Finite Group Databasebeta

← #920678

Subgroups — #920678

Order 512.

3692 conjugacy classes of subgroups.

The subgroup lattice has 791 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2404C211yesmaximal, characteristic
2C22×#397C211yesmaximal, characteristic, complemented
3#17201C212yesmaximal, complemented
4#17147C212yesmaximal, complemented
5#16831C212yesmaximal, complemented
6#16808C212yesmaximal, complemented
7#16380C214yesmaximal, complemented
8#15494C214yesmaximal, complemented
9#15091C212yesmaximal, complemented
10#14919C212yesmaximal, complemented
11#14718C212yesmaximal, complemented
12#14259C212yesmaximal, complemented
13#14134C212yesmaximal, complemented
14#13956C212yesmaximal, complemented
15#13869C211yesmaximal, characteristic, complemented
16C23×#41C2211yescharacteristic
17C22×#120C2212yes—
18C23×C4⋊C4C2211yescharacteristic
19C23×C22⋊C4C2212yes—
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22#2404—28no—
23C2×#397—28no—
24C2×#397C2212yes—
25C2×#397C2212yes—
26C2×#397C2212yes—
27C2×#397C2212yes—
28C2×#394C2214yes—
29C2×#393C2211yescharacteristic
30C2×#393C2211yescharacteristic
31C2×#392C2211yescharacteristic
32C2×#392C2211yescharacteristic
33C2×#386C2212yes—
34C2×#386C2212yes—
35C2×#386C2212yes—
36C2×#386C2212yes—
37C2×#386C2212yes—
38C2×#386C2212yes—
39C2×#386C2214yes—
40C2×#386C2212yes—
41C2×#385C2212yes—
42C2×#381C2211yescharacteristic
43C2×#381C2212yes—
44C2×#379C2211yescharacteristic
45C22×#95C2212yes—
46C22×#95C2211yescharacteristic
47#1446C2214yescomplemented
48#1445C2214yescomplemented
49#1443C2214yescomplemented
50#1443C2214yescomplemented
51#1443C2214yescomplemented
52#1442C2214yescomplemented
53#1442C2214yescomplemented
54#1442C2214yescomplemented
55#1442C2214yescomplemented
56#1440C2212yes—
57#1440C2212yes—
58#1440C2214yes—
59#1440C2214yes—
60#1440C2214yescomplemented
61#1439C2214yes—
62#1439C2214yescomplemented
63#1439C2214yescomplemented
64#1439C2214yescomplemented
65#1438C2214yes—
66#1438C2214yes—
67#1438C2214yescomplemented
68C22⋊#95C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2212yes—
71C22⋊#95C2212yes—
72C22⋊#95C2212yes—
73C22⋊#95C2214yes—
74C22⋊#95C2212yes—
75C22⋊#95C2214yescomplemented
76C22⋊#95C2214yescomplemented
77C26C2311yescharacteristic
78C22×#41—28no—
79C22×#41—24no—
80C22×#41—42no—
81C22×#41—42no—
82C22×#41—44no—
83C23×Q8C2311yescharacteristic
84C23×D4C2312yes—
85C23×D4C2312yes—
86C23×D4C2311yescharacteristic
87C23×D4C2312yes—
88C24×C4C2311yescharacteristic
89C24×C4C2312yes—
90C24×C4C2312yes—
91C24×C4C2312yes—
92C24×C4C2311yescharacteristic
93C24×C4C2312yes—
94C2×#120—28no—
95C2×#120—28no—
96C2×#120—44no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—42no—
99C22×C4⋊C4—42no—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C22⋊C4—28no—
104C22×C22⋊C4—44no—
105C22×C22⋊C4—42no—
106C22×C22⋊C4—48no—
107C22×C22⋊C4—42no—
108C22×C22⋊C4—42no—
109C22×C22⋊C4D418yes—
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
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⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2311yescharacteristic
129C22×C22⋊C4C2311yescharacteristic
130#397—28no—
131#397—28no—
132#397—28no—
133#397—28no—
