Finite Group Databasebeta

← #826754

Subgroups — #826754

Order 512.

2432 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2369C211yesmaximal, characteristic, complemented
2C2×#2362C211yesmaximal, characteristic
3#16564C212yesmaximal
4#16062C214yesmaximal, complemented
5#15362C214yesmaximal
6#15193C211yesmaximal, characteristic, complemented
7#15191C211yesmaximal, characteristic, complemented
8#15125C211yesmaximal, characteristic
9#15111C212yesmaximal, complemented
10#15100C212yesmaximal
11#14150C212yesmaximal, complemented
12#14129C212yesmaximal
13#14046C212yesmaximal, complemented
14#14017C212yesmaximal, complemented
15#13997C212yesmaximal
16#13918C211yesmaximal, characteristic, complemented
17C4×#399C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×#117C2211yescharacteristic
22C2×#399C2211yescharacteristic
23C2×#393C2212yes—
24#2369—28no—
25#2362—28no—
26C2×#391C2214yes—
27C2×#391C2211yescharacteristic
28C2×#391C2212yes—
29C2×#391C2212yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2211yescharacteristic
33C2×#389C2212yes—
34C2×#389C2214yes—
35C2×#389C2212yes—
36C2×#388C2212yes—
37C2×#388C2212yes—
38C2×#387C2212yes—
39C2×#382C2212yes—
40C2×#380C2212yes—
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49#1446C2214yes—
50#1446C2212yes—
51#1445C2214yes—
52#1445C2211yescharacteristic
53#1445C2212yes—
54#1445C2214yes—
55#1445C2212yes—
56#1444C2211yescharacteristic
57#1444C2212yes—
58#1444C2214yes—
59#1442C2214yes—
60#1442C2214yes—
61#1442C2212yes—
62#1442C2212yes—
63#1442C2212yes—
64#1442C2214yes—
65#1441C2214yes—
66#1439C2212yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2212yes—
71#1439C2211yescharacteristic
72#1437C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C4⋊#95C2214yes—
83#1434C2212yes—
84C4×#95C2211yescharacteristic
85C23×Q8C2311yescharacteristic
86C23×Q8C2312yes—
87C24×C4C2311yescharacteristic
88C2×C4⋊Q8—28no—
89C2×C4⋊Q8—24no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8Q818yescomplemented
93C2×C4×Q8—24no—
94C2×C4×Q8—44no—
95C2×#117—24no—
96C2×#117—44no—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2312yes—
122C22×C42C2312yes—
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2312yes—
127C22×C42C2314yes—
128#399—24no—
129#393—44no—
130#391—28no—
131#391—24no—
132#391—216no—
133#391—44no—
134#389—24no—
135#389—28no—
136#389—28no—
137#389—28no—
138#389—48no—
139#389—44no—
140#388—28no—
141#388—28no—
142#387—28no—
143#382—28no—
144#380—28no—
145C4×C4⋊C4—28no—
146C4×C4⋊C4—28no—
147C4×C4⋊C4—28no—
148C4×C4⋊C4—24no—
149C4×C4⋊C4—44no—
150C4×C4⋊C4D418yes—
151C4×C22⋊C4—24no—
152C4×C22⋊C4—44no—
153C2×#95C2314yes—
154C2×#95C2311yescharacteristic
155C2×#95C2312yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2311yescharacteristic
166C2×#95C2312yes—
167C2×#95C2311yescharacteristic
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2311yescharacteristic
184C25C2411yescharacteristic
185C22×Q8—22no—
186C22×Q8—24no—
187C22×Q8—22no—
188C22×Q8—24no—
189C22×Q8—28no—
190C22×Q8—24no—
191C22×Q8—42no—
192C22×Q8—44no—
193C22×Q8—42no—
194C22×Q8C2×Q814yes—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4C2×Q814yes—
200C23×C4C2411yescharacteristic
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2414yes—
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2414yes—
211C23×C4C2411yescharacteristic
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C4⋊Q8—232no—
217C22⋊Q8—232no—
218C4×Q8—416no—
219#117—416no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—28no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4C2×D414yes—
258C2×C4⋊C4C2×D414yes—
259C2×C4⋊C4C2×Q818yes—
260C2×C4⋊C4C2×Q814yes—
261C2×C4⋊C4#4118yes—
262C2×C4⋊C4#4114yes—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—41no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4C2×Q818yes—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—22no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—28no—
297C2×C42—44no—
298C2×C42—44no—
299C2×C42—44no—
300C2×C42—44no—
301C2×C42C2×D414yes—
302C2×C42C2×D418yes—
303#95—22no—
304#95—22no—
305#95—24no—
306#95—28no—
307#95—28no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—28no—
312#95—28no—
313#95—24no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—22no—
319#95—44no—
320#95—42no—
321#95—44no—
322#95—44no—
323#95—42no—
324#95—42no—
325#95—42no—
