Finite Group Databasebeta

← #732585

Subgroups — #732585

Order 512.

2582 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic
2C2×#2349C211yesmaximal, characteristic
3#17058C212yesmaximal, complemented
4#16366C212yesmaximal, complemented
5#15470C214yesmaximal, complemented
6#15385C214yesmaximal, complemented
7#15187C212yesmaximal, complemented
8#15139C212yesmaximal, complemented
9#15019C212yesmaximal, complemented
10#14934C212yesmaximal, complemented
11#14348C211yesmaximal, characteristic, complemented
12#13925C211yesmaximal, characteristic, complemented
13#13916C212yesmaximal
14#13916C212yesmaximal
15#13911C211yesmaximal, characteristic, complemented
16#13850C211yesmaximal, characteristic, complemented
17#13785C211yesmaximal, characteristic
18C22×C22⋊Q8C2212yes—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23#2538—28no—
24C2×#400C2214yes—
25C2×#396C2214yes—
26#2349—28no—
27C2×#390C2212yes—
28C2×#390C2212yes—
29C2×#389C2211yescharacteristic
30C2×#389C2212yes—
31C2×#389C2211yescharacteristic
32C2×#385C2212yes—
33C2×#385C2212yes—
34C2×#384C2211yescharacteristic
35C2×#382C2211yescharacteristic
36C2×#382C2212yes—
37C2×#381C2214yes—
38C2×#381C2214yes—
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×#376C2211yescharacteristic
44C22×#95C2212yes—
45#1446C2211yescharacteristic
46#1446C2211yescharacteristic
47#1446C2212yes—
48#1445C2214yes—
49#1445C2211yescharacteristic
50#1445C2211yescharacteristic
51#1445C2212yes—
52#1443C2214yes—
53#1443C2214yes—
54#1442C2214yes—
55#1442C2212yes—
56#1442C2212yes—
57#1442C2214yes—
58#1442C2214yes—
59#1442C2214yes—
60#1442C2212yes—
61#1442C2212yes—
62#1441C2212yes—
63#1441C2214yes—
64#1441C2212yes—
65#1440C2214yes—
66#1440C2214yes—
67#1440C2214yes—
68#1440C2214yes—
69#1437C2214yes—
70#1436C2214yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2212yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C22⋊#95C2214yes—
80C22⋊#95C2212yes—
81C22⋊#95C2214yes—
82C22⋊#95C2212yes—
83C4×#95C2212yes—
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C23×D4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2312yes—
89C24×C4C2312yes—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—48no—
92C2×C4×Q8—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8—24no—
95C2×C4×D4—24no—
96C2×C4×D4—24no—
97C2×C4×D4—28no—
98C2×C4×D4—28no—
99C2×C4×D4—24no—
100C2×C4×D4—24no—
101C2×C4×D4D418yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129#400—48no—
130#396—216no—
131#390—28no—
132#390—28no—
133#389—24no—
134#389—24no—
135#389—28no—
136#385—28no—
137#385—28no—
138#384—24no—
139#382—24no—
140#382—44no—
141#381—48no—
142#381—48no—
143#380—28no—
144#380—28no—
145C4×C4⋊C4—24no—
146C4×C4⋊C4—24no—
147#376—24no—
148C2×#95—28no—
149C2×#95C2314yes—
150C2×#95C2312yes—
151C2×#95C2314yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2311yescharacteristic
156C2×#95C2311yescharacteristic
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2311yescharacteristic
164C2×#95C2312yes—
165C2×#95C2311yescharacteristic
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C25C2412yes—
183C22×Q8—22no—
184C22×Q8—22no—
185C22×Q8—22no—
186C22×Q8—24no—
187C22×Q8—42no—
188C22×D4—22no—
189C22×D4—22no—
190C22×D4—24no—
191C22×D4C2×D414yes—
192C22×D4#4118yes—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—28no—
198C23×C4—28no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4—44no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—44no—
207C23×C4C2×D418yes—
208C23×C4C2414yes—
209C23×C4C2411yescharacteristic
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C22⋊Q8—432no—
225C4×Q8—416no—
226C4×D4—232no—
227C4×D4—416no—
228C4×D4—416no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4C2×D414yes—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
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×C22⋊C4—44no—
284C2×C22⋊C4C2×D418yes—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42C2×D414yes—
299C2×C42#4114yes—
300C2×C42#4114yes—
301#95—22no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—22no—
306#95—28no—
307#95—24no—
308#95—24no—
309#95—22no—
310#95—22no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—28no—