134#394—48no—
135#393—24no—
136#393—24no—
137#392—24no—
138#392—24no—
139#386—28no—
140#386—28no—
141#386—28no—
142#386—28no—
143#386—44no—
144#386—44no—
145#386—48no—
146#386—44no—
147#385—44no—
148#381—24no—
149#381—44no—
150#379—42no—
151C2×#95—28no—
152C2×#95—24no—
153C2×#95C2311yescharacteristic
154C2×#95C2311yescharacteristic
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C25—24no—
178C25—24no—
179C25—22no—
180C25—24no—
181C25—22no—
182C25—42no—
183C25—44no—
184C25C2×D414yes—
185C25C2411yescharacteristic
186C25C2412yes—
187C25C2411yescharacteristic
188C25C2411yescharacteristic
189C2×#41—432no—
190C2×#41—416no—
191C2×#41—416no—
192C2×#41—432no—
193C2×#41—416no—
194C22×Q8—22no—
195C22×Q8—24no—
196C22×Q8—42no—
197C22×Q8—41no—
198C22×Q8—41no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—24no—
205C22×D4—24no—
206C22×D4—24no—
207C22×D4—24no—
208C22×D4—24no—
209C22×D4—28no—
210C22×D4—24no—
211C22×D4—22no—
212C22×D4—42no—
213C22×D4—41no—
214C22×D4—41no—
215C22×D4—42no—
216C22×D4—42no—
217C22×D4—44no—
218C22×D4—44no—
219C22×D4—44no—
220C22×D4#4118yes—
221C23×C4—24no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—24no—
230C23×C4—24no—
231C23×C4—24no—
232C23×C4—24no—
233C23×C4—24no—
234C23×C4—24no—
235C23×C4—24no—
236C23×C4—24no—
237C23×C4—44no—
238C23×C4—42no—
239C23×C4—42no—
240C23×C4—42no—
241C23×C4—42no—
242C23×C4—44no—
243C23×C4—44no—
244C23×C4—42no—
245C23×C4—44no—
246C23×C4—44no—
247C23×C4—44no—
248C23×C4—44no—
249C23×C4—42no—
250C23×C4C2×D414yes—
251C23×C4C2×D414yes—
252C23×C4C2414yes—
253C23×C4C2414yes—
254C23×C4C2414yes—
255C23×C4C2414yes—
256C23×C4C2412yes—
257C23×C4C2412yes—
258C23×C4C2411yescharacteristic
259C23×C4C2411yescharacteristic
260C23×C4C2411yescharacteristic
261C23×C4C2411yescharacteristic
262C23×C4C2412yes—
263#120—432no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—416no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
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—416no—
296C2×C22⋊C4—41no—
297C2×C22⋊C4—41no—
298C2×C22⋊C4—41no—
299C2×C22⋊C4—41no—
300C2×C22⋊C4—41no—
301C2×C22⋊C4—41no—
302C2×C22⋊C4—41no—
303C2×C22⋊C4—41no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—432no—
311C2×C22⋊C4—416no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—42no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—42no—
321C2×C22⋊C4—42no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4C2×D414yes—
330C2×C22⋊C4C2×D414yes—
331C2×C22⋊C4C2×D414yes—
332C2×C22⋊C4C2×D414yes—
333#95—24no—
334#95—24no—
335#95—28no—
336#95—28no—
337#95—24no—
338#95—22no—
339#95—22no—
340#95—24no—
341#95—24no—
342#95—28no—
343#95—44no—
344#95—44no—
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—
354#95—44no—
355#95—44no—
356#95—44no—
357C24—22no—
358C24—22no—
359C24—24no—
360C24—22no—
361C24—21no—
362C24—21no—
363C24—22no—
364C24—22no—
365C24—24no—
366C24—28no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—24no—
371C24—21no—
372C24—21no—
373C24—22no—
374C24—22no—
375C24—22no—
376C24—22no—
377C24—22no—
378C24—22no—
379C24—24no—
380C24—24no—
381C24—24no—
382C24—24no—
383C24—22no—
384C24—28no—
385C24—216no—
386C24—44no—
387C24—48no—
388C24—416no—
389C24—48no—
390C24—48no—
