Finite Group Databasebeta

← #779433

Subgroups — #779433

Order 512.

3970 conjugacy classes of subgroups.

The subgroup lattice has 766 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2445C212yesmaximal, complemented
2C2×#2455C212yesmaximal, complemented
3C2×#2361C212yesmaximal, complemented
4#16282C212yesmaximal, complemented
5#15961C212yesmaximal, complemented
6#15780C212yesmaximal, complemented
7#15397C214yesmaximal, complemented
8#15130C212yesmaximal, complemented
9#15124C211yesmaximal, characteristic
10#14806C214yesmaximal, complemented
11#14151C212yesmaximal, complemented
12#14130C211yesmaximal, characteristic, complemented
13#14018C212yesmaximal, complemented
14#14001C212yesmaximal, complemented
15C4×#399C211yesmaximal, characteristic, complemented
16C22×#127C2211yescharacteristic
17C22×#123C2214yes—
18C22×#121C2212yes—
19C22×C4×D4C2214yes—
20C22×C4×D4C2212yes—
21C23×C4⋊C4C2211yescharacteristic
22#2455—216no—
23#2445—216no—
24C2×#399C2211yescharacteristic
25C2×#394C2214yes—
26C2×#394C2214yes—
27C2×#394C2214yes—
28C2×#394C2214yes—
29C2×#392C2212yes—
30C2×#392C2212yes—
31#2361—216no—
32C2×#390C2212yes—
33C2×#390C2212yes—
34C2×#390C2212yes—
35C2×#390C2212yes—
36C2×#389C2214yes—
37C2×#389C2212yes—
38C2×#389C2212yes—
39C2×#388C2214yes—
40C2×#388C2214yes—
41C2×#388C2214yes—
42C2×#386C2214yes—
43C2×#386C2214yes—
44C2×#385C2214yes—
45C2×#385C2212yes—
46C2×#385C2214yes—
47C2×#384C2212yes—
48C2×#380C2211yescharacteristic
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2212yes—
53C2×C4×C22⋊C4C2211yescharacteristic
54#1445C2212yes—
55#1445C2211yescharacteristic
56#1444C2212yes—
57#1444C2211yescharacteristic
58#1442C2212yes—
59#1441C2214yes—
60#1441C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2211yescharacteristic
66#1439C2212yes—
67#1439C2214yes—
68#1439C2214yes—
69#1438C2212yes—
70#1438C2212yes—
71C4⋊#95C2211yescharacteristic
72C4⋊#95C2212yes—
73C4⋊#95C2212yes—
74C4⋊#95C2214yes—
75C22⋊#95C2214yes—
76C22⋊#95C2212yes—
77C4×#95C2211yescharacteristic
78C23×D4C2312yes—
79C23×D4C2314yes—
80C23×D4C2312yes—
81C23×D4C2312yes—
82C24×C4C2311yescharacteristic
83C24×C4C2312yes—
84C24×C4C2314yes—
85C2×#127—28no—
86C2×#127—42no—
87C2×#123—216no—
88C2×#123—216no—
89C2×#123—48no—
90C2×#121—28no—
91C2×#121—28no—
92C2×#121—28no—
93C2×C4×D4—28no—
94C2×C4×D4—28no—
95C2×C4×D4—216no—
96C2×C4×D4—216no—
97C2×C4×D4—48no—
98C2×C4×D4—44no—
99C22×C4⋊C4—28no—
100C22×C4⋊C4—42no—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2314yes—
128C22×C42C2312yes—
129C22×C42C2314yes—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133#399D418yescomplemented
134#394—216no—
135#394—216no—
136#394—216no—
137#394—48no—
138#392—28no—
139#392—44no—
140#390—28no—
141#390—28no—
142#390—28no—
143#390D4116yescomplemented
144#389—28no—
145#389—28no—
146#389—216no—
147#388—216no—
148#388—216no—
149#388—48no—
150#386—216no—
151#386—216no—
152#385—216no—
153#385—216no—
154#385—28no—
155#384—28no—
156#380—24no—
157C4×C4⋊C4—28no—
158C4×C4⋊C4—28no—
159C4×C4⋊C4—42no—
160C4×C22⋊C4—28no—
161C4×C22⋊C4—42no—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2311yescharacteristic
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2311yescharacteristic
180C2×#95C2311yescharacteristic
181C2×#95C2311yescharacteristic
182C25C2414yes—
183C25C2411yescharacteristic
184C22×D4—28no—
185C22×D4—28no—
186C22×D4—28no—
187C22×D4—28no—
188C22×D4—28no—
189C22×D4—28no—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—28no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—24no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—24no—
199C22×D4—44no—
200C22×D4—44no—
201C22×D4—44no—
202C22×D4—42no—
203C22×D4—42no—
204C22×D4—44no—
205C22×D4—42no—
206C22×D4—42no—
207C22×D4C2×D418yes—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—28no—
213C23×C4—24no—
214C23×C4—28no—
215C23×C4—28no—
216C23×C4—28no—
217C23×C4—42no—
218C23×C4—44no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4—42no—
