Finite Group Databasebeta

← #4416477

Subgroups — #4416477

Order 512.

2337 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2853C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic, complemented
3#17067C211yesmaximal, characteristic, complemented
4#17066C211yesmaximal, characteristic, complemented
5#17061C211yesmaximal, characteristic, complemented
6#17060C211yesmaximal, characteristic, complemented
7#16704C212yesmaximal, complemented
8#16700C212yesmaximal, complemented
9#16395C212yesmaximal
10#16393C212yesmaximal, complemented
11#15114C212yesmaximal, complemented
12#15113C212yesmaximal
13#15088C212yesmaximal, complemented
14#15083C212yesmaximal, complemented
15#15081C212yesmaximal, complemented
16#15079C212yesmaximal, complemented
17#14685C211yesmaximal, characteristic, complemented
18#14675C212yesmaximal, complemented
19#14658C212yesmaximal
20C22×#123C2211yescharacteristic
21C22×#117C2212yes—
22C23×C42C2211yescharacteristic
23#2853—28no—
24C2×#398C2212yes—
25C2×#397C2212yes—
26C2×#388C2211yescharacteristic
27C2×#388C2211yescharacteristic
28C2×#388C2211yescharacteristic
29C2×#388C2211yescharacteristic
30C2×#388C2214yes—
31C2×#387C2211yescharacteristic
32C2×#387C2211yescharacteristic
33C2×#387C2211yescharacteristic
34C2×#387C2211yescharacteristic
35C2×#387C2214yes—
36C2×#386C2211yescharacteristic
37C2×#386C2211yescharacteristic
38C2×#385C2211yescharacteristic
39C2×#385C2211yescharacteristic
40#2285—28no—
41C2×#384C2211yescharacteristic
42C2×#384C2211yescharacteristic
43C2×#382C2212yes—
44C2×#381C2211yescharacteristic
45C2×#381C2212yes—
46C2×#380C2212yes—
47C2×#380C2212yes—
48C2×#380C2212yes—
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2212yes—
51#1445C2214yes—
52#1445C2212yes—
53#1445C2212yes—
54#1444C2212yes—
55#1444C2212yes—
56#1444C2212yes—
57#1444C2212yes—
58#1444C2212yes—
59#1443C2214yes—
60#1442C2212yes—
61#1442C2214yes—
62#1442C2212yes—
63#1442C2212yes—
64#1442C2212yes—
65#1442C2212yes—
66#1441C2212yes—
67#1441C2212yes—
68#1441C2212yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2214yes—
72#1441C2212yes—
73#1441C2212yes—
74#1440C2214yes—
75#1439C2214yes—
76#1439C2212yes—
77#1439C2212yes—
78#1439C2214yes—
79#1439C2214yes—
80#1439C2212yes—
81#1439C2214yes—
82#1439C2212yes—
83#1439C2212yes—
84#1438C2214yes—
85C4⋊#95C2212yes—
86C4⋊#95C2212yes—
87#1434C2214yes—
88#1434C2214yes—
89#1434C2211yescharacteristic
90#1434C2211yescharacteristic
91#1434C2212yes—
92C22⋊#95C2212yes—
93C23×D4C2311yescharacteristic
94C24×C4C2311yescharacteristic
95C24×C4C2311yescharacteristic
96C24×C4C2311yescharacteristic
97C2×#123—24no—
98C2×#123—28no—
99C2×#117—28no—
100C2×#117—48no—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2314yes—
129C22×C42—28no—
130C22×C42D418yes—
131C22×C42C2314yes—
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136#398—44no—
137#397—44no—
138#388—24no—
139#388—24no—
140#388—24no—
141#388—24no—
142#388—48no—
143#387—24no—
144#387—24no—
145#387—24no—
146#387—24no—
147#387—48no—
148#386—24no—
149#386—24no—
150#385—24no—
151#385—24no—
152#384—24no—
153#384—24no—
154#382—28no—
155#381—24no—
156#381—44no—
157#380—28no—
158#380—28no—
159#380—28no—
160C4×C4⋊C4—28no—
161C4×C4⋊C4—28no—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2311yescharacteristic
168C2×#95C2311yescharacteristic
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2311yescharacteristic
174C2×#95C2311yescharacteristic
175C2×#95C2311yescharacteristic
176C2×#95C2311yescharacteristic
177C2×#95C2311yescharacteristic
178C2×#95C2311yescharacteristic
179C2×#95C2311yescharacteristic
180C2×#95C2311yescharacteristic
181C2×#95C2311yescharacteristic
182C2×#95C2311yescharacteristic
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C25C2411yescharacteristic
197C25C2411yescharacteristic
198C22×D4—22no—
199C22×D4—24no—
200C22×D4—42no—
201C22×D4—42no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4C2×D414yes—
212C23×C4C2×D414yes—
213C23×C4C2×D414yes—
214C23×C4#4118yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2414yes—
218C23×C4C2414yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2414yes—
222C23×C4C2414yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232#123—416no—
233#117—432no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C42—232no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—28no—
306C2×C42—28no—
307C2×C42—28no—
308C2×C42C2×D414yes—
