Finite Group Databasebeta

← #920684

Subgroups — #920684

Order 512.

2954 conjugacy classes of subgroups.

The subgroup lattice has 731 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2409C211yesmaximal, characteristic
2C2×#2398C211yesmaximal, characteristic
3#17211C214yesmaximal, complemented
4#16831C212yesmaximal, complemented
5#16823C212yesmaximal, complemented
6#16481C212yesmaximal, complemented
7#16470C212yesmaximal, complemented
8#15494C214yesmaximal
9#15201C212yesmaximal, complemented
10#15179C212yesmaximal, complemented
11#15132C212yesmaximal, complemented
12#14261C212yesmaximal, complemented
13#14139C212yesmaximal, complemented
14#13922C212yesmaximal, complemented
15#13872C211yesmaximal, characteristic
16C24×Q8C2211yescharacteristic
17C22×C22⋊Q8C2212yes—
18C22×C4×Q8C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21#2409—28no—
22#2398—28no—
23C2×#400C2212yes—
24C2×#397C2211yescharacteristic
25C2×#397C2211yescharacteristic
26C2×#397C2214yes—
27C2×#395C2212yes—
28C2×#394C2212yes—
29C2×#393C2211yescharacteristic
30C2×#393C2212yes—
31C2×#393C2211yescharacteristic
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#391C2212yes—
37C2×#389C2212yes—
38C2×#386C2212yes—
39C2×#386C2212yes—
40C2×#385C2212yes—
41C2×#380C2211yescharacteristic
42C2×#380C2212yes—
43C2×#380C2211yescharacteristic
44C2×C4×C22⋊C4C2212yes—
45C2×C4×C22⋊C4C2212yes—
46C2×#376C2211yescharacteristic
47#1446C2214yes—
48#1446C2214yes—
49#1446C2214yes—
50#1443C2214yes—
51#1443C2214yes—
52#1443C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1442C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1442C2214yes—
59#1442C2214yes—
60#1441C2212yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1440C2214yes—
65#1440C2214yes—
66#1439C2212yes—
67#1439C2214yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C23×Q8—28no—
78C23×Q8—24no—
79C23×Q8—42no—
80C23×Q8—42no—
81C23×Q8—44no—
82C23×Q8C2312yes—
83C23×Q8C2312yes—
84C23×Q8C2311yescharacteristic
85C23×D4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2312yes—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—44no—
93C2×C4×Q8—24no—
94C2×C4×Q8—42no—
95C2×C4×Q8—42no—
96C2×C4×D4—42no—
97C2×C4×D4—42no—
98C2×C4×D4D418yes—
99C22×C4⋊C4—44no—
100C22×C4⋊C4—42no—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2311yescharacteristic
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2311yescharacteristic
123#400—28no—
124#397—24no—
125#397—24no—
126#397—48no—
127#395—28no—
128#394—28no—
129#393—24no—
130#393—24no—
131#393—28no—
132#391—28no—
133#391—28no—
134#391—44no—
135#391—44no—
136#391—44no—
137#389—44no—
138#386—28no—
139#386—44no—
140#385—28no—
141#380—24no—
142#380—44no—
143#380—42no—
144C4×C22⋊C4—28no—
145C4×C22⋊C4—28no—
146#376—24no—
147C2×#95C2312yes—
148C2×#95C2311yescharacteristic
149C2×#95C2311yescharacteristic
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
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×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C25C2411yescharacteristic
177C25C2411yescharacteristic
178C22×Q8—24no—
179C22×Q8—22no—
180C22×Q8—24no—
181C22×Q8—24no—
182C22×Q8—28no—
183C22×Q8—24no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—41no—
189C22×Q8—41no—
190C22×Q8—44no—
191C22×Q8—42no—
192C22×Q8—42no—
193C22×Q8—42no—
194C22×Q8—432no—
195C22×Q8—416no—
196C22×Q8—416no—
197C22×Q8—432no—
198C22×Q8—416no—
199C22×Q8#4118yes—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—41no—
205C22×D4—41no—
206C22×D4C2×D414yes—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—28no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—44no—
221C23×C4—42no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—42no—
225C23×C4C2×D414yes—
226C23×C4C2×D414yes—
227C23×C4C2414yes—
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2414yes—
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2414yes—
238C23×C4C2412yes—
239C23×C4C2411yescharacteristic
240C22⋊Q8—432no—
241C4×Q8—416no—
242C4×D4—416no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—41no—
263C2×C4⋊C4—41no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—416no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4C2×D414yes—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
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—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—41no—
290C2×C22⋊C4—41no—
291C2×C22⋊C4—41no—
292C2×C22⋊C4—41no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4C2×D414yes—
299C2×C22⋊C4C2×D414yes—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42C2×D414yes—
309#95—22no—
310#95—22no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—44no—
