Finite Group Databasebeta

← #764902

Subgroups — #764902

Order 512.

3488 conjugacy classes of subgroups.

The subgroup lattice has 751 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2487C211yesmaximal, characteristic, complemented
2C22×#399C211yesmaximal, characteristic, complemented
3C2×#2321C211yesmaximal, characteristic
4C2×#2312C211yesmaximal, characteristic, complemented
5#15758C214yesmaximal, complemented
6#15696C214yesmaximal, complemented
7#15394C214yesmaximal, complemented
8#15351C214yesmaximal, complemented
9#14727C212yesmaximal, complemented
10#14718C212yesmaximal
11#14149C211yesmaximal, characteristic, complemented
12#14043C212yesmaximal, complemented
13#14019C212yesmaximal, complemented
14#13960C211yesmaximal, characteristic, complemented
15#13924C211yesmaximal, characteristic, complemented
16C22×#120C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2212yes—
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2212yes—
22#2487—28no—
23C2×#400C2212yescomplemented
24C2×#399—28no—
25C2×#399C2214yes—
26C2×#399C2214yes—
27C2×#399C2212yescomplemented
28C2×#397C2214yes—
29C2×#394C2214yes—
30#2321—28no—
31#2312—28no—
32C2×#390C2212yes—
33C2×#389C2212yescomplemented
34C2×#389C2214yescomplemented
35C2×#388C2212yes—
36C2×#388C2212yes—
37C2×#388C2214yes—
38C2×#388C2214yes—
39C2×#388C2214yescomplemented
40C2×#386C2212yes—
41C2×#386C2211yescharacteristic
42C2×#386C2212yes—
43C2×#386C2212yes—
44C2×#385C2211yescharacteristic
45C2×#385C2212yes—
46C2×#385C2212yes—
47C2×#380C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×#380C2212yescomplemented
50C2×#379C2211yescharacteristic
51C2×#379C2212yes—
52C2×C4×C22⋊C4C2211yescharacteristic
53C2×C4×C22⋊C4C2212yes—
54C22×#95C2211yescharacteristic
55#1445C2214yescomplemented
56#1444C2214yescomplemented
57#1442C2214yescomplemented
58#1441C2214yescomplemented
59#1441C2214yescomplemented
60#1441C2212yescomplemented
61#1440C2214yes—
62#1440C2214yes—
63#1440C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yescomplemented
67#1439C2212yescomplemented
68#1438C2214yes—
69#1436C2214yescomplemented
70#1436C2214yescomplemented
71#1436C2214yescomplemented
72C4⋊#95C2214yescomplemented
73#1434C2212yescomplemented
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yescomplemented
79C26C2311yescharacteristic
80C23×D4C2311yescharacteristic
81C23×D4C2312yes—
82C24×C4C2312yes—
83C24×C4C2312yes—
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C2×#120—24no—
88C2×#120—24no—
89C2×#120—24no—
90C2×C4×D4—24no—
91C2×C4×D4—24no—
92C2×C4×D4—24no—
93C22×C4⋊C4—28no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—48no—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C22⋊C4—24no—
109C22×C22⋊C4—24no—
110C22×C22⋊C4—28no—
111C22×C22⋊C4—48no—
112C22×C22⋊C4D418yes—
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C42C2311yescharacteristic
132C22×C42C2312yes—
133C22×C42C2314yes—
134#400—28no—
135#399—216no—
136#399—216no—
137#399—28no—
138#397—48no—
139#394—48no—
140#390—28no—
141#389—28no—
142#389—48no—
143#388—216no—
144#388—28no—
145#388—28no—
146#388—216no—
147#388—48no—
148#386—28no—
149#386—24no—
150#386—28no—
151#386—28no—
152#385—24no—
153#385—28no—
154#385—28no—
155#380—24no—
156#380—28no—
157#380D418yescomplemented
158#379—24no—
159#379—28no—
160C4×C22⋊C4—24no—
161C4×C22⋊C4—28no—
162C2×#95—24no—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C25—22no—
184C25—22no—
185C25—22no—
186C25—22no—
187C25—22no—
188C25—22no—
189C25—44no—
190C25—44no—
191C25C2×D414yes—
192C25C2412yes—
193C25C2411yescharacteristic
194C25C2411yescharacteristic
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—22no—
199C22×D4—22no—
200C22×D4—22no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—44no—
205C22×D4—44no—
