Finite Group Databasebeta

← #769360

Subgroups — #769360

Order 512.

3834 conjugacy classes of subgroups.

The subgroup lattice has 782 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#394C211yesmaximal, characteristic
2C2×#2312C211yesmaximal, characteristic, complemented
3#15709C214yesmaximal, complemented
4#15705C214yesmaximal, complemented
5#15647C214yesmaximal, complemented
6#15643C214yesmaximal, complemented
7#14918C212yesmaximal, complemented
8#14724C212yesmaximal
9#13992C212yesmaximal, complemented
10#13991C212yesmaximal, complemented
11#13903C211yesmaximal, characteristic, complemented
12#13903C211yesmaximal, characteristic, complemented
13#13881C211yesmaximal, characteristic, complemented
14#13880C211yesmaximal, characteristic, complemented
15#13775C211yesmaximal, characteristic, complemented
16C24×D4C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C23×C22⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22C2×#394—28no—
23C2×#394C2214yes—
24C2×#394C2214yes—
25C2×#394C2214yes—
26C2×#394C2214yes—
27#2312—28no—
28C2×#388C2212yescomplemented
29C2×#388C2212yescomplemented
30C2×#388C2212yescomplemented
31C2×#388C2212yescomplemented
32C2×#388C2214yescomplemented
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#384C2212yescomplemented
36C2×#382C2212yescomplemented
37C2×#381C2212yes—
38C2×#381C2212yes—
39C2×#380C2211yescharacteristic
40C2×#380C2211yescharacteristic
41C2×#379C2212yes—
42C2×#379C2212yes—
43C2×#379C2212yes—
44C2×#379C2212yes—
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2212yes—
48C22×#95C2211yescharacteristic
49#1444C2214yescomplemented
50#1443C2214yescomplemented
51#1442C2214yescomplemented
52#1442C2214yescomplemented
53#1440C2214yescomplemented
54#1440C2214yescomplemented
55#1440C2214yescomplemented
56#1439C2212yescomplemented
57#1439C2214yescomplemented
58#1439C2214yescomplemented
59#1439C2214yescomplemented
60#1439C2212yescomplemented
61#1439C2212yescomplemented
62#1439C2212yescomplemented
63#1438C2214yes—
64#1438C2214yes—
65#1437C2214yescomplemented
66#1437C2214yescomplemented
67#1437C2214yescomplemented
68#1437C2214yescomplemented
69#1437C2214yescomplemented
70#1436C2214yescomplemented
71C4⋊#95C2212yescomplemented
72C4⋊#95C2212yescomplemented
73#1434C2214yescomplemented
74C22⋊#95C2212yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C26C2311yescharacteristic
78C26C2311yescharacteristic
79C23×D4—24no—
80C23×D4—44no—
81C23×D4—44no—
82C23×D4—44no—
83C23×D4C2314yes—
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C22×C4⋊C4—24no—
90C22×C4⋊C4—44no—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2314yes—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2311yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2314yes—
99C22×C22⋊C4—24no—
100C22×C22⋊C4—24no—
101C22×C22⋊C4—24no—
102C22×C22⋊C4—24no—
103C22×C22⋊C4—24no—
104C22×C22⋊C4—24no—
105C22×C22⋊C4—24no—
106C22×C22⋊C4—44no—
107C22×C22⋊C4—44no—
108C22×C22⋊C4D418yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2314yes—
129C22×C42C2312yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132C22×C42C2312yes—
133#394—216no—
134#394—216no—
135#394—48no—
136#394—48no—
137#388—28no—
138#388—28no—
139#388—28no—
140#388—28no—
141#388—48no—
142#385—44no—
143#385—44no—
144#384—44no—
145#382—44no—
146#381—28no—
147#381—44no—
148#380—24no—
149#380—24no—
150#379—28no—
151#379—28no—
152#379—28no—
153#379—44no—
154C4×C22⋊C4—28no—
155C4×C22⋊C4—24no—
156C4×C22⋊C4—24no—
157C2×#95—24no—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C25—22no—
178C25—24no—
179C25—24no—
180C25—22no—
181C25—22no—
182C25—24no—
183C25—22no—
184C25—22no—
185C25—24no—
186C25—24no—
187C25—42no—
188C25—42no—
189C25—42no—
190C25—42no—
191C25—42no—
192C25—42no—
193C25—42no—
194C25C2×D414yes—
195C25C2412yes—
196C25C2412yes—
197C25C2411yescharacteristic
198C22×D4—28no—
199C22×D4—28no—
200C22×D4—28no—
201C22×D4—44no—
202C22×D4—44no—
203C22×D4—44no—
204C22×D4—44no—
205C22×D4—416no—
206C22×D4—416no—
207C22×D4—416no—
208C22×D4—464no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4C2×D414yes—
229C23×C4C2×D414yes—
230C23×C4#4114yes—