323#95—44no—
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—
336#95—44no—
337#95—44no—
338C24—22no—
339C24—22no—
340C24—21no—
341C24—22no—
342C24—22no—
343C24—21no—
344C24—21no—
345C24—21no—
346C24—21no—
347C24—22no—
348C24—22no—
349C24—21no—
350C24—21no—
351C24—21no—
352C24—22no—
353C24—22no—
354C24C22×D412yes—
355C24C22×D412yes—
356C24H(5,𝔽2)12yes—
357C242−1+412yes—
358C242−1+412yes—
359C242−1+412yes—
360C24C2511yescharacteristic
361C2×Q8—216no—
362C2×Q8—216no—
363C2×Q8—216no—
364C2×Q8—48no—
365C2×Q8—44no—
366C2×Q8—416no—
367C2×Q8—48no—
368C2×Q8—44no—
369C2×Q8—48no—
370C2×Q8—48no—
371C2×Q8—416no—
372C2×Q8—44no—
373C2×Q8—4128no—
374C2×Q8—48no—
375C2×Q8—48no—
376C2×Q8—48no—
377C2×Q8—48no—
378C2×Q8—48no—
379C2×D4—28no—
380C2×D4—28no—
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—44no—
385C2×D4—44no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—216no—
419C22×C4—216no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—48no—
463C22×C4—416no—
464C22×C4—416no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—48no—
474C22×C4—84no—
475C22×C4—88no—
476C22×C4—84no—
477C22×C4—88no—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C2×#4118yes—
484C22×C4C2×#4114yes—
485C22×C4H(5,𝔽2)12yes—
486C22×C4H(5,𝔽2)12yes—
487C22×C4H(5,𝔽2)14yes—
488C22×C42−1+414yes—
489C4⋊C4—44no—
490C4⋊C4—416no—
491C4⋊C4—44no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—88no—
498C22⋊C4—44no—
499C22⋊C4—44no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—416no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—84no—
507C22⋊C4—88no—
508C42—28no—
509C42—48no—
510C42—48no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—22no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—42no—
536C23—42no—
537C23—41no—
538C23—41no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—41no—
544C23—41no—
545C23—41no—
546C23—41no—
547C23—41no—
548C23—41no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23#53614yes—
556C23#53714yes—
557C23C23×D411yescharacteristic
558C23C22×#4112yes—
559C23C2×H(5,𝔽2)11yescharacteristic
560C23C2×H(5,𝔽2)11yescharacteristic
561C23C2×H(5,𝔽2)11yescharacteristic
562C23C2×H(5,𝔽2)12yes—
563C23C2×2−1+411yescharacteristic
564C23C2×2−1+411yescharacteristic
565C23C2×2−1+411yescharacteristic
566C23C2×2−1+412yes—
567C23#58512yes—
568Q8—416no—
569Q8—432no—
570Q8—432no—
571D4—416no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—28no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—432no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—44no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—416no—
610C2×C4—416no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—816no—
654C2×C4#53414yes—
655C2×C4#53818yes—
656C2×C4#54618yes—
657C2×C4#55114yes—
658C2×C4H(3,𝔽4)14yes—
659C22—24no—
660C22—24no—
661C22—28no—
662C22—28no—
663C22—42no—
664C22—42no—
665C22—44no—
666C22—44no—
667C22—44no—
668C22—44no—
669C22—42no—
670C22—42no—
671C22—42no—
672C22—42no—
673C22—42no—
674C22—42no—
675C22—44no—
676C22—44no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—42no—
681C22C2×#53411yescharacteristic
682C22C2×#53611yescharacteristic
683C22C2×#53711yescharacteristic
684C22C2×#53812yes—
685C22C2×#54612yes—
686C22C2×#55111yescharacteristic
687C22#348912yes—
688C22#351612yes—
689C22#351612yes—
690C22#352212yes—
691C22#352312yes—
692C22C2×H(3,𝔽4)11yescharacteristic
693C22#353611yescharacteristic
694C22#353611yescharacteristic
695C22#353811yescharacteristic
696C22#353811yescharacteristic
697C22#353811yescharacteristic
698C22#354011yescharacteristic
699C22#354112yes—
700C22#355212yes—
701C22#355212yes—
702C22#355812yes—
703C22#356112yes—
704C4—42no—
705C4—42no—
706C4—42no—
707C4—48no—
708C4—42no—
709C4—44no—
710C4—82no—
711C4—84no—
712C4—82no—
713C4—84no—
714C4—82no—
715C4—84no—
716C4#268418yes—
717C2—44no—
718C2—44no—
719C2#3737411yescharacteristic
720C2#3744211yescharacteristic
721C2#3802512yes—
722C2#3802712yes—
723C2#4282912yes—
724C2#4284412yes—
725C2#4541911yescharacteristic
726C2#4542011yescharacteristic
727C2C2×#268411yescharacteristic
728C2#5299011yescharacteristic
729C2#5303211yescharacteristic
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. #17211C2semidirect
4. #16831C2semidirect
5. #16823C2semidirect
6. #16481C2semidirect
7. #16470C2semidirect
9. #15201C2semidirect
10. #15179C2semidirect
11. #15132C2semidirect
12. #14261C2semidirect
13. #14139C2semidirect
14. #13922C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24092
C2×#23982
#172112
#168312
#168232
#164812
#164702
#154942
#152012
#151792
#151322
#142612
#141392
#139222
#138722