Finite Group Databasebeta

← #645202

Subgroups — #645202

Order 512.

2466 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2863C211yesmaximal, characteristic, complemented
2C2×C4×#124C211yesmaximal, characteristic
3#17080C211yesmaximal, characteristic, complemented
4#17079C211yesmaximal, characteristic, complemented
5#17077C211yesmaximal, characteristic
6#17071C211yesmaximal, characteristic
7#16515C214yesmaximal, complemented
8#15204C212yesmaximal, complemented
9#15161C212yesmaximal, complemented
10#14878C212yesmaximal, complemented
11#14870C212yesmaximal, complemented
12#14866C212yesmaximal, complemented
13#14812C212yesmaximal
14#14356C212yesmaximal, complemented
15#14288C212yesmaximal
16#14198C214yesmaximal, complemented
17C4×#384C211yesmaximal, characteristic, complemented
18C22×#124C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21#2863—28no—
22C2×#399C2212yes—
23C2×#398C2214yes—
24C2×#396C2214yes—
25C2×#396C2214yes—
26C2×#393C2212yes—
27C4×#124—28no—
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#391C2211yescharacteristic
31C2×#391C2211yescharacteristic
32C2×#390C2211yescharacteristic
33C2×#389C2211yescharacteristic
34C2×#389C2211yescharacteristic
35C2×#389C2211yescharacteristic
36C2×#388C2212yes—
37C2×#388C2212yes—
38C2×#387C2212yes—
39C2×#387C2212yes—
40C2×#387C2212yes—
41C2×#385C2214yes—
42C2×#384C2212yes—
43C2×#383C2212yes—
44C2×#382C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2212yes—
50C2×C43C2211yescharacteristic
51#1446C2212yes—
52#1446C2212yes—
53#1446C2212yes—
54#1445C2212yes—
55#1444C2212yes—
56#1444C2212yes—
57#1444C2212yes—
58#1442C2212yes—
59#1442C2212yes—
60#1442C2212yes—
61#1441C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1439C2212yes—
67#1439C2214yes—
68#1439C2212yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1438C2212yes—
73#1438C2212yes—
74#1437C2214yes—
75#1437C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78C4⋊#95C2214yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2211yescharacteristic
82C4⋊#95C2211yescharacteristic
83#1434C2214yes—
84C22⋊#95C2214yes—
85C23×Q8C2311yescharacteristic
86C23×Q8C2311yescharacteristic
87C23×D4C2311yescharacteristic
88C24×C4C2312yes—
89C2×#124—24no—
90C2×#124—24no—
91C2×#124Q818yescomplemented
92C2×C4×Q8—24no—
93C2×C4×Q8—44no—
94C2×C4×D4—24no—
95C2×C4×D4—44no—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2314yes—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2314yes—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129#399—44no—
130#398—48no—
131#396—216no—
132#396—48no—
133#393—44no—
134#391—24no—
135#391—24no—
136#391—24no—
137#391—24no—
138#390—24no—
139#389—24no—
140#389—24no—
141#389—24no—
142#388—28no—
143#388—28no—
144#387—28no—
145#387—28no—
146#387—28no—
147#385—48no—
148#384—28no—
149#383—28no—
150#382—24no—
151C4×C4⋊C4—28no—
152C4×C4⋊C4—44no—
153C4×C4⋊C4—44no—
154C4×C22⋊C4—28no—
155C4×C22⋊C4—28no—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C43D418yes—
185C25C2412yes—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—42no—
192C22×Q8—42no—
193C22×Q8—42no—
194C22×Q8—42no—
195C22×Q8C2×Q814yes—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—42no—
199C22×D4—42no—
200C22×D4C2×Q814yes—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—44no—
205C23×C4—44no—
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2414yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222#124—232no—
223C4×Q8—416no—
224C4×D4—416no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4#4118yes—
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—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4C2×Q818yes—
287C2×C22⋊C4C2×Q818yes—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—22no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—24no—
295C2×C42—28no—
296C2×C42—44no—
297C2×C42—42no—
298C2×C42—42no—
299C2×C42—44no—
300C2×C42—44no—
301C2×C42C2×D414yes—
302C2×C42C2×D414yes—
303C2×C42C2×D414yes—
304C2×C42C2×D414yes—
305C2×C42C2×D414yes—
306C2×C42C2×D414yes—
307C2×C42C2×D414yes—
308C2×C42C2×Q814yes—
309C2×C42#4114yes—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
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—24no—