206C22×D4#4114yes—
207C23×C4—24no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—28no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—44no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4—44no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4C2×D414yes—
226C23×C4C2×D414yes—
227C23×C4C2×D414yes—
228C23×C4#4114yes—
229C23×C4C2411yescharacteristic
230C23×C4C2414yes—
231C23×C4C2412yes—
232C23×C4C2414yes—
233C23×C4C2411yescharacteristic
234C23×C4C2414yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2411yescharacteristic
238C23×C4C2414yes—
239C23×C4C2411yescharacteristic
240C23×C4C2411yescharacteristic
241#120—232no—
242C4×D4—416no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—432no—
266C2×C4⋊C4—416no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4C2×D418yes—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—232no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—28no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—432no—
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⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4C2×D418yes—
322C2×C22⋊C4C2×D418yes—
323C2×C22⋊C4C2×D418yes—
324C2×C22⋊C4#4114yes—
325C2×C42—22no—
326C2×C42—22no—
327C2×C42—22no—
328C2×C42—28no—
329C2×C42—24no—
330C2×C42—44no—
331C2×C42—44no—
332C2×C42—44no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—28no—
337#95—28no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—28no—
342#95—28no—
343#95—28no—
344#95—28no—
345#95—28no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95C2×D418yes—
353C24—24no—
354C24—24no—
355C24—28no—
356C24—22no—
357C24—21no—
358C24—21no—
359C24—22no—
360C24—21no—
361C24—22no—
362C24—22no—
363C24—21no—
364C24—24no—
365C24—24no—
366C24—24no—
367C24—21no—
368C24—21no—
369C24—22no—
370C24—22no—
371C24—24no—
372C24—24no—
373C24—24no—
374C24—28no—
375C24—28no—
376C24—48no—
377C24—48no—
378C24—416no—
379C24—416no—
380C24—44no—
381C24—44no—
382C24—44no—
383C24—44no—
384C24—84no—
385C24—84no—
386C24—84no—
387C24—84no—
388C24—88no—
389C24—88no—
390C24C22×D414yes—
391C24C22×D412yes—
392C24C22×D412yes—
393C24C2×#4112yes—
394C24C2×#4112yes—
395C24C2×#4112yes—
396C24C2×#4112yes—
397C242−1+412yes—
398C242−1+412yes—
399C24C2511yescharacteristic
400C2×D4—28no—
401C2×D4—28no—
402C2×D4—28no—
403C2×D4—216no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—44no—
407C2×D4—44no—
408C2×D4—88no—
409C2×D4—84no—
410C2×D4—84no—
411C2×D4—88no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—216no—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—28no—
442C22×C4—21no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—48no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—416no—
474C22×C4—416no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4—44no—
482C22×C4—84no—
483C22×C4—84no—
484C22×C4—84no—
485C22×C4—84no—
486C22×C4—88no—
487C22×C4—88no—
488C22×C4#121116yes—
489C22×C4C22×D418yes—
490C22×C4C22×D414yes—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C2×#4112yes—
496C22×C4C2×#4114yes—
497C22×C4C2×#4112yes—
498C22×C4C2×#4112yes—
499C22×C4H(5,𝔽2)18yes—
500C4⋊C4—48no—
501C4⋊C4—416no—
502C4⋊C4—416no—
503C4⋊C4—416no—
504C4⋊C4—48no—
505C4⋊C4—48no—
506C4⋊C4—44no—
507C4⋊C4—84no—
508C4⋊C4—84no—
509C4⋊C4—88no—
510C22⋊C4—216no—
511C22⋊C4—28no—
512C22⋊C4—216no—
513C22⋊C4—216no—
514C22⋊C4—216no—
515C22⋊C4—416no—
516C22⋊C4—416no—
517C22⋊C4—416no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—48no—
521C22⋊C4—44no—
522C22⋊C4—84no—
523C22⋊C4—88no—
524C22⋊C4—88no—
525C22⋊C4—88no—
526C22⋊C4—84no—
527C22⋊C4—84no—
528C42—44no—
529C42—84no—
530C42—88no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—28no—
538C23—28no—
539C23—28no—
540C23—28no—
541C23—24no—
542C23—24no—
543C23—22no—
544C23—24no—
545C23—22no—