223C23×C4—44no—
224C23×C4#4114yes—
225C23×C4C2414yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2414yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2414yes—
235C23×C4C2411yescharacteristic
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238#127—416no—
239#123—464no—
240#121—264no—
241C4×D4—464no—
242C4×D4—432no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—416no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—41no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—41no—
271C2×C4⋊C4—41no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4C2×D418yes—
275C2×C4⋊C4C2×D418yes—
276C2×C4⋊C4C2×D414yes—
277C2×C4⋊C4#4118yes—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—41no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—41no—
311C2×C22⋊C4—41no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4C2×D418yes—
320C2×C22⋊C4C2×D4116yes—
321C2×C22⋊C4C2×D414yes—
322C2×C22⋊C4C2×D4116yes—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—28no—
327C2×C42—24no—
328C2×C42—28no—
329C2×C42—24no—
330C2×C42—24no—
331C2×C42—24no—
332C2×C42—28no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42C2×D418yes—
336#95—24no—
337#95—24no—
338#95—28no—
339#95—24no—
340#95—24no—
341#95—28no—
342#95—28no—
343#95—28no—
344#95—28no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—41no—
349#95—41no—
350#95—41no—
351#95—42no—
352#95—44no—
353#95—42no—
354#95—44no—
355#95C2×D414yes—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—21no—
360C24—21no—
361C24—21no—
362C24—21no—
363C24—22no—
364C24—24no—
365C24—24no—
366C24—24no—
367C24—24no—
368C24—24no—
369C24—24no—
370C24—24no—
371C24—24no—
372C24—24no—
373C24—24no—
374C24—24no—
375C24—24no—
376C24—24no—
377C24—24no—
378C24C22×D418yes—
379C24C22×D412yes—
380C24C2×#4112yes—
381C24H(5,𝔽2)12yes—
382C24C2511yescharacteristic
383C2×D4—232no—
384C2×D4—216no—
385C2×D4—216no—
386C2×D4—216no—
387C2×D4—416no—
388C2×D4—416no—
389C2×D4—416no—
390C2×D4—416no—
391C2×D4—416no—
392C2×D4—416no—
393C2×D4—416no—
394C2×D4—416no—
395C2×D4—48no—
396C2×D4—48no—
397C2×D4—48no—
398C2×D4—48no—
399C2×D4—48no—
400C2×D4—48no—
401C2×D4—84no—
402C2×D4—84no—
403C2×D4—88no—
404C2×D4—88no—
405C2×D4—84no—
406C2×D4—84no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—21no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—216no—
452C22×C4—28no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—48no—
473C22×C4—416no—
474C22×C4—416no—
475C22×C4—416no—
476C22×C4—416no—
477C22×C4—44no—
478C22×C4—48no—
479C22×C4—44no—
480C22×C4—88no—
481C22×C4—88no—
482C22×C4—88no—
483C22×C4—84no—
484C22×C4—88no—
485C22×C4—84no—
486C22×C4—88no—
487C22×C4C22×D414yes—
488C22×C4C22×D412yes—
489C22×C4C22×D414yes—
490C22×C4C22×D418yes—
491C22×C4C22×D412yes—
492C22×C4C22×D414yes—
493C22×C4C22×D414yes—
494C22×C4C22×D414yes—
495C22×C4C2×#4112yes—
496C22×C4C2×#4118yes—
497C22×C4C2×#4112yes—
498C22×C4C2×#4114yes—
499C22×C4H(5,𝔽2)14yes—
500C22×C4H(5,𝔽2)12yes—
501C22×C42−1+412yes—
502C22×C42−1+412yes—
503C4⋊C4—216no—
504C4⋊C4—216no—
505C4⋊C4—216no—
506C4⋊C4—44no—
507C4⋊C4—416no—
508C4⋊C4—48no—
509C4⋊C4—48no—
510C4⋊C4—416no—
511C4⋊C4—416no—
512C4⋊C4—44no—
513C4⋊C4—48no—
514C4⋊C4—88no—
515C22⋊C4—232no—
516C22⋊C4—216no—
517C22⋊C4—232no—
518C22⋊C4—416no—
519C22⋊C4—44no—
520C22⋊C4—416no—
521C22⋊C4—48no—
522C22⋊C4—44no—
523C22⋊C4—416no—
524C22⋊C4—416no—
525C22⋊C4—48no—
526C22⋊C4—416no—
527C22⋊C4—88no—
528C22⋊C4—88no—
529C22⋊C4—88no—
530C42—216no—
531C42—416no—
532C42—48no—
533C42—44no—
534C42—44no—
535C23—24no—
536C23—22no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—22no—
541C23—22no—
542C23—24no—
543C23—22no—
544C23—24no—
545C23—22no—
546C23—22no—
547C23—28no—
548C23—28no—
549C23—28no—
550C23—28no—
551C23—28no—
552C23—28no—
553C23—28no—
554C23—41no—
555C23—42no—
556C23—41no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—41no—
562C23—41no—
563C23—42no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—44no—