328#95—24no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338C24—22no—
339C24—24no—
340C24—22no—
341C24—22no—
342C24—24no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—24no—
347C24—24no—
348C24C22×Q814yes—
349C24C2511yescharacteristic
350C2×Q8—28no—
351C2×Q8—48no—
352C2×Q8—44no—
353C2×Q8—44no—
354C2×Q8—48no—
355C2×Q8—44no—
356C2×Q8—44no—
357C2×Q8—44no—
358C2×Q8—84no—
359C2×Q8—84no—
360C2×D4—28no—
361C2×D4—48no—
362C2×D4—44no—
363C2×D4—44no—
364C2×D4—48no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—24no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—48no—
431C22×C4—48no—
432C22×C4—48no—
433C22×C4—88no—
434C22×C4—88no—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×Q814yes—
443C22×C4C22×Q812yes—
444C22×C4C22×Q812yes—
445C22×C4C22×Q812yes—
446C22×C4C2×#4118yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4112yes—
450C22×C4C2×#4114yes—
451C22×C4H(5,𝔽2)14yes—
452C22×C4H(5,𝔽2)12yes—
453C22×C4H(5,𝔽2)12yes—
454C22×C42−1+412yes—
455C22×C42−1+412yes—
456C4⋊C4—48no—
457C4⋊C4—416no—
458C4⋊C4—416no—
459C4⋊C4—44no—
460C4⋊C4—44no—
461C4⋊C4—48no—
462C4⋊C4—44no—
463C4⋊C4—44no—
464C4⋊C4—48no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—44no—
468C4⋊C4—48no—
469C4⋊C4—88no—
470C4⋊C4—88no—
471C22⋊C4—216no—
472C22⋊C4—216no—
473C22⋊C4—416no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C22⋊C4—88no—
478C22⋊C4—88no—
479C22⋊C4—88no—
480C42—28no—
481C42—28no—
482C42—28no—
483C42—28no—
484C42—44no—
485C42—44no—
486C42—48no—
487C42—88no—
488C42C22⋊Q8116yes—
489C23—24no—
490C23—28no—
491C23—28no—
492C23—24no—
493C23—24no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—42no—
505C23—42no—
506C23—44no—
507C23—44no—
508C23—44no—
509C23—44no—
510C23—44no—
511C23—44no—
512C23C23×D411yescharacteristic
513C23C23×Q811yescharacteristic
514C23C22×#4111yescharacteristic
515C23C22×#4112yes—
516C23C2×H(5,𝔽2)12yes—
517C23C2×H(5,𝔽2)11yescharacteristic
518C23C2×H(5,𝔽2)11yescharacteristic
519C23C2×2−1+411yescharacteristic
520C23C2×2−1+411yescharacteristic
521C23#58512yes—
522C23#58512yes—
523Q8—416no—
524Q8—88no—
525D4—416no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—28no—
558C2×C4—28no—
559C2×C4—28no—
560C2×C4—28no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—816no—
606C2×C4C2×C22⋊Q814yes—
607C2×C4C2×C22⋊Q814yes—
608C2×C4C2×C22⋊Q814yes—
609C2×C4#53214yes—
610C2×C4#53514yes—
611C2×C4#53714yes—
612C2×C4#53714yes—
613C22—48no—
614C22—48no—
615C22—44no—
616C22—44no—
617C22—44no—
618C22—82no—
619C22—84no—
620C22—84no—
621C22—82no—
622C22—82no—
623C22—82no—
624C22C22×C22⋊Q811yescharacteristic
625C22C2×#53211yescharacteristic
626C22C2×#53511yescharacteristic
627C22C2×#53711yescharacteristic
628C22C2×#53711yescharacteristic
629C22#344911yescharacteristic
630C22#344911yescharacteristic
631C22#345912yes—
632C22#346112yes—
633C22#347512yes—
634C22Q8×#4112yes—
635C22#350112yes—
636C22#350311yescharacteristic
637C22#350311yescharacteristic
638C22#350612yes—
639C22#350811yescharacteristic
640C22#350911yescharacteristic
641C22#351912yes—
642C22#352212yes—
643C22#355812yes—
644C22#356012yes—
645C22#356112yes—
646C22#356412yes—
647C4—24no—
648C4—24no—
649C4—24no—
650C4—24no—
651C4—24no—
652C4—24no—
653C4—24no—
654C4—44no—
655C4—48no—
656C4—82no—
657C4—84no—
658C4—82no—
659C4—82no—
660C4—82no—
661C4—82no—
662C4—82no—
663C2—84no—
664C2#3459411yescharacteristic
665C2#3494711yescharacteristic
666C2#3517211yescharacteristic
667C2#3523811yescharacteristic
668C2#3528011yescharacteristic
669C2#3753811yescharacteristic
670C2#3754611yescharacteristic
671C2#3957212yes—
672C2#3957412yes—
673C2#3970012yes—
674C2#3970212yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2863C2semidirect
3. #17080C2semidirect
4. #17079C2semidirect
7. #16515C2semidirect
8. #15204C2semidirect
9. #15161C2semidirect
10. #14878C2semidirect
11. #14870C2semidirect
12. #14866C2semidirect
14. #14356C2semidirect
16. #14198C2semidirect
17. C4×#384C2semidirect
91. C2×#124Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#28632
C2×C4×#1242
#170802
#170792
#170772
#170712
#165152
#152042
#151612
#148782
#148702
#148662
#148122
#143562
#142882
#141982
C4×#3842