteristic
658C22#344911yescharacteristic
659C22#344911yescharacteristic
660C22#344911yescharacteristic
661C22#349012yes—
662C22#349012yes—
663C22#351512yes—
664C22#351812yes—
665C22#352112yes—
666C22#352412yes—
667C22#352914yes—
668C22#354114yes—
669C22#355914yes—
670C22#356014yes—
671C4—24no—
672C4—24no—
673C4—24no—
674C4—42no—
675C4—42no—
676C4—42no—
677C4—44no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4—84no—
682C4—84no—
683C4—82no—
684C4—82no—
685C4—82no—
686C2—44no—
687C2—84no—
688C2#3505011yescharacteristic
689C2#3520011yescharacteristic
690C2#3521011yescharacteristic
691C2#3665211yescharacteristic
692C2#3743811yescharacteristic
693C2#3753911yescharacteristic
694C2#3755611yescharacteristic
695C2#4573614yes—
696C2#4573914yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
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×#2851C2semidirect
3. #17011C2semidirect
4. #17007C2semidirect
5. #16993C2semidirect
6. #16989C2semidirect
7. #16884C2semidirect
8. #16807C2semidirect
9. #16798C2semidirect
10. #15017C2semidirect
11. #14925C2semidirect
12. #14916C2semidirect
13. #14892C2semidirect
15. #14683C2semidirect
16. #14626C2semidirect
17. #14624C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28512
C2×#22972
#170112
#170072
#169932
#169892
#168842
#168072
#167982
#150172
#149252
#149162
#148922
#148892
#146832
#146262
#146242