326#95—42no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—42no—
332#95C2×D418yes—
333#95#4114yes—
334C24—21no—
335C24—21no—
336C24—21no—
337C24—22no—
338C24—21no—
339C24—22no—
340C24—22no—
341C24—21no—
342C24—21no—
343C24—22no—
344C24C22×Q812yes—
345C24C2511yescharacteristic
346C2×Q8—216no—
347C2×Q8—28no—
348C2×Q8—216no—
349C2×Q8—44no—
350C2×Q8—44no—
351C2×Q8—48no—
352C2×Q8—84no—
353C2×Q8—84no—
354C2×Q8—88no—
355C2×Q8—88no—
356C22×C4—22no—
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—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—24no—
399C22×C4—21no—
400C22×C4—28no—
401C22×C4—216no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—44no—
434C22×C4—44no—
435C22×C4—48no—
436C22×C4C22×D412yes—
437C22×C4C22×D414yes—
438C22×C4C22×D414yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×Q814yes—
442C22×C4C22×Q812yes—
443C22×C4C22×Q814yes—
444C22×C4C22×Q812yes—
445C22×C4C2×#4118yes—
446C22×C4C2×#4114yes—
447C22×C4C2×#4114yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4112yes—
450C22×C4C2×#4114yes—
451C22×C4H(5,𝔽2)12yes—
452C22×C4H(5,𝔽2)14yes—
453C22×C42−1+414yes—
454C22×C42−1+412yes—
455C4⋊C4—28no—
456C4⋊C4—216no—
457C4⋊C4—216no—
458C4⋊C4—216no—
459C4⋊C4—28no—
460C4⋊C4—416no—
461C4⋊C4—48no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—48no—
468C4⋊C4—88no—
469C4⋊C4—88no—
470C4⋊C4C22⋊Q8116yes—
471C22⋊C4—216no—
472C22⋊C4—44no—
473C22⋊C4—48no—
474C22⋊C4—48no—
475C22⋊C4—48no—
476C22⋊C4—44no—
477C22⋊C4—88no—
478C42—28no—
479C42—216no—
480C42—48no—
481C42—48no—
482C42—48no—
483C42—44no—
484C42—48no—
485C42—88no—
486C23—22no—
487C23—24no—
488C23—22no—
489C23—24no—
490C23—22no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—42no—
495C23—41no—
496C23—42no—
497C23—41no—
498C23—41no—
499C23—41no—
500C23—42no—
501C23—42no—
502C23—41no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23C23×D411yescharacteristic
510C23C23×Q811yescharacteristic
511C23C22×#4111yescharacteristic
512C23C22×#4112yes—
513C23C22×#4111yescharacteristic
514C23C2×H(5,𝔽2)12yes—
515C23C2×H(5,𝔽2)11yescharacteristic
516C23C2×2−1+411yescharacteristic
517C23C2×2−1+412yes—
518C23#58511yescharacteristic
519C23#58512yes—
520Q8—416no—
521Q8—816no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—28no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—28no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—41no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—44no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—416no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4C2×C22⋊Q818yes—
611C2×C4C2×C22⋊Q814yes—
612C2×C4#52514yes—
613C2×C4#53514yes—
614C2×C4#54714yes—
615C2×C4#55418yes—
616C2×C4#55918yes—
617C22—44no—
618C22—42no—
619C22—44no—
620C22—42no—
621C22—42no—
622C22—81no—
623C22—81no—
624C22—81no—
625C22—82no—
626C22—81no—
627C22—82no—
628C22C22×C22⋊Q811yescharacteristic
629C22C2×#52511yescharacteristic
630C22C2×#53511yescharacteristic
631C22C2×#54711yescharacteristic
632C22C2×#55412yes—
633C22Q8×#4111yescharacteristic
634C22C2×#55912yes—
635C22#348111yescharacteristic
636C22#348212yes—
637C22#348611yescharacteristic
638C22#348811yescharacteristic
639C22#350011yescharacteristic
640C22#350212yes—
641C22#350311yescharacteristic
642C22#350612yes—
643C22#351412yes—
644C22#351511yescharacteristic
645C22#351812yes—
646C22#352112yes—
647C22#352312yes—
648C22#353612yes—
649C22#354112yes—
650C22#354712yes—
651C4—24no—
652C4—28no—
653C4—44no—
654C4—42no—
655C4—44no—
656C4—42no—
657C4—44no—
658C4—48no—
659C4—42no—
660C4—44no—
661C4—82no—
662C4—81no—
663C4—82no—
664C4—82no—
665C4—84no—
666C4#258018yes—
667C2—82no—
668C2C2×#258011yescharacteristic
669C2#3571111yescharacteristic
670C2#3607511yescharacteristic
671C2#3627511yescharacteristic
672C2#3642311yescharacteristic
673C2#3949111yescharacteristic
674C2#4281011yescharacteristic
675C2#4522512yes—
676C2#4540912yes—
677C2#5963012yes—
678C2#5992012yes—
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
1. C2×#2369C2semidirect
4. #16062C2semidirect
6. #15193C2semidirect
7. #15191C2semidirect
9. #15111C2semidirect
11. #14150C2semidirect
13. #14046C2semidirect
14. #14017C2semidirect
16. #13918C2semidirect
92. C2×C22⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23692
C2×#23622
#165642
#160622
#153622
#151932
#151912
#151252
#151112
#151002
#141502
#141292
#140462
#140172
#139972
#139182
C4×#3992