Finite Group Databasebeta

← #5955975

Subgroups — #5955975

Order 512.

2994 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#388C211yesmaximal, characteristic
2C2×#2283C211yesmaximal, characteristic, complemented
3#15175C212yesmaximal, complemented
4#15174C212yesmaximal, complemented
5#15115C214yesmaximal, complemented
6#15025C212yesmaximal, complemented
7#15023C214yesmaximal, complemented
8#15020C212yesmaximal, complemented
9#14919C212yesmaximal, complemented
10#14918C212yesmaximal
11#14822C211yesmaximal, characteristic, complemented
12#14821C211yesmaximal, characteristic, complemented
13#14820C211yesmaximal, characteristic
14#14734C211yesmaximal, characteristic
15#13782C212yesmaximal, complemented
16#13775C212yesmaximal
17C4×#381C211yesmaximal, characteristic, complemented
18C23×C4⋊C4C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22C23×C42C2211yescharacteristic
23C2×#400C2212yescomplemented
24C2×#399C2212yes—
25C2×#398C2212yescomplemented
26C2×#397C2212yes—
27C2×#388—28no—
28C2×#388C2212yes—
29C2×#388C2212yes—
30C2×#388C2212yes—
31C2×#388C2212yes—
32C2×#385C2212yes—
33C2×#385C2212yes—
34#2283—28no—
35C2×#383C2212yescomplemented
36C2×#382C2211yescharacteristic, complemented
37C2×#382C2211yescharacteristic, complemented
38C2×#381C2212yes—
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×C4×C22⋊C4C2212yes—
48C2×#376C2212yescomplemented
49C22×#95C2211yescharacteristic
50C22×#95C2211yescharacteristic
51#1445C2214yescomplemented
52#1443C2214yescomplemented
53#1443C2214yescomplemented
54#1442C2214yescomplemented
55#1442C2214yescomplemented
56#1441C2214yescomplemented
57#1441C2212yescomplemented
58#1441C2212yescomplemented
59#1440C2214yes—
60#1440C2214yes—
61#1440C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2212yes—
66#1439C2212yes—
67#1437C2214yes—
68#1437C2214yes—
69#1437C2214yes—
70#1437C2214yes—
71#1436C2214yescomplemented
72#1436C2214yescomplemented
73C4⋊#95C2214yescomplemented
74#1434C2212yescomplemented
75#1434C2212yescomplemented
76#1434C2214yescomplemented
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C22⋊#95C2212yes—
80C22⋊#95C2212yes—
81C22⋊#95C2211yescharacteristic
82C22⋊#95C2211yescharacteristic
83C4×#95C2212yescomplemented
84C4×#95C2212yescomplemented
85C26C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C22×C4⋊C4—24no—
93C22×C4⋊C4—44no—
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2314yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C22⋊C4—24no—
103C22×C22⋊C4—24no—
104C22×C22⋊C4—24no—
105C22×C22⋊C4—24no—
106C22×C22⋊C4—24no—
107C22×C22⋊C4—28no—
108C22×C22⋊C4—44no—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C42—28no—
124C22×C42D418yes—
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2314yes—
130C22×C42C2312yes—
131C22×C42C2312yes—
132#400—44no—
133#399—44no—
134#398—44no—
135#397—44no—
136#388—28no—
137#388—28no—
138#388—28no—
139#388—28no—
140#385—28no—
141#385—28no—
142#383—28no—
143#382—24no—
144#382—24no—
145#381—28no—
146#381—28no—
147#380—24no—
148#380—24no—
149#379—28no—
150#379—28no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—24no—
153C4×C22⋊C4—28no—
154C4×C22⋊C4—28no—
155#376—28no—
156C2×#95—24no—
157C2×#95—24no—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C25—22no—
188C25—22no—
189C25—24no—
190C25—24no—
191C25—42no—
192C25—42no—
193C25—42no—
194C25—42no—
195C25#4114yes—
196C25C2412yes—
197C25C2411yescharacteristic
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—22no—
207C23×C4—22no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—42no—
218C23×C4—42no—
219C23×C4C2×D414yes—
220C23×C4C2×D414yes—
221C23×C4C2×D414yes—
222C23×C4#4114yes—
223C23×C4#4118yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2414yes—
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—416no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C22⋊C4—232no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
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—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—28no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—42no—
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—44no—
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—44no—
312C2×C42—44no—
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—28no—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—24no—
337#95—24no—
338#95—42no—
339#95—44no—
340#95—42no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—28no—
353C24—22no—
354C24—22no—
355C24—24no—
356C24—24no—
357C24—22no—
358C24—22no—
359C24—24no—
360C24—24no—
361C24—22no—
362C24—22no—
363C24—216no—
364C24—48no—
365C24—48no—
366C24—44no—
367C24—44no—
368C24—44no—
369C24—44no—
370C24—48no—
371C24—48no—
372C24—44no—
373C24—48no—
374C24—44no—
375C24—88no—
376C24—88no—
377C24—88no—
378C24—88no—
379C24C22×D412yes—
380C24C2×#4114yes—