546C23—22no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—28no—
552C23—432no—
553C23—41no—
554C23—416no—
555C23—41no—
556C23—41no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—41no—
561C23—42no—
562C23—42no—
563C23—44no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—44no—
568C23—42no—
569C23—44no—
570C23—42no—
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—816no—
584C23—88no—
585C23—88no—
586C23—816no—
587C23—816no—
588C23—816no—
589C23—816no—
590C23C2×#12114yes—
591C23#52514yes—
592C23#53614yes—
593C23#54714yes—
594C23#54814yes—
595C23#55214yes—
596C23#55614yes—
597C23C23×D411yescharacteristic
598C23C23×D411yescharacteristic
599C23C22×#4111yescharacteristic
600C23C22×#4111yescharacteristic
601C23C22×#4111yescharacteristic
602C23C2×H(5,𝔽2)14yes—
603C23C2×2−1+411yescharacteristic
604C23C2×2−1+411yescharacteristic
605C23#58514yes—
606D4—416no—
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—24no—
616C2×C4—22no—
617C2×C4—28no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—28no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—28no—
624C2×C4—28no—
625C2×C4—28no—
626C2×C4—28no—
627C2×C4—28no—
628C2×C4—416no—
629C2×C4—432no—
630C2×C4—41no—
631C2×C4—44no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—44no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—44no—
673C2×C4—44no—
674C2×C4—88no—
675C2×C4—88no—
676C2×C4—816no—
677C2×C4C2×#12114yes—
678C2×C4C2×#12114yes—
679C22—28no—
680C22—28no—
681C22—28no—
682C22—28no—
683C22—28no—
684C22—28no—
685C22—24no—
686C22—24no—
687C22—44no—
688C22—44no—
689C22—44no—
690C22—44no—
691C22—44no—
692C22—44no—
693C22—44no—
694C22—44no—
695C22—42no—
696C22—44no—
697C22—44no—
698C22—832no—
699C22—84no—
700C22—82no—
701C22—84no—
702C22—82no—
703C22—84no—
704C22—84no—
705C22—82no—
706C22—81no—
707C22—81no—
708C22#241518yes—
709C22C22×#12111yescharacteristic
710C22C2×#52511yescharacteristic
711C22C2×#53611yescharacteristic
712C22C2×#54711yescharacteristic
713C22C2×#54811yescharacteristic
714C22C2×#55211yescharacteristic
715C22C2×#55611yescharacteristic
716C22#348412yes—
717C22#348512yes—
718C22#348512yes—
719C22#349112yes—
720C22#349812yes—
721C22#350812yes—
722C22#351314yes—
723C22#351314yes—
724C22#354114yes—
725C22#354114yes—
726C4—24no—
727C4—24no—
728C4—48no—
729C4—44no—
730C4—48no—
731C4—42no—
732C4—44no—
733C4—84no—
734C4—82no—
735C4—81no—
736C4—81no—
737C4—84no—
738C2—28no—
739C2—82no—
740C2—84no—
741C2C2×#241511yescharacteristic
742C2#3535412yes—
743C2#3544711yescharacteristic
744C2#3552811yescharacteristic
745C2#3670512yes—
746C2#4408112yes—
747C2#4408212yes—
748C2#4410212yes—
749C2#4410312yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#120C22
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×#2487C2semidirect
2. C22×#399C2semidirect
4. C2×#2312C2semidirect
5. #15758C2semidirect
6. #15696C2semidirect
7. #15394C2semidirect
8. #15351C2semidirect
9. #14727C2semidirect
11. #14149C2semidirect
12. #14043C2semidirect
13. #14019C2semidirect
14. #13960C2semidirect
15. #13924C2semidirect
23. C2×#400C22semidirect
27. C2×#399C22semidirect
33. C2×#389C22semidirect
34. C2×#389C22semidirect
39. C2×#388C22semidirect
49. C2×#380C22semidirect
55. #1445C22semidirect
56. #1444C22semidirect
57. #1442C22semidirect
58. #1441C22semidirect
59. #1441C22semidirect
60. #1441C22semidirect
66. #1439C22semidirect
67. #1439C22semidirect
69. #1436C22semidirect
70. #1436C22semidirect
71. #1436C22semidirect
72. C4⋊#95C22semidirect
73. #1434C22semidirect
78. C22⋊#95C22semidirect
157. #380D4semidirect

Maximal subgroups

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

subgroupindex
C2×#24872
C22×#3992
C2×#23212
C2×#23122
#157582
#156962
#153942
#153512
#147272
#147182
#141492
#140432
#140192
#139602
#139242