Finite Group Databasebeta

← #627246

Subgroups — #627246

Order 512.

3802 conjugacy classes of subgroups.

The subgroup lattice has 713 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2627C211yesmaximal, characteristic, complemented
2C22×#385C211yesmaximal, characteristic
3#15680C214yesmaximal, complemented
4#15622C214yesmaximal, complemented
5#15545C214yesmaximal, complemented
6#15515C214yesmaximal, complemented
7#14822C211yesmaximal, characteristic, complemented
8#14734C211yesmaximal, characteristic
9#14720C211yesmaximal, characteristic, complemented
10#14717C211yesmaximal, characteristic
11#14693C212yesmaximal
12#14265C212yesmaximal, complemented
13#13962C212yesmaximal, complemented
14#13868C212yesmaximal
15#13795C211yesmaximal, characteristic, complemented
16C25×C4C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×#117C2212yes—
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23#2627—28no—
24C2×#400C2212yescomplemented
25C2×#400C2212yescomplemented
26C2×#399C2212yes—
27C2×#398C2212yescomplemented
28C2×#398C2212yescomplemented
29C2×#395C2212yescomplemented
30C2×#395C2212yescomplemented
31C2×#393C2212yes—
32C2×#385—28no—
33C2×#385C2214yes—
34C2×#385C2212yes—
35C2×#385C2212yes—
36C2×#385C2214yes—
37C2×#385C2214yes—
38C2×#385C2214yes—
39C2×#385C2212yes—
40C2×#381C2212yes—
41C2×#380C2211yescharacteristic
42C2×#379C2212yes—
43C2×#379C2212yes—
44C22×#95C2211yescharacteristic
45C22×#95C2211yescharacteristic
46#1445C2214yescomplemented
47#1444C2214yescomplemented
48#1443C2214yescomplemented
49#1442C2214yescomplemented
50#1442C2214yescomplemented
51#1441C2214yescomplemented
52#1441C2214yescomplemented
53#1441C2212yescomplemented
54#1441C2214yescomplemented
55#1441C2212yescomplemented
56#1439C2214yes—
57#1439C2214yes—
58#1439C2214yes—
59#1439C2212yes—
60#1439C2214yes—
61#1439C2212yes—
62#1437C2214yes—
63#1437C2214yes—
64C4⋊#95C2214yescomplemented
65C4⋊#95C2214yescomplemented
66C4⋊#95C2214yescomplemented
67#1434C2214yescomplemented
68C22⋊#95C2212yes—
69C22⋊#95C2212yes—
70C22⋊#95C2214yes—
71C22⋊#95C2214yes—
72C22⋊#95C2212yes—
73C22⋊#95C2212yes—
74C22⋊#95C2212yes—
75C22⋊#95C2211yescharacteristic
76C22⋊#95C2211yescharacteristic
77C26C2311yescharacteristic
78C23×Q8C2311yescharacteristic
79C24×C4—24no—
80C24×C4—44no—
81C24×C4—44no—
82C24×C4—44no—
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2314yes—
86C24×C4C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—44no—
92C2×#117—28no—
93C2×#117—28no—
94C2×#117—44no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—44no—
97C22×C4⋊C4—44no—
98C22×C4⋊C4D418yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C22⋊C4—24no—
113C22×C22⋊C4—24no—
114C22×C22⋊C4—24no—
115C22×C22⋊C4—44no—
116C22×C22⋊C4Q818yescomplemented
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2314yes—
128C22×C42C2314yes—
129#400—28no—
130#400—44no—
131#399—44no—
132#398—28no—
133#398—28no—
134#395—28no—
135#395—44no—
136#393—44no—
137#385—28no—
138#385—216no—
139#385—216no—
140#385—28no—
141#385—48no—
142#385—44no—
143#385—48no—
144#381—28no—
145#380—24no—
146#379—28no—
147#379—44no—
148C2×#95—24no—
149C2×#95—24no—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C25—22no—
176C25—22no—
177C25—24no—
178C25—24no—
179C25—24no—
180C25—24no—
181C25—42no—
182C25—42no—
183C25C2×Q814yes—
184C25C2412yes—
185C25C2411yescharacteristic
186C22×Q8—22no—
187C22×Q8—42no—
188C22×Q8—42no—
