Finite Group Databasebeta

← #824079

Subgroups — #824079

Order 512.

2934 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2442C212yesmaximal, complemented
2C2×#2370C211yesmaximal, characteristic, complemented
3C2×D4×C4⋊C4C211yesmaximal, characteristic
4#17010C212yesmaximal, complemented
5#15796C214yesmaximal, complemented
6#15339C214yesmaximal, complemented
7#15242C211yesmaximal, characteristic, complemented
8#15196C211yesmaximal, characteristic, complemented
9#15166C212yesmaximal, complemented
10#15129C212yesmaximal
11#15129C211yesmaximal, characteristic
12#14176C212yesmaximal
13#13974C212yesmaximal, complemented
14#13933C211yesmaximal, characteristic, complemented
15#13907C212yesmaximal, complemented
16#13875C212yesmaximal, complemented
17C4×#394C211yesmaximal, characteristic
18C22×#125C2211yescharacteristic
19C22×C4×D4C2211yescharacteristic
20C22×C4×D4C2212yes—
21C23×C4⋊C4C2212yes—
22#2442—216no—
23C2×#400C2214yes—
24C2×#398C2214yes—
25C2×#394C2211yescharacteristic
26C2×#394C2212yes—
27#2370—28no—
28D4×C4⋊C4—28no—
29C2×#389C2212yes—
30C2×#389C2211yescharacteristic
31C2×#389C2212yes—
32C2×#389C2211yescharacteristic
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#388C2214yes—
36C2×#387C2212yes—
37C2×#387C2212yes—
38C2×#385C2214yes—
39C2×#385C2214yes—
40C2×#384C2211yescharacteristic
41C2×#382C2212yes—
42C2×#380C2212yes—
43C2×#380C2212yes—
44C2×#380C2214yes—
45C2×#380C2212yes—
46C2×#380C2214yes—
47C2×C4×C4⋊C4C2211yescharacteristic
48C2×C4×C4⋊C4C2211yescharacteristic
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2212yes—
53C2×C4×C22⋊C4C2212yes—
54#1445C2214yes—
55#1444C2212yes—
56#1444C2212yes—
57#1443C2214yes—
58#1443C2214yes—
59#1442C2211yescharacteristic
60#1442C2214yes—
61#1441C2212yes—
62#1441C2212yes—
63#1441C2212yes—
64#1441C2211yescharacteristic
65#1441C2214yes—
66#1441C2214yes—
67#1441C2212yes—
68#1441C2212yes—
69#1440C2214yes—
70#1440C2214yes—
71#1440C2214yes—
72#1440C2212yes—
73#1439C2212yes—
74#1438C2211yescharacteristic
75#1437C2214yes—
76#1436C2214yes—
77C4⋊#95C2212yes—
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82#1434C2214yes—
83#1434C2212yes—
84C22⋊#95C2214yes—
85C4×#95C2211yescharacteristic
86C23×D4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2314yes—
89C2×#125—24no—
90C2×#125—24no—
91C2×#125D418yescomplemented
92C2×C4×D4—28no—
93C2×C4×D4—24no—
94C2×C4×D4—28no—
95C2×C4×D4—48no—
96C22×C4⋊C4—28no—
97C22×C4⋊C4—28no—
98C22×C4⋊C4—28no—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131C22×C42C2312yes—
132#400—216no—
133#398—216no—
134#394—24no—
135#394—28no—
136#389—24no—
137#389—28no—
138#389—28no—
139#389—28no—
140#389—24no—
141#389—28no—
142#388—48no—
143#387—28no—
144#387—28no—
145#385—216no—
146#385—216no—
147#384—24no—
148#382—28no—
149#380—28no—
150#380—28no—
151#380—216no—
152#380—28no—
153#380—48no—
154C4×C4⋊C4—24no—
155C4×C4⋊C4—24no—
156C4×C4⋊C4—24no—
157C4×C4⋊C4—28no—
158C4×C4⋊C4D418yes—
159C4×C22⋊C4—28no—
160C4×C22⋊C4—28no—
161C2×#95C2311yescharacteristic
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2311yescharacteristic
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2311yescharacteristic
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2311yescharacteristic
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2312yes—
188C2×#95C2314yes—
189C2×#95C2312yes—
190C25C2412yes—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—22no—
194C22×D4—22no—
195C22×D4#4114yes—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—28no—
203C23×C4—28no—
204C23×C4—28no—
205C23×C4—28no—
206C23×C4—44no—
207C23×C4—44no—
208C23×C4—44no—
209C23×C4#4118yes—
210C23×C4C2414yes—
211C23×C4C2411yescharacteristic
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226#125—232no—
227C4×D4—232no—
228C4×D4—432no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—41no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—432no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4C2×D414yes—
272C2×C4⋊C4C2×D418yes—
273C2×C4⋊C4C2×D418yes—
274C2×C4⋊C4C2×D414yes—
275C2×C4⋊C4C2×D414yes—
276C2×C4⋊C4C2×D414yes—
277C2×C4⋊C4#4114yes—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—44no—
302C2×C42—22no—
303C2×C42—24no—
304C2×C42—22no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—44no—
314C2×C42—42no—
315C2×C42—44no—
316C2×C42C2×D414yes—
317C2×C42C2×D414yes—
