Finite Group Databasebeta

← #920871

Subgroups — #920871

Order 512.

2554 conjugacy classes of subgroups.

The subgroup lattice has 716 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2486C211yesmaximal, characteristic
2C2×#2398C211yesmaximal, characteristic
3#17220C214yesmaximal, complemented
4#17059C212yesmaximal, complemented
5#16831C212yesmaximal, complemented
6#16675C212yesmaximal, complemented
7#16666C212yesmaximal, complemented
8#15269C214yesmaximal
9#15202C212yesmaximal, complemented
10#15183C212yesmaximal, complemented
11#15132C212yesmaximal, complemented
12#14266C212yesmaximal, complemented
13#14130C212yesmaximal, complemented
14#13907C212yesmaximal, complemented
15#13872C211yesmaximal, characteristic
16C22×#123C2212yes—
17C22×C4×Q8C2211yescharacteristic
18C22×C4×D4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21#2486—28no—
22#2398—28no—
23C2×#400C2212yes—
24C2×#398C2212yes—
25C2×#397C2211yescharacteristic
26C2×#397C2211yescharacteristic
27C2×#397C2212yes—
28C2×#397C2214yes—
29C2×#394C2212yes—
30C2×#393C2211yescharacteristic
31C2×#393C2211yescharacteristic
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#386C2212yes—
37C2×#385C2214yes—
38C2×#381C2212yes—
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×#380C2211yescharacteristic
42C2×#380C2211yescharacteristic
43C2×#380C2214yes—
44C2×C4×C22⋊C4C2212yes—
45C2×C4×C22⋊C4C2212yes—
46C2×#376C2211yescharacteristic
47#1446C2214yes—
48#1445C2214yes—
49#1445C2214yes—
50#1444C2214yes—
51#1444C2214yes—
52#1444C2214yes—
53#1444C2214yes—
54#1443C2214yes—
55#1443C2214yes—
56#1443C2214yes—
57#1442C2214yes—
58#1442C2214yes—
59#1442C2214yes—
60#1441C2214yes—
61#1441C2214yes—
62#1441C2212yes—
63#1441C2214yes—
64#1440C2214yes—
65#1440C2214yes—
66#1439C2212yes—
67#1439C2214yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C22⋊#95C2214yes—
75C22⋊#95C2212yes—
76C22⋊#95C2214yes—
77C23×Q8C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2312yes—
82C24×C4C2312yes—
83C2×#123—28no—
84C2×#123—44no—
85C2×#123—44no—
86C2×C4×Q8—24no—
87C2×C4×Q8—42no—
88C2×C4×Q8—42no—
89C2×C4×D4—42no—
90C2×C4×D4—42no—
91C2×C4×D4D418yes—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—28no—
94C22×C4⋊C4—42no—
95C22×C4⋊C4—42no—
96C22×C4⋊C4—42no—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123#400—28no—
124#398—28no—
125#397—24no—
126#397—24no—
127#397—28no—
128#397—48no—
129#394—28no—
130#393—24no—
131#393—24no—
132#389—28no—
133#389—28no—
134#389—44no—
135#389—44no—
136#386—44no—
137#385—48no—
138#381—28no—
139#380—28no—
140#380—24no—
141#380—28no—
142#380—48no—
143#380—42no—
144C4×C22⋊C4—28no—
145C4×C22⋊C4—28no—
146#376—24no—
147C2×#95C2311yescharacteristic
148C2×#95C2311yescharacteristic
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C25C2411yescharacteristic
175C25C2411yescharacteristic
176C22×Q8—22no—
177C22×Q8—24no—
178C22×Q8—41no—
179C22×Q8—41no—
180C22×Q8—42no—
181C22×D4—22no—
182C22×D4—22no—
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—41no—
186C22×D4—41no—
187C22×D4C2×D414yes—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—22no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—28no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—44no—
207C23×C4—42no—
208C23×C4C2×D414yes—
209C23×C4C2×D414yes—
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2414yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223#123—432no—
224C4×Q8—416no—
225C4×D4—416no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—24no—
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—28no—
243C2×C4⋊C4—416no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—41no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—41no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—416no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4C2×D414yes—
265C2×C4⋊C4#4118yes—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—41no—
287C2×C22⋊C4—41no—
288C2×C22⋊C4—41no—
289C2×C22⋊C4—41no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4C2×D414yes—
295C2×C22⋊C4C2×D414yes—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42C2×D414yes—
305#95—22no—
306#95—22no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—28no—
313#95—24no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—44no—
319#95—44no—
320#95—44no—
321#95—42no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332C24—22no—
333C24—22no—
334C24—21no—
335C24—21no—
336C24—21no—
337C24—21no—
338C24—21no—
