Finite Group Databasebeta

← #800425

Subgroups — #800425

Order 512.

2508 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×#2512C211yesmaximal, characteristic
2C2×#2365C211yesmaximal, characteristic, complemented
3#16673C212yesmaximal, complemented
4#15904C214yesmaximal, complemented
5#15869C214yesmaximal, complemented
6#15243C211yesmaximal, characteristic
7#15180C211yesmaximal, characteristic, complemented
8#15147C211yesmaximal, characteristic
9#15100C212yesmaximal, complemented
10#14894C212yesmaximal, complemented
11#14166C212yesmaximal, complemented
12#14153C212yesmaximal
13#14026C212yesmaximal
14#14017C212yesmaximal, complemented
15#14000C212yesmaximal, complemented
16C4×#398C211yesmaximal, characteristic, complemented
17#13810C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×#126C2212yes—
20C22×#124C2212yes—
21C22×C4×Q8C2212yes—
22#2512—28no—
23C2×#398C2211yescharacteristic
24C2×#397C2212yes—
25C2×#397C2212yes—
26C2×#396C2212yes—
27#2365—28no—
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#390C2211yescharacteristic
33C2×#390C2211yescharacteristic
34C2×#390C2211yescharacteristic
35C2×#390C2212yes—
36C2×#389C2212yes—
37C2×#389C2211yescharacteristic
38C2×#388C2212yes—
39C2×#387C2214yes—
40C2×#384C2211yescharacteristic
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×#376C2212yes—
48#1445C2211yescharacteristic
49#1444C2214yes—
50#1444C2211yescharacteristic
51#1442C2212yes—
52#1442C2212yes—
53#1442C2214yes—
54#1442C2214yes—
55#1442C2212yes—
56#1442C2214yes—
57#1441C2212yes—
58#1441C2211yescharacteristic
59#1441C2212yes—
60#1441C2212yes—
61#1440C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2212yes—
67#1439C2212yes—
68#1439C2212yes—
69#1439C2214yes—
70#1437C2214yes—
71#1437C2214yes—
72#1437C2214yes—
73#1437C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81#1434C2214yes—
82#1434C2212yes—
83C4×#95C2211yescharacteristic
84C4×#95C2212yes—
85C23×Q8C2312yes—
86C23×D4C2311yescharacteristic
87C2×C4⋊Q8—28no—
88C2×C4⋊Q8—24no—
89C2×#126—28no—
90C2×#126—28no—
91C2×#126—44no—
92C2×#124—28no—
93C2×#124—28no—
94C2×#124—44no—
95C2×C4×Q8—28no—
96C2×C4×Q8—44no—
97C2×C4×Q8—44no—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2314yes—
131C22×C42C2314yes—
132#398—24no—
133#397—28no—
134#397—44no—
135#396—28no—
136#391—28no—
137#391—28no—
138#391—28no—
139#391—44no—
140#390—24no—
141#390—24no—
142#390—24no—
143#390—44no—
144#389—24no—
145#389—28no—
146#388—28no—
147#387—216no—
148#384—24no—
149C4×C4⋊C4—24no—
150C4×C4⋊C4—28no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—28no—
153C4×C4⋊C4—44no—
154C4×C4⋊C4D418yes—
155#376—44no—
156C2×#95C2311yescharacteristic
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2311yescharacteristic
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2311yescharacteristic
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2311yescharacteristic
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C25C2412yes—
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8#4118yes—
193C22×D4—22no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—22no—
197C22×D4—22no—
198C23×C4C2412yes—
199C23×C4C2412yes—
200C23×C4C2411yescharacteristic
201C23×C4C2414yes—
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C23×C4C2412yes—
205C23×C4C2411yescharacteristic
206C23×C4C2414yes—
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C4⋊Q8—232no—
215#126—432no—
216#124—432no—
217C4×Q8—432no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
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—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4C2×D414yes—
259C2×C4⋊C4#4114yes—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—28no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—28no—
294C2×C42—24no—
295C2×C42—22no—
296C2×C42—22no—
297C2×C42—28no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—44no—
304C2×C42—44no—
305C2×C42—44no—
306C2×C42—44no—
307C2×C42C2×D418yes—
308C2×C42C2×D414yes—
309C2×C42#4118yes—
310#95—24no—
311#95—28no—
312#95—22no—
313#95—22no—
314#95—28no—
315#95—24no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—22no—
322#95—24no—
323#95—22no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—44no—
328#95—42no—
329#95—42no—
330#95—42no—
331#95—44no—
332#95—44no—
333#95—42no—
