Finite Group Databasebeta

← #2033743

Subgroups — #2033743

Order 512.

3756 conjugacy classes of subgroups.

The subgroup lattice has 722 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2683C211yesmaximal, characteristic, complemented
2C2×#2672C214yesmaximal, complemented
3C2×#2856C211yesmaximal, characteristic
4#17071C212yesmaximal, complemented
5#16261C214yesmaximal, complemented
6#15673C212yesmaximal, complemented
7#15625C212yesmaximal, complemented
8#15615C214yesmaximal, complemented
9#15325C214yesmaximal, complemented
10#15313C212yesmaximal, complemented
11#15260C212yesmaximal, complemented
12#14697C212yesmaximal, complemented
13#14445C211yesmaximal, characteristic, complemented
14C22×#127C2211yescharacteristic
15C22×#124C2212yes—
16C22×#123C2212yes—
17C22×C22⋊Q8C2214yes—
18C22×#121C2214yes—
19C22×#121C2211yescharacteristic
20C22×#121C2211yescharacteristic
21C23×C42C2211yescharacteristic
22#2856—28no—
23#2683—28no—
24#2672—232no—
25C2×#399C2214yes—
26C2×#396C2214yes—
27C2×#396C2214yes—
28C2×#396C2214yes—
29C2×#395C2212yes—
30C2×#395C2214yes—
31C2×#394C2214yes—
32C2×#394C2214yes—
33C2×#394C2214yes—
34C2×#394C2214yes—
35C2×#393C2212yes—
36C2×#390C2211yescharacteristic
37C2×#390C2214yes—
38C2×#390C2211yescharacteristic
39C2×#389C2214yes—
40C2×#389C2212yes—
41C2×#389C2212yes—
42C2×#388C2214yes—
43C2×#386C2214yes—
44C2×#386C2212yes—
45C2×#385C2214yes—
46C2×#385C2212yes—
47C2×#384C2211yescharacteristic
48C2×#380C2212yes—
49C2×#380C2212yes—
50#1444C2214yes—
51#1444C2214yes—
52#1441C2214yes—
53#1440C2212yes—
54#1439C2214yes—
55#1439C2214yes—
56#1439C2214yes—
57#1439C2214yes—
58#1438C2212yes—
59#1438C2214yes—
60#1438C2212yes—
61#1438C2212yes—
62#1438C2214yes—
63#1437C2212yes—
64#1436C2212yes—
65#1436C2214yes—
66#1436C2214yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C22⋊#95C2214yes—
70C23×Q8C2312yes—
71C23×D4C2312yes—
72C23×D4C2312yes—
73C23×D4C2314yes—
74C23×D4C2311yescharacteristic
75C23×D4C2311yescharacteristic
76C24×C4C2311yescharacteristic
77C24×C4C2312yes—
78C24×C4C2312yes—
79C2×#127—24no—
80C2×#127—44no—
81C2×#124—216no—
82C2×#124—44no—
83C2×#123—216no—
84C2×#123—44no—
85C2×C22⋊Q8—216no—
86C2×C22⋊Q8—216no—
87C2×C22⋊Q8—48no—
88C2×#121—24no—
89C2×#121—216no—
90C2×#121—24no—
91C2×#121—216no—
92C2×#121—48no—
93C2×#121—42no—
94C2×#121—42no—
95C2×#121—42no—
96C2×#121—42no—
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2314yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C22⋊C4C2314yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C42—24no—
125C22×C42—44no—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130#399—216no—
131#396—216no—
132#396—216no—
133#396—48no—
134#395—28no—
135#395—216no—
136#394—216no—
137#394—216no—
138#394—216no—
139#394—48no—
140#393—44no—
141#390—216no—
142#390—24no—
143#390D418yes—
144#389—28no—
145#389—216no—
146#389—44no—
147#388—216no—
148#386—28no—
149#386—216no—
150#385—216no—
151#385—44no—
152#384—24no—
153#380—44no—
154#380D4116yescomplemented
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C25C2412yes—
169C25C2412yes—
170C25C2411yescharacteristic
171C22×Q8—24no—
172C22×Q8—28no—
173C22×Q8—44no—
174C22×Q8—42no—
175C22×Q8—42no—
176C22×D4—24no—
177C22×D4—28no—
178C22×D4—28no—
179C22×D4—28no—
180C22×D4—24no—
181C22×D4—24no—
182C22×D4—24no—
183C22×D4—22no—
184C22×D4—28no—
185C22×D4—28no—
186C22×D4—24no—
187C22×D4—28no—
188C22×D4—24no—
189C22×D4—41no—
190C22×D4—41no—
191C22×D4—44no—
