Finite Group Databasebeta

← #4307548

Subgroups — #4307548

Order 512.

2946 conjugacy classes of subgroups.

The subgroup lattice has 719 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2362C212yesmaximal, complemented
2C2×#2355C211yesmaximal, characteristic, complemented
3C2×#2354C211yesmaximal, characteristic
4#15185C211yesmaximal, characteristic, complemented
5#15179C212yesmaximal, complemented
6#15177C211yesmaximal, characteristic
7#15163C212yesmaximal
8#15151C211yesmaximal, characteristic, complemented
9#15131C212yesmaximal, complemented
10#15083C212yesmaximal, complemented
11#15074C214yesmaximal, complemented
12#14995C214yesmaximal, complemented
13#14982C212yesmaximal
14C4×#398C212yesmaximal, complemented
15C4×#394C211yesmaximal, characteristic
16#13818C211yesmaximal, characteristic, complemented
17#13801C212yesmaximal, complemented
18C22×C22⋊Q8C2212yes—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×#117C2212yes—
22C2×#398C2212yes—
23C2×#398C2212yes—
24C2×#396C2214yes—
25C2×#394C2211yescharacteristic
26#2362—216no—
27#2355—28no—
28#2354—28no—
29C2×#391C2211yescharacteristic
30C2×#391C2211yescharacteristic
31C2×#391C2212yes—
32C2×#390C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#388C2214yes—
36C2×#388C2214yes—
37C2×#388C2214yes—
38C2×#387C2212yes—
39C2×#387C2212yes—
40C2×#387C2214yes—
41C2×#384C2211yescharacteristic
42C2×#384C2211yescharacteristic
43C2×#382C2212yes—
44C2×#380C2212yes—
45C2×#380C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2214yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2212yes—
54C2×C4×C22⋊C4C2212yes—
55#1445C2212yes—
56#1445C2212yes—
57#1445C2212yes—
58#1445C2212yes—
59#1444C2212yes—
60#1442C2211yescharacteristic
61#1442C2214yes—
62#1442C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2211yescharacteristic
66#1441C2214yes—
67#1440C2212yes—
68#1440C2214yes—
69#1439C2214yes—
70#1439C2212yes—
71#1439C2214yes—
72#1439C2214yes—
73#1438C2211yescharacteristic
74#1438C2212yes—
75#1437C2214yes—
76#1437C2214yes—
77#1436C2214yes—
78C4⋊#95C2212yes—
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82#1434C2214yes—
83C4×#95C2211yescharacteristic
84C4×#95C2212yes—
85C4×#95C2214yes—
86C23×Q8C2311yescharacteristic
87C23×D4C2311yescharacteristic
88C24×C4C2312yes—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—28no—
92C2×C4×Q8—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8D418yescomplemented
95C2×C4×D4—28no—
96C2×C4×D4—24no—
97C2×#117—28no—
98C2×#117—28no—
99C2×#117—28no—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C42C2314yes—
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2314yes—
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2312yes—
132C22×C42C2312yes—
133C22×C42C2312yes—
134#398—28no—
135#398—44no—
136#396—48no—
137#394—24no—
138#391—28no—
139#391—24no—
140#391—24no—
141#390—28no—
142#389—28no—
143#389—28no—
144#388—216no—
145#388—216no—
146#388—216no—
147#387—216no—
148#387—28no—
149#387—28no—
150#384—24no—
151#384—24no—
152#382—28no—
153#380—28no—
154#380—28no—
155C4×C4⋊C4—24no—
156C4×C4⋊C4—28no—
157C4×C4⋊C4—28no—
158C4×C4⋊C4—24no—
159C4×C4⋊C4—28no—
160C4×C4⋊C4—216no—
161C4×C4⋊C4D418yes—
162C4×C22⋊C4—28no—
163C4×C22⋊C4—28no—
164C2×#95C2311yescharacteristic
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2311yescharacteristic
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2311yescharacteristic
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2311yescharacteristic
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2312yes—
190C2×#95C2314yes—
191C2×#95C2312yes—
192C25C2412yes—
193C22×Q8—22no—
194C22×Q8—22no—
195C22×Q8—24no—
196C22×Q8—24no—
197C22×Q8C2×D414yes—
198C22×D4—24no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—22no—
202C22×D4—42no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4#4118yes—
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2414yes—
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C22⋊Q8—264no—
227C4×Q8—232no—
228C4×D4—232no—
229#117—432no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—41no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4C2×D418yes—
268C2×C4⋊C4C2×D414yes—
269C2×C4⋊C4C2×D414yes—
270C2×C4⋊C4C2×D414yes—
271C2×C4⋊C4#4118yes—
272C2×C4⋊C4#4114yes—
273C2×C4⋊C4#4114yes—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C42—24no—
302C2×C42—22no—
303C2×C42—24no—
304C2×C42—22no—
305C2×C42—24no—
306C2×C42—28no—
307C2×C42—28no—
308C2×C42—24no—
309C2×C42—28no—
310C2×C42—22no—
311C2×C42—28no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—28no—
316C2×C42—24no—
317C2×C42—44no—
318C2×C42—44no—
319C2×C42—44no—
320C2×C42C2×D414yes—
321C2×C42C2×D414yes—
322C2×C42C2×D418yes—
323C2×C42C2×D418yes—
324#95—24no—
325#95—24no—
326#95—28no—
327#95—24no—
328#95—22no—
329#95—22no—
330#95—28no—
331#95—24no—
332#95—24no—
333#95—28no—
334#95—24no—