391C24—416no—
392C24—48no—
393C24—44no—
394C24—44no—
395C24—88no—
396C24—84no—
397C24—84no—
398C24—88no—
399C24C22×D412yes—
400C24C22×D412yes—
401C24C22×D412yes—
402C24C2×#4114yes—
403C24C2×#4114yes—
404C24H(5,𝔽2)14yes—
405C24H(5,𝔽2)12yes—
406C24H(5,𝔽2)12yes—
407C24H(5,𝔽2)14yes—
408C24C2511yescharacteristic
409#41—4128no—
410C2×Q8—48no—
411C2×Q8—44no—
412C2×Q8—44no—
413C2×Q8—48no—
414C2×Q8—44no—
415C2×D4—216no—
416C2×D4—216no—
417C2×D4—216no—
418C2×D4—48no—
419C2×D4—44no—
420C2×D4—44no—
421C2×D4—48no—
422C2×D4—44no—
423C2×D4—48no—
424C2×D4—48no—
425C2×D4—48no—
426C2×D4—48no—
427C2×D4—48no—
428C2×D4—48no—
429C2×D4—416no—
430C2×D4—48no—
431C2×D4—48no—
432C2×D4—84no—
433C2×D4—84no—
434C2×D4—88no—
435C2×D4—88no—
436C2×D4—88no—
437C22×C4—28no—
438C22×C4—28no—
439C22×C4—216no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—24no—
483C22×C4—24no—
484C22×C4—22no—
485C22×C4—24no—
486C22×C4—24no—
487C22×C4—24no—
488C22×C4—24no—
489C22×C4—24no—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—24no—
494C22×C4—24no—
495C22×C4—24no—
496C22×C4—24no—
497C22×C4—24no—
498C22×C4—48no—
499C22×C4—48no—
500C22×C4—48no—
501C22×C4—48no—
502C22×C4—48no—
503C22×C4—48no—
504C22×C4—48no—
505C22×C4—48no—
506C22×C4—48no—
507C22×C4—48no—
508C22×C4—48no—
509C22×C4—44no—
510C22×C4—44no—
511C22×C4—48no—
512C22×C4—48no—
513C22×C4—48no—
514C22×C4—48no—
515C22×C4—48no—
516C22×C4—48no—
517C22×C4—48no—
518C22×C4—88no—
519C22×C4—84no—
520C22×C4—88no—
521C22×C4—88no—
522C22×C4—88no—
523C22×C4—88no—
524C22×C4—84no—
525C22×C4—88no—
526C22×C4C22×D412yes—
527C22×C4C22×D412yes—
528C22×C4C22×D412yes—
529C22×C4C22×D412yes—
530C22×C4C2×#4114yes—
531C4⋊C4—48no—
532C4⋊C4—48no—
533C4⋊C4—88no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—44no—
538C22⋊C4—48no—
539C22⋊C4—48no—
540C22⋊C4—416no—
541C22⋊C4—48no—
542C22⋊C4—416no—
543C22⋊C4—48no—
544C22⋊C4—48no—
545C22⋊C4—48no—
546C22⋊C4—84no—
547C22⋊C4—84no—
548C22⋊C4—88no—
549C22⋊C4—84no—
550C22⋊C4—84no—
551C22⋊C4—88no—
552C22⋊C4—88no—
553C22⋊C4—88no—
554C23—22no—
555C23—22no—
556C23—24no—
557C23—28no—
558C23—28no—
559C23—24no—
560C23—24no—
561C23—24no—
562C23—24no—
563C23—28no—
564C23—28no—
565C23—22no—
566C23—22no—
567C23—24no—
568C23—24no—
569C23—28no—
570C23—24no—
571C23—28no—
572C23—24no—
573C23—22no—
574C23—22no—
575C23—22no—
576C23—24no—
577C23—24no—
578C23—24no—
579C23—24no—
580C23—24no—
581C23—28no—
582C23—28no—
583C23—416no—
584C23—432no—
585C23—42no—
586C23—41no—
587C23—44no—
588C23—41no—
589C23—42no—
590C23—42no—
591C23—44no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—42no—
597C23—416no—
598C23—42no—
599C23—41no—
600C23—42no—
601C23—42no—
602C23—44no—
603C23—44no—
604C23—44no—
605C23—44no—
606C23—41no—
607C23—42no—
608C23—42no—
609C23—44no—
610C23—44no—
611C23—42no—
612C23—44no—
613C23—44no—
614C23—44no—
615C23—44no—
616C23—42no—
617C23—42no—
618C23—44no—
619C23—44no—
620C23—44no—
621C23—44no—
622C23—88no—
623C23—816no—
624C23—816no—
625C23—816no—
626C23—832no—
627C23#53414yes—
628C23#53514yes—
629C23#53818yes—
630C23#54618yes—
631C23#55114yes—
632C23H(3,𝔽4)14yes—
633C23C23×D411yescharacteristic
634C23C22×#4112yes—
635C23C2×H(5,𝔽2)11yescharacteristic
