Finite Group Databasebeta

← #5955447

Subgroups — #5955447

Order 512.

3902 conjugacy classes of subgroups.

The subgroup lattice has 711 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2329C211yesmaximal, characteristic, complemented
2C22×#380C211yesmaximal, characteristic
3#14822C211yesmaximal, characteristic, complemented
4#14775C214yesmaximal, complemented
5#14774C214yesmaximal, complemented
6#14771C214yesmaximal, complemented
7#14770C214yesmaximal, complemented
8#14734C211yesmaximal, characteristic
9#14728C212yesmaximal, complemented
10#14726C212yesmaximal
11#14720C211yesmaximal, characteristic, complemented
12#14717C211yesmaximal, characteristic
13#13782C212yesmaximal, complemented
14#13778C211yesmaximal, characteristic, complemented
15C4×#379C212yesmaximal
16C25×C4C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C22⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22C2×#400C2212yescomplemented
23C2×#399C2212yes—
24C2×#398C2214yescomplemented
25C2×#396C2214yes—
26C2×#395C2212yescomplemented
27C2×#393C2212yes—
28#2329—28no—
29C2×#391C2212yescomplemented
30C2×#389C2212yescomplemented
31C2×#387C2212yescomplemented
32C2×#387C2212yescomplemented
33C2×#385C2212yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#380—28no—
37C2×#380C2211yescharacteristic
38C2×#380C2212yes—
39C2×#380C2214yes—
40C2×#380C2214yes—
41C2×#379C2212yes—
42C2×#379C2212yes—
43C2×C4×C22⋊C4C2212yes—
44C2×C4×C22⋊C4C2212yes—
45C2×C4×C22⋊C4C2214yes—
46C2×C4×C22⋊C4C2214yes—
47C22×#95C2211yescharacteristic
48C22×#95C2211yescharacteristic
49#1444C2214yescomplemented
50#1444C2214yescomplemented
51#1442C2214yescomplemented
52#1442C2214yescomplemented
53#1441C2212yescomplemented
54#1441C2214yescomplemented
55#1441C2212yescomplemented
56#1439C2214yes—
57#1439C2212yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2212yes—
63#1436C2214yescomplemented
64#1436C2214yescomplemented
65#1436C2214yescomplemented
66#1436C2214yescomplemented
67C4⋊#95C2214yescomplemented
68#1434C2214yescomplemented
69C22⋊#95C2212yes—
70C22⋊#95C2211yescharacteristic
71C22⋊#95C2212yes—
72C22⋊#95C2211yescharacteristic
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C26C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C24×C4—24no—
80C24×C4—24no—
81C24×C4—44no—
82C24×C4—44no—
83C24×C4Q818yescomplemented
84C24×C4C2311yescharacteristic
85C24×C4C2314yes—
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—28no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—44no—
96C22×C4⋊C4—44no—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C22⋊C4—24no—
110C22×C22⋊C4—24no—
111C22×C22⋊C4—44no—
112C22×C22⋊C4—44no—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C42C2314yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127#400—44no—
128#399—44no—
129#398—48no—
130#396—48no—
131#395—44no—
132#393—44no—
133#391—28no—
134#389—28no—
135#387—28no—
136#387—28no—
137#385—28no—
138#385—28no—
139#385—28no—
140#380—24no—
141#380—28no—
142#380—216no—
143#380—216no—
144#379—28no—
145#379—28no—
146C4×C22⋊C4—28no—
147C4×C22⋊C4—28no—
148C4×C22⋊C4—48no—
149C4×C22⋊C4—48no—
150C2×#95—24no—
151C2×#95D418yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C25—22no—
177C25—22no—
178C25—24no—
179C25—24no—
180C25—24no—
181C25—24no—
182C25—42no—
183C25—42no—
184C25C2×Q814yes—
185C25C2412yes—
186C25C2411yescharacteristic
187C22×Q8—22no—
188C22×Q8—24no—
189C22×Q8—42no—
190C22×Q8—42no—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—232no—
198C23×C4—28no—
199C23×C4—28no—
200C23×C4—28no—
