Finite Group Databasebeta

← #793569

Subgroups — #793569

Order 512.

2588 conjugacy classes of subgroups.

The subgroup lattice has 685 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2604C212yesmaximal
2C2×#2496C211yesmaximal, characteristic
3C2×#2365C211yesmaximal, characteristic, complemented
4#16656C212yesmaximal
5#15454C214yesmaximal, complemented
6#15443C214yesmaximal
7#15156C211yesmaximal, characteristic, complemented
8#15143C211yesmaximal, characteristic, complemented
9#15125C211yesmaximal, characteristic
10#14901C212yesmaximal, complemented
11#14876C212yesmaximal, complemented
12#14151C212yesmaximal, complemented
13#13994C212yesmaximal
14#13884C212yesmaximal, complemented
15#13876C212yesmaximal
16C4×#399C211yesmaximal, characteristic
17#13810C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×#126C2212yes—
20C22×C4×Q8C2212yes—
21#2604—216no—
22#2496—28no—
23C2×#400C2212yes—
24C2×#400C2212yes—
25C2×#399C2211yescharacteristic
26C2×#397C2212yes—
27C2×#395C2212yes—
28C2×#395C2212yes—
29C2×#393C2212yes—
30#2365—28no—
31C2×#391C2212yes—
32C2×#389C2211yescharacteristic
33C2×#389C2214yes—
34C2×#389C2211yescharacteristic
35C2×#389C2212yes—
36C2×#388C2214yes—
37C2×#388C2214yes—
38C2×#387C2212yes—
39C2×#387C2214yes—
40C2×#387C2214yes—
41C2×#384C2212yes—
42C2×#384C2211yescharacteristic
43C2×#380C2212yes—
44C2×#380C2211yescharacteristic
45C2×#380C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C4⋊C4C2212yes—
48C2×C4×C4⋊C4C2212yes—
49C2×C4×C4⋊C4C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2212yes—
53C2×#376C2212yes—
54#1446C2212yes—
55#1445C2211yescharacteristic
56#1445C2212yes—
57#1444C2211yescharacteristic
58#1444C2214yes—
59#1443C2214yes—
60#1442C2214yes—
61#1442C2212yes—
62#1442C2214yes—
63#1442C2214yes—
64#1441C2212yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1440C2214yes—
69#1439C2211yescharacteristic
70#1439C2214yes—
71#1439C2212yes—
72#1439C2212yes—
73#1436C2214yes—
74#1436C2214yes—
75C4⋊#95C2211yescharacteristic
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2214yes—
82#1434C2212yes—
83#1434C2214yes—
84#1434C2214yes—
85C4×#95C2211yescharacteristic
86C23×Q8C2312yes—
87C24×C4C2311yescharacteristic
88C2×C4⋊Q8—28no—
89C2×C4⋊Q8—24no—
90C2×#126—28no—
91C2×#126—28no—
92C2×#126—44no—
93C2×C4×Q8—28no—
94C2×C4×Q8—28no—
95C2×C4×Q8—44no—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C42C2314yes—
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2314yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128#400—28no—
129#400—28no—
130#399D418yes—
131#397—28no—
132#395—28no—
133#395—44no—
134#393—28no—
135#391—28no—
136#389—216no—
137#389—28no—
138#389—24no—
139#389—24no—
140#388—216no—
141#388—216no—
142#387—216no—
143#387—28no—
144#387—216no—
145#384—24no—
146#384—28no—
147#380—24no—
148#380—24no—
149#380—28no—
150C4×C4⋊C4—28no—
151C4×C4⋊C4—24no—
152C4×C4⋊C4—28no—
153C4×C4⋊C4—28no—
154C4×C4⋊C4D418yes—
155C4×C22⋊C4—24no—
156C4×C22⋊C4—28no—
157#376—28no—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2311yescharacteristic
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2311yescharacteristic
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2311yescharacteristic
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2311yescharacteristic
188C2×#95C2312yes—
189C25C2411yescharacteristic
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8—28no—
193C22×Q8—24no—
194C22×Q8—44no—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2414yes—
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2412yes—
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C4⋊Q8—232no—
217#126—432no—
218C4×Q8—432no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—28no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4C2×D414yes—
265C2×C4⋊C4C2×D414yes—
266C2×C4⋊C4C2×D414yes—
267C2×C4⋊C4C2×D418yes—
268C2×C4⋊C4#4118yes—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—41no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4C2×D414yes—
289C2×C22⋊C4C2×D418yes—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—28no—
296C2×C42—28no—
297C2×C42—28no—
298C2×C42—22no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—28no—
304C2×C42—28no—
305C2×C42—44no—
306C2×C42C2×D418yes—
307C2×C42C2×D414yes—
308C2×C42#4118yes—
309#95—28no—
310#95—24no—
311#95—22no—
312#95—22no—
313#95—24no—
314#95—28no—
315#95—24no—
316#95—24no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—24no—
322#95—28no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—28no—
