Finite Group Databasebeta

← #612924

Subgroups — #612924

Order 512.

2842 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×#2442C211yesmaximal, characteristic
2C22×C4×C4⋊C4C211yesmaximal, characteristic
3#15015C212yesmaximal
4#14920C212yesmaximal
5#14863C214yesmaximal, complemented
6#14862C214yesmaximal, complemented
7#14705C212yesmaximal
8#14685C212yesmaximal
9#14685C211yesmaximal, characteristic
10#14130C211yesmaximal, characteristic
11#14054C212yesmaximal, complemented
12#14030C212yesmaximal, complemented
13#14006C212yesmaximal, complemented
14#13982C212yesmaximal, complemented
15#13907C211yesmaximal, characteristic
16#13881C211yesmaximal, characteristic
17C4×#381C211yesmaximal, characteristic
18C22×#125C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C42C2212yes—
22C23×C42C2211yescharacteristic
23#2442—28no—
24C2×#400C2212yes—
25C2×#398C2212yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2211yescharacteristic
29C2×#389C2211yescharacteristic
30C2×#385C2212yes—
31C2×#385C2212yes—
32C2×#382C2212yes—
33C2×#382C2211yescharacteristic
34C2×#381C2212yes—
35C2×#381C2211yescharacteristic
36C2×#381C2211yescharacteristic
37C2×#380C2211yescharacteristic
38C2×#380C2211yescharacteristic
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×C4×C4⋊C4—28no—
42C2×C4×C4⋊C4C2214yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2211yescharacteristic
47C22×#95C2212yes—
48#1445C2214yes—
49#1445C2214yes—
50#1444C2212yes—
51#1443C2214yes—
52#1443C2214yes—
53#1442C2214yes—
54#1442C2212yes—
55#1442C2214yes—
56#1442C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1441C2214yes—
60#1436C2214yes—
61#1436C2214yes—
62#1436C2214yes—
63#1436C2214yes—
64C4⋊#95C2214yes—
65C4⋊#95C2214yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71#1434C2212yes—
72#1434C2214yes—
73#1434C2214yes—
74#1434C2212yes—
75#1434C2214yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C22⋊#95C2211yescharacteristic
80C22⋊#95C2212yes—
81C22⋊#95C2212yes—
82C22⋊#95C2211yescharacteristic
83C4×#95C2212yes—
84C4×#95C2212yes—
85C24×C4C2312yes—
86C24×C4C2312yes—
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C2×#125—24no—
91C2×#125—28no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—28no—
94C22×C4⋊C4—28no—
95C22×C4⋊C4D418yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4C2311yescharacteristic
111C22×C22⋊C4C2311yescharacteristic
112C22×C22⋊C4C2311yescharacteristic
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C42—24no—
117C22×C42—24no—
118C22×C42—28no—
119C22×C42—28no—
120C22×C42—44no—
121C22×C42Q818yes—
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2314yes—
126C22×C42C2314yes—
127C22×C42C2312yes—
128C22×C42C2312yes—
129#400—28no—
130#398—28no—
131#389—24no—
132#389—24no—
133#389—28no—
134#389—28no—
135#385—28no—
136#385—44no—
137#382—24no—
138#382—28no—
139#381—28no—
140#381—24no—
141#381—24no—
142#380—28no—
143#380—24no—
144#380—24no—
145#380—44no—
146C4×C4⋊C4—28no—
147C4×C4⋊C4—216no—
148C4×C4⋊C4—28no—
149C4×C22⋊C4—24no—
150C4×C22⋊C4—24no—
151C2×#95—28no—
152C2×#95C2311yescharacteristic
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2311yescharacteristic
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C25C2411yescharacteristic
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—22no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—22no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—28no—
198C23×C4—28no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—44no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—44no—
205C23×C4C2×D418yes—
206C23×C4C2×D414yes—
207C23×C4C2×Q814yes—
208C23×C4C2×Q814yes—
209C23×C4C2×Q814yes—
210C23×C4#4114yes—
211C23×C4#4114yes—
212C23×C4#4114yes—
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2414yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2411yescharacteristic
229#125—416no—
230C2×C4⋊C4—232no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—232no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4C2×D414yes—
260C2×C4⋊C4C2×D418yes—
261C2×C4⋊C4C2×D414yes—
262C2×C4⋊C4#4118yes—
263C2×C4⋊C4#4118yes—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—22no—
282C2×C42—232no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—22no—
288C2×C42—22no—
289C2×C42—28no—
290C2×C42—28no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—28no—
297C2×C42—28no—
298C2×C42—44no—
299C2×C42—44no—
300C2×C42—432no—
301C2×C42C2×Q818yes—
302C2×C42C2×Q818yes—
303#95—28no—
304#95—28no—
305#95—22no—
306#95—22no—
307#95—24no—