201C23×C4—28no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—432no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—44no—
210C23×C4—44no—
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×C4—416no—
220C23×C4—416no—
221C23×C4—432no—
222C23×C4C2×D414yes—
223C23×C4C2×D414yes—
224C23×C4C2×D414yes—
225C23×C4C2×Q814yes—
226C23×C4C2×Q8116yes—
227C23×C4C2×Q814yes—
228C23×C4#4114yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2414yes—
232C23×C4C2412yes—
233C23×C4C2412yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2412yes—
238C23×C4C2412yes—
239C23×C4C2414yes—
240C23×C4C2412yes—
241C23×C4C2412yes—
242C22⋊Q8—232no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—416no—
265C2×C4⋊C4—416no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—42no—
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—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
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—42no—
301C2×C22⋊C4—42no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—416no—
308C2×C22⋊C4—416no—
309C2×C22⋊C4#4118yes—
310C2×C22⋊C4#4118yes—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—28no—
314C2×C42—44no—
315C2×C42—44no—
316C2×C42—44no—
317C2×C42—44no—
318#95—28no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—42no—
333#95—44no—
334#95—42no—
335#95—44no—
336#95—44no—
337#95—42no—
338#95—42no—
339#95—44no—
340#95C2×D418yes—
341#95C2×D418yes—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—28no—
347C24—28no—
348C24—216no—
349C24—28no—
350C24—216no—
351C24—22no—
352C24—24no—
353C24—24no—
354C24—22no—
355C24—22no—
356C24—24no—
357C24—24no—
358C24—22no—
359C24—216no—
360C24—48no—
361C24—416no—
362C24—416no—
363C24—416no—
364C24—416no—
365C24—44no—
366C24—44no—
367C24—48no—
368C24—44no—
369C24—48no—
370C24—44no—
371C24C22×D412yes—
372C24C22×Q814yes—
373C24C22×Q812yes—
374C24C2×#4112yes—
375C24C2×#4114yes—
376C24C2×#4114yes—
377C24H(5,𝔽2)14yes—
378C242−1+414yes—
379C24C2511yescharacteristic
380C2×Q8—28no—
381C2×Q8—82no—
382C2×Q8—82no—
383C2×Q8—88no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—216no—
387C22×C4—28no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—216no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—28no—
420C22×C4—28no—
421C22×C4—232no—
422C22×C4—232no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—44no—
451C22×C4—44no—
452C22×C4—48no—
453C22×C4—416no—
454C22×C4—48no—
455C22×C4—416no—
456C22×C4—416no—
457C22×C4—416no—
458C22×C4—416no—
459C22×C4—44no—
460C22×C4—44no—
461C22×C4—44no—
462C22×C4—44no—
463C22×C4—4128no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—44no—
469C22×C4—44no—
470C22×C4—88no—
471C22×C4—88no—
472C22×C4—88no—
473C22×C4—88no—
474C22×C4C22⋊Q8116yes—
475C22×C4C22×D414yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D414yes—
479C22×C4C22×Q814yes—
480C22×C4C22×Q814yes—
481C22×C4C2×#4114yes—
482C22×C4C2×#4118yes—
483C22×C4C2×#4114yes—
484C22×C4C2×#4114yes—
485C22×C4H(5,𝔽2)14yes—
486C22×C4H(5,𝔽2)14yes—
487C4⋊C4—28no—
488C4⋊C4—216no—
489C4⋊C4—416no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—84no—
493C4⋊C4—82no—
494C4⋊C4—84no—
495C4⋊C4—82no—
496C4⋊C4—88no—
497C4⋊C4—88no—
498C4⋊C4—88no—
499C22⋊C4—216no—
500C22⋊C4—216no—
501C22⋊C4—416no—
502C22⋊C4—416no—
503C22⋊C4—416no—
504C22⋊C4—416no—
505C22⋊C4—416no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C42—84no—
511C42—84no—
512C42—88no—
513C23—24no—
514C23—24no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—24no—
520C23—24no—
521C23—28no—
522C23—28no—
523C23—28no—
