Finite Group Databasebeta

← #764630

Subgroups — #764630

Order 512.

2802 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2484C211yesmaximal, characteristic
2C22×#398C211yesmaximal, characteristic
3C2×#2317C211yesmaximal, characteristic
4C2×#2313C211yesmaximal, characteristic
5#15812C214yesmaximal, complemented
6#15778C214yesmaximal, complemented
7#15738C214yesmaximal, complemented
8#15629C214yesmaximal, complemented
9#14935C212yesmaximal
10#14728C212yesmaximal
11#14128C211yesmaximal, characteristic
12#14043C212yesmaximal
13#13972C212yesmaximal
14#13958C211yesmaximal, characteristic
15#13870C211yesmaximal, characteristic
16C22×#117C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C23×C4⋊C4C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22#2484—28no—
23C2×#400C2212yes—
24C2×#398—28no—
25C2×#398C2212yes—
26C2×#398C2214yes—
27C2×#398C2214yes—
28C2×#398C2214yes—
29#2317—28no—
30#2313—28no—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#389C2214yes—
34C2×#387C2214yes—
35C2×#387C2214yes—
36C2×#387C2214yes—
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#385C2212yes—
40C2×#385C2212yes—
41C2×#384C2212yes—
42C2×#382C2212yes—
43C2×#381C2212yes—
44C2×#381C2211yescharacteristic
45C2×#381C2212yes—
46C2×#381C2211yescharacteristic
47C2×#380C2212yes—
48C2×C4×C22⋊C4C2212yes—
49C22×#95C2211yescharacteristic
50C22×#95C2211yescharacteristic
51C22×#95C2211yescharacteristic
52#1445C2212yes—
53#1445C2212yes—
54#1444C2214yes—
55#1444C2214yes—
56#1444C2212yes—
57#1443C2214yes—
58#1443C2214yes—
59#1442C2214yes—
60#1442C2212yes—
61#1442C2214yes—
62#1441C2214yes—
63#1441C2214yes—
64#1441C2212yes—
65#1441C2212yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1434C2214yes—
74#1434C2214yes—
75C22⋊#95C2212yes—
76C22⋊#95C2212yes—
77C22⋊#95C2212yes—
78C22⋊#95C2212yes—
79C24×C4C2311yescharacteristic
80C24×C4C2311yescharacteristic
81C24×C4C2311yescharacteristic
82C24×C4C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C2×#117—24no—
87C2×#117—24no—
88C2×#117—24no—
89C2×#117—24no—
90C2×#117—28no—
91C22×C4⋊C4—24no—
92C22×C4⋊C4—24no—
93C22×C4⋊C4—24no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—44no—
99C22×C4⋊C4—44no—
100C22×C4⋊C4D418yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C42C2312yes—
123C22×C42C2312yes—
124C22×C42C2314yes—
125#400—28no—
126#398—216no—
127#398—216no—
128#398—28no—
129#398—48no—
130#389—28no—
131#389—28no—
132#389—48no—
133#387—216no—
134#387—216no—
135#387—216no—
136#385—28no—
137#385—28no—
138#385—28no—
139#385—28no—
140#384—28no—
141#382—28no—
142#381—24no—
143#381—28no—
144#381—28no—
145#381D418yes—
146#380—28no—
147C4×C22⋊C4—28no—
148C2×#95—24no—
149C2×#95—24no—
150C2×#95—24no—
151C2×#95C2312yes—
152C2×#95C2314yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C25C2411yescharacteristic
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—22no—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—22no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4C2×D414yes—
209C23×C4C2×D414yes—
210C23×C4C2×D414yes—
211C23×C4C2×D414yes—
212C23×C4#4114yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2414yes—
216C23×C4C2412yes—
217C23×C4C2414yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227#117—232no—
228#117—416no—
229C2×C4⋊C4—232no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—416no—
268C2×C4⋊C4—416no—
269C2×C4⋊C4—416no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4C2×D418yes—
277C2×C4⋊C4C2×D418yes—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4C2×D418yes—
289C2×C22⋊C4#4118yes—
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—
299#95—24no—
300#95—24no—
301#95—28no—
302#95—28no—
303#95—28no—
304#95—28no—
305#95—24no—
306#95—24no—
307#95—28no—
308#95—28no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—44no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95C2×D418yes—
326#95C2×D418yes—
327C24—22no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24C22×D412yes—
