Finite Group Databasebeta

← #729380

Subgroups — #729380

Order 512.

2334 conjugacy classes of subgroups.

The subgroup lattice has 665 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2568C211yesmaximal, characteristic
2C2×#2568C211yesmaximal, characteristic
3C2×#2399C211yesmaximal, characteristic
4C2×#2355C211yesmaximal, characteristic
5#16804C212yesmaximal
6#16705C212yesmaximal
7#15738C214yesmaximal
8#15503C214yesmaximal
9#15237C212yesmaximal
10#15184C212yesmaximal
11#15164C212yesmaximal
12#15085C212yesmaximal
13#14274C211yesmaximal, characteristic
14#14021C212yesmaximal
15#14021C212yesmaximal
16#13925C211yesmaximal, characteristic
17#13783C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21#2568—28no—
22#2568—28no—
23#2399—28no—
24C2×#400C2212yes—
25C2×#400C2212yes—
26C2×#400C2214yes—
27C2×#398C2211yescharacteristic
28C2×#398C2212yes—
29C2×#398C2211yescharacteristic
30C2×#395C2211yescharacteristic
31C2×#395C2212yes—
32C2×#395C2211yescharacteristic
33#2355—28no—
34C2×#391C2212yes—
35C2×#391C2212yes—
36C2×#391C2212yes—
37C2×#391C2212yes—
38C2×#391C2212yes—
39C2×#391C2212yes—
40C2×#391C2212yes—
41C2×#391C2212yes—
42C2×#391C2212yes—
43C2×#389C2212yes—
44C2×#387C2212yes—
45C2×#387C2212yes—
46C2×#384C2211yescharacteristic
47C2×#382C2212yes—
48C2×#382C2211yescharacteristic
49C2×#382C2211yescharacteristic
50C2×#382C2212yes—
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2211yescharacteristic
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2212yes—
55C2×C4×C4⋊C4C2211yescharacteristic
56C2×C4×C4⋊C4C2212yes—
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×#376C2211yescharacteristic
59#1446C2212yes—
60#1445C2212yes—
61#1445C2212yes—
62#1445C2214yes—
63#1445C2212yes—
64#1443C2214yes—
65#1443C2214yes—
66#1443C2214yes—
67#1443C2214yes—
68#1442C2214yes—
69#1442C2214yes—
70#1441C2214yes—
71#1441C2214yes—
72#1441C2214yes—
73#1441C2214yes—
74#1441C2214yes—
75#1441C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78#1436C2214yes—
79#1436C2214yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83C4⋊#95C2212yes—
84#1434C2214yes—
85#1434C2214yes—
86#1434C2212yes—
87C23×Q8C2312yes—
88C23×Q8C2311yescharacteristic
89C23×Q8C2311yescharacteristic
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—24no—
92C2×C4⋊Q8—24no—
93C2×C4×Q8—24no—
94C2×C4×Q8—24no—
95C2×C4×Q8—24no—
96C2×C4×Q8—24no—
97C2×C4×Q8—24no—
98C2×C4×Q8D418yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2312yes—
118C22×C42C2312yes—
119C22×C42C2312yes—
120C22×C42C2311yescharacteristic
121C22×C42C2311yescharacteristic
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127#400—28no—
128#400—28no—
129#400—216no—
130#398—28no—
131#398—24no—
132#398—24no—
133#395—28no—
134#395—24no—
135#395—24no—
136#391—28no—
137#391—28no—
138#391—28no—
139#391—28no—
140#391—28no—
141#391—28no—
142#391—28no—
143#391—28no—
144#391—28no—
145#389—28no—
146#387—28no—
147#387—28no—
148#384—24no—
149#382—28no—
150#382—24no—
151#382—28no—
152#382—24no—
153C4×C4⋊C4—24no—
154C4×C4⋊C4—24no—
155C4×C4⋊C4—24no—
156C4×C4⋊C4—28no—
157C4×C4⋊C4—28no—
158C4×C4⋊C4—28no—
159C4×C4⋊C4D418yes—
160#376—24no—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2311yescharacteristic
164C2×#95C2311yescharacteristic
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2312yes—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—24no—
199C22×Q8—24no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—24no—
203C22×Q8—24no—
204C22×Q8—24no—
205C22×Q8—24no—
206C22×Q8—22no—
207C22×Q8—22no—
208C22×Q8—24no—
209C22×Q8—24no—
210C22×Q8—41no—
211C22×Q8—41no—
212C22×Q8—42no—
213C22×Q8C2×D414yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2414yes—
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2412yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C4⋊Q8—416no—
232C4×Q8—232no—
233C4×Q8—416no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—28no—
273C2×C4⋊C4—42no—
274C2×C4⋊C4—42no—
275C2×C4⋊C4—42no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4C2×D414yes—
282C2×C4⋊C4C2×D414yes—