524C23—28no—
525C23—232no—
526C23—24no—
527C23—28no—
528C23—28no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—24no—
534C23—28no—
535C23—28no—
536C23—28no—
537C23—28no—
538C23—464no—
539C23—416no—
540C23—416no—
541C23—416no—
542C23—464no—
543C23—416no—
544C23—416no—
545C23—416no—
546C23—42no—
547C23—44no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23C2×C22⋊Q814yes—
557C23#52514yes—
558C23#52918yes—
559C23#53214yes—
560C23#53318yes—
561C23#54418yes—
562C23C23×D411yescharacteristic
563C23C23×Q811yescharacteristic
564C23C22×#4111yescharacteristic
565C23C22×#4112yes—
566C23C22×#4112yes—
567C23C2×H(5,𝔽2)12yes—
568C23C2×H(5,𝔽2)12yes—
569C23C2×H(5,𝔽2)12yes—
570C23C2×2−1+412yes—
571Q8—88no—
572C2×C4—232no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—28no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—28no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—28no—
588C2×C4—24no—
589C2×C4—22no—
590C2×C4—24no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—24no—
594C2×C4—22no—
595C2×C4—28no—
596C2×C4—28no—
597C2×C4—28no—
598C2×C4—28no—
599C2×C4—416no—
600C2×C4—464no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
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—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
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—88no—
643C2×C4—88no—
644C2×C4—88no—
645C2×C4—88no—
646C2×C4C2×C22⋊Q818yes—
647C2×C4#54318yes—
648C22—216no—
649C22—24no—
650C22—24no—
651C22—216no—
652C22—24no—
653C22—24no—
654C22—24no—
655C22—24no—
656C22—216no—
657C22—28no—
658C22—464no—
659C22—48no—
660C22—44no—
661C22—44no—
662C22—48no—
663C22—48no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—44no—
668C22#239618yes—
669C22C22×C22⋊Q811yescharacteristic
670C22C2×#52511yescharacteristic
671C22C2×#52912yes—
672C22C2×#53211yescharacteristic
673C22C2×#53312yes—
674C22C2×#54312yes—
675C22C2×#54412yes—
676C22#348411yescharacteristic
677C22#348411yescharacteristic
678C22#348411yescharacteristic
679C22#348611yescharacteristic
680C22#349012yes—
681C22#349212yes—
682C22#351914yes—
683C22#351914yes—
684C22#352014yes—
685C22#352014yes—
686C4—28no—
687C4—28no—
688C4—42no—
689C4—42no—
690C4—44no—
691C4—44no—
692C4—48no—
693C4—44no—
694C4—82no—
695C4—84no—
696C4—82no—
697C4—82no—
698C4—82no—
699C2—28no—
700C2—48no—
701C2C2×#239611yescharacteristic
702C2#3498211yescharacteristic
703C2#3507914yes—
704C2#3522411yescharacteristic
705C2#3535511yescharacteristic
706C2#3535711yescharacteristic
707C2#3537414yes—
708C2#3539511yescharacteristic
709C2#3539711yescharacteristic
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
1. C2×#2329C2semidirect
3. #14822C2semidirect
4. #14775C2semidirect
5. #14774C2semidirect
6. #14771C2semidirect
7. #14770C2semidirect
9. #14728C2semidirect
11. #14720C2semidirect
13. #13782C2semidirect
14. #13778C2semidirect
22. C2×#400C22semidirect
24. C2×#398C22semidirect
26. C2×#395C22semidirect
29. C2×#391C22semidirect
30. C2×#389C22semidirect
31. C2×#387C22semidirect
32. C2×#387C22semidirect
49. #1444C22semidirect
50. #1444C22semidirect
51. #1442C22semidirect
52. #1442C22semidirect
53. #1441C22semidirect
54. #1441C22semidirect
55. #1441C22semidirect
63. #1436C22semidirect
64. #1436C22semidirect
65. #1436C22semidirect
66. #1436C22semidirect
67. C4⋊#95C22semidirect
68. #1434C22semidirect
83. C24×C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23292
C22×#3802
#148222
#147752
#147742
#147712
#147702
#147342
#147282
#147262
#147202
#147172
#137822
#137782
C4×#3792