Finite Group Databasebeta

← #4418511

Subgroups — #4418511

Order 512.

2721 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2562C211yesmaximal, characteristic, complemented
2C2×#2664C211yesmaximal, characteristic, complemented
3C2×#2636C211yesmaximal, characteristic
4C2×#2349C211yesmaximal, characteristic, complemented
5#17039C212yesmaximal, complemented
6#17010C212yesmaximal, complemented
7#16399C212yesmaximal, complemented
8#16398C212yesmaximal, complemented
9#15590C214yesmaximal, complemented
10#15577C214yesmaximal, complemented
11#15159C212yesmaximal
12#14868C212yesmaximal, complemented
13#14826C212yesmaximal, complemented
14#14735C212yesmaximal, complemented
15#14664C211yesmaximal, characteristic, complemented
16#14291C211yesmaximal, characteristic, complemented
17#13955C211yesmaximal, characteristic, complemented
18C22×#126C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2212yes—
22#2664—28no—
23#2636—28no—
24#2562—28no—
25C2×#400C2211yescharacteristic
26C2×#400C2211yescharacteristic
27C2×#400C2212yes—
28C2×#399C2211yescharacteristic
29C2×#399C2211yescharacteristic
30C2×#398C2211yescharacteristic
31C2×#398C2212yes—
32C2×#397C2211yescharacteristic
33C2×#397C2214yes—
34C2×#396C2212yes—
35C2×#396C2212yes—
36C2×#394C2211yescharacteristic
37C2×#394C2212yes—
38C2×#394C2212yes—
39C2×#392C2211yescharacteristic
40#2349—28no—
41C2×#389C2212yes—
42C2×#389C2212yes—
43C2×#388C2212yes—
44C2×#388C2214yes—
45C2×#386C2212yes—
46C2×#386C2214yes—
47C2×#385C2212yes—
48C2×#385C2214yes—
49C2×#382C2211yescharacteristic
50C2×#382C2211yescharacteristic
51C2×#380C2212yes—
52C2×#380C2212yes—
53C2×#380C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C4⋊C4C2211yescharacteristic
56C22×#95C2212yes—
57#1445C2212yes—
58#1445C2212yes—
59#1443C2214yes—
60#1443C2214yes—
61#1442C2212yes—
62#1442C2212yes—
63#1442C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2212yes—
68#1441C2212yes—
69#1441C2212yes—
70#1441C2212yes—
71#1441C2214yes—
72#1440C2214yes—
73#1440C2214yes—
74#1439C2214yes—
75#1439C2212yes—
76#1439C2214yes—
77#1439C2212yes—
78#1438C2214yes—
79#1437C2212yes—
80#1436C2212yes—
81C4⋊#95C2214yes—
82#1434C2212yes—
83#1434C2212yes—
84C22⋊#95C2214yes—
85C22⋊#95C2214yes—
86C22⋊#95C2214yes—
87C23×D4C2311yescharacteristic
88C23×D4C2312yes—
89C24×C4C2314yes—
90C24×C4C2312yes—
91C2×#126—24no—
92C2×#126—24no—
93C2×#126—24no—
94C2×#125—24no—
95C2×#125—24no—
96C2×#125—24no—
97C2×C4×D4—24no—
98C2×C4×D4—28no—
99C2×C4×D4—24no—
100C2×C4×D4—48no—
101C2×C4×D4D418yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2314yes—
132C22×C42C2312yes—
133C22×C42C2312yes—
134C22×C42C2311yescharacteristic
135#400—28no—
136#400—24no—
137#400—24no—
138#399—24no—
139#399—24no—
140#398—28no—
141#398—24no—
142#397—24no—
143#397—48no—
144#396—28no—
145#396—28no—
146#394—28no—
147#394—24no—
148#394—28no—
149#392—24no—
150#389—28no—
151#389—28no—
152#388—216no—
153#388—28no—
154#386—28no—
155#386—48no—
156#385—28no—
157#385—216no—
158#382—24no—
159#382—24no—
160#380—28no—
161#380—28no—
162#380—28no—
163C4×C4⋊C4—24no—
164C4×C4⋊C4D418yescomplemented
165C2×#95—28no—
166C2×#95C2311yescharacteristic
167C2×#95C2311yescharacteristic
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2311yescharacteristic
193C2×#95C2311yescharacteristic
194C25C2411yescharacteristic
195C25C2412yes—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—44no—
204C22×D4—44no—
205C22×D4C2×D414yes—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—28no—
209C23×C4—28no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4C2×D418yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2414yes—
225C23×C4C2414yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233#126—416no—
234#125—416no—
235C4×D4—232no—
236C4×D4—432no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4C2×D414yes—
279C2×C4⋊C4C2×D418yes—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4C2×D418yes—
318C2×C42—24no—
319C2×C42—44no—
320C2×C42—44no—
321C2×C42C2×D418yes—