283C2×C4⋊C4C2×D414yes—
284C2×C4⋊C4C2×D414yes—
285C2×C4⋊C4C2×D414yes—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—24no—
289C2×C42—22no—
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—42no—
301C2×C42—42no—
302C2×C42—42no—
303C2×C42—44no—
304C2×C42C2×D414yes—
305C2×C42C2×D414yes—
306C2×C42C2×D418yes—
307C2×C42C2×D414yes—
308C2×C42C2×D414yes—
309C2×C42#4118yes—
310C2×C42#4114yes—
311#95—22no—
312#95—22no—
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—28no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—28no—
332#95—28no—
333#95—28no—
334#95—28no—
335#95—28no—
336#95—28no—
337#95—28no—
338#95—28no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95C2×D418yes—
345C24C2511yescharacteristic
346C2×Q8—28no—
347C2×Q8—28no—
348C2×Q8—28no—
349C2×Q8—44no—
350C2×Q8—44no—
351C2×Q8—44no—
352C2×Q8—48no—
353C2×Q8—44no—
354C2×Q8—44no—
355C2×Q8—48no—
356C2×Q8—44no—
357C2×Q8—44no—
358C2×Q8—48no—
359C2×Q8—48no—
360C2×Q8—48no—
361C2×Q8—48no—
362C2×Q8—48no—
363C2×Q8—48no—
364C2×Q8—48no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—21no—
386C22×C4—22no—
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—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4C22×D414yes—
447C22×C4C22×D414yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4C22×D412yes—
458C22×C4C2×#4114yes—
459C22×C4C2×#4114yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4114yes—
464C22×C42−1+414yes—
465C22×C42−1+412yes—
466C22×C42−1+412yes—
467C22×C42−1+414yes—
468C4⋊C4—28no—
469C4⋊C4—28no—
470C4⋊C4—28no—
471C4⋊C4—28no—
472C4⋊C4—28no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—44no—
479C4⋊C4—48no—
480C4⋊C4—48no—
481C4⋊C4—44no—
482C4⋊C4—44no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—84no—
486C4⋊C4—88no—
487C42—28no—
488C42—28no—
489C42—216no—
490C42—48no—
491C42—48no—
492C42—48no—
493C42—48no—
494C42—84no—
495C42#121116yes—
496C23C23×D411yescharacteristic
497C23C23×D411yescharacteristic
498C23C22×#4111yescharacteristic
499C23C22×#4112yes—
500C23C2×2−1+412yes—
501C23C2×2−1+411yescharacteristic
502C23C2×2−1+411yescharacteristic
503C23C2×2−1+412yes—
504C23#58511yescharacteristic
505C23#58511yescharacteristic
506C23#58512yes—
507Q8—416no—
508Q8—416no—
509Q8—816no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—24no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
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—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—42no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—44no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
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×C4C2×#12114yes—
606C2×C4C2×#12114yes—
607C2×C4C2×#12114yes—
608C2×C4#53214yes—
609C2×C4#53614yes—
610C2×C4#53614yes—
611C2×C4#54818yes—
612C2×C4#55618yes—
613C22C22×#12111yescharacteristic
614C22C2×#53211yescharacteristic
615C22C2×#53611yescharacteristic
616C22C2×#53611yescharacteristic
617C22#344911yescharacteristic
618C22C2×#54812yes—
619C22C2×#55612yes—
620C22#348511yescharacteristic
621C22#348611yescharacteristic
622C22#348912yes—
623C22#348911yescharacteristic
624C22#348911yescharacteristic
625C22#349512yes—
626C22#349511yescharacteristic
627C22#349511yescharacteristic
628C22#350712yes—
629C22#350712yes—
630C22#350912yes—
631C22#351012yes—
632C22#355512yes—
633C22#355512yes—
634C22#355612yes—
635C22#355712yes—
636C4—24no—
637C4—24no—
638C4—24no—
639C4—44no—
640C4—44no—
641C4—42no—
642C4—42no—
643C4—44no—
644C4—44no—
645C4—44no—
646C4—42no—
647C4—84no—
648C4—84no—
649C4—82no—
650C4—82no—
651C4—82no—
652C4#254518yes—
653C2C2×#254511yescharacteristic
654C2#3530611yescharacteristic
655C2#3563311yescharacteristic
656C2#3593811yescharacteristic
657C2#3595911yescharacteristic
658C2#3971912yes—
659C2#3972412yes—
660C2#4267611yescharacteristic
661C2#4281111yescharacteristic
662C2#4515212yes—
663C2#4543112yes—
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#25682
C2×#23992
C2×#23552
#168042
#167052
#157382
#155032
#152372
#151842
#151642
#150852
#142742
#140212
#139252
#137832