Finite Group Databasebeta

← #1410004

Subgroups — #1410004

Order 512.

2612 conjugacy classes of subgroups.

The subgroup lattice has 683 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic, complemented
2C2×#2550C212yesmaximal, complemented
3C2×#2489C212yesmaximal
4C2×#2492C211yesmaximal, characteristic
5#17062C212yesmaximal
6#16964C214yesmaximal, complemented
7#16813C212yesmaximal, complemented
8#16812C214yesmaximal
9#15240C211yesmaximal, characteristic, complemented
10#15240C212yesmaximal, complemented
11#15223C212yesmaximal
12#15193C211yesmaximal, characteristic, complemented
13#15193C212yesmaximal, complemented
14#15135C211yesmaximal, characteristic
15#14322C212yesmaximal
16C4×#399C211yesmaximal, characteristic
17#13812C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×#125C2212yes—
20C22×C22⋊Q8C2211yescharacteristic
21#2552—28no—
22#2550—216no—
23#2492—28no—
24#2489—216no—
25C2×#400C2212yes—
26C2×#400C2214yes—
27C2×#399C2212yes—
28C2×#399C2211yescharacteristic
29C2×#398C2214yes—
30C2×#398C2214yes—
31C2×#397C2214yes—
32C2×#395C2214yes—
33C2×#393C2212yes—
34C2×#391C2214yes—
35C2×#391C2211yescharacteristic
36C2×#391C2212yes—
37C2×#391C2211yescharacteristic
38C2×#391C2212yes—
39C2×#391C2212yes—
40C2×#389C2212yes—
41C2×#389C2214yes—
42C2×#389C2212yes—
43C2×#389C2212yes—
44C2×#389C2212yes—
45C2×#388C2212yes—
46C2×#388C2214yes—
47C2×#384C2211yescharacteristic
48C2×#384C2212yes—
49C2×#382C2214yes—
50C2×#382C2212yes—
51C2×#380C2212yes—
52C2×#380C2211yescharacteristic
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C4⋊C4C2212yes—
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C22⋊C4C2211yescharacteristic
58C2×C4×C22⋊C4C2212yes—
59#1445C2212yes—
60#1445C2212yes—
61#1445C2211yescharacteristic
62#1445C2214yes—
63#1445C2214yes—
64#1445C2214yes—
65#1445C2214yes—
66#1445C2214yes—
67#1445C2212yes—
68#1445C2214yes—
69#1445C2214yes—
70#1445C2212yes—
71#1445C2212yes—
72#1444C2212yes—
73#1444C2211yescharacteristic
74#1444C2214yes—
75#1444C2212yes—
76#1439C2212yes—
77#1439C2212yes—
78#1439C2211yescharacteristic
79#1439C2214yes—
80#1439C2214yes—
81C4⋊#95C2211yescharacteristic
82C4⋊#95C2212yes—
83#1434C2214yes—
84#1434C2212yes—
85C4×#95C2211yescharacteristic
86C4×#95C2212yes—
87C23×Q8C2311yescharacteristic
88C23×Q8C2312yes—
89C24×C4C2311yescharacteristic
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—28no—
92C2×#125—28no—
93C2×#125—28no—
94C2×#125—28no—
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8—24no—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126C22×C42C2314yes—
127C22×C42C2312yes—
128#400—216no—
129#400—28no—
130#399—28no—
131#399—24no—
132#398—216no—
133#398—216no—
134#397—216no—
135#395—216no—
136#393—28no—
137#391—28no—
138#391—24no—
139#391—28no—
140#391—28no—
141#391—24no—
142#391—216no—
143#389—28no—
144#389—28no—
145#389—28no—
146#389—216no—
147#389—28no—
148#388—216no—
149#388—28no—
150#384—24no—
151#384—28no—
152#382—216no—
153#382—28no—
154#380—28no—
155#380—24no—
156C4×C4⋊C4—28no—
157C4×C4⋊C4—24no—
158C4×C4⋊C4D4116yes—
159C4×C4⋊C4D418yes—
160C4×C22⋊C4—28no—
161C4×C22⋊C4—24no—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2311yescharacteristic
172C2×#95C2312yes—
173C2×#95C2311yescharacteristic
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2311yescharacteristic
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2311yescharacteristic
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2314yes—
189C2×#95C2312yes—
190C2×#95C2314yes—
191C2×#95C2312yes—
192C2×#95C2314yes—
193C25C2411yescharacteristic
194C22×Q8—24no—
195C22×Q8—22no—
196C22×Q8—24no—
197C22×Q8—28no—
198C22×Q8—24no—
199C22×Q8—22no—
200C22×Q8—24no—
201C22×Q8—22no—
202C22×Q8—42no—
203C22×Q8—44no—
204C23×C4—24no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—24no—
208C23×C4—22no—
209C23×C4C2414yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C4⋊Q8—416no—
226#125—432no—
227C22⋊Q8—416no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4C2×D418yes—
273C2×C4⋊C4C2×D414yes—
274C2×C4⋊C4C2×D418yes—
275C2×C4⋊C4C2×D414yes—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—41no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4#4114yes—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—22no—
302C2×C42—24no—
303C2×C42—44no—
304C2×C42—44no—
305C2×C42—44no—
306C2×C42C2×D418yes—
307C2×C42C2×D414yes—
308C2×C42C2×D4116yes—
309C2×C42C2×D418yes—
