Finite Group Databasebeta

← #644946

Subgroups — #644946

Order 512.

3478 conjugacy classes of subgroups.

The subgroup lattice has 743 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2848C211yesmaximal, characteristic
2C2×#2280C211yesmaximal, characteristic
3#17059C211yesmaximal, characteristic, complemented
4#17058C211yesmaximal, characteristic, complemented
5#16391C214yesmaximal
6#16366C211yesmaximal, characteristic, complemented
7#16365C211yesmaximal, characteristic, complemented
8#15114C212yesmaximal, complemented
9#15063C212yesmaximal, complemented
10#15001C212yesmaximal, complemented
11#14906C212yesmaximal, complemented
12#14841C212yesmaximal, complemented
13#14840C212yesmaximal, complemented
14#14355C212yesmaximal, complemented
15#14292C212yesmaximal, complemented
16#14202C214yesmaximal, complemented
17C4×#379C211yesmaximal, characteristic
18C25×C4C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C22⋊Q8C2211yescharacteristic
21C22×C22⋊Q8C2211yescharacteristic
22C22×#117C2211yescharacteristic
23C22×#117C2211yescharacteristic
24#2848—28no—
25C2×#400C2212yescomplemented
26C2×#398C2212yescomplemented
27C2×#397C2214yes—
28C2×#387C2214yescomplemented
29C2×#385C2212yes—
30C2×#385C2212yes—
31C2×#385C2212yes—
32#2280—28no—
33C2×#381C2212yes—
34C2×#380C2211yescharacteristic
35C2×#380C2211yescharacteristic
36C2×#380C2211yescharacteristic
37C2×#380C2211yescharacteristic
38C2×#380C2211yescharacteristic
39C2×#380C2211yescharacteristic
40C2×#380C2211yescharacteristic
41C2×#379C2212yes—
42C2×#379C2212yes—
43C2×#379C2212yes—
44C2×C4×C22⋊C4C2212yes—
45C2×C4×C22⋊C4C2212yes—
46C2×C4×C22⋊C4C2212yes—
47C2×C4×C22⋊C4C2212yes—
48C2×C4×C22⋊C4C2212yes—
49#1446C2212yescomplemented
50#1446C2212yescomplemented
51#1445C2212yescomplemented
52#1445C2212yescomplemented
53#1445C2212yescomplemented
54#1445C2212yescomplemented
55#1444C2212yescomplemented
56#1444C2212yescomplemented
57#1444C2212yescomplemented
58#1442C2212yescomplemented
59#1442C2212yescomplemented
60#1442C2212yescomplemented
61#1441C2212yescomplemented
62#1441C2212yescomplemented
63#1440C2214yes—
64#1440C2214yes—
65#1440C2214yes—
66#1440C2214yes—
67#1440C2214yes—
68#1440C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1439C2214yes—
74#1436C2214yescomplemented
75#1436C2214yescomplemented
76#1436C2214yescomplemented
77#1436C2214yescomplemented
78C4⋊#95C2214yescomplemented
79#1434C2214yescomplemented
80#1434C2214yescomplemented
81C22⋊#95C2212yes—
82C22⋊#95C2212yes—
83C22⋊#95C2211yescharacteristic
84C22⋊#95C2211yescharacteristic
85C26C2311yescharacteristic
86C23×Q8C2311yescharacteristic
87C24×C4—28no—
88C24×C4—44no—
89C24×C4—44no—
90C24×C4D418yes—
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C24×C4C2311yescharacteristic
94C24×C4C2311yescharacteristic
95C24×C4C2311yescharacteristic
96C24×C4C2311yescharacteristic
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8—24no—
99C2×C22⋊Q8—24no—
100C2×C22⋊Q8—28no—
101C2×C22⋊Q8—44no—
102C2×C22⋊Q8—44no—
103C2×#117—24no—
104C2×#117—24no—
105C2×#117—44no—
106C2×#117—44no—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2312yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2312yes—
133C22×C22⋊C4C2312yes—
134C22×C22⋊C4C2312yes—
135C22×C42C2312yes—
136C22×C42C2312yes—
137C22×C42C2314yes—
138#400—44no—
139#398—44no—
140#397—216no—
141#387—48no—
142#385—28no—
143#385—28no—
144#385—28no—
145#381—28no—
146#380—24no—
147#380—24no—
148#380—24no—
149#380—24no—
150#380—24no—
151#380—24no—
152#380—24no—
153#379—28no—
154#379—28no—
155#379—28no—
156C4×C22⋊C4—28no—