315#95—28no—
316#95—24no—
317#95—24no—
318#95—42no—
319#95—44no—
320#95—42no—
321#95—44no—
322#95—42no—
323#95—42no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334C24—22no—
335C24—22no—
336C24—24no—
337C24—22no—
338C24—22no—
339C24—24no—
340C24—22no—
341C24—22no—
342C24—24no—
343C24C22×D414yes—
344C24C2×#4118yes—
345C24C2511yescharacteristic
346C2×Q8—28no—
347C2×Q8—48no—
348C2×Q8—48no—
349C2×Q8—44no—
350C2×Q8—44no—
351C2×D4—28no—
352C2×D4—28no—
353C2×D4—216no—
354C2×D4—216no—
355C2×D4#124116yes—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—216no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—416no—
441C22×C4—416no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4—88no—
445C22×C4—88no—
446C22×C4—84no—
447C22×C4—84no—
448C22×C4—88no—
449C22×C4—88no—
450C22×C4C22×D414yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4114yes—
461C22×C4H(5,𝔽2)12yes—
462C22×C4H(5,𝔽2)14yes—
463C22×C42−1+412yes—
464C22×C42−1+414yes—
465C4⋊C4—28no—
466C4⋊C4—44no—
467C4⋊C4—44no—
468C4⋊C4—416no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—84no—
477C4⋊C4—84no—
478C4⋊C4—84no—
479C22⋊C4—216no—
480C22⋊C4—416no—
481C22⋊C4—48no—
482C22⋊C4—48no—
483C22⋊C4—88no—
484C22⋊C4—88no—
485C22⋊C4—88no—
486C42—48no—
487C42—48no—
488C42—44no—
489C42—44no—
490C42—48no—
491C42#123116yes—
492C23—24no—
493C23—24no—
494C23—28no—
495C23—28no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—28no—
501C23—28no—
502C23—28no—
503C23—28no—
504C23—42no—
505C23—42no—
506C23—44no—
507C23—44no—
508C23—42no—
509C23—42no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—44no—
514C23C2×#12418yes—
515C23C23×D411yescharacteristic
516C23C22×#4111yescharacteristic
517C23C22×#4111yescharacteristic
518C23C22×#4112yes—
519C23C2×H(5,𝔽2)11yescharacteristic
520C23C2×H(5,𝔽2)12yes—
521C23C2×2−1+412yes—
522C23C2×2−1+411yescharacteristic
523C23#58511yescharacteristic
524C23#58511yescharacteristic
525C23#58512yes—
526Q8—416no—
527D4—232no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—42no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
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—44no—
593C2×C4—42no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—816no—
616C2×C4—816no—
617C2×C4—816no—
618C2×C4C2×#12314yes—
619C2×C4C2×#12314yes—
620C2×C4C2×#12314yes—
621C2×C4C2×#12414yes—
622C2×C4#53614yes—
623C2×C4#55218yes—
624C2×C4#55218yes—
625C22—28no—
626C22—28no—
627C22—28no—
628C22—48no—
629C22—48no—
630C22—48no—
631C22—44no—
632C22—44no—
633C22—44no—
634C22—44no—
635C22—48no—
636C22C22×#12311yescharacteristic
637C22C22×#12411yescharacteristic
638C22C2×#53611yescharacteristic
639C22#346011yescharacteristic
640C22#346111yescharacteristic
641C22C2×#55212yes—
642C22C2×#55212yes—
643C22#348411yescharacteristic
644C22#348511yescharacteristic
645C22#348811yescharacteristic
646C22#348911yescharacteristic
647C22#349412yes—
648C22#349412yes—
649C22#351111yescharacteristic
650C22#351211yescharacteristic
651C22#351312yes—
652C22#351412yes—
653C22#352012yes—
654C22#352112yes—
655C22#354812yes—
656C22#355412yes—
657C22#355512yes—
658C22#355712yes—
659C4—24no—
660C4—24no—
661C4—42no—
662C4—42no—
663C4—44no—
664C4—44no—
665C4—81no—
666C4—82no—
667C4—84no—
668C4—81no—
669C4—82no—
670C4—82no—
671C4—82no—
672C4—84no—
673C4—82no—
674C4#250318yes—
675C2—48no—
676C2C2×#250311yescharacteristic
677C2#3525511yescharacteristic
678C2#3554011yescharacteristic
679C2#3584711yescharacteristic
680C2#3593911yescharacteristic
681C2#3822412yes—
682C2#3822812yes—
683C2#4247911yescharacteristic
684C2#4267611yescharacteristic
685C2#4505012yes—
686C2#4520612yes—
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
3. #17058C2semidirect
4. #16366C2semidirect
5. #15470C2semidirect
6. #15385C2semidirect
7. #15187C2semidirect
8. #15139C2semidirect
9. #15019C2semidirect
10. #14934C2semidirect
11. #14348C2semidirect
12. #13925C2semidirect
15. #13911C2semidirect
16. #13850C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
C2×#23492
#170582
#163662
#154702
#153852
#151872
#151392
#150192
#149342
#143482
#139252
#139162
#139112
#138502
#137852