192C22×D4—41no—
193C22×D4—41no—
194C22×D4—42no—
195C22×D4—44no—
196C22×D4—44no—
197C22×D4—42no—
198C22×D4—42no—
199C22×D4—44no—
200C22×D4—42no—
201C22×D4—44no—
202C22×D4C2×D414yes—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4—22no—
206C23×C4—22no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—28no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—44no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4C2×D418yes—
220C23×C4C2414yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2414yes—
224C23×C4C2414yes—
225C23×C4C2412yes—
226C23×C4C2411yescharacteristic
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230#127—416no—
231#124—432no—
232#123—432no—
233C22⋊Q8—464no—
234#121—464no—
235#121—416no—
236#121—416no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—41no—
257C2×C4⋊C4—41no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—41no—
260C2×C4⋊C4—41no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4C2×D418yes—
271C2×C4⋊C4C2×D414yes—
272C2×C4⋊C4C2×D418yes—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—28no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
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⋊C4C2×D4116yes—
314C2×C22⋊C4C2×D4116yes—
315C2×C42—22no—
316C2×C42—24no—
317C2×C42—28no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—416no—
321C2×C42—42no—
322C2×C42—42no—
323C2×C42—42no—
324C2×C42C2×D414yes—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—24no—
331#95—28no—
332#95—28no—
333#95—28no—
334#95—44no—
335#95—42no—
336#95—44no—
337#95C2×D4116yes—
338C24—21no—
339C24—22no—
340C24—22no—
341C24—21no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—24no—
346C24—24no—
347C24—22no—
348C24—22no—
349C24—24no—
350C24—24no—
351C24—24no—
352C24—24no—
353C24—24no—
354C24—22no—
355C24—24no—
356C24—22no—
357C24—22no—
358C24—24no—
359C24—24no—
360C24—24no—
361C24—24no—
362C24C22×D414yes—
363C24C22×D414yes—
364C24H(5,𝔽2)12yes—
365C24C2511yescharacteristic
366C2×Q8—416no—
367C2×Q8—48no—
368C2×Q8—84no—
369C2×Q8—88no—
370C2×Q8—84no—
371C2×D4—28no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—416no—
375C2×D4—416no—
376C2×D4—48no—
377C2×D4—416no—
378C2×D4—416no—
379C2×D4—44no—
380C2×D4—44no—
381C2×D4—416no—
382C2×D4—48no—
383C2×D4—48no—
384C2×D4—48no—
385C2×D4—44no—
386C2×D4—416no—
387C2×D4—88no—
388C2×D4—88no—
389C2×D4—84no—
390C2×D4—84no—
391C2×D4—88no—
392C2×D4—84no—
393C2×D4—84no—
394C2×D4—88no—
395C2×D4—84no—
396C2×D4—88no—
397C2×D4—84no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—28no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—216no—
431C22×C4—216no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—44no—
460C22×C4—416no—
461C22×C4—44no—
462C22×C4—88no—
463C22×C4—84no—
464C22×C4—84no—
465C22×C4C22×D418yes—
466C22×C4C22×D414yes—
467C22×C4C22×D418yes—
468C22×C4C22×D412yes—
469C22×C4C22×D414yes—
470C22×C4C22×D414yes—
471C22×C4C22×D414yes—
472C22×C4H(5,𝔽2)12yes—
473C22×C4H(5,𝔽2)14yes—
474C22×C4H(5,𝔽2)14yes—
475C22×C42−1+414yes—
476C22×C42−1+414yes—
477C4⋊C4—216no—
478C4⋊C4—48no—
479C4⋊C4—44no—
480C4⋊C4—416no—
481C4⋊C4—416no—
482C4⋊C4—416no—
483C4⋊C4—44no—
484C4⋊C4—416no—
485C4⋊C4—416no—
486C4⋊C4—84no—
487C4⋊C4—84no—
488C4⋊C4—88no—
489C22⋊C4—232no—
490C22⋊C4—232no—
491C22⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—416no—
494C22⋊C4—48no—
495C22⋊C4—416no—
496C22⋊C4—48no—
497C22⋊C4—416no—
498C22⋊C4—88no—
499C22⋊C4—88no—
500C22⋊C4—88no—
501C22⋊C4—84no—
502C22⋊C4—88no—
