Finite Group Databasebeta

← #1395850

Subgroups — #1395850

Order 512.

2304 conjugacy classes of subgroups.

The subgroup lattice has 675 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic, complemented
2C2×#2362C211yesmaximal, characteristic
3#16682C212yesmaximal
4#16679C212yesmaximal
5#16622C214yesmaximal, complemented
6#16536C214yesmaximal
7#15193C211yesmaximal, characteristic, complemented
8#15185C211yesmaximal, characteristic, complemented
9#15182C212yesmaximal
10#15150C211yesmaximal, characteristic
11#15133C212yesmaximal, complemented
12#15106C212yesmaximal, complemented
13#14289C212yesmaximal, complemented
14#14129C212yesmaximal
15#14017C212yesmaximal, complemented
16#13921C211yesmaximal, characteristic, complemented
17C4×#399C211yesmaximal, characteristic
18C22×#125C2212yes—
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×#117C2211yescharacteristic
22#2552—28no—
23C2×#399C2211yescharacteristic
24C2×#398C2214yes—
25C2×#398C2214yes—
26C2×#397C2212yes—
27C2×#397C2212yes—
28#2362—28no—
29C2×#391C2212yes—
30C2×#391C2211yescharacteristic
31C2×#391C2212yes—
32C2×#389C2212yes—
33C2×#389C2211yescharacteristic
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#388C2212yes—
37C2×#388C2212yes—
38C2×#387C2212yes—
39C2×#384C2212yes—
40C2×#384C2211yescharacteristic
41C2×#383C2212yes—
42C2×#380C2212yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49#1445C2212yes—
50#1445C2212yes—
51#1445C2211yescharacteristic
52#1445C2212yes—
53#1445C2212yes—
54#1445C2212yes—
55#1445C2212yes—
56#1445C2214yes—
57#1444C2211yescharacteristic
58#1444C2212yes—
59#1444C2214yes—
60#1443C2214yes—
61#1442C2214yes—
62#1442C2214yes—
63#1441C2214yes—
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1441C2214yes—
68#1439C2211yescharacteristic
69#1439C2212yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1439C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2211yescharacteristic
82C4⋊#95C2212yes—
83#1434C2212yes—
84C4×#95C2211yescharacteristic
85C23×Q8C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C2×#125—28no—
88C2×#125—28no—
89C2×#125—44no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8—44no—
95C2×#117—24no—
96C2×#117—44no—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2312yes—
129#399—24no—
130#398—216no—
131#398—48no—
132#397—44no—
133#397—44no—
134#391—28no—
135#391—24no—
136#391—44no—
137#389—24no—
138#389—28no—
139#389—28no—
140#389—44no—
141#388—28no—
142#388—28no—
143#387—28no—
144#384—28no—
145#384—24no—
146#383—44no—
147#380—28no—
148C4×C4⋊C4—28no—
149C4×C4⋊C4—24no—
150C4×C4⋊C4—44no—
151C4×C4⋊C4D418yes—
152C4×C22⋊C4—24no—
153C4×C22⋊C4—44no—
154C2×#95C2314yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2311yescharacteristic
159C2×#95C2312yes—
160C2×#95C2311yescharacteristic
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2311yescharacteristic
168C2×#95C2312yes—
169C2×#95C2311yescharacteristic
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C25C2411yescharacteristic
186C22×Q8—22no—
187C22×Q8—22no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8#4114yes—
191C23×C4—22no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4#4114yes—
196C23×C4C2414yes—
197C23×C4C2411yescharacteristic
198C23×C4C2412yes—
199C23×C4C2414yes—
200C23×C4C2412yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2411yescharacteristic
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2411yescharacteristic
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212#125—432no—
213C22⋊Q8—232no—
214C4×Q8—416no—
215#117—416no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—28no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
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—28no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4C2×D414yes—
264C2×C4⋊C4C2×D414yes—
265C2×C4⋊C4#4114yes—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—41no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—22no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—44no—
300C2×C42—44no—
301C2×C42C2×D418yes—
302C2×C42C2×D414yes—
303#95—22no—
304#95—22no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—22no—
314#95—22no—
315#95—24no—
316#95—28no—
317#95—24no—