335#95—24no—
336#95—28no—
337#95—28no—
338#95—24no—
339#95—24no—
340#95—28no—
341#95—28no—
342#95—22no—
343#95—24no—
344#95—28no—
345#95—22no—
346#95—24no—
347#95—44no—
348#95—44no—
349#95—42no—
350#95—44no—
351#95C2×D418yes—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24C2×#4114yes—
367C24C2511yescharacteristic
368C2×Q8—28no—
369C2×Q8—216no—
370C2×Q8—44no—
371C2×Q8—48no—
372C2×Q8—44no—
373C2×D4—28no—
374C2×D4—48no—
375C2×D4—48no—
376C2×D4—44no—
377C2×D4—44no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—216no—
425C22×C4—216no—
426C22×C4—216no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4C22×D412yes—
456C22×C4C22×D414yes—
457C22×C4C22×D414yes—
458C22×C4C22×D412yes—
459C22×C4C22×D414yes—
460C22×C4C22×D414yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C2×#4112yes—
466C22×C4C2×#4118yes—
467C22×C4C2×#4114yes—
468C22×C4C2×#4114yes—
469C22×C4C2×#4112yes—
470C22×C4C2×#4112yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4114yes—
473C22×C4C2×#4114yes—
474C22×C4H(5,𝔽2)12yes—
475C22×C42−1+414yes—
476C22×C42−1+412yes—
477C4⋊C4—28no—
478C4⋊C4—232no—
479C4⋊C4—216no—
480C4⋊C4—216no—
481C4⋊C4—28no—
482C4⋊C4—28no—
483C4⋊C4—48no—
484C4⋊C4—44no—
485C4⋊C4—48no—
486C4⋊C4—44no—
487C4⋊C4—48no—
488C4⋊C4—416no—
489C4⋊C4—84no—
490C4⋊C4—84no—
491C4⋊C4—88no—
492C4⋊C4#121116yes—
493C22⋊C4—216no—
494C22⋊C4—232no—
495C22⋊C4—48no—
496C22⋊C4—48no—
497C22⋊C4—48no—
498C22⋊C4—416no—
499C22⋊C4—48no—
500C22⋊C4—88no—
501C22⋊C4—88no—
502C42—28no—
503C42—216no—
504C42—28no—
505C42—48no—
506C42—416no—
507C42—416no—
508C42—44no—
509C42—48no—
510C42—84no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
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—42no—
545C23—42no—
546C23C23×D411yescharacteristic
547C23C23×D411yescharacteristic
548C23C22×#4111yescharacteristic
549C23C22×#4112yes—
550C23C22×#4112yes—
551C23C22×#4111yescharacteristic
552C23C2×H(5,𝔽2)11yescharacteristic
553C23C2×2−1+411yescharacteristic
554C23C2×2−1+412yes—
555C23#58512yes—
556C23#58511yescharacteristic
557Q8—416no—
558D4—416no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—22no—
567C2×C4—28no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—24no—
571C2×C4—28no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—22no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—28no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—28no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—28no—
593C2×C4—24no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—41no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—432no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—41no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—41no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—41no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—44no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4C2×#12118yes—
645C2×C4C2×#12114yes—
646C2×C4#52514yes—
647C2×C4#52518yes—
648C2×C4#52918yes—
649C2×C4#53214yes—
650C2×C4#54814yes—
651C2×C4#55014yes—
652C22—44no—
653C22—44no—
654C22—44no—
655C22—44no—
656C22—44no—
657C22—44no—
658C22—44no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22—82no—
663C22—82no—
664C22—82no—
665C22—82no—
666C22—82no—
667C22C22×#12111yescharacteristic
668C22C2×#52512yes—
669C22C2×#52511yescharacteristic
670C22C2×#52912yes—
671C22C2×#53211yescharacteristic
672C22C2×#54811yescharacteristic
673C22C2×#55011yescharacteristic
674C22#347512yes—
675C22#348011yescharacteristic
676C22#348212yes—
677C22#348511yescharacteristic
678C22#348811yescharacteristic
679C22#348912yes—
680C22#349512yes—
681C22#349512yes—
682C22#349711yescharacteristic
683C22#349911yescharacteristic
684C22#351112yes—
685C22#351412yes—
686C22#351512yes—
687C22#352211yescharacteristic
688C22#352312yes—
689C22#352412yes—
690C4—24no—
691C4—28no—
692C4—24no—
693C4—48no—
694C4—42no—
695C4—42no—
696C4—44no—
697C4—44no—
698C4—44no—
699C4—44no—
700C4—42no—
701C4—82no—
702C4—82no—
703C4—84no—
704C4—81no—
705C4#239218yes—
706C2—84no—
707C2C2×#239211yescharacteristic
708C2#3583212yes—
709C2#3600612yes—
710C2#3602011yescharacteristic
711C2#3605111yescharacteristic
712C2#3608911yescharacteristic
713C2#3650412yes—
714C2#3650612yes—
715C2#3650811yescharacteristic
716C2#3651611yescharacteristic
717C2#3657611yescharacteristic
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×#2362C2semidirect
2. C2×#2355C2semidirect
4. #15185C2semidirect
5. #15179C2semidirect
8. #15151C2semidirect
9. #15131C2semidirect
10. #15083C2semidirect
11. #15074C2semidirect
12. #14995C2semidirect
14. C4×#398C2semidirect
16. #13818C2semidirect
17. #13801C2semidirect
94. C2×C4×Q8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#23622
C2×#23552
C2×#23542
#151852
#151792
#151772
#151632
#151512
#151312
#150832
#150742
#149952
#149822
C4×#3982
C4×#3942
#138182
#138012