Finite Group Databasebeta

← #712181

Subgroups — #712181

Order 512.

2550 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2499C211yesmaximal, characteristic
2C2×#2625C211yesmaximal, characteristic
3C2×#2347C211yesmaximal, characteristic
4C2×#2316C211yesmaximal, characteristic
5#16678C212yesmaximal, complemented
6#16677C212yesmaximal, complemented
7#15486C214yesmaximal
8#15290C214yesmaximal
9#15225C212yesmaximal, complemented
10#15189C212yesmaximal, complemented
11#14869C212yesmaximal, complemented
12#14818C212yesmaximal, complemented
13#14142C212yesmaximal, complemented
14#14139C211yesmaximal, characteristic
15#14130C211yesmaximal, characteristic
16#14005C212yesmaximal, complemented
17#13853C211yesmaximal, characteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C22×#117C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22#2625—28no—
23#2499—28no—
24C2×#400C2212yes—
25C2×#398C2212yes—
26C2×#397C2214yes—
27C2×#397C2212yes—
28C2×#397C2212yes—
29#2347—28no—
30#2316—28no—
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#391C2211yescharacteristic
34C2×#391C2211yescharacteristic
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#389C2212yes—
42C2×#389C2214yes—
43C2×#388C2214yes—
44C2×#387C2212yes—
45C2×#387C2212yes—
46C2×#385C2212yes—
47C2×#384C2211yescharacteristic
48C2×#383C2212yes—
49C2×#382C2211yescharacteristic
50C2×#380C2214yes—
51C2×#380C2211yescharacteristic
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C22⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2211yescharacteristic
56#1445C2211yescharacteristic
57#1445C2211yescharacteristic
58#1445C2211yescharacteristic
59#1445C2211yescharacteristic
60#1445C2214yes—
61#1444C2214yes—
62#1444C2212yes—
63#1443C2214yes—
64#1443C2214yes—
65#1442C2212yes—
66#1442C2214yes—
67#1442C2214yes—
68#1441C2212yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2212yes—
72#1441C2214yes—
73#1441C2214yes—
74#1441C2214yes—
75#1441C2214yes—
76#1440C2212yes—
77#1440C2212yes—
78#1436C2214yes—
79#1436C2214yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82#1434C2214yes—
83C22⋊#95C2212yes—
84C22⋊#95C2214yes—
85C4×#95C2212yes—
86C4×#95C2212yes—
87C23×Q8C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2312yes—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—24no—
93C2×#117—24no—
94C2×#117—24no—
95C2×#117—24no—
96C2×#117—24no—
97C2×#117—24no—
98C2×#117D418yes—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—28no—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C42C2312yes—
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129C22×C42C2314yes—
130#400—28no—
131#398—28no—
132#397—216no—
133#397—28no—
134#397—28no—
135#391—28no—
136#391—24no—
137#391—24no—
138#391—28no—
139#389—28no—
140#389—28no—
141#389—28no—
142#389—28no—
143#389—28no—
144#389—28no—
145#389—28no—
146#389—48no—
147#388—216no—
148#387—28no—
149#387—28no—
150#385—44no—
151#384—24no—
152#383—28no—
153#382—24no—
154#380—216no—
155#380D418yes—
156C4×C4⋊C4—28no—
157C4×C4⋊C4—28no—
158C4×C22⋊C4—24no—
159C4×C22⋊C4—24no—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2311yescharacteristic
179C2×#95C2311yescharacteristic
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2311yescharacteristic
183C2×#95C2311yescharacteristic
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2312yes—
193C25C2411yescharacteristic
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—24no—
198C22×Q8—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—28no—
203C23×C4—42no—
204C23×C4—44no—
205C23×C4—42no—
206C23×C4C2×D414yes—
207C23×C4C2×D414yes—
208C23×C4#4114yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2414yes—
219C23×C4C2414yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C22⋊Q8—416no—
226#117—232no—
227#117—416no—
228C2×C4⋊C4—232no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4C2×D414yes—
273C2×C4⋊C4C2×D414yes—
274C2×C4⋊C4C2×D418yes—
275C2×C4⋊C4#4118yes—
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—24no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—41no—
292C2×C22⋊C4—41no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4C2×D418yes—
296C2×C22⋊C4C2×D418yes—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—28no—
310C2×C42—44no—
311C2×C42—44no—