381C24C2×#4114yes—
382C24C2×#4112yes—
383C24C2×#4112yes—
384C24C2×#4114yes—
385C24C2511yescharacteristic
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—28no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—28no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
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—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—24no—
457C22×C4—24no—
458C22×C4—44no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—416no—
462C22×C4—48no—
463C22×C4—416no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4—48no—
469C22×C4—44no—
470C22×C4—44no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—88no—
476C22×C4—88no—
477C22×C4#123116yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C2×#4118yes—
485C22×C4C2×#4114yes—
486C22×C4C2×#4114yes—
487C22×C4C2×#4114yes—
488C22×C4H(5,𝔽2)14yes—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—48no—
492C4⋊C4—44no—
493C4⋊C4—44no—
494C4⋊C4—48no—
495C4⋊C4—88no—
496C4⋊C4—88no—
497C22⋊C4—216no—
498C22⋊C4—216no—
499C22⋊C4—216no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—48no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—88no—
508C22⋊C4—88no—
509C22⋊C4—88no—
510C22⋊C4—88no—
511C42—28no—
512C42—28no—
513C42—28no—
514C42—28no—
515C42—48no—
516C42—48no—
517C42—88no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—28no—
523C23—24no—
524C23—24no—
525C23—28no—
526C23—24no—
527C23—24no—
528C23—28no—
529C23—28no—
530C23—24no—
531C23—28no—
532C23—28no—
533C23—24no—
534C23—24no—
535C23—416no—
536C23—416no—
537C23—416no—
538C23—42no—
539C23—44no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—42no—
551C23—42no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—832no—
557C23—88no—
558C23—88no—
559C23—88no—
560C23—832no—
561C23—88no—
562C23C2×#12314yes—
563C23#52514yes—
564C23#52918yes—
565C23#52918yes—
566C23#53214yes—
567C23C23×D411yescharacteristic
568C23C22×#4112yes—
569C23C22×#4111yescharacteristic
570C23C22×#4111yescharacteristic
571C23C22×#4112yes—
572C23C2×H(5,𝔽2)12yes—
573C23#58512yes—
574C23#58512yes—
575C23#58512yes—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—28no—
583C2×C4—28no—
584C2×C4—28no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—28no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—28no—
599C2×C4—28no—
600C2×C4—416no—
601C2×C4—416no—
602C2×C4—416no—
603C2×C4—416no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
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—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—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—88no—
655C2×C4—88no—
656C2×C4C2×#12314yes—
657C2×C4C2×#12314yes—
658C2×C4#53514yes—
659C22—24no—
660C22—24no—
661C22—24no—
662C22—24no—
663C22—24no—
664C22—216no—
665C22—216no—
666C22—24no—
667C22—44no—
668C22—44no—
669C22—48no—
670C22—48no—
671C22—44no—
672C22—832no—
673C22—82no—
674C22—82no—
675C22—82no—
676C22—84no—
677C22—84no—
678C22—82no—
679C22#239518yes—
680C22C22×#12311yescharacteristic
681C22C2×#52511yescharacteristic
682C22C2×#52912yes—
683C22C2×#52912yes—
684C22C2×#53211yescharacteristic
685C22C2×#53511yescharacteristic
686C22#344911yescharacteristic
687C22#344911yescharacteristic
688C22#344911yescharacteristic
689C22#349114yes—
690C22#349112yes—
691C22#349312yes—
692C22#349414yes—
693C22#350512yes—
694C22#350512yes—
695C22#351414yes—
696C22#352114yes—
697C4—24no—
698C4—24no—
699C4—24no—
700C4—44no—
701C4—48no—
702C4—42no—
703C4—42no—
704C4—42no—
705C4—82no—
706C4—82no—
707C4—82no—
708C4—82no—
709C4—82no—
710C4—84no—
711C4—82no—
712C2—28no—
713C2—84no—
714C2C2×#239511yescharacteristic
715C2#3509711yescharacteristic
716C2#3510211yescharacteristic
717C2#3511811yescharacteristic
718C2#3513111yescharacteristic
719C2#3514511yescharacteristic
720C2#3526211yescharacteristic
721C2#3545714yes—
722C2#3546214yes—
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. #15175C2semidirect
4. #15174C2semidirect
5. #15115C2semidirect
6. #15025C2semidirect
7. #15023C2semidirect
8. #15020C2semidirect
9. #14919C2semidirect
11. #14822C2semidirect
12. #14821C2semidirect
15. #13782C2semidirect
17. C4×#381C2semidirect
23. C2×#400C22semidirect
25. C2×#398C22semidirect
35. C2×#383C22semidirect
36. C2×#382C22semidirect
37. C2×#382C22semidirect
44. C2×C4×C4⋊C4C22semidirect
45. C2×C4×C4⋊C4C22semidirect
48. C2×#376C22semidirect
51. #1445C22semidirect
52. #1443C22semidirect
53. #1443C22semidirect
54. #1442C22semidirect
55. #1442C22semidirect
56. #1441C22semidirect
57. #1441C22semidirect
58. #1441C22semidirect
71. #1436C22semidirect
72. #1436C22semidirect
73. C4⋊#95C22semidirect
74. #1434C22semidirect
75. #1434C22semidirect
76. #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
#151752
#151742
#151152
#150252
#150232
#150202
#149192
#149182
#148222
#148212
#148202
#147342
#137822
#137752
C4×#3812