328#95—24no—
329#95—28no—
330#95—22no—
331#95—24no—
332#95—44no—
333#95—42no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95C2×D414yes—
339#95C2×D418yes—
340C24—21no—
341C24—22no—
342C24—22no—
343C24—21no—
344C24—22no—
345C24—21no—
346C24—21no—
347C24—21no—
348C24—22no—
349C24—21no—
350C24C22×D412yes—
351C24C2511yescharacteristic
352C2×Q8—216no—
353C2×Q8—416no—
354C2×Q8—416no—
355C2×Q8—48no—
356C2×Q8—48no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—21no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—21no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4C22×D412yes—
440C22×C4C22×D414yes—
441C22×C4C22×D414yes—
442C22×C4C22×D412yes—
443C22×C4C22×D414yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D414yes—
447C22×C4C22×D412yes—
448C22×C4C2×#4114yes—
449C22×C4C2×#4118yes—
450C22×C4C2×#4114yes—
451C22×C4C2×#4114yes—
452C22×C4C2×#4114yes—
453C22×C4H(5,𝔽2)12yes—
454C22×C42−1+412yes—
455C22×C42−1+414yes—
456C22×C42−1+412yes—
457C22×C42−1+412yes—
458C4⋊C4—216no—
459C4⋊C4—216no—
460C4⋊C4—28no—
461C4⋊C4—28no—
462C4⋊C4—28no—
463C4⋊C4—216no—
464C4⋊C4—416no—
465C4⋊C4—48no—
466C4⋊C4—48no—
467C4⋊C4—416no—
468C4⋊C4—48no—
469C4⋊C4—48no—
470C4⋊C4—44no—
471C4⋊C4—44no—
472C4⋊C4—44no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—88no—
476C22⋊C4—28no—
477C22⋊C4—216no—
478C22⋊C4—44no—
479C22⋊C4—416no—
480C22⋊C4—48no—
481C22⋊C4—48no—
482C22⋊C4—48no—
483C42—28no—
484C42—216no—
485C42—216no—
486C42—416no—
487C42—44no—
488C42—48no—
489C42—416no—
490C23—22no—
491C23—24no—
492C23—24no—
493C23—22no—
494C23—22no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—42no—
499C23—42no—
500C23—41no—
501C23—41no—
502C23—42no—
503C23—42no—
504C23—41no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—41no—
510C23—41no—
511C23—42no—
512C23—42no—
513C23C23×D411yescharacteristic
514C23C23×D411yescharacteristic
515C23C22×#4112yes—
516C23C22×#4112yes—
517C23C2×H(5,𝔽2)11yescharacteristic
518C23C2×2−1+411yescharacteristic
519C23C2×2−1+411yescharacteristic
520C23C2×2−1+412yes—
521C23C2×2−1+411yescharacteristic
522C23#58512yes—
523C23#58511yescharacteristic
524Q8—432no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—22no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—22no—
536C2×C4—28no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—44no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—41no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—44no—
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—41no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—88no—
615C2×C4#52514yes—
616C2×C4#53214yes—
617C2×C4#53614yes—
618C2×C4D4214yes—
619C2×C4#54718yes—
620C2×C4#55218yes—
621C2×C4#55618yes—
622C22—42no—
623C22—42no—
624C22—44no—
625C22—44no—
626C22—42no—
627C22—82no—
628C22—81no—
629C22—81no—
630C22—81no—
631C22—82no—
632C22—81no—
633C22C2×#52511yescharacteristic
634C22C2×#53211yescharacteristic
635C22C2×#53611yescharacteristic
636C22C2×D4211yescharacteristic
637C22C2×#54712yes—
638C22D4×#4112yes—
639C22C2×#55212yes—
640C22C2×#55612yes—
641C22#348711yescharacteristic
642C22#348911yescharacteristic
643C22#349312yes—
644C22#349512yes—
645C22#349711yescharacteristic
646C22#349911yescharacteristic
647C22#349911yescharacteristic
648C22#351112yes—
649C22#351312yes—
650C22#351812yes—
651C22#352312yes—
652C22#354711yescharacteristic
653C22#355011yescharacteristic
654C22#355512yes—
655C22#356312yes—
656C4—24no—
657C4—28no—
658C4—48no—
659C4—44no—
660C4—44no—
661C4—42no—
662C4—42no—
663C4—44no—
664C4—44no—
665C4—42no—
666C4—82no—
667C4—81no—
668C4—84no—
669C4—82no—
670C4—82no—
671C4#260418yescomplemented
672C2—82no—
673C2C2×#260411yescharacteristic
674C2#3600711yescharacteristic
675C2#3606511yescharacteristic
676C2#3610211yescharacteristic
677C2#3612711yescharacteristic
678C2#3934311yescharacteristic
679C2#4287911yescharacteristic
680C2#4501712yes—
681C2#4506512yes—
682C2#5924412yes—
683C2#5929012yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×#2365C2semidirect
5. #15454C2semidirect
7. #15156C2semidirect
8. #15143C2semidirect
10. #14901C2semidirect
11. #14876C2semidirect
12. #14151C2semidirect
14. #13884C2semidirect
17. #13810C2semidirect
671. C4#2604semidirect

Maximal subgroups

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

subgroupindex
C2×#26042
C2×#24962
C2×#23652
#166562
#154542
#154432
#151562
#151432
#151252
#149012
#148762
#141512
#139942
#138842
#138762
C4×#3992
#138102