332C24C22×D412yes—
333C24C2×#4112yes—
334C24C2×#4114yes—
335C242−1+414yes—
336C24C2511yescharacteristic
337C22×C4—28no—
338C22×C4—28no—
339C22×C4—216no—
340C22×C4—216no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—28no—
346C22×C4—28no—
347C22×C4—28no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—28no—
385C22×C4—28no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—44no—
411C22×C4—44no—
412C22×C4—48no—
413C22×C4—48no—
414C22×C4—416no—
415C22×C4—416no—
416C22×C4—44no—
417C22×C4—44no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—44no—
421C22×C4—44no—
422C22×C4—44no—
423C22×C4—44no—
424C22×C4—44no—
425C22×C4—48no—
426C22×C4—48no—
427C22×C4—84no—
428C22×C4—84no—
429C22×C4—84no—
430C22×C4—84no—
431C22×C4#121116yes—
432C22×C4C22×D418yes—
433C22×C4C22×D414yes—
434C22×C4C22×D414yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C2×#4114yes—
440C22×C4C2×#4114yes—
441C22×C4C2×#4112yes—
442C22×C4C2×#4112yes—
443C22×C4H(5,𝔽2)14yes—
444C22×C42−1+414yes—
445C22×C42−1+414yes—
446C4⋊C4—28no—
447C4⋊C4—28no—
448C4⋊C4—216no—
449C4⋊C4—216no—
450C4⋊C4—416no—
451C4⋊C4—416no—
452C4⋊C4—416no—
453C4⋊C4—416no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—44no—
457C4⋊C4—44no—
458C4⋊C4—84no—
459C4⋊C4—84no—
460C4⋊C4—84no—
461C4⋊C4—84no—
462C4⋊C4—88no—
463C22⋊C4—216no—
464C22⋊C4—216no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C42—216no—
468C42—48no—
469C42—88no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—24no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—28no—
479C23—28no—
480C23—28no—
481C23—28no—
482C23C2×#12114yes—
483C23#52514yes—
484C23#53614yes—
485C23#54718yes—
486C23#55218yes—
487C23C23×D411yescharacteristic
488C23C23×D411yescharacteristic
489C23C22×#4111yescharacteristic
490C23C22×#4112yes—
491C23C2×H(5,𝔽2)12yes—
492C23C2×2−1+412yes—
493C23C2×2−1+412yes—
494C23C2×2−1+412yes—
495C23#58512yes—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—232no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—28no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—28no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—28no—
523C2×C4—28no—
524C2×C4—28no—
525C2×C4—416no—
526C2×C4—416no—
527C2×C4—416no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—44no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—44no—
551C2×C4—44no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—44no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—44no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—88no—
575C2×C4—88no—
576C2×C4—88no—
577C2×C4C2×#12114yes—
578C2×C4C2×#12114yes—
579C22—216no—
580C22—24no—
581C22—24no—
582C22—24no—
583C22—24no—
584C22—24no—
585C22—24no—
586C22—216no—
587C22#241318yes—
588C22C22×#12111yescharacteristic
589C22C2×#52511yescharacteristic
590C22C2×#53611yescharacteristic
591C22C2×#54712yes—
592C22C2×#55212yes—
593C22#348411yescharacteristic
594C22#348511yescharacteristic
595C22#348611yescharacteristic
596C22#348611yescharacteristic
597C22#348812yes—
598C22#349412yes—
599C22#349912yes—
600C22#350912yes—
601C22#351514yes—
602C22#352014yes—
603C22#353814yes—
604C22#356314yes—
605C4—24no—
606C4—24no—
607C4—48no—
608C4—44no—
609C4—44no—
610C4—44no—
611C4—44no—
612C4—44no—
613C4—44no—
614C4—44no—
615C4—84no—
616C4—82no—
617C4—82no—
618C4—82no—
619C2—28no—
620C2C2×#241311yescharacteristic
621C2#3535711yescharacteristic
622C2#3545311yescharacteristic
623C2#3551411yescharacteristic
624C2#3566711yescharacteristic
625C2#3670811yescharacteristic
626C2#3695911yescharacteristic
627C2#4397714yes—
628C2#4404214yes—
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
5. #15812C2semidirect
6. #15778C2semidirect
7. #15738C2semidirect
8. #15629C2semidirect

Maximal subgroups

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

subgroupindex
C2×#24842
C22×#3982
C2×#23172
C2×#23132
#158122
#157782
#157382
#156292
#149352
#147282
#141282
#140432
#139722
#139582
#138702