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—28no—
317#95—28no—
318#95—24no—
319#95—24no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—42no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—42no—
329#95—42no—
330#95—44no—
331#95—42no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24C22×D412yes—
341C24C22×Q812yes—
342C24C2×#4112yes—
343C24C2×#4112yes—
344C24C2×#4112yes—
345C24H(5,𝔽2)12yes—
346C242−1+412yes—
347C24C2511yescharacteristic
348C22×C4—28no—
349C22×C4—28no—
350C22×C4—28no—
351C22×C4—28no—
352C22×C4—28no—
353C22×C4—28no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—28no—
364C22×C4—28no—
365C22×C4—216no—
366C22×C4—216no—
367C22×C4—216no—
368C22×C4—28no—
369C22×C4—216no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—216no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—416no—
441C22×C4—416no—
442C22×C4—416no—
443C22×C4—416no—
444C22×C4—44no—
445C22×C4—48no—
446C22×C4—44no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4C22⋊Q8116yes—
450C22×C4#125116yescomplemented
451C22×C4C22×D414yes—
452C22×C4C22×D412yes—
453C22×C4C22×D414yes—
454C22×C4C22×D412yes—
455C22×C4C22×Q814yes—
456C22×C4C22×Q812yes—
457C22×C4C22×Q814yes—
458C22×C4C22×Q812yes—
459C22×C4C2×#4118yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4118yes—
462C22×C4C2×#4114yes—
463C22×C4C2×#4114yes—
464C22×C4C2×#4112yes—
465C22×C4C2×#4114yes—
466C22×C4C2×#4114yes—
467C4⋊C4—216no—
468C4⋊C4—216no—
469C4⋊C4—28no—
470C4⋊C4—216no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—48no—
476C4⋊C4—84no—
477C4⋊C4—88no—
478C4⋊C4—84no—
479C4⋊C4—88no—
480C4⋊C4#123116yes—
481C22⋊C4—44no—
482C22⋊C4—44no—
483C22⋊C4—48no—
484C22⋊C4—44no—
485C22⋊C4—44no—
486C22⋊C4—48no—
487C42—216no—
488C42—216no—
489C42—416no—
490C42—416no—
491C42—44no—
492C42—44no—
493C42—48no—
494C23—22no—
495C23—22no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—22no—
504C23—22no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23C2×C22⋊Q814yes—
511C23C2×#12514yes—
512C23#54414yes—
513C23#54614yes—
514C23#54814yes—
515C23#55214yes—
516C23#55314yes—
517C23C23×D411yescharacteristic
518C23C23×Q811yescharacteristic
519C23C22×#4111yescharacteristic
520C23C22×#4112yes—
521C23C22×#4112yes—
522C23C22×#4111yescharacteristic
523C23C22×#4111yescharacteristic
524C23C2×H(5,𝔽2)11yescharacteristic
525C23C2×2−1+411yescharacteristic
526C23#58512yes—
527C23#58512yes—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—232no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—24no—
542C2×C4—28no—
543C2×C4—232no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—28no—
553C2×C4—432no—
554C2×C4—432no—
555C2×C4—416no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4C2×C22⋊Q814yes—
618C2×C4C2×C22⋊Q814yes—
619C2×C4C2×#12318yes—
620C2×C4C2×#12314yes—
621C2×C4#52514yes—
622C2×C4C2×#12518yes—
623C2×C4#52918yes—
624C2×C4#53318yes—
625C22—24no—
626C22—24no—
627C22—24no—
628C22—24no—
629C22—28no—
630C22—28no—
631C22—28no—
632C22—24no—
633C22—24no—
634C22—28no—
635C22#244218yes—
636C22C22×C22⋊Q811yescharacteristic
637C22C22×#12311yescharacteristic
638C22C2×#52511yescharacteristic
639C22C22×#12511yescharacteristic
640C22C2×#52912yes—
641C22C2×#53312yes—
642C22C2×#54411yescharacteristic
643C22#345812yes—
644C22C2×#54611yescharacteristic
645C22C2×#54811yescharacteristic
646C22C2×#55211yescharacteristic
647C22C2×#55311yescharacteristic
648C22#347512yes—
649C22#347512yes—
650C22#348012yes—
651C22#348012yes—
652C22#348511yescharacteristic
653C22#348711yescharacteristic
654C22#349312yes—
655C22#351312yes—
656C22#351312yes—
657C22#351612yes—
658C22#351612yes—
659C4—24no—
660C4—28no—
661C4—28no—
662C4—44no—
663C4—48no—
664C4—44no—
665C4—81no—
666C4—81no—
667C4—82no—
668C4—81no—
669C4—82no—
670C4—81no—
671C4—82no—
672C4—82no—
673C4—84no—
674C4#285518yes—
675C2—28no—
676C2C2×#244211yescharacteristic
677C2#3494812yes—
678C2#3515111yescharacteristic
679C2#3523512yes—
680C2#3615511yescharacteristic
681C2#3616211yescharacteristic
682C2#3617812yes—
683C2#3657811yescharacteristic
684C2#3658211yescharacteristic
685C2#3658812yes—
686C2C2×#285511yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
5. #14863C2semidirect
6. #14862C2semidirect
11. #14054C2semidirect
12. #14030C2semidirect
13. #14006C2semidirect
14. #13982C2semidirect
450. C22×C4#125semidirect

Maximal subgroups

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

subgroupindex
C2×#24422
C22×C4×C4⋊C42
#150152
#149202
#148632
#148622
#147052
#146852
#141302
#140542
#140302
#140062
#139822
#139072
#138812
C4×#3812