Finite Group Databasebeta

← #760606

Subgroups — #760606

Order 512.

3146 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
1C22×#388C211yesmaximal, characteristic
2C2×#2283C211yesmaximal, characteristic, complemented
3#16704C212yesmaximal, complemented
4#16701C212yesmaximal, complemented
5#15117C214yesmaximal, complemented
6#15023C214yesmaximal, complemented
7#15015C211yesmaximal, characteristic, complemented
8#15014C211yesmaximal, characteristic, complemented
9#14820C211yesmaximal, characteristic
10#14734C211yesmaximal, characteristic
11#14249C212yesmaximal, complemented
12#14231C212yesmaximal, complemented
13#14039C212yesmaximal, complemented
14#13991C212yesmaximal
15#13962C212yesmaximal, complemented
16#13868C212yesmaximal
17#13756C211yesmaximal, characteristic, complemented
18C22×#117C2212yes—
19C23×C4⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23C23×C42C2211yescharacteristic
24C2×#399C2212yes—
25C2×#397C2212yes—
26C2×#388—28no—
27C2×#388C2212yes—
28C2×#388C2212yes—
29C2×#388C2212yes—
30C2×#388C2212yes—
31C2×#388C2214yes—
32C2×#387C2214yescomplemented
33C2×#385C2212yes—
34C2×#385C2212yes—
35#2283—28no—
36C2×#383C2212yescomplemented
37C2×#382C2211yescharacteristic, complemented
38C2×#382C2211yescharacteristic, complemented
39C2×#381C2212yes—
40C2×#380C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×#379C2212yes—
43C2×#379C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic, complemented
45C2×C4×C4⋊C4C2211yescharacteristic, complemented
46C2×C4×C22⋊C4C2212yes—
47C2×#376C2212yescomplemented
48C22×#95C2211yescharacteristic
49C22×#95C2211yescharacteristic
50#1445C2212yescomplemented
51#1445C2212yescomplemented
52#1444C2212yescomplemented
53#1443C2214yescomplemented
54#1442C2214yescomplemented
55#1442C2212yescomplemented
56#1442C2214yescomplemented
57#1441C2214yescomplemented
58#1441C2214yescomplemented
59#1441C2214yescomplemented
60#1440C2214yes—
61#1440C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2212yes—
66#1439C2212yes—
67#1439C2214yes—
68#1437C2214yes—
69#1437C2214yes—
70#1437C2214yes—
71#1436C2214yescomplemented
72C4⋊#95C2212yescomplemented
73C4⋊#95C2212yescomplemented
74C4⋊#95C2214yescomplemented
75#1434C2214yescomplemented
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C22⋊#95C2212yes—
80C22⋊#95C2211yescharacteristic
81C22⋊#95C2212yes—
82C22⋊#95C2211yescharacteristic
83C4×#95C2212yescomplemented
84C4×#95C2212yescomplemented
85C26C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C2×#117—28no—
93C2×#117—28no—
94C2×#117—44no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—44no—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2314yes—
106C22×C22⋊C4—24no—
107C22×C22⋊C4—28no—
108C22×C22⋊C4—24no—
109C22×C22⋊C4—24no—
110C22×C22⋊C4—28no—
111C22×C22⋊C4—44no—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C42—28no—
127C22×C42D418yes—
128C22×C42C2312yes—
129C22×C42C2314yes—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2312yes—
135#399—28no—
136#397—28no—
137#388—28no—
138#388—28no—
139#388—28no—
140#388—28no—
141#388—48no—
142#387—48no—
143#385—28no—
144#385—44no—
145#383—28no—
146#382—24no—
147#382—24no—
148#381—28no—
149#380—24no—
150#380—24no—
151#379—28no—
152#379—44no—
153C4×C4⋊C4—24no—
154C4×C4⋊C4—24no—
155C4×C22⋊C4—28no—
156#376—28no—
157C2×#95—24no—
158C2×#95—24no—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C25—22no—
188C25—24no—
189C25—24no—
190C25—24no—
191C25—22no—
192C25—22no—
193C25—42no—
194C25—42no—
195C25—42no—
196C25C2412yes—
197C25C2411yescharacteristic
198C23×C4—24no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4C2×D414yes—
221C23×C4C2×D414yes—
222C23×C4C2×D414yes—
223C23×C4#4118yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2414yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239#117—432no—
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—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—416no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C22⋊C4—232no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—24no—
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—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—416no—
293C2×C22⋊C4—416no—
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×C22⋊C4—44no—
301C2×C22⋊C4#4118yes—
302C2×C22⋊C4#4118yes—
303C2×C42—232no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—28no—
311C2×C42—28no—
312C2×C42—28no—
313C2×C42—44no—
314C2×C42—44no—
315C2×C42C2×D414yes—
316C2×C42C2×D414yes—
317C2×C42C2×D414yes—
318C2×C42C2×D414yes—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—28no—
332#95—24no—
333#95—28no—
334#95—24no—
335#95—28no—
336#95—44no—
337#95—44no—
338#95—42no—
339#95—42no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—28no—
352C24—22no—
353C24—22no—
354C24—24no—
355C24—24no—
356C24—24no—
357C24—22no—
358C24—22no—
359C24—24no—
360C24—22no—
361C24—22no—
362C24—416no—
363C24—416no—
364C24—48no—
365C24—48no—
366C24—44no—
367C24—44no—
368C24—44no—