189C22×Q8—42no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—28no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—28no—
203C23×C4—28no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—42no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—416no—
218C23×C4—416no—
219C23×C4—416no—
220C23×C4—464no—
221C23×C4C2×D414yes—
222C23×C4C2×D414yes—
223C23×C4C2×D414yes—
224C23×C4C2×Q814yes—
225C23×C4C2×Q814yes—
226C23×C4#4114yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2414yes—
232C23×C4C2412yes—
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2414yes—
238C23×C4C2412yes—
239C23×C4C2412yes—
240C22⋊Q8—416no—
241#117—432no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—416no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—416no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4C2×D418yes—
277C2×C4⋊C4C2×D418yes—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—232no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
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—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—28no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—42no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—416no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4C2×Q8116yes—
312C2×C22⋊C4#4118yes—
313C2×C42—28no—
314C2×C42—28no—
315C2×C42—44no—
316#95—28no—
317#95—28no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—28no—
326#95—28no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—42no—
335#95—44no—
336#95—42no—
337#95—44no—
338#95—42no—
339#95—44no—
340#95—42no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—28no—
346C24—28no—
347C24—216no—
348C24—28no—
349C24—216no—
350C24—22no—
351C24—22no—
352C24—24no—
353C24—24no—
354C24—22no—
355C24—24no—
356C24—22no—
357C24—24no—
358C24—216no—
359C24—48no—
360C24—416no—
361C24—416no—
362C24—416no—
363C24—416no—
364C24—44no—
365C24—48no—
366C24—44no—
367C24—44no—
368C24—44no—
369C24—48no—
370C24C22×D412yes—
371C24C22×Q814yes—
372C24C22×Q812yes—
373C24C2×#4112yes—
374C24C2×#4114yes—
375C24H(5,𝔽2)14yes—
376C24H(5,𝔽2)14yes—
377C242−1+414yes—
378C24C2511yescharacteristic
379C2×Q8—44no—
380C2×Q8—82no—
381C2×Q8—82no—
382C2×Q8—88no—
383C22×C4—28no—
384C22×C4—28no—
385C22×C4—216no—
386C22×C4—216no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—28no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—28no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—44no—
454C22×C4—44no—
455C22×C4—48no—
456C22×C4—416no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—416no—
460C22×C4—416no—
461C22×C4—416no—
462C22×C4—416no—
463C22×C4—416no—
464C22×C4—44no—
465C22×C4—48no—
466C22×C4—44no—
467C22×C4—44no—
468C22×C4—416no—
469C22×C4—416no—
470C22×C4—4128no—
471C22×C4—416no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—44no—
476C22×C4—88no—
477C22×C4—88no—
478C22×C4C22⋊Q8116yes—
479C22×C4C22×D414yes—
480C22×C4C22×D412yes—
481C22×C4C22×D412yes—
482C22×C4C22×D414yes—
483C22×C4C22×Q814yes—
484C22×C4C22×Q814yes—
485C22×C4C2×#4118yes—
486C22×C4C2×#4114yes—
487C22×C4H(5,𝔽2)14yes—
488C4⋊C4—48no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—416no—
493C4⋊C4—48no—
494C4⋊C4—44no—
495C4⋊C4—82no—
496C4⋊C4—82no—
497C4⋊C4—84no—
498C4⋊C4—84no—
499C4⋊C4—88no—
500C4⋊C4—88no—
501C22⋊C4—216no—
502C22⋊C4—232no—
503C22⋊C4—416no—
504C22⋊C4—416no—
505C22⋊C4—48no—
506C22⋊C4—48no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—88no—
511C22⋊C4—88no—
512C42—416no—
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—24no—