636C23C2×H(5,𝔽2)11yescharacteristic
637C23C2×H(5,𝔽2)12yes—
638C23C2×H(5,𝔽2)12yes—
639C23#58512yes—
640C23#58514yes—
641Q8—416no—
642D4—416no—
643D4—432no—
644D4—816no—
645D4—816no—
646C2×C4—22no—
647C2×C4—22no—
648C2×C4—22no—
649C2×C4—22no—
650C2×C4—24no—
651C2×C4—24no—
652C2×C4—24no—
653C2×C4—24no—
654C2×C4—24no—
655C2×C4—24no—
656C2×C4—24no—
657C2×C4—416no—
658C2×C4—432no—
659C2×C4—42no—
660C2×C4—42no—
661C2×C4—42no—
662C2×C4—42no—
663C2×C4—416no—
664C2×C4—42no—
665C2×C4—41no—
666C2×C4—41no—
667C2×C4—41no—
668C2×C4—41no—
669C2×C4—42no—
670C2×C4—42no—
671C2×C4—42no—
672C2×C4—42no—
673C2×C4—42no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4—44no—
677C2×C4—42no—
678C2×C4—44no—
679C2×C4—42no—
680C2×C4—44no—
681C2×C4—42no—
682C2×C4—42no—
683C2×C4—44no—
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—44no—
703C2×C4—44no—
704C2×C4—44no—
705C2×C4—44no—
706C2×C4—44no—
707C2×C4—44no—
708C2×C4—44no—
709C2×C4—44no—
710C2×C4—44no—
711C2×C4—44no—
712C2×C4—816no—
713C2×C4—816no—
714C2×C4—816no—
715C22—28no—
716C22—24no—
717C22—24no—
718C22—28no—
719C22—28no—
720C22—28no—
721C22—216no—
722C22—24no—
723C22—28no—
724C22—44no—
725C22—42no—
726C22—42no—
727C22—44no—
728C22—44no—
729C22—48no—
730C22—42no—
731C22—42no—
732C22—44no—
733C22—44no—
734C22—44no—
735C22—42no—
736C22—44no—
737C22—44no—
738C22—48no—
739C22—48no—
740C22—832no—
741C22—84no—
742C22—82no—
743C22—82no—
744C22—82no—
745C22—82no—
746C22—82no—
747C22—82no—
748C22—84no—
749C22—84no—
750C22#268418yes—
751C22C2×#53411yescharacteristic
752C22C2×#53511yescharacteristic
753C22#344911yescharacteristic
754C22C2×#53812yes—
755C22C2×#54612yes—
756C22C2×#55111yescharacteristic
757C22#348512yes—
758C22#351314yes—
759C22#352114yes—
760C22C2×H(3,𝔽4)11yescharacteristic
761C22#353312yes—
762C22#353312yes—
763C22#353412yes—
764C22#353414yes—
765C22#353412yes—
766C22#356014yes—
767C4—42no—
768C4—42no—
769C4—42no—
770C4—42no—
771C4—44no—
772C4—84no—
773C4—84no—
774C4—84no—
775C4—82no—
776C4—82no—
777C4—84no—
778C2—28no—
779C2—44no—
780C2—84no—
781C2—82no—
782C2#3695511yescharacteristic
783C2#3744611yescharacteristic
784C2#3802312yes—
785C2#3803012yes—
786C2#4069214yes—
787C2#4324412yes—
788C2C2×#268411yescharacteristic
789C2#5253012yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2404C2
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C22×#397C2semidirect
3. #17201C2semidirect
4. #17147C2semidirect
5. #16831C2semidirect
6. #16808C2semidirect
7. #16380C2semidirect
8. #15494C2semidirect
9. #15091C2semidirect
10. #14919C2semidirect
11. #14718C2semidirect
12. #14259C2semidirect
13. #14134C2semidirect
14. #13956C2semidirect
15. #13869C2semidirect
47. #1446C22semidirect
48. #1445C22semidirect
49. #1443C22semidirect
50. #1443C22semidirect
51. #1443C22semidirect
52. #1442C22semidirect
53. #1442C22semidirect
54. #1442C22semidirect
55. #1442C22semidirect
60. #1440C22semidirect
62. #1439C22semidirect
63. #1439C22semidirect
64. #1439C22semidirect
67. #1438C22semidirect
75. C22⋊#95C22semidirect
76. C22⋊#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#24042
C22×#3972
#172012
#171472
#168312
#168082
#163802
#154942
#150912
#149192
#147182
#142592
#141342
#139562
#138692