369C24—44no—
370C24—44no—
371C24—44no—
372C24—48no—
373C24—48no—
374C24—88no—
375C24—88no—
376C24—84no—
377C24—84no—
378C24C22×D412yes—
379C24C2×#4114yes—
380C24C2×#4114yes—
381C24C2×#4114yes—
382C24H(5,𝔽2)12yes—
383C242−1+412yes—
384C24C2511yescharacteristic
385C22×C4—28no—
386C22×C4—28no—
387C22×C4—216no—
388C22×C4—216no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—28no—
419C22×C4—28no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—416no—
460C22×C4—416no—
461C22×C4—416no—
462C22×C4—44no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—88no—
471C22×C4—84no—
472C22×C4—84no—
473C22×C4#123116yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C2×#4114yes—
481C22×C4C2×#4118yes—
482C22×C4C2×#4114yes—
483C22×C4C2×#4114yes—
484C22×C4C2×#4114yes—
485C22×C4H(5,𝔽2)14yes—
486C4⋊C4—416no—
487C4⋊C4—416no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—48no—
491C4⋊C4—44no—
492C4⋊C4—44no—
493C4⋊C4—48no—
494C4⋊C4—88no—
495C22⋊C4—216no—
496C22⋊C4—216no—
497C22⋊C4—216no—
498C22⋊C4—416no—
499C22⋊C4—48no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—88no—
507C22⋊C4—88no—
508C22⋊C4—88no—
509C42—28no—
510C42—28no—
511C42—28no—
512C42—28no—
513C42—416no—
514C42—48no—
515C42—48no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—28no—
521C23—28no—
522C23—28no—
523C23—24no—
524C23—24no—
525C23—28no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23—28no—
530C23—28no—
531C23—24no—
532C23—24no—
533C23—416no—
534C23—416no—
535C23—416no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—44no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—42no—
547C23—44no—
548C23—44no—
549C23—42no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—832no—
555C23—832no—
556C23—88no—
557C23—88no—
558C23—88no—
559C23—88no—
560C23C2×#12314yes—
561C23#53214yes—
562C23#53714yes—
563C23#55118yes—
564C23#55218yes—
565C23C23×D411yescharacteristic
566C23C22×#4112yes—
567C23C22×#4112yes—
568C23C22×#4112yes—
569C23C2×H(5,𝔽2)12yes—
570C23C2×H(5,𝔽2)11yescharacteristic
571C23C2×2−1+411yescharacteristic
572C23#58512yes—
573C23#58512yes—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—28no—
587C2×C4—28no—
588C2×C4—232no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—28no—
597C2×C4—28no—
598C2×C4—28no—
599C2×C4—28no—
600C2×C4—416no—
601C2×C4—416no—
602C2×C4—416no—
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—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
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—44no—
648C2×C4—44no—
649C2×C4—88no—
650C2×C4—88no—
651C2×C4C2×#12314yes—
652C2×C4C2×#12314yes—
653C2×C4#52514yes—
654C2×C4#53514yes—
655C22—24no—
656C22—216no—
657C22—216no—
658C22—24no—
659C22—24no—
660C22—24no—
661C22—24no—
662C22—24no—
663C22—48no—
664C22—48no—
665C22—44no—
666C22—44no—
667C22—44no—
668C22—832no—
669C22—82no—
670C22—84no—
671C22—84no—
672C22—82no—
673C22—82no—
674C22—82no—
675C22#249818yes—
676C22C22×#12311yescharacteristic
677C22C2×#52511yescharacteristic
678C22C2×#53211yescharacteristic
679C22C2×#53511yescharacteristic
680C22C2×#53711yescharacteristic
681C22#344911yescharacteristic
682C22#344911yescharacteristic
683C22C2×#55112yes—
684C22C2×#55212yes—
685C22#347514yes—
686C22#348114yes—
687C22#349412yes—
688C22#349414yes—
689C22#349512yes—
690C22#352114yes—
691C22#355812yes—
692C22#356112yes—
693C4—24no—
694C4—24no—
695C4—24no—
696C4—24no—
697C4—48no—
698C4—44no—
699C4—42no—
700C4—42no—
701C4—82no—
702C4—84no—
703C4—82no—
704C4—82no—
705C4—82no—
706C4—82no—
707C4—82no—
708C2—28no—
709C2—84no—
710C2C2×#249811yescharacteristic
711C2#3480911yescharacteristic
712C2#3496511yescharacteristic
713C2#3514511yescharacteristic
714C2#3526211yescharacteristic
715C2#3530211yescharacteristic
716C2#3754611yescharacteristic
717C2#5926514yes—
718C2#5928314yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
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×#2283C2semidirect
3. #16704C2semidirect
4. #16701C2semidirect
5. #15117C2semidirect
6. #15023C2semidirect
7. #15015C2semidirect
8. #15014C2semidirect
11. #14249C2semidirect
12. #14231C2semidirect
13. #14039C2semidirect
15. #13962C2semidirect
17. #13756C2semidirect
32. C2×#387C22semidirect
36. C2×#383C22semidirect
37. C2×#382C22semidirect
38. C2×#382C22semidirect
44. C2×C4×C4⋊C4C22semidirect
45. C2×C4×C4⋊C4C22semidirect
47. C2×#376C22semidirect
50. #1445C22semidirect
51. #1445C22semidirect
52. #1444C22semidirect
53. #1443C22semidirect
54. #1442C22semidirect
55. #1442C22semidirect
56. #1442C22semidirect
57. #1441C22semidirect
58. #1441C22semidirect
59. #1441C22semidirect
71. #1436C22semidirect
72. C4⋊#95C22semidirect
73. C4⋊#95C22semidirect
74. C4⋊#95C22semidirect
75. #1434C22semidirect
83. C4×#95C22semidirect
84. C4×#95C22semidirect

Maximal subgroups

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

subgroupindex
C22×#3882
C2×#22832
#167042
#167012
#151172
#150232
#150152
#150142
#148202
#147342
#142492
#142312
#140392
#139912
#139622
#138682
#137562