Finite Group Databasebeta

← #644909

Subgroups — #644909

Order 512.

2488 conjugacy classes of subgroups.

The subgroup lattice has 715 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#123C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic, complemented
3#17065C211yesmaximal, characteristic, complemented
4#17064C211yesmaximal, characteristic, complemented
5#17063C211yesmaximal, characteristic, complemented
6#17062C211yesmaximal, characteristic, complemented
7#16699C212yesmaximal, complemented
8#16360C212yesmaximal
9#15110C212yesmaximal, complemented
10#15101C212yesmaximal, complemented
11#15085C212yesmaximal, complemented
12#15016C212yesmaximal, complemented
13#14906C212yesmaximal, complemented
14#14905C212yesmaximal
15#14684C211yesmaximal, characteristic, complemented
16#14344C212yesmaximal
17#14295C212yesmaximal, complemented
18#14240C212yesmaximal, complemented
19#14239C212yesmaximal, complemented
20C22×#123C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×#117C2212yes—
23C23×C42C2211yescharacteristic
24C2×#400C2212yes—
25C2×#400C2212yes—
26C2×#399C2212yes—
27C2×#399C2212yes—
28C2×#398C2212yes—
29C2×#397C2212yes—
30C4×#123—28no—
31C2×#388C2211yescharacteristic
32C2×#388C2211yescharacteristic
33C2×#387C2211yescharacteristic
34C2×#387C2211yescharacteristic
35C2×#386C2211yescharacteristic
36C2×#385C2211yescharacteristic
37#2285—28no—
38C2×#384C2211yescharacteristic
39C2×#384C2211yescharacteristic
40C2×#382C2212yes—
41C2×#381C2212yes—
42C2×#381C2211yescharacteristic
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C22⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52#1445C2212yes—
53#1445C2212yes—
54#1445C2212yes—
55#1445C2212yes—
56#1445C2212yes—
57#1445C2212yes—
58#1444C2212yes—
59#1444C2212yes—
60#1444C2212yes—
61#1443C2214yes—
62#1442C2212yes—
63#1442C2212yes—
64#1442C2212yes—
65#1442C2214yes—
66#1441C2212yes—
67#1441C2212yes—
68#1441C2212yes—
69#1441C2214yes—
70#1440C2214yes—
71#1440C2214yes—
72#1439C2212yes—
73#1439C2212yes—
74#1439C2214yes—
75#1439C2214yes—
76#1439C2212yes—
77#1438C2214yes—
78#1437C2214yes—
79#1436C2214yes—
80#1436C2214yes—
81#1436C2214yes—
82#1436C2214yes—
83#1436C2214yes—
84#1434C2212yes—
85#1434C2214yes—
86#1434C2212yes—
87#1434C2211yescharacteristic
88#1434C2211yescharacteristic
89#1434C2212yes—
90C22⋊#95C2214yes—
91C22⋊#95C2212yes—
92C22⋊#95C2212yes—
93C23×D4C2311yescharacteristic
94C24×C4C2311yescharacteristic
95C24×C4C2311yescharacteristic
96C24×C4C2311yescharacteristic
97C24×C4C2311yescharacteristic
98C2×#123—24no—
99C2×#123—24no—
100C2×#123—24no—
101C2×C4×D4—24no—
102C2×C4×D4—28no—
103C2×#117—28no—
104C2×#117—44no—
105C2×#117—44no—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2314yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2311yescharacteristic
130C22×C22⋊C4C2314yes—
131C22×C42—28no—
132C22×C42D418yes—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138C22×C42C2314yes—
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141#400—28no—
142#400—44no—
143#399—28no—
144#399—44no—
145#398—44no—
146#397—44no—
147#388—24no—
148#388—24no—
149#387—24no—
150#387—24no—
151#386—24no—
152#385—24no—
153#384—24no—
154#384—24no—
155#382—28no—
156#381—24no—
157#381—28no—
158#380—28no—
159#380—28no—
160C4×C4⋊C4—28no—
161C4×C4⋊C4—28no—
162C4×C4⋊C4—24no—
163C4×C4⋊C4—24no—
164C4×C22⋊C4—24no—
165C4×C22⋊C4—24no—
166C4×C22⋊C4—24no—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2311yescharacteristic
179C2×#95C2311yescharacteristic
180C2×#95C2314yes—
181C2×#95C2311yescharacteristic
182C2×#95C2311yescharacteristic
183C2×#95C2311yescharacteristic
184C2×#95C2311yescharacteristic
185C2×#95C2311yescharacteristic
186C2×#95C2311yescharacteristic
187C2×#95C2311yescharacteristic
188C2×#95C2311yescharacteristic
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2314yes—
200C25C2411yescharacteristic
201C25C2411yescharacteristic
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—24no—
205C22×D4—22no—
206C22×D4—42no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—24no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4C2×D414yes—
221C23×C4C2×D414yes—
222C23×C4C2×D414yes—
223C23×C4#4118yes—
224C23×C4C2411yescharacteristic
225C23×C4C2414yes—
226C23×C4C2414yes—
227C23×C4C2411yescharacteristic
228C23×C4C2414yes—
229C23×C4C2414yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241#123—232no—
242C4×D4—416no—
243#117—432no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4#4114yes—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4#4114yes—
