Finite Group Databasebeta

← #1393949

Subgroups — #1393949

Order 512.

2540 conjugacy classes of subgroups.

The subgroup lattice has 691 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2496C211yesmaximal, characteristic
2C2×#2637C212yesmaximal
3C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
4#16673C212yesmaximal
5#16517C212yesmaximal, complemented
6#16329C214yesmaximal, complemented
7#16314C214yesmaximal
8#15158C211yesmaximal, characteristic, complemented
9#15150C211yesmaximal, characteristic
10#15145C211yesmaximal, characteristic, complemented
11#15145C212yesmaximal, complemented
12#15131C212yesmaximal
13#14879C212yesmaximal, complemented
14#14133C212yesmaximal, complemented
15#13918C211yesmaximal, characteristic, complemented
16#13876C212yesmaximal
17C4×#393C211yesmaximal, characteristic
18C22×#126C2212yes—
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2212yes—
21#2637—216no—
22#2496—28no—
23C2×#400C2212yes—
24C2×#398C2212yes—
25C2×#398C2214yes—
26C2×#398C2212yes—
27C2×#398C2214yes—
28C2×#397C2212yes—
29C2×#397C2212yes—
30C2×#397C2212yes—
31C2×#396C2214yes—
32C2×#395C2212yes—
33C2×#393C2211yescharacteristic
34Q8×C4⋊C4—28no—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2211yescharacteristic
38C2×#389C2211yescharacteristic
39C2×#389C2214yes—
40C2×#389C2212yes—
41C2×#388C2214yes—
42C2×#387C2212yes—
43C2×#387C2214yes—
44C2×#384C2212yes—
45C2×#384C2211yescharacteristic
46C2×#384C2211yescharacteristic
47C2×#384C2212yes—
48C2×#382C2212yes—
49C2×#380C2211yescharacteristic
50C2×#380C2211yescharacteristic
51C2×#380C2212yes—
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C22⋊C4C2212yes—
55C2×C4×C22⋊C4C2211yescharacteristic
56#1446C2211yescharacteristic
57#1445C2212yes—
58#1445C2212yes—
59#1444C2214yes—
60#1444C2214yes—
61#1444C2214yes—
62#1443C2214yes—
63#1442C2214yes—
64#1442C2211yescharacteristic
65#1442C2212yes—
66#1442C2214yes—
67#1442C2214yes—
68#1441C2214yes—
69#1441C2214yes—
70#1441C2212yes—
71#1441C2214yes—
72#1439C2214yes—
73#1439C2211yescharacteristic
74#1439C2214yes—
75#1439C2212yes—
76#1436C2214yes—
77#1436C2214yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2211yescharacteristic
82C4⋊#95C2212yes—
83#1434C2212yes—
84C4×#95C2212yes—
85C4×#95C2211yescharacteristic
86C23×Q8C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C2×#126—28no—
89C2×#126—28no—
90C2×#126—44no—
91C2×C4×Q8—28no—
92C2×C4×Q8—24no—
93C2×C4×Q8—28no—
94C2×C4×Q8—28no—
95C2×C4×Q8—44no—
96C22×C4⋊C4C2314yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2314yes—
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2314yes—
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129#400—28no—
130#398—28no—
131#398—216no—
132#398—48no—
133#398—44no—
134#397—28no—
135#397—28no—
136#397—44no—
137#396—216no—
138#395—28no—
139#393D418yes—
140#389—24no—
141#389—28no—
142#389—24no—
143#389—28no—
144#389—216no—
145#389—44no—
146#388—216no—
147#387—28no—
148#387—216no—
149#384—28no—
150#384—24no—
151#384—24no—
152#384—44no—
153#382—28no—
154#380—24no—
155#380—24no—
156#380—44no—
157C4×C4⋊C4—28no—
158C4×C4⋊C4D418yes—
159C4×C22⋊C4—24no—
160C4×C22⋊C4—28no—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2311yescharacteristic
169C2×#95C2312yes—
170C2×#95C2311yescharacteristic
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2311yescharacteristic
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2311yescharacteristic
189C2×#95C2312yes—
190C2×#95C2312yes—
191C25C2411yescharacteristic
192C22×Q8—22no—
193C22×Q8—24no—
194C22×Q8—22no—
195C22×Q8—24no—
196C22×Q8C2×D414yes—
197C23×C4—22no—
198C23×C4—24no—
199C23×C4—22no—
200C23×C4—24no—
201C23×C4—22no—
202C23×C4C2412yes—
203C23×C4C2414yes—
204C23×C4C2412yes—
205C23×C4C2414yes—
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2412yes—
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218#126—432no—
219C4×Q8—232no—
220C4×Q8—432no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—24no—
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—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—28no—
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—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—42no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—42no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4C2×D414yes—
277C2×C4⋊C4C2×D414yes—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—41no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4C2×D414yes—
298C2×C22⋊C4C2×D418yes—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—28no—
305C2×C42—28no—