157C4×C22⋊C4—28no—
158C4×C22⋊C4—28no—
159C4×C22⋊C4—44no—
160C4×C22⋊C4—44no—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C25—24no—
188C25—24no—
189C25—24no—
190C25—42no—
191C25—42no—
192C25—42no—
193C25—41no—
194C25—41no—
195C25C2×D414yes—
196C25C2411yescharacteristic
197C25C2411yescharacteristic
198C25C2411yescharacteristic
199C22×Q8—22no—
200C22×Q8—24no—
201C22×Q8—42no—
202C22×Q8—42no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—232no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—416no—
216C23×C4—416no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4—42no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—464no—
224C23×C4C2×D414yes—
225C23×C4C2×D414yes—
226C23×C4C2×D414yes—
227C23×C4C2×D414yes—
228C23×C4C2×D414yes—
229C23×C4C2×D414yes—
230C23×C4#4118yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2414yes—
240C23×C4C2414yes—
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C22⋊Q8—416no—
246C22⋊Q8—416no—
247C22⋊Q8—416no—
248#117—416no—
249#117—416no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—42no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
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—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—42no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—42no—
313C2×C22⋊C4—42no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—42no—
316C2×C22⋊C4#4118yes—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—28no—
320C2×C42—28no—
321C2×C42—44no—
322C2×C42—44no—
323C2×C42—44no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—28no—
337#95—24no—
338#95—24no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—21no—
364C24—21no—
365C24—21no—
366C24—21no—
367C24—22no—
368C24—21no—
369C24—21no—
370C24—22no—
371C24—28no—
372C24—28no—
373C24—28no—
374C24—44no—
375C24—44no—
376C24—44no—
377C24—44no—
378C24—416no—
379C24—48no—
380C24—48no—
381C24—416no—
382C24—416no—
383C24—48no—
384C24—48no—
385C24—416no—
386C24—48no—
387C24—48no—
388C24C22×D412yes—
389C24C22×D412yes—
390C24C22×D412yes—
391C24C2×#4114yes—
392C24C2×#4114yes—
393C242−1+414yes—
394C24C2511yescharacteristic
395C2×Q8—44no—
396C2×Q8—44no—
397C2×Q8—44no—
398C2×Q8—88no—
399C22×C4—232no—
400C22×C4—28no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—28no—
418C22×C4—28no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—28no—
422C22×C4—28no—
423C22×C4—28no—
424C22×C4—28no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—24no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—416no—
479C22×C4—416no—
480C22×C4—416no—
481C22×C4—416no—
482C22×C4—416no—
483C22×C4—4128no—
484C22×C4#123116yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C2×#4118yes—
490C22×C4C2×#4114yes—
491C22×C4C2×#4114yes—
492C22×C4H(5,𝔽2)14yes—
493C22×C4H(5,𝔽2)14yes—
494C4⋊C4—416no—
495C4⋊C4—44no—
496C4⋊C4—44no—
497C4⋊C4—44no—
498C4⋊C4—44no—
499C4⋊C4—44no—
500C4⋊C4—48no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—48no—
504C4⋊C4—48no—
505C4⋊C4—88no—
506C22⋊C4—48no—
507C22⋊C4—416no—
508C22⋊C4—416no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—48no—
513C22⋊C4—48no—
514C22⋊C4—48no—
515C22⋊C4—48no—
516C22⋊C4—48no—
517C22⋊C4—48no—
518C22⋊C4—48no—
519C22⋊C4—48no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C42—48no—
523C42—48no—
524C42—88no—
525C23—22no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—24no—
544C23—24no—
545C23—28no—
546C23—28no—
547C23—28no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—28no—
553C23—28no—
554C23—28no—
555C23—416no—
556C23—416no—
557C23—416no—
558C23—416no—