231C23×C4#4114yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2414yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C23×C4C2414yes—
241C23×C4C2412yes—
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—416no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—232no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
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—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—416no—
305C2×C22⋊C4—416no—
306C2×C22⋊C4—416no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
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⋊C4—44no—
322C2×C22⋊C4C2×D418yes—
323C2×C22⋊C4C2×D418yes—
324C2×C22⋊C4#4118yes—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—24no—
328C2×C42—24no—
329C2×C42—42no—
330C2×C42—42no—
331C2×C42—42no—
332C2×C42—42no—
333C2×C42—44no—
334C2×C42—44no—
335#95—28no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—28no—
343#95—28no—
344#95—28no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95—44no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—28no—
359C24—28no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24—24no—
368C24—24no—
369C24—24no—
370C24—24no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—24no—
375C24—24no—
376C24—24no—
377C24—22no—
378C24—22no—
379C24—24no—
380C24—28no—
381C24—28no—
382C24—44no—
383C24—44no—
384C24—44no—
385C24—44no—
386C24—416no—
387C24—48no—
388C24—48no—
389C24—416no—
390C24—44no—
391C24—416no—
392C24—44no—
393C24—44no—
394C24—416no—
395C24—44no—
396C24—44no—
397C24—44no—
398C24—48no—
399C24—48no—
400C24—48no—
401C24—48no—
402C24—84no—
403C24—84no—
404C24—84no—
405C24—84no—
406C24—84no—
407C24—84no—
408C24—88no—
409C24—88no—
410C24—88no—
411C24—88no—
412C24C22×D414yes—
413C24C22×D412yes—
414C24C2×#4112yes—
415C24C2×#4112yes—
416C24C2×#4114yes—
417C24H(5,𝔽2)14yes—
418C24H(5,𝔽2)14yes—
419C24C2511yescharacteristic
420C2×D4—416no—
421C2×D4—416no—
422C2×D4—416no—
423C2×D4—416no—
424C2×D4—416no—
425C2×D4—416no—
426C2×D4—4128no—
427C2×D4—416no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—28no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—28no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—22no—
469C22×C4—24no—
470C22×C4—22no—
471C22×C4—24no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—24no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—24no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—48no—
488C22×C4—48no—
489C22×C4—44no—
490C22×C4—44no—
491C22×C4—416no—
492C22×C4—44no—
493C22×C4—44no—
494C22×C4—44no—
495C22×C4—44no—
496C22×C4—44no—
497C22×C4—44no—
498C22×C4—44no—
499C22×C4—44no—
500C22×C4—48no—
501C22×C4—48no—
502C22×C4—84no—
503C22×C4—84no—
504C22×C4—88no—
505C22×C4#123116yes—
506C22×C4C22×D414yes—
507C22×C4C22×D412yes—
508C22×C4C22×D412yes—
509C22×C4C2×#4118yes—
510C22×C4C2×#4112yes—
511C22×C4C2×#4112yes—
512C22×C4C2×#4112yes—
513C22×C4C2×#4112yes—
514C22×C4H(5,𝔽2)14yes—
515C4⋊C4—48no—
516C4⋊C4—48no—
517C4⋊C4—82no—
518C4⋊C4—82no—
519C4⋊C4—82no—
520C4⋊C4—82no—
521C4⋊C4—88no—
522C4⋊C4—88no—
523C22⋊C4—216no—
524C22⋊C4—216no—
525C22⋊C4—216no—
526C22⋊C4—416no—
527C22⋊C4—416no—
528C22⋊C4—416no—
529C22⋊C4—48no—
530C22⋊C4—48no—
531C22⋊C4—48no—
532C22⋊C4—48no—
533C22⋊C4—48no—
534C22⋊C4—88no—
535C22⋊C4—88no—
536C22⋊C4—88no—
537C22⋊C4—84no—
538C22⋊C4—84no—
539C22⋊C4—84no—
540C22⋊C4—84no—
541C22⋊C4—88no—
542C22⋊C4—88no—
543C42—48no—
544C42—48no—
545C42—84no—
546C42—84no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—22no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—28no—
556C23—28no—
557C23—28no—
558C23—28no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—24no—
563C23—24no—
564C23—24no—
565C23—24no—
566C23—24no—
567C23—24no—
568C23—24no—
569C23—28no—
570C23—28no—
571C23—416no—
572C23—416no—
573C23—416no—
574C23—416no—
575C23—416no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—44no—
580C23—42no—
581C23—42no—
582C23—44no—
583C23—42no—
584C23—42no—
585C23—42no—
586C23—42no—
587C23—42no—
588C23—42no—
589C23—42no—
590C23—44no—
591C23—44no—
592C23—44no—
593C23—44no—
594C23—44no—
595C23—44no—
596C23—44no—
597C23—44no—
598C23—44no—
599C23—44no—