310#95—24no—
311#95—22no—
312#95—22no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—28no—
318#95—24no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—24no—
324#95—24no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—24no—
330#95—24no—
331#95—28no—
332#95—22no—
333#95—24no—
334#95—22no—
335#95—42no—
336#95—44no—
337#95—42no—
338#95—44no—
339#95C2×D418yes—
340#95C2×D4116yes—
341C24—21no—
342C24—22no—
343C24—21no—
344C24—21no—
345C24—21no—
346C24—21no—
347C24—22no—
348C24—22no—
349C24—21no—
350C24—22no—
351C24C2×#4112yes—
352C24C2511yescharacteristic
353C2×Q8—48no—
354C2×Q8—416no—
355C2×Q8—44no—
356C2×Q8—48no—
357C2×Q8—48no—
358C2×Q8—44no—
359C2×Q8—44no—
360C2×Q8—48no—
361C2×Q8—84no—
362C2×Q8—88no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—44no—
439C22×C4—48no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—48no—
443C22×C4C22×D414yes—
444C22×C4C22×D414yes—
445C22×C4C22×D412yes—
446C22×C4C22×D418yes—
447C22×C4C22×D414yes—
448C22×C4C22×D414yes—
449C22×C4C22×D418yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D414yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C22×C42−1+412yes—
456C22×C42−1+412yes—
457C22×C42−1+412yes—
458C22×C42−1+414yes—
459C22×C42−1+412yes—
460C4⋊C4—28no—
461C4⋊C4—216no—
462C4⋊C4—48no—
463C4⋊C4—416no—
464C4⋊C4—48no—
465C4⋊C4—44no—
466C4⋊C4—48no—
467C4⋊C4—416no—
468C4⋊C4—416no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—48no—
474C4⋊C4—88no—
475C4⋊C4—88no—
476C4⋊C4—84no—
477C4⋊C4#127116yes—
478C22⋊C4—28no—
479C22⋊C4—44no—
480C22⋊C4—48no—
481C22⋊C4—416no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C22⋊C4—84no—
485C42—28no—
486C42—216no—
487C42—48no—
488C42—416no—
489C42—48no—
490C42—44no—
491C23—22no—
492C23—24no—
493C23—22no—
494C23—22no—
495C23—24no—
496C23—41no—
497C23—41no—
498C23—42no—
499C23—41no—
500C23—42no—
501C23—42no—
502C23—41no—
503C23—41no—
504C23—42no—
505C23—42no—
506C23—41no—
507C23—41no—
508C23—42no—
509C23—41no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23C23×D411yescharacteristic
515C23C23×D412yes—
516C23C22×#4111yescharacteristic
517C23C2×2−1+411yescharacteristic
518C23C2×2−1+411yescharacteristic
519C23C2×2−1+412yes—
520C23C2×2−1+411yescharacteristic
521C23C2×2−1+411yescharacteristic
522C23#58512yes—
523C23#58511yescharacteristic
524C23#58512yes—
525Q8—816no—
526Q8—88no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—28no—
540C2×C4—22no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—44no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—44no—
573C2×C4—42no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—88no—
613C2×C4C2×#12718yes—
614C2×C4C2×#12714yes—
615C2×C4#53618yes—
616C2×C4#53614yes—
617C2×C4#53614yes—
618C2×C4#53618yes—
619C2×C4#54814yes—
620C22—42no—
621C22—42no—
622C22—44no—
623C22—42no—
624C22—44no—
625C22—82no—
626C22—81no—
627C22—82no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22C22×#12711yescharacteristic
632C22C2×#53611yescharacteristic
633C22C2×#53612yes—
634C22C2×#53612yes—
635C22C2×#53611yescharacteristic
636C22C2×#54811yescharacteristic
637C22D4×#4112yes—
638C22#348512yes—
639C22#348511yescharacteristic
640C22#348911yescharacteristic
641C22#348912yes—
642C22#349712yes—
643C22#350011yescharacteristic
644C22#350711yescharacteristic
645C22#350911yescharacteristic
646C22#351512yes—
647C22#355511yescharacteristic
648C22#355512yes—
649C22#355611yescharacteristic
650C22#355712yes—
651C22#356312yes—
652C22#356512yes—
653C22#356512yes—
654C4—24no—
655C4—28no—
656C4—42no—
657C4—44no—
658C4—44no—
659C4—42no—
660C4—48no—
661C4—44no—
662C4—42no—
663C4—81no—
664C4—82no—
665C4—82no—
666C4—84no—
667C4—82no—
668C4—82no—
669C4#265818yes—
670C2—82no—
671C2C2×#265811yescharacteristic
672C2#3606011yescharacteristic
673C2#3646511yescharacteristic
674C2#3912211yescharacteristic
675C2#3972411yescharacteristic
676C2#3988012yes—
677C2#3989012yes—
678C2#4264412yes—
679C2#4264711yescharacteristic
680C2#4280112yes—
681C2#4291511yescharacteristic
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
1. C2×#2552C2semidirect
2. C2×#2550C2semidirect
6. #16964C2semidirect
7. #16813C2semidirect
9. #15240C2semidirect
10. #15240C2semidirect
12. #15193C2semidirect
13. #15193C2semidirect
17. #13812C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#25502
C2×#24892
C2×#24922
#170622
#169642
#168132
#168122
#152402
#152232
#151932
#151352
#143222
C4×#3992
#138122