306C2×C42—22no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—24no—
311C2×C42—44no—
312C2×C42C2×D418yes—
313C2×C42C2×D414yes—
314C2×C42#4118yes—
315#95—22no—
316#95—22no—
317#95—28no—
318#95—24no—
319#95—24no—
320#95—28no—
321#95—24no—
322#95—24no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—28no—
330#95—28no—
331#95—22no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—28no—
337#95—44no—
338#95—44no—
339#95—42no—
340#95—44no—
341#95—44no—
342#95C2×D418yes—
343#95C2×D414yes—
344#95C2×D418yes—
345C24—21no—
346C24—22no—
347C24—21no—
348C24—21no—
349C24—21no—
350C24—22no—
351C24—22no—
352C24—21no—
353C24—22no—
354C24—21no—
355C24C22×D412yes—
356C24C2511yescharacteristic
357C2×Q8—28no—
358C2×Q8—28no—
359C2×Q8—216no—
360C2×Q8—28no—
361C2×Q8—216no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—21no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—44no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4C22×D412yes—
445C22×C4C22×D414yes—
446C22×C4C22×D414yes—
447C22×C4C22×D412yes—
448C22×C4C22×D414yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D414yes—
452C22×C4C22×D412yes—
453C22×C4C2×#4114yes—
454C22×C4C2×#4114yes—
455C22×C4C2×#4114yes—
456C22×C4H(5,𝔽2)12yes—
457C22×C4H(5,𝔽2)12yes—
458C22×C4H(5,𝔽2)12yes—
459C22×C42−1+414yes—
460C22×C42−1+414yes—
461C22×C42−1+414yes—
462C22×C42−1+412yes—
463C4⋊C4—28no—
464C4⋊C4—216no—
465C4⋊C4—28no—
466C4⋊C4—28no—
467C4⋊C4—48no—
468C4⋊C4—416no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—44no—
472C4⋊C4—48no—
473C4⋊C4—44no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—416no—
478C4⋊C4—84no—
479C4⋊C4—88no—
480C4⋊C4—88no—
481C4⋊C4—88no—
482C4⋊C4—88no—
483C22⋊C4—28no—
484C22⋊C4—216no—
485C22⋊C4—44no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—88no—
490C42—28no—
491C42—216no—
492C42—216no—
493C42—416no—
494C42—44no—
495C42—48no—
496C23—22no—
497C23—24no—
498C23—22no—
499C23—22no—
500C23—24no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—42no—
505C23—42no—
506C23—41no—
507C23—41no—
508C23—42no—
509C23—42no—
510C23—41no—
511C23—42no—
512C23—42no—
513C23—41no—
514C23—41no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23C23×D411yescharacteristic
520C23C23×D411yescharacteristic
521C23C22×#4112yes—
522C23C2×H(5,𝔽2)11yescharacteristic
523C23C2×H(5,𝔽2)11yescharacteristic
524C23C2×H(5,𝔽2)11yescharacteristic
525C23C2×2−1+412yes—
526C23C2×2−1+411yescharacteristic
527C23C2×2−1+412yes—
528C23C2×2−1+412yes—
529C23#58511yescharacteristic
530Q8—232no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
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—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—88no—
621C2×C4#52514yes—
622C2×C4#53614yes—
623C2×C4#53714yes—
624C2×C4#54118yes—
625C2×C4#54218yes—
626C2×C4D4214yes—
627C2×C4#54718yes—
628C22—42no—
629C22—42no—
630C22—44no—
631C22—42no—
632C22—44no—
633C22—82no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—82no—
638C22—81no—
639C22C2×#52511yescharacteristic
640C22C2×#53611yescharacteristic
641C22C2×#53711yescharacteristic
642C22C2×#54112yes—
643C22C2×#54212yes—
644C22C2×D4211yescharacteristic
645C22C2×#54712yes—
646C22#348611yescharacteristic
647C22#348811yescharacteristic
648C22#349212yes—
649C22#349512yes—
650C22#349711yescharacteristic
651C22#349811yescharacteristic
652C22#349911yescharacteristic
653C22#349912yes—
654C22#351112yes—
655C22#354412yes—
656C22#354611yescharacteristic
657C22#354811yescharacteristic
658C22#355212yes—
659C22#355212yes—
660C22#355512yes—
661C22#355612yes—
662C4—24no—
663C4—28no—
664C4—44no—
665C4—42no—
666C4—44no—
667C4—44no—
668C4—42no—
669C4—44no—
670C4—42no—
671C4—44no—
672C4—82no—
673C4—81no—
674C4—84no—
675C4—82no—
676C4—84no—
677C4#263718yescomplemented
678C2—82no—
679C2C2×#263711yescharacteristic
680C2#3608911yescharacteristic
681C2#3610311yescharacteristic
682C2#3934111yescharacteristic
683C2#3935312yes—
684C2#3936011yescharacteristic
685C2#3937712yes—
686C2#4245512yes—
687C2#4251312yes—
688C2#4279111yescharacteristic
689C2#4281311yescharacteristic
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×Q8×C4⋊C4C2semidirect
5. #16517C2semidirect
6. #16329C2semidirect
8. #15158C2semidirect
10. #15145C2semidirect
11. #15145C2semidirect
13. #14879C2semidirect
14. #14133C2semidirect
15. #13918C2semidirect
677. C4#2637semidirect

Maximal subgroups

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

subgroupindex
C2×#24962
C2×#26372
C2×Q8×C4⋊C42
#166732
#165172
#163292
#163142
#151582
#151502
#151452
#151312
#148792
#141332
#139182
#138762
C4×#3932