309C2×C42C2×D414yes—
310C2×C42C2×D414yes—
311C2×C42C2×D414yes—
312#95—28no—
313#95—28no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—28no—
331#95—24no—
332#95—24no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346C24—22no—
347C24—21no—
348C24—22no—
349C24—21no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—21no—
354C24—22no—
355C24—22no—
356C24—21no—
357C24—22no—
358C24—21no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24C22×D412yes—
363C24C2×#4114yes—
364C24C2511yescharacteristic
365C2×D4—44no—
366C2×D4—44no—
367C2×D4—44no—
368C2×D4—88no—
369C22×C4—232no—
370C22×C4—28no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—28no—
382C22×C4—28no—
383C22×C4—28no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—28no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
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—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—44no—
449C22×C4—44no—
450C22×C4—44no—
451C22×C4—416no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4#123116yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C2×#4114yes—
462C22×C4C2×#4114yes—
463C22×C4H(5,𝔽2)14yes—
464C22×C4H(5,𝔽2)12yes—
465C22×C42−1+414yes—
466C22×C42−1+412yes—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—44no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—416no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—44no—
478C4⋊C4—48no—
479C4⋊C4—44no—
480C4⋊C4—88no—
481C4⋊C4—88no—
482C22⋊C4—44no—
483C22⋊C4—44no—
484C22⋊C4—44no—
485C22⋊C4—416no—
486C22⋊C4—416no—
487C22⋊C4—44no—
488C22⋊C4—44no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—44no—
493C22⋊C4—88no—
494C22⋊C4—88no—
495C42—28no—
496C42—28no—
497C42—28no—
498C42—28no—
499C42—416no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—28no—
507C23—28no—
508C23—24no—
509C23—24no—
510C23—44no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—44no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—44no—
524C23—44no—
525C23—42no—
526C23—42no—
527C23—44no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—44no—
533C23C2×#12314yes—
534C23C23×D411yescharacteristic
535C23C22×#4112yes—
536C23C2×H(5,𝔽2)12yes—
537C23C2×H(5,𝔽2)11yescharacteristic
538C23C2×2−1+412yes—
539C23C2×2−1+411yescharacteristic
540C23#58512yes—
541C23#58512yes—
542C23#58512yes—
543D4—88no—
544C2×C4—232no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—416no—
565C2×C4—416no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—44no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—44no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4C2×#12314yes—
628C2×C4C2×#12314yes—
629C2×C4#53514yes—
630C2×C4#53614yes—
631C2×C4#53714yes—
632C22—24no—
633C22—24no—
634C22—24no—
635C22—42no—
636C22—42no—
637C22—42no—
638C22—48no—
639C22—48no—
640C22—42no—
641C22—81no—
642C22—81no—
643C22—81no—
644C22—81no—
645C22—81no—
646C22—81no—
647C22—84no—
648C22—84no—
649C22—81no—
650C22C22×#12311yescharacteristic
651C22C2×#53511yescharacteristic
652C22C2×#53611yescharacteristic
653C22C2×#53711yescharacteristic
654C22#344911yescharacteristic
655C22#344911yescharacteristic
656C22#344911yescharacteristic
657C22#349114yes—
658C22#349214yes—
659C22#349412yes—
660C22#349512yes—
661C22#352914yes—
662C22#354612yes—
663C22#354712yes—
664C22#354712yes—
665C22#355512yes—
666C22#356514yes—
667C4—24no—
668C4—24no—
669C4—24no—
670C4—24no—
671C4—24no—
672C4—42no—
673C4—81no—
674C4—81no—
675C4—84no—
676C4—81no—
677C4—81no—
678C4—81no—
679C4—81no—
680C4—84no—
681C4—84no—
682C4—84no—
683C4—81no—
684C2—44no—
685C2—161no—
686C2#3509711yescharacteristic
687C2#3522711yescharacteristic
688C2#3526311yescharacteristic
689C2#3668711yescharacteristic
690C2#3754511yescharacteristic
691C2#3762111yescharacteristic
692C2#3763311yescharacteristic
693C2#4577414yes—
694C2#4577514yes—
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. #17067C2semidirect
4. #17066C2semidirect
5. #17061C2semidirect
6. #17060C2semidirect
7. #16704C2semidirect
8. #16700C2semidirect
10. #16393C2semidirect
11. #15114C2semidirect
13. #15088C2semidirect
14. #15083C2semidirect
15. #15081C2semidirect
16. #15079C2semidirect
17. #14685C2semidirect
18. #14675C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28532
C2×#22852
#170672
#170662
#170612
#170602
#167042
#167002
#163952
#163932
#151142
#151132
#150882
#150832
#150812
#150792
#146852
#146752
#146582