600C23—44no—
601C23—44no—
602C23—44no—
603C23—416no—
604C23—44no—
605C23—44no—
606C23—44no—
607C23—44no—
608C23—44no—
609C23—44no—
610C23—44no—
611C23—88no—
612C23—88no—
613C23—88no—
614C23—88no—
615C23—832no—
616C23—832no—
617C23—88no—
618C23—88no—
619C23—88no—
620C23—832no—
621C23—832no—
622C23—88no—
623C23C2×#12314yes—
624C23#53214yes—
625C23#53514yes—
626C23#55318yes—
627C23#55518yes—
628C23C23×D411yescharacteristic
629C23C22×#4111yescharacteristic
630C23C22×#4111yescharacteristic
631C23C22×#4112yes—
632C23C2×H(5,𝔽2)12yes—
633C23C2×H(5,𝔽2)12yes—
634C23C2×H(5,𝔽2)12yes—
635C23#58512yes—
636C23#58512yes—
637D4—464no—
638C2×C4—22no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—24no—
643C2×C4—24no—
644C2×C4—24no—
645C2×C4—24no—
646C2×C4—24no—
647C2×C4—28no—
648C2×C4—24no—
649C2×C4—24no—
650C2×C4—24no—
651C2×C4—28no—
652C2×C4—28no—
653C2×C4—28no—
654C2×C4—416no—
655C2×C4—416no—
656C2×C4—42no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—42no—
666C2×C4—42no—
667C2×C4—42no—
668C2×C4—42no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—42no—
677C2×C4—42no—
678C2×C4—42no—
679C2×C4—42no—
680C2×C4—42no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—42no—
684C2×C4—42no—
685C2×C4—42no—
686C2×C4—44no—
687C2×C4—44no—
688C2×C4—44no—
689C2×C4—44no—
690C2×C4—44no—
691C2×C4—44no—
692C2×C4—44no—
693C2×C4—44no—
694C2×C4—44no—
695C2×C4—44no—
696C2×C4—44no—
697C2×C4—44no—
698C2×C4—44no—
699C2×C4—44no—
700C2×C4—44no—
701C2×C4—44no—
702C2×C4—88no—
703C2×C4—88no—
704C2×C4—88no—
705C2×C4C2×#12314yes—
706C2×C4C2×#12314yes—
707C22—24no—
708C22—24no—
709C22—24no—
710C22—24no—
711C22—24no—
712C22—216no—
713C22—216no—
714C22—24no—
715C22—44no—
716C22—44no—
717C22—44no—
718C22—44no—
719C22—44no—
720C22—44no—
721C22—44no—
722C22—48no—
723C22—48no—
724C22—44no—
725C22—44no—
726C22—44no—
727C22—832no—
728C22—832no—
729C22—82no—
730C22—82no—
731C22—82no—
732C22—84no—
733C22—84no—
734C22—84no—
735C22—84no—
736C22—84no—
737C22—84no—
738C22—82no—
739C22#250018yes—
740C22C22×#12311yescharacteristic
741C22C2×#53211yescharacteristic
742C22C2×#53511yescharacteristic
743C22C2×#55312yes—
744C22C2×#55512yes—
745C22#348411yescharacteristic
746C22#348411yescharacteristic
747C22#348511yescharacteristic
748C22#348511yescharacteristic
749C22#349412yes—
750C22#349412yes—
751C22#350512yes—
752C22#350512yes—
753C22#351314yes—
754C22#351414yes—
755C22#353214yes—
756C22#353314yes—
757C4—24no—
758C4—24no—
759C4—44no—
760C4—48no—
761C4—44no—
762C4—82no—
763C4—82no—
764C4—82no—
765C4—82no—
766C4—82no—
767C4—84no—
768C4—82no—
769C2—28no—
770C2—84no—
771C2—84no—
772C2C2×#250011yescharacteristic
773C2#3535111yescharacteristic
774C2#3539411yescharacteristic
775C2#3546611yescharacteristic
776C2#3554011yescharacteristic
777C2#3670311yescharacteristic
778C2#3695611yescharacteristic
779C2#4393814yes—
780C2#4394114yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC24×D4C22
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×#2312C2semidirect
3. #15709C2semidirect
4. #15705C2semidirect
5. #15647C2semidirect
6. #15643C2semidirect
7. #14918C2semidirect
9. #13992C2semidirect
10. #13991C2semidirect
11. #13903C2semidirect
12. #13903C2semidirect
13. #13881C2semidirect
14. #13880C2semidirect
15. #13775C2semidirect
28. C2×#388C22semidirect
29. C2×#388C22semidirect
30. C2×#388C22semidirect
31. C2×#388C22semidirect
32. C2×#388C22semidirect
35. C2×#384C22semidirect
36. C2×#382C22semidirect
49. #1444C22semidirect
50. #1443C22semidirect
51. #1442C22semidirect
52. #1442C22semidirect
53. #1440C22semidirect
54. #1440C22semidirect
55. #1440C22semidirect
56. #1439C22semidirect
57. #1439C22semidirect
58. #1439C22semidirect
59. #1439C22semidirect
60. #1439C22semidirect
61. #1439C22semidirect
62. #1439C22semidirect
65. #1437C22semidirect
66. #1437C22semidirect
67. #1437C22semidirect
68. #1437C22semidirect
69. #1437C22semidirect
70. #1436C22semidirect
71. C4⋊#95C22semidirect
72. C4⋊#95C22semidirect
73. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C22×#3942
C2×#23122
#157092
#157052
#156472
#156432
#149182
#147242
#139922
#139912
#139032
#138812
#138802
#137752