572C23—44no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23—44no—
578C23—44no—
579C23—44no—
580C23—44no—
581C23—44no—
582C23—44no—
583C23—44no—
584C23—44no—
585C23—44no—
586C23—44no—
587C23—44no—
588C23—44no—
589C23—44no—
590C23—44no—
591C23—44no—
592C23#55014yes—
593C23C23×D412yes—
594C23C23×D411yescharacteristic
595C23C22×#4111yescharacteristic
596C23C22×#4112yes—
597C23C2×H(5,𝔽2)11yescharacteristic
598C23C2×H(5,𝔽2)12yes—
599C23C2×H(5,𝔽2)11yescharacteristic
600C23C2×2−1+411yescharacteristic
601C23C2×2−1+411yescharacteristic
602C23#58512yes—
603C23#58511yescharacteristic
604D4—464no—
605D4—432no—
606D4—816no—
607D4—816no—
608C2×C4—24no—
609C2×C4—24no—
610C2×C4—24no—
611C2×C4—24no—
612C2×C4—24no—
613C2×C4—24no—
614C2×C4—24no—
615C2×C4—28no—
616C2×C4—22no—
617C2×C4—24no—
618C2×C4—28no—
619C2×C4—24no—
620C2×C4—28no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—24no—
624C2×C4—28no—
625C2×C4—24no—
626C2×C4—24no—
627C2×C4—24no—
628C2×C4—22no—
629C2×C4—28no—
630C2×C4—22no—
631C2×C4—24no—
632C2×C4—28no—
633C2×C4—22no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—44no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—44no—
648C2×C4—42no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—41no—
656C2×C4—41no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—41no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—416no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—41no—
669C2×C4—42no—
670C2×C4—44no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—44no—
675C2×C4—42no—
676C2×C4—44no—
677C2×C4—42no—
678C2×C4—41no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—41no—
682C2×C4—816no—
683C2×C4—832no—
684C2×C4#52518yes—
685C2×C4#53914yes—
686C2×C4D4218yes—
687C2×C4D4214yes—
688C2×C4#54718yes—
689C2×C4#55914yes—
690C22—28no—
691C22—24no—
692C22—42no—
693C22—44no—
694C22—42no—
695C22—42no—
696C22—42no—
697C22—44no—
698C22—44no—
699C22—42no—
700C22—42no—
701C22—48no—
702C22—48no—
703C22—48no—
704C22—48no—
705C22—48no—
706C22—48no—
707C22—48no—
708C22—84no—
709C22—84no—
710C22—84no—
711C22—84no—
712C22—84no—
713C22—84no—
714C22—84no—
715C22—84no—
716C22C2×#52512yes—
717C22C2×#53911yescharacteristic
718C22C2×D4212yes—
719C22C2×D4211yescharacteristic
720C22C2×#54712yes—
721C22C2×#55011yescharacteristic
722C22D4×#4112yes—
723C22C2×#55911yescharacteristic
724C22#348812yes—
725C22#348912yes—
726C22#349612yes—
727C22#349611yescharacteristic
728C22#349911yescharacteristic
729C22#350612yes—
730C22#350711yescharacteristic
731C22#351912yes—
732C22#352212yes—
733C22#352412yes—
734C22#353611yescharacteristic
735C22#354011yescharacteristic
736C22#354911yescharacteristic
737C22#355912yes—
738C22#355911yescharacteristic
739C4—28no—
740C4—48no—
741C4—44no—
742C4—42no—
743C4—42no—
744C4—44no—
745C4—48no—
746C4—42no—
747C4—42no—
748C4—84no—
749C4—82no—
750C4—81no—
751C4#245518yescomplemented
752C2—44no—
753C2—88no—
754C2C2×#245511yescharacteristic
755C2#3577812yes—
756C2#3610112yes—
757C2#3938411yescharacteristic
758C2#4251711yescharacteristic
759C2#4504711yescharacteristic
760C2#4510612yes—
761C2#4542211yescharacteristic
762C2#5911212yes—
763C2#5962811yescharacteristic
764C2#5981611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radical#15124C2
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×#2445C2semidirect
2. C2×#2455C2semidirect
3. C2×#2361C2semidirect
4. #16282C2semidirect
5. #15961C2semidirect
6. #15780C2semidirect
7. #15397C2semidirect
8. #15130C2semidirect
10. #14806C2semidirect
11. #14151C2semidirect
12. #14130C2semidirect
13. #14018C2semidirect
14. #14001C2semidirect
15. C4×#399C2semidirect
133. #399D4semidirect
143. #390D4semidirect
751. C4#2455semidirect

Maximal subgroups

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

subgroupindex
C2×#24452
C2×#24552
C2×#23612
#162822
#159612
#157802
#153972
#151302
#151242
#148062
#141512
#141302
#140182
#140012
C4×#3992