503C22⋊C4—88no—
504C42—28no—
505C42—44no—
506C42—48no—
507C42—48no—
508C23—24no—
509C23—24no—
510C23—22no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—22no—
517C23—24no—
518C23—24no—
519C23—28no—
520C23—24no—
521C23—24no—
522C23—28no—
523C23—41no—
524C23—41no—
525C23—42no—
526C23—41no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—41no—
532C23—42no—
533C23—44no—
534C23—42no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—42no—
540C23—44no—
541C23—44no—
542C23—44no—
543C23—44no—
544C23—42no—
545C23—44no—
546C23—42no—
547C23—44no—
548C23—44no—
549C23—42no—
550C23—44no—
551C23—42no—
552C23—44no—
553C23—42no—
554C23—42no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23—44no—
560C23—42no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—42no—
566C23—44no—
567C23—42no—
568C23—44no—
569C23—44no—
570C23D4214yes—
571C23C23×D411yescharacteristic
572C23C23×D412yes—
573C23C2×H(5,𝔽2)12yes—
574C23C2×H(5,𝔽2)11yescharacteristic
575C23C2×H(5,𝔽2)12yes—
576C23C2×H(5,𝔽2)11yescharacteristic
577C23C2×2−1+412yes—
578C23C2×2−1+412yes—
579C23#58512yes—
580Q8—816no—
581D4—88no—
582D4—832no—
583D4—88no—
584D4—816no—
585D4—816no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—28no—
603C2×C4—28no—
604C2×C4—28no—
605C2×C4—28no—
606C2×C4—22no—
607C2×C4—24no—
608C2×C4—28no—
609C2×C4—28no—
610C2×C4—28no—
611C2×C4—24no—
612C2×C4—432no—
613C2×C4—416no—
614C2×C4—41no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—41no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
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—42no—
646C2×C4—44no—
647C2×C4—816no—
648C2×C4#53614yes—
649C2×C4#53718yes—
650C2×C4D4218yes—
651C22—28no—
652C22—24no—
653C22—44no—
654C22—42no—
655C22—42no—
656C22—44no—
657C22—44no—
658C22—44no—
659C22—44no—
660C22—48no—
661C22—44no—
662C22—44no—
663C22—48no—
664C22—44no—
665C22—84no—
666C22—84no—
667C22—84no—
668C22—84no—
669C22—84no—
670C22—82no—
671C22—82no—
672C22—84no—
673C22—84no—
674C22—84no—
675C22—84no—
676C22—84no—
677C22—82no—
678C22C2×#53611yescharacteristic
679C22C2×#53712yes—
680C22C2×D4212yes—
681C22C2×D4211yescharacteristic
682C22#348411yescharacteristic
683C22#348411yescharacteristic
684C22#348912yes—
685C22#349612yes—
686C22#349712yes—
687C22#349812yes—
688C22#349912yes—
689C22#352911yescharacteristic
690C22#353412yes—
691C22#353612yes—
692C22#353612yes—
693C22#353612yes—
694C22#354311yescharacteristic
695C22#354712yes—
696C22#355011yescharacteristic
697C22#355012yes—
698C22#355912yes—
699C4—28no—
700C4—28no—
701C4—24no—
702C4—42no—
703C4—48no—
704C4—48no—
705C4—44no—
706C4—84no—
707C4—84no—
708C4—82no—
709C2—44no—
710C2—84no—
711C2—162no—
712C2#3671811yescharacteristic
713C2#3729912yes—
714C2#3767312yes—
715C2#3934312yes—
716C2#3935912yes—
717C2#4578612yes—
718C2#4622811yescharacteristic
719C2#5319512yes—
720C2#5912611yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2856C2
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×#2683C2semidirect
2. C2×#2672C2semidirect
4. #17071C2semidirect
5. #16261C2semidirect
6. #15673C2semidirect
7. #15625C2semidirect
8. #15615C2semidirect
9. #15325C2semidirect
10. #15313C2semidirect
11. #15260C2semidirect
12. #14697C2semidirect
13. #14445C2semidirect
154. #380D4semidirect

Maximal subgroups

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

subgroupindex
C2×#26832
C2×#26722
C2×#28562
#170712
#162612
#156732
#156252
#156152
#153252
#153132
#152602
#146972
#144452