318#95—28no—
319#95—44no—
320#95—42no—
321#95—42no—
322#95—42no—
323#95—42no—
324#95—44no—
325#95—44no—
326#95—44no—
327#95—44no—
328#95—44no—
329#95—42no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95C2×D418yes—
334C24—21no—
335C24—21no—
336C24—21no—
337C24—21no—
338C24—21no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—21no—
344C24C2×#4112yes—
345C24C2511yescharacteristic
346C2×Q8—28no—
347C2×Q8—216no—
348C2×Q8—44no—
349C2×Q8—44no—
350C2×Q8—48no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—28no—
398C22×C4—216no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—21no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—48no—
433C22×C4C22×D412yes—
434C22×C4C22×D414yes—
435C22×C4C22×D414yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C2×#4114yes—
439C22×C4C2×#4112yes—
440C22×C4C2×#4114yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4112yes—
443C22×C4C2×#4112yes—
444C22×C4H(5,𝔽2)12yes—
445C22×C4H(5,𝔽2)12yes—
446C22×C4H(5,𝔽2)14yes—
447C22×C42−1+414yes—
448C22×C42−1+414yes—
449C22×C42−1+414yes—
450C4⋊C4—28no—
451C4⋊C4—216no—
452C4⋊C4—28no—
453C4⋊C4—28no—
454C4⋊C4—28no—
455C4⋊C4—48no—
456C4⋊C4—48no—
457C4⋊C4—48no—
458C4⋊C4—416no—
459C4⋊C4—416no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—44no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—84no—
466C4⋊C4—88no—
467C4⋊C4—88no—
468C22⋊C4—216no—
469C22⋊C4—44no—
470C22⋊C4—48no—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C22⋊C4—44no—
474C22⋊C4—88no—
475C42—28no—
476C42—216no—
477C42—48no—
478C42—48no—
479C42—48no—
480C42—44no—
481C42—48no—
482C23—22no—
483C23—24no—
484C23—22no—
485C23—22no—
486C23—24no—
487C23—41no—
488C23—41no—
489C23—42no—
490C23—41no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—42no—
496C23—41no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23C23×D411yescharacteristic
506C23C22×#4111yescharacteristic
507C23C22×#4111yescharacteristic
508C23C22×#4111yescharacteristic
509C23C2×H(5,𝔽2)12yes—
510C23C2×H(5,𝔽2)11yescharacteristic
511C23C2×H(5,𝔽2)11yescharacteristic
512C23C2×2−1+412yes—
513C23C2×2−1+412yes—
514C23C2×2−1+412yes—
515C23#58511yescharacteristic
516Q8—416no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—22no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—28no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—416no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—41no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4#52514yes—
607C2×C4#53514yes—
608C2×C4#53614yes—
609C2×C4#53918yes—
610C2×C4#53918yes—
611C2×C4#54614yes—
612C22—42no—
613C22—42no—
614C22—44no—
615C22—42no—
616C22—44no—
617C22—81no—
618C22—81no—
619C22—81no—
620C22—82no—
621C22—81no—
622C22—82no—
623C22C2×#52511yescharacteristic
624C22C2×#53511yescharacteristic
625C22C2×#53611yescharacteristic
626C22C2×#53912yes—
627C22C2×#53912yes—
628C22C2×#54611yescharacteristic
629C22#348011yescharacteristic
630C22#348311yescharacteristic
631C22#348611yescharacteristic
632C22#348811yescharacteristic
633C22#349911yescharacteristic
634C22#350811yescharacteristic
635C22#350912yes—
636C22#350912yes—
637C22#351711yescharacteristic
638C22#352012yes—
639C22#352312yes—
640C22#353612yes—
641C22#354112yes—
642C22#354412yes—
643C22#354812yes—
644C22#355212yes—
645C22#355512yes—
646C4—24no—
647C4—42no—
648C4—44no—
649C4—44no—
650C4—44no—
651C4—44no—
652C4—44no—
653C4—42no—
654C4—42no—
655C4—44no—
656C4—81no—
657C4—84no—
658C4—82no—
659C4—82no—
660C4—84no—
661C4#263318yes—
662C2—82no—
663C2C2×#263311yescharacteristic
664C2#3574911yescharacteristic
665C2#3641011yescharacteristic
666C2#3895111yescharacteristic
667C2#3946511yescharacteristic
668C2#4002812yes—
669C2#4012412yes—
670C2#4281011yescharacteristic
671C2#4284912yes—
672C2#4291811yescharacteristic
673C2#4293912yes—
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×#2552C2semidirect
5. #16622C2semidirect
7. #15193C2semidirect
8. #15185C2semidirect
11. #15133C2semidirect
12. #15106C2semidirect
13. #14289C2semidirect
15. #14017C2semidirect
16. #13921C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#23622
#166822
#166792
#166222
#165362
#151932
#151852
#151822
#151502
#151332
#151062
#142892
#141292
#140172
#139212
C4×#3992