312C2×C42C2×D418yes—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—24no—
321#95—24no—
322#95—28no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—42no—
338#95—44no—
339#95—42no—
340#95—44no—
341#95—42no—
342#95—42no—
343#95—44no—
344#95—44no—
345#95C2×D418yes—
346C24—22no—
347C24—21no—
348C24—22no—
349C24—21no—
350C24—22no—
351C24—21no—
352C24—22no—
353C24—21no—
354C24C22×D412yes—
355C24C22×D412yes—
356C24C2×#4112yes—
357C24C2511yescharacteristic
358C2×Q8—44no—
359C2×Q8—44no—
360C2×Q8—44no—
361C2×Q8—48no—
362C2×Q8—48no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—28no—
366C22×C4—28no—
367C22×C4—216no—
368C22×C4—216no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—216no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—21no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—88no—
450C22×C4—88no—
451C22×C4—84no—
452C22×C4—84no—
453C22×C4#121116yes—
454C22×C4C22×D414yes—
455C22×C4C22×D418yes—
456C22×C4C22×D414yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C2×#4118yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4112yes—
464C22×C4C2×#4114yes—
465C22×C4H(5,𝔽2)12yes—
466C22×C42−1+414yes—
467C22×C42−1+412yes—
468C22×C42−1+414yes—
469C4⋊C4—216no—
470C4⋊C4—28no—
471C4⋊C4—28no—
472C4⋊C4—216no—
473C4⋊C4—216no—
474C4⋊C4—48no—
475C4⋊C4—416no—
476C4⋊C4—48no—
477C4⋊C4—416no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—48no—
482C4⋊C4—88no—
483C4⋊C4—88no—
484C4⋊C4—88no—
485C22⋊C4—216no—
486C22⋊C4—48no—
487C22⋊C4—44no—
488C22⋊C4—48no—
489C22⋊C4—44no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C42—216no—
493C42—48no—
494C42—44no—
495C42—44no—
496C42—48no—
497C42—88no—
498C23—22no—
499C23—22no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—22no—
504C23—24no—
505C23—22no—
506C23—22no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23C2×#12114yes—
521C23C23×D411yescharacteristic
522C23C23×D411yescharacteristic
523C23C22×#4111yescharacteristic
524C23C22×#4112yes—
525C23C2×H(5,𝔽2)11yescharacteristic
526C23C2×2−1+412yes—
527C23C2×2−1+412yes—
528C23C2×2−1+411yescharacteristic
529C23#58512yes—
530C23#58511yescharacteristic
531C23#58511yescharacteristic
532Q8—88no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—232no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—28no—
541C2×C4—28no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—28no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—28no—
553C2×C4—24no—
554C2×C4—28no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—44no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—816no—
625C2×C4C2×#12114yes—
626C2×C4C2×#12114yes—
627C2×C4#52514yes—
628C2×C4#53614yes—
629C2×C4#54718yes—
630C2×C4#55218yes—
631C22—24no—
632C22—24no—
633C22—24no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—44no—
641C22—42no—
642C22C22×#12111yescharacteristic
643C22C2×#52511yescharacteristic
644C22C2×#53611yescharacteristic
645C22#346112yes—
646C22C2×#54712yes—
647C22C2×#55212yes—
648C22#348811yescharacteristic
649C22#348912yes—
650C22#348911yescharacteristic
651C22#349611yescharacteristic
652C22#349711yescharacteristic
653C22#349911yescharacteristic
654C22#350011yescharacteristic
655C22#351111yescharacteristic
656C22#351211yescharacteristic
657C22#351312yes—
658C22#351612yes—
659C22#352112yes—
660C22#352312yes—
661C22#355212yes—
662C22#355512yes—
663C22#355512yes—
664C22#356512yes—
665C4—24no—
666C4—48no—
667C4—44no—
668C4—42no—
669C4—48no—
670C4—42no—
671C4—44no—
672C4—44no—
673C4—82no—
674C4—81no—
675C4—81no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4#241318yes—
681C2—44no—
682C2C2×#241311yescharacteristic
683C2#3530811yescharacteristic
684C2#3556411yescharacteristic
685C2#3598211yescharacteristic
686C2#3599711yescharacteristic
687C2#3826712yes—
688C2#3907812yes—
689C2#4291511yescharacteristic
690C2#4291811yescharacteristic
691C2#4510712yes—
692C2#4532612yes—
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. #16678C2semidirect
6. #16677C2semidirect
9. #15225C2semidirect
10. #15189C2semidirect
11. #14869C2semidirect
12. #14818C2semidirect
13. #14142C2semidirect
16. #14005C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24992
C2×#26252
C2×#23472
C2×#23162
#166782
#166772
#154862
#152902
#152252
#151892
#148692
#148182
#141422
#141392
#141302
#140052
#138532