318C2×C42C2×D418yes—
319#95—24no—
320#95—24no—
321#95—22no—
322#95—22no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—28no—
336#95—22no—
337#95—24no—
338#95—22no—
339#95—44no—
340#95—44no—
341#95—42no—
342#95—42no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95C2×D418yes—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24C2×#4114yes—
361C24C2×#4114yes—
362C24H(5,𝔽2)14yes—
363C24C2511yescharacteristic
364C2×D4—28no—
365C2×D4—28no—
366C2×D4—216no—
367C2×D4—216no—
368C2×D4—28no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—216no—
408C22×C4—216no—
409C22×C4—216no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—21no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—416no—
449C22×C4—416no—
450C22×C4—416no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—416no—
456C22×C4—88no—
457C22×C4—88no—
458C22×C4—88no—
459C22×C4C22×D412yes—
460C22×C4C22×D414yes—
461C22×C4C22×D414yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D414yes—
465C22×C4C22×D414yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C2×#4114yes—
470C22×C4C2×#4114yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4114yes—
474C22×C42−1+412yes—
475C22×C42−1+412yes—
476C22×C42−1+414yes—
477C4⋊C4—28no—
478C4⋊C4—216no—
479C4⋊C4—216no—
480C4⋊C4—28no—
481C4⋊C4—28no—
482C4⋊C4—44no—
483C4⋊C4—48no—
484C4⋊C4—416no—
485C4⋊C4—48no—
486C4⋊C4—44no—
487C4⋊C4—44no—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—84no—
491C4⋊C4—88no—
492C4⋊C4—88no—
493C4⋊C4—84no—
494C4⋊C4#121116yes—
495C22⋊C4—216no—
496C22⋊C4—48no—
497C22⋊C4—416no—
498C22⋊C4—48no—
499C22⋊C4—48no—
500C22⋊C4—416no—
501C22⋊C4—48no—
502C22⋊C4—416no—
503C42—28no—
504C42—28no—
505C42—48no—
506C42—48no—
507C42—44no—
508C42—48no—
509C42—84no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23#55318yes—
545C23C23×D411yescharacteristic
546C23C23×D411yescharacteristic
547C23C22×#4111yescharacteristic
548C23C22×#4112yes—
549C23C22×#4111yescharacteristic
550C23C2×H(5,𝔽2)12yes—
551C23C2×2−1+412yes—
552C23C2×2−1+411yescharacteristic
553C23C2×2−1+411yescharacteristic
554C23#58511yescharacteristic
555C23#58512yes—
556D4—232no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—28no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—432no—
607C2×C4—44no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—41no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—41no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—832no—
646C2×C4C2×#12118yes—
647C2×C4C2×#12114yes—
648C2×C4#52514yes—
649C2×C4#53614yes—
650C2×C4#54814yes—
651C2×C4#54818yes—
652C2×C4#55218yes—
653C22—28no—
654C22—28no—
655C22—28no—
656C22—44no—
657C22—44no—
658C22—44no—
659C22—44no—
660C22—44no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—44no—
668C22C22×#12111yescharacteristic
669C22C2×#52511yescharacteristic
670C22C2×#53611yescharacteristic
671C22C2×#54812yes—
672C22C2×#54811yescharacteristic
673C22D4×#4111yescharacteristic
674C22C2×#55212yes—
675C22C2×#55312yes—
676C22#348211yescharacteristic
677C22#348411yescharacteristic
678C22#348812yes—
679C22#348911yescharacteristic
680C22#349212yes—
681C22#349512yes—
682C22#349911yescharacteristic
683C22#350011yescharacteristic
684C22#351212yes—
685C22#351312yes—
686C22#351512yes—
687C22#351811yescharacteristic
688C22#354412yes—
689C22#355512yes—
690C22#355912yes—
691C4—28no—
692C4—24no—
693C4—44no—
694C4—42no—
695C4—44no—
696C4—42no—
697C4—42no—
698C4—42no—
699C4—44no—
700C4—82no—
701C4—84no—
702C4—84no—
703C4—82no—
704C4—81no—
705C4—82no—
706C4#253618yes—
707C2—48no—
708C2C2×#253611yescharacteristic
709C2#3572811yescharacteristic
710C2#3608011yescharacteristic
711C2#3644411yescharacteristic
712C2#3658111yescharacteristic
713C2#3917311yescharacteristic
714C2#4248211yescharacteristic
715C2#4497112yes—
716C2#4528812yes—
717C2#5913912yes—
718C2#5920912yes—
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
1. C2×#2442C2semidirect
2. C2×#2370C2semidirect
4. #17010C2semidirect
5. #15796C2semidirect
6. #15339C2semidirect
7. #15242C2semidirect
8. #15196C2semidirect
9. #15166C2semidirect
13. #13974C2semidirect
14. #13933C2semidirect
15. #13907C2semidirect
16. #13875C2semidirect
91. C2×#125D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24422
C2×#23702
C2×D4×C4⋊C42
#170102
#157962
#153392
#152422
#151962
#151662
#151292
#141762
#139742
#139332
#139072
#138752
C4×#3942