339C24—21no—
340C24—21no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—21no—
346C24—22no—
347C24—22no—
348C24C22×D412yes—
349C24C22×D412yes—
350C24H(5,𝔽2)12yes—
351C24H(5,𝔽2)12yes—
352C24H(5,𝔽2)12yes—
353C242−1+412yes—
354C24C2511yescharacteristic
355C2×Q8—48no—
356C2×Q8—44no—
357C2×Q8—44no—
358C2×Q8—48no—
359C2×Q8—44no—
360C2×D4—28no—
361C2×D4—28no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—44no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—21no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—216no—
400C22×C4—216no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—48no—
444C22×C4—416no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—44no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—88no—
455C22×C4—84no—
456C22×C4—84no—
457C22×C4—84no—
458C22×C4—84no—
459C22×C4—84no—
460C22×C4—84no—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C2×#4118yes—
467C22×C4C2×#4114yes—
468C22×C4H(5,𝔽2)12yes—
469C22×C4H(5,𝔽2)12yes—
470C22×C42−1+414yes—
471C22×C42−1+414yes—
472C4⋊C4—216no—
473C4⋊C4—44no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—44no—
477C4⋊C4—48no—
478C4⋊C4—416no—
479C4⋊C4—48no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—88no—
483C4⋊C4—88no—
484C4⋊C4—88no—
485C22⋊C4—44no—
486C22⋊C4—44no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—416no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C22⋊C4—416no—
493C22⋊C4—48no—
494C22⋊C4—84no—
495C42—28no—
496C42—48no—
497C42—48no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—42no—
523C23—42no—
524C23—41no—
525C23—41no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—41no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—41no—
535C23—41no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23#53514yes—
543C23#53714yes—
544C23C23×D411yescharacteristic
545C23C22×#4112yes—
546C23C2×H(5,𝔽2)11yescharacteristic
547C23C2×H(5,𝔽2)11yescharacteristic
548C23C2×H(5,𝔽2)11yescharacteristic
549C23C2×H(5,𝔽2)11yescharacteristic
550C23C2×H(5,𝔽2)11yescharacteristic
551C23C2×2−1+411yescharacteristic
552C23C2×2−1+412yes—
553C23C2×2−1+412yes—
554C23#58512yes—
555Q8—416no—
556D4—416no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—28no—
569C2×C4—28no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—44no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—416no—
594C2×C4—416no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
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—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—816no—
638C2×C4—816no—
639C2×C4#53414yes—
640C2×C4#54218yes—
641C2×C4#54718yes—
642C2×C4#55114yes—
643C2×C4#56314yes—
644C22—24no—
645C22—24no—
646C22—28no—
647C22—28no—
648C22—42no—
649C22—42no—
650C22—44no—
651C22—44no—
652C22—42no—
653C22—42no—
654C22—44no—
655C22—44no—
656C22—42no—
657C22—42no—
658C22—44no—
659C22—44no—
660C22—42no—
661C22—42no—
662C22—42no—
663C22—42no—
664C22—42no—
665C22—42no—
666C22C2×#53411yescharacteristic
667C22C2×#53511yescharacteristic
668C22C2×#53711yescharacteristic
669C22C2×#54212yes—
670C22C2×#54712yes—
671C22C2×#55111yescharacteristic
672C22#348912yes—
673C22#351312yes—
674C22#351312yes—
675C22#352212yes—
676C22#352312yes—
677C22C2×#56311yescharacteristic
678C22#353611yescharacteristic
679C22#353611yescharacteristic
680C22#353711yescharacteristic
681C22#353711yescharacteristic
682C22#353811yescharacteristic
683C22#354011yescharacteristic
684C22#354112yes—
685C22#354412yes—
686C22#354412yes—
687C22#355912yes—
688C22#356312yes—
689C4—42no—
690C4—48no—
691C4—42no—
692C4—42no—
693C4—42no—
694C4—44no—
695C4—84no—
696C4—82no—
697C4—84no—
698C4—84no—
699C4—82no—
700C4—82no—
701C4#271718yes—
702C2—44no—
703C2—44no—
704C2#3737411yescharacteristic
705C2#3743911yescharacteristic
706C2#3804112yes—
707C2#3804312yes—
708C2#4253712yes—
709C2#4256812yes—
710C2#4538011yescharacteristic
711C2#4541911yescharacteristic
712C2C2×#271711yescharacteristic
713C2#5299111yescharacteristic
714C2#5303111yescharacteristic
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. #17220C2semidirect
4. #17059C2semidirect
5. #16831C2semidirect
6. #16675C2semidirect
7. #16666C2semidirect
9. #15202C2semidirect
10. #15183C2semidirect
11. #15132C2semidirect
12. #14266C2semidirect
13. #14130C2semidirect
14. #13907C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24862
C2×#23982
#172202
#170592
#168312
#166752
#166662
#152692
#152022
#151832
#151322
#142662
#141302
#139072
#138722