308C2×C42—232no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—28no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—22no—
317C2×C42—24no—
318C2×C42—44no—
319C2×C42—44no—
320C2×C42—42no—
321C2×C42—42no—
322C2×C42C2×D414yes—
323C2×C42C2×D414yes—
324C2×C42C2×D414yes—
325C2×C42C2×D414yes—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—28no—
333#95—28no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—28no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—28no—
344#95—28no—
345#95—24no—
346#95—24no—
347#95—28no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359C24—21no—
360C24—21no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—21no—
367C24—21no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24C22×D412yes—
375C24C2×#4112yes—
376C24C2×#4114yes—
377C24C2511yescharacteristic
378C2×D4—28no—
379C2×D4—48no—
380C2×D4—48no—
381C2×D4—44no—
382C2×D4—44no—
383C22×C4—232no—
384C22×C4—28no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—28no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
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—28no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—416no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—88no—
469C22×C4#123116yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4C2×#4114yes—
480C22×C4C2×#4114yes—
481C22×C4C2×#4112yes—
482C22×C4C2×#4112yes—
483C22×C4H(5,𝔽2)14yes—
484C22×C42−1+414yes—
485C4⋊C4—28no—
486C4⋊C4—28no—
487C4⋊C4—48no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—44no—
491C4⋊C4—48no—
492C4⋊C4—44no—
493C4⋊C4—48no—
494C4⋊C4—48no—
495C4⋊C4—44no—
496C4⋊C4—88no—
497C4⋊C4—88no—
498C22⋊C4—28no—
499C22⋊C4—28no—
500C22⋊C4—28no—
501C22⋊C4—44no—
502C22⋊C4—416no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—44no—
507C22⋊C4—88no—
508C22⋊C4—88no—
509C22⋊C4—88no—
510C42—28no—
511C42—28no—
512C42—28no—
513C42—28no—
514C42—44no—
515C42—44no—
516C42—416no—
517C42—44no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—24no—
525C23—24no—
526C23—28no—
527C23—28no—
528C23—22no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—44no—
537C23—44no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—42no—
545C23—42no—
546C23—44no—
547C23—44no—
548C23—42no—
549C23—44no—
550C23—44no—
551C23C2×#12314yes—
552C23C23×D411yescharacteristic
553C23C22×#4112yes—
554C23C22×#4111yescharacteristic
555C23C22×#4111yescharacteristic
556C23C2×H(5,𝔽2)12yes—
557C23C2×2−1+412yes—
558C23#58512yes—
559C23#58512yes—
560C23#58512yes—
561D4—416no—
562C2×C4—232no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—416no—
589C2×C4—416no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—88no—
646C2×C4C2×#12314yes—
647C2×C4C2×#12314yes—
648C2×C4#52514yes—
649C2×C4#53514yes—
650C2×C4#53614yes—
651C22—24no—
652C22—24no—
653C22—24no—
654C22—42no—
655C22—42no—
656C22—48no—
657C22—48no—
658C22—42no—
659C22—42no—
660C22—42no—
661C22—42no—
662C22—48no—
663C22—42no—
664C22—81no—
665C22—84no—
666C22—81no—
667C22—81no—
668C22—81no—
669C22C22×#12311yescharacteristic
670C22C2×#52511yescharacteristic
671C22C2×#53511yescharacteristic
672C22C2×#53611yescharacteristic
673C22#344911yescharacteristic
674C22#344911yescharacteristic
675C22#344911yescharacteristic
676C22#346112yes—
677C22#346112yes—
678C22#347512yes—
679C22#347512yes—
680C22#349114yes—
681C22#349214yes—
682C22#351412yes—
683C22#351712yes—
684C22#355714yes—
685C22#356214yes—
686C4—24no—
687C4—24no—
688C4—24no—
689C4—24no—
690C4—24no—
691C4—42no—
692C4—42no—
693C4—42no—
694C4—42no—
695C4—81no—
696C4—84no—
697C4—84no—
698C4—84no—
699C4—81no—
700C4—84no—
701C4—81no—
702C4—81no—
703C2—44no—
704C2—82no—
705C2#3462711yescharacteristic
706C2#3480911yescharacteristic
707C2#3509711yescharacteristic
708C2#3519011yescharacteristic
709C2#3522711yescharacteristic
710C2#3754411yescharacteristic
711C2#3755111yescharacteristic
712C2#3800814yes—
713C2#3801014yes—
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×#2285C2semidirect
3. #17065C2semidirect
4. #17064C2semidirect
5. #17063C2semidirect
6. #17062C2semidirect
7. #16699C2semidirect
9. #15110C2semidirect
10. #15101C2semidirect
11. #15085C2semidirect
12. #15016C2semidirect
13. #14906C2semidirect
15. #14684C2semidirect
17. #14295C2semidirect
18. #14240C2semidirect
19. #14239C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1232
C2×#22852
#170652
#170642
#170632
#170622
#166992
#163602
#151102
#151012
#150852
#150162
#149062
#149052
#146842
#143442
#142952
#142402
#142392