334#95—42no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95C2×D418yes—
339C24—22no—
340C24—22no—
341C24—24no—
342C24—24no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—24no—
347C24—22no—
348C24—24no—
349C24—22no—
350C24C2511yescharacteristic
351C2×Q8—216no—
352C2×Q8—216no—
353C2×Q8—216no—
354C2×Q8—48no—
355C2×Q8—48no—
356C2×Q8—48no—
357C2×Q8—48no—
358C2×D4—44no—
359C2×D4—44no—
360C2×D4—44no—
361C2×D4—48no—
362C2×D4—48no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4C22×D414yes—
442C22×C4C22×D412yes—
443C22×C4C22×D414yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C2×#4114yes—
447C22×C4C2×#4114yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4118yes—
450C22×C4C2×#4112yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4114yes—
453C22×C4C2×#4114yes—
454C22×C4H(5,𝔽2)12yes—
455C22×C4H(5,𝔽2)12yes—
456C22×C4H(5,𝔽2)12yes—
457C22×C42−1+412yes—
458C22×C42−1+414yes—
459C4⋊C4—28no—
460C4⋊C4—216no—
461C4⋊C4—216no—
462C4⋊C4—28no—
463C4⋊C4—28no—
464C4⋊C4—28no—
465C4⋊C4—44no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—416no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—48no—
474C4⋊C4—84no—
475C22⋊C4—48no—
476C22⋊C4—416no—
477C22⋊C4—416no—
478C22⋊C4—48no—
479C22⋊C4—48no—
480C22⋊C4—48no—
481C22⋊C4—84no—
482C22⋊C4—84no—
483C22⋊C4—88no—
484C22⋊C4—88no—
485C42—28no—
486C42—216no—
487C42—216no—
488C42—416no—
489C42—416no—
490C42—44no—
491C42—44no—
492C42—48no—
493C23—42no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—44no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—44no—
506C23—44no—
507C23—44no—
508C23—42no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23—44no—
513C23—42no—
514C23—42no—
515C23—44no—
516C23C23×D411yescharacteristic
517C23C22×#4112yes—
518C23C22×#4112yes—
519C23C22×#4111yescharacteristic
520C23C2×H(5,𝔽2)11yescharacteristic
521C23C2×H(5,𝔽2)11yescharacteristic
522C23C2×H(5,𝔽2)11yescharacteristic
523C23C2×2−1+412yes—
524C23C2×2−1+411yescharacteristic
525C23#58512yes—
526C23#58511yescharacteristic
527Q8—432no—
528D4—88no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—28no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—28no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—44no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—41no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—41no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—41no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4#52514yes—
620C2×C4#53214yes—
621C2×C4#53514yes—
622C2×C4#54614yes—
623C2×C4#55118yes—
624C2×C4#55318yes—
625C22—84no—
626C22—82no—
627C22—82no—
628C22—82no—
629C22—82no—
630C22—82no—
631C22—82no—
632C22—82no—
633C22—84no—
634C22—84no—
635C22—84no—
636C22C2×#52511yescharacteristic
637C22C2×#53211yescharacteristic
638C22C2×#53511yescharacteristic
639C22C2×#54611yescharacteristic
640C22C2×#55112yes—
641C22C2×#55312yes—
642C22#348012yes—
643C22#348212yes—
644C22#348611yescharacteristic
645C22#348911yescharacteristic
646C22#350011yescharacteristic
647C22#350611yescharacteristic
648C22#350612yes—
649C22#350712yes—
650C22#350912yes—
651C22#351111yescharacteristic
652C22#351612yes—
653C22#352212yes—
654C22#352412yes—
655C22#354112yes—
656C22#354611yescharacteristic
657C22#354811yescharacteristic
658C22#356112yes—
659C4—24no—
660C4—44no—
661C4—48no—
662C4—44no—
663C4—42no—
664C4—44no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—44no—
669C4—44no—
670C4—81no—
671C4—82no—
672C4—82no—
673C4—84no—
674C4#260318yes—
675C2—162no—
676C2C2×#260311yescharacteristic
677C2#3593811yescharacteristic
678C2#3598511yescharacteristic
679C2#3641911yescharacteristic
680C2#3642211yescharacteristic
681C2#3947111yescharacteristic
682C2#4278811yescharacteristic
683C2#4496212yes—
684C2#4496912yes—
685C2#5968212yes—
686C2#5990212yes—
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
2. C2×#2365C2semidirect
3. #16673C2semidirect
4. #15904C2semidirect
5. #15869C2semidirect
7. #15180C2semidirect
9. #15100C2semidirect
10. #14894C2semidirect
11. #14166C2semidirect
14. #14017C2semidirect
15. #14000C2semidirect
16. C4×#398C2semidirect
17. #13810C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25122
C2×#23652
#166732
#159042
#158692
#152432
#151802
#151472
#151002
#148942
#141662
#141532
#140262
#140172
#140002
C4×#3982
#138102