322C2×C42C2×D414yes—
323C2×C42C2×D414yes—
324C2×C42#4114yes—
325#95—24no—
326#95—28no—
327#95—24no—
328#95—28no—
329#95—28no—
330#95—24no—
331#95—24no—
332#95—22no—
333#95—22no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95C2×D414yes—
352#95C2×D414yes—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—24no—
358C24—24no—
359C24—24no—
360C24—24no—
361C24—21no—
362C24—21no—
363C24—21no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—24no—
368C24—22no—
369C24C22×D414yes—
370C24H(5,𝔽2)14yes—
371C242−1+414yes—
372C24C2511yescharacteristic
373C2×D4—28no—
374C2×D4—28no—
375C2×D4—216no—
376C2×D4—216no—
377C2×D4—28no—
378C2×D4—416no—
379C2×D4—48no—
380C2×D4—84no—
381C2×D4—84no—
382C2×D4—88no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—216no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—416no—
454C22×C4—416no—
455C22×C4—88no—
456C22×C4—88no—
457C22×C4—88no—
458C22×C4—88no—
459C22×C4—88no—
460C22×C4—88no—
461C22×C4—88no—
462C22×C4C22×D414yes—
463C22×C4C22×D414yes—
464C22×C4C22×D414yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4112yes—
474C4⋊C4—28no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C4⋊C4—48no—
480C4⋊C4—48no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—44no—
485C4⋊C4—44no—
486C4⋊C4—84no—
487C4⋊C4—84no—
488C4⋊C4—84no—
489C4⋊C4—88no—
490C22⋊C4—216no—
491C22⋊C4—416no—
492C22⋊C4—48no—
493C22⋊C4—416no—
494C22⋊C4—48no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—84no—
499C22⋊C4—84no—
500C22⋊C4—84no—
501C22⋊C4—88no—
502C22⋊C4—84no—
503C22⋊C4—88no—
504C42—48no—
505C42—84no—
506C42#121116yes—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—28no—
518C23—28no—
519C23—28no—
520C23—28no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23—44no—
529C23—41no—
530C23—42no—
531C23—42no—
532C23—41no—
533C23—42no—
534C23—41no—
535C23—42no—
536C23—41no—
537C23—41no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23#53718yes—
546C23C23×D411yescharacteristic
547C23C23×D411yescharacteristic
548C23C22×#4111yescharacteristic
549C23C2×H(5,𝔽2)12yes—
550C23C2×2−1+412yes—
551C23#58512yes—
552C23#58512yes—
553C23#58514yes—
554D4—232no—
555D4—816no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
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—816no—
633C2×C4—832no—
634C2×C4C2×#12114yes—
635C2×C4C2×#12114yes—
636C2×C4C2×#12114yes—
637C22—28no—
638C22—28no—
639C22—28no—
640C22—44no—
641C22—44no—
642C22—48no—
643C22—48no—
644C22—44no—
645C22—44no—
646C22—48no—
647C22—48no—
648C22—81no—
649C22—81no—
650C22—82no—
651C22—82no—
652C22—81no—
653C22—82no—
654C22—82no—
655C22—84no—
656C22C22×#12111yescharacteristic
657C22C2×#53712yes—
658C22#344912yes—
659C22#348411yescharacteristic
660C22#348512yes—
661C22#348611yescharacteristic
662C22#349212yes—
663C22#350011yescharacteristic
664C22#350011yescharacteristic
665C22#350012yes—
666C22#351012yes—
667C22#351212yes—
668C22#353314yes—
669C22#355314yes—
670C22#356014yes—
671C22#356414yes—
672C4—24no—
673C4—24no—
674C4—24no—
675C4—44no—
676C4—44no—
677C4—44no—
678C4—81no—
679C4—82no—
680C4—81no—
681C4—84no—
682C4—81no—
683C4—82no—
684C4—82no—
685C4—82no—
686C4—84no—
687C2—48no—
688C2—161no—
689C2#3522311yescharacteristic
690C2#3532811yescharacteristic
691C2#3561711yescharacteristic
692C2#4050212yes—
693C2#4136912yes—
694C2#4137212yes—
695C2#4238912yes—
696C2#5093514yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#392C22
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×#2562C2semidirect
2. C2×#2664C2semidirect
4. C2×#2349C2semidirect
5. #17039C2semidirect
6. #17010C2semidirect
7. #16399C2semidirect
8. #16398C2semidirect
9. #15590C2semidirect
10. #15577C2semidirect
12. #14868C2semidirect
13. #14826C2semidirect
14. #14735C2semidirect
15. #14664C2semidirect
16. #14291C2semidirect
17. #13955C2semidirect
164. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25622
C2×#26642
C2×#26362
C2×#23492
#170392
#170102
#163992
#163982
#155902
#155772
#151592
#148682
#148262
#147352
#146642
#142912
#139552