559C23—416no—
560C23—416no—
561C23—416no—
562C23—464no—
563C23—464no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—44no—
569C23—44no—
570C23—44no—
571C23—42no—
572C23—42no—
573C23—44no—
574C23—42no—
575C23—42no—
576C23—42no—
577C23—42no—
578C23—42no—
579C23—42no—
580C23—44no—
581C23—44no—
582C23C2×#12314yes—
583C23#53214yes—
584C23#53614yes—
585C23C23×D411yescharacteristic
586C23C22×#4112yes—
587C23C22×#4112yes—
588C23C2×H(5,𝔽2)12yes—
589C23C2×H(5,𝔽2)12yes—
590C23C2×2−1+412yes—
591C23#58511yescharacteristic
592C23#58511yescharacteristic
593C23#58512yes—
594Q8—88no—
595C2×C4—232no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—28no—
605C2×C4—28no—
606C2×C4—28no—
607C2×C4—28no—
608C2×C4—28no—
609C2×C4—28no—
610C2×C4—28no—
611C2×C4—416no—
612C2×C4—416no—
613C2×C4—416no—
614C2×C4—416no—
615C2×C4—416no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
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—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—48no—
658C2×C4—42no—
659C2×C4—44no—
660C2×C4—42no—
661C2×C4—44no—
662C2×C4—42no—
663C2×C4—44no—
664C2×C4—42no—
665C2×C4—48no—
666C2×C4—44no—
667C2×C4C2×#12314yes—
668C2×C4C2×#12314yes—
669C2×C4#53414yes—
670C2×C4#53414yes—
671C22—24no—
672C22—24no—
673C22—24no—
674C22—24no—
675C22—24no—
676C22—24no—
677C22—24no—
678C22—24no—
679C22—24no—
680C22—464no—
681C22—42no—
682C22—42no—
683C22—42no—
684C22—42no—
685C22—42no—
686C22—42no—
687C22—42no—
688C22—42no—
689C22—48no—
690C22—48no—
691C22—48no—
692C22—48no—
693C22—48no—
694C22—48no—
695C22—42no—
696C22—42no—
697C22—42no—
698C22—42no—
699C22C22×#12311yescharacteristic
700C22C2×#53211yescharacteristic
701C22C2×#53411yescharacteristic
702C22C2×#53411yescharacteristic
703C22C2×#53611yescharacteristic
704C22#344911yescharacteristic
705C22#344911yescharacteristic
706C22#345912yes—
707C22#346012yes—
708C22#347514yes—
709C22#349014yes—
710C22#349512yes—
711C22#349512yes—
712C22#352014yes—
713C22#353912yes—
714C22#354012yes—
715C22#354114yes—
716C4—24no—
717C4—24no—
718C4—24no—
719C4—24no—
720C4—48no—
721C4—82no—
722C4—82no—
723C4—82no—
724C4—82no—
725C4—82no—
726C4—82no—
727C4—84no—
728C4—82no—
729C4—82no—
730C2—44no—
731C2—44no—
732C2—44no—
733C2#3457511yescharacteristic
734C2#3481011yescharacteristic
735C2#3499211yescharacteristic
736C2#3508511yescharacteristic
737C2#3531111yescharacteristic
738C2#3743811yescharacteristic
739C2#3744511yescharacteristic
740C2#3801214yes—
741C2#3801514yes—
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
3. #17059C2semidirect
4. #17058C2semidirect
6. #16366C2semidirect
7. #16365C2semidirect
8. #15114C2semidirect
9. #15063C2semidirect
10. #15001C2semidirect
11. #14906C2semidirect
12. #14841C2semidirect
13. #14840C2semidirect
14. #14355C2semidirect
15. #14292C2semidirect
16. #14202C2semidirect
25. C2×#400C22semidirect
26. C2×#398C22semidirect
28. C2×#387C22semidirect
49. #1446C22semidirect
50. #1446C22semidirect
51. #1445C22semidirect
52. #1445C22semidirect
53. #1445C22semidirect
54. #1445C22semidirect
55. #1444C22semidirect
56. #1444C22semidirect
57. #1444C22semidirect
58. #1442C22semidirect
59. #1442C22semidirect
60. #1442C22semidirect
61. #1441C22semidirect
62. #1441C22semidirect
74. #1436C22semidirect
75. #1436C22semidirect
76. #1436C22semidirect
77. #1436C22semidirect
78. C4⋊#95C22semidirect
79. #1434C22semidirect
80. #1434C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28482
C2×#22802
#170592
#170582
#163912
#163662
#163652
#151142
#150632
#150012
#149062
#148412
#148402
#143552
#142922
#142022
C4×#3792