528C23—28no—
529C23—24no—
530C23—24no—
531C23—24no—
532C23—24no—
533C23—28no—
534C23—28no—
535C23—28no—
536C23—28no—
537C23—28no—
538C23—464no—
539C23—416no—
540C23—416no—
541C23—416no—
542C23—416no—
543C23—416no—
544C23—464no—
545C23—416no—
546C23—42no—
547C23—42no—
548C23—44no—
549C23—44no—
550C23—42no—
551C23—42no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23C2×C22⋊Q814yes—
557C23#53214yes—
558C23#53414yes—
559C23#54418yes—
560C23#55118yes—
561C23#55418yes—
562C23C23×D411yescharacteristic
563C23C23×Q811yescharacteristic
564C23C22×#4111yescharacteristic
565C23C22×#4112yes—
566C23C2×H(5,𝔽2)12yes—
567C23C2×H(5,𝔽2)12yes—
568C23C2×H(5,𝔽2)12yes—
569C23C2×2−1+412yes—
570C23#58512yes—
571Q8—88no—
572C2×C4—232no—
573C2×C4—24no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—28no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—24no—
588C2×C4—22no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—28no—
596C2×C4—416no—
597C2×C4—416no—
598C2×C4—416no—
599C2×C4—464no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
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—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
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—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—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—88no—
648C2×C4—88no—
649C2×C4C2×C22⋊Q818yes—
650C22—24no—
651C22—24no—
652C22—24no—
653C22—24no—
654C22—24no—
655C22—216no—
656C22—24no—
657C22—216no—
658C22—28no—
659C22—216no—
660C22—464no—
661C22—44no—
662C22—44no—
663C22—44no—
664C22—44no—
665C22—44no—
666C22—44no—
667C22—48no—
668C22—48no—
669C22—48no—
670C22#256418yes—
671C22C22×C22⋊Q811yescharacteristic
672C22C2×#53211yescharacteristic
673C22C2×#53411yescharacteristic
674C22C2×#54412yes—
675C22#346012yes—
676C22C2×#55112yes—
677C22C2×#55412yes—
678C22#348411yescharacteristic
679C22#348411yescharacteristic
680C22#348611yescharacteristic
681C22#348711yescharacteristic
682C22#349212yes—
683C22#350112yes—
684C22#351314yes—
685C22#352014yes—
686C22#353214yes—
687C22#353514yes—
688C4—28no—
689C4—44no—
690C4—44no—
691C4—44no—
692C4—42no—
693C4—42no—
694C4—48no—
695C4—82no—
696C4—82no—
697C4—82no—
698C4—82no—
699C4—82no—
700C4—84no—
701C2—28no—
702C2—48no—
703C2C2×#256411yescharacteristic
704C2#3517411yescharacteristic
705C2#3522411yescharacteristic
706C2#3539611yescharacteristic
707C2#3539711yescharacteristic
708C2#3671411yescharacteristic
709C2#3718611yescharacteristic
710C2#3809714yes—
711C2#3903714yes—
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×#2627C2semidirect
3. #15680C2semidirect
4. #15622C2semidirect
5. #15545C2semidirect
6. #15515C2semidirect
7. #14822C2semidirect
9. #14720C2semidirect
12. #14265C2semidirect
13. #13962C2semidirect
15. #13795C2semidirect
24. C2×#400C22semidirect
25. C2×#400C22semidirect
27. C2×#398C22semidirect
28. C2×#398C22semidirect
29. C2×#395C22semidirect
30. C2×#395C22semidirect
46. #1445C22semidirect
47. #1444C22semidirect
48. #1443C22semidirect
49. #1442C22semidirect
50. #1442C22semidirect
51. #1441C22semidirect
52. #1441C22semidirect
53. #1441C22semidirect
54. #1441C22semidirect
55. #1441C22semidirect
64. C4⋊#95C22semidirect
65. C4⋊#95C22semidirect
66. C4⋊#95C22semidirect
67. #1434C22semidirect
116. C22×C22⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26272
C22×#3852
#156802
#156222
#155452
#155152
#148222
#147342
#147202
#147172
#146932
#142652
#139622
#138682
#137952