Finite Group Databasebeta

← #4389780

Subgroups — #4389780

Order 512.

2570 conjugacy classes of subgroups.

The subgroup lattice has 625 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2817C211yesmaximal, characteristic
2C2×#2583C211yesmaximal, characteristic
3C2×#2484C211yesmaximal, characteristic
4C2×#2499C211yesmaximal, characteristic
5C2×#2625C211yesmaximal, characteristic
6C2×#2627C211yesmaximal, characteristic
7#16964C212yesmaximal, complemented
8#16813C212yesmaximal, complemented
9#15856C214yesmaximal, complemented
10#15804C214yesmaximal, complemented
11#15539C214yesmaximal
12#15539C214yesmaximal
13#15173C212yesmaximal, complemented
14#15153C212yesmaximal, complemented
15#13780C211yesmaximal, characteristic
16C22×C22⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C4⋊C4C2211yescharacteristic
20#2817—28no—
21#2627—28no—
22#2625—28no—
23#2583—28no—
24#2499—28no—
25#2484—28no—
26C2×#400C2212yes—
27C2×#400C2212yes—
28C2×#400C2212yes—
29C2×#400C2212yes—
30C2×#400C2212yes—
31C2×#398C2212yes—
32C2×#398C2212yes—
33C2×#398C2212yes—
34C2×#398C2214yes—
35C2×#397C2214yes—
36C2×#397C2214yes—
37C2×#397C2214yes—
38C2×#395C2212yes—
39C2×#395C2212yes—
40C2×#395C2212yes—
41C2×#395C2212yes—
42C2×#391C2212yes—
43C2×#391C2211yescharacteristic
44C2×#391C2211yescharacteristic
45C2×#389C2212yes—
46C2×#389C2212yes—
47C2×#389C2212yes—
48C2×#387C2214yes—
49C2×#387C2214yes—
50C2×#385C2212yes—
51C2×#385C2212yes—
52C2×#385C2212yes—
53C2×#385C2212yes—
54C2×#384C2211yescharacteristic
55C2×#382C2211yescharacteristic
56C2×#381C2212yes—
57C2×#381C2212yes—
58C2×#381C2211yescharacteristic
59C2×#380C2211yescharacteristic
60C2×C4×C22⋊C4C2211yescharacteristic
61#1445C2214yes—
62#1445C2214yes—
63#1445C2214yes—
64#1445C2214yes—
65#1445C2214yes—
66#1445C2214yes—
67#1445C2214yes—
68#1445C2214yes—
69#1444C2214yes—
70#1444C2214yes—
71#1444C2214yes—
72#1444C2214yes—
73#1437C2214yes—
74#1436C2214yes—
75C4⋊#95C2214yes—
76C4⋊#95C2214yes—
77C4⋊#95C2214yes—
78C4⋊#95C2214yes—
79#1434C2212yes—
80C4×#95C2212yes—
81C23×Q8C2311yescharacteristic
82C23×Q8C2311yescharacteristic
83C24×C4C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C24×C4C2311yescharacteristic
86C2×C22⋊Q8—24no—
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8—24no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—24no—
92C2×#117—24no—
93C2×#117—24no—
94C2×#117—24no—
95C22×C4⋊C4—24no—
96C22×C4⋊C4—24no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C42C2312yes—
123C22×C42C2314yes—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126#400—28no—
127#400—28no—
128#400—28no—
129#400—28no—
130#400—28no—
131#398—216no—
132#398—28no—
133#398—28no—
134#398—28no—
135#397—216no—
136#397—216no—
137#397—216no—
138#395—28no—
139#395—28no—
140#395—28no—
141#395—28no—
142#391—28no—
143#391—24no—
144#391—24no—
145#389—28no—
146#389—28no—
147#389—28no—
148#387—216no—
149#387—216no—
150#385—28no—
151#385—28no—
152#385—28no—
153#385—28no—
154#384—24no—
155#382—24no—
156#381—28no—
157#381—28no—
158#381D418yes—
159#380D418yes—
160C4×C22⋊C4D418yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C25C2411yescharacteristic
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—22no—
192C22×Q8—22no—
193C22×Q8—22no—
194C22×Q8—42no—
195C22×Q8—42no—
196C22×Q8—42no—
197C22×Q8—42no—
198C23×C4—22no—
199C23×C4—24no—
200C23×C4—22no—
201C23×C4—22no—
202C23×C4—22no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—42no—
206C23×C4—42no—
207C23×C4—42no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4C2×D414yes—
211C23×C4C2×D414yes—
212C23×C4C2×D414yes—
213C23×C4C2412yes—
214C23×C4C2412yes—
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2414yes—
221C23×C4C2414yes—
222C23×C4C2412yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C22⋊Q8—416no—
228C22⋊Q8—416no—
229#117—416no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
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—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—216no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—42no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—416no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4C2×D418yes—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4C2×D418yes—
296C2×C22⋊C4C2×D418yes—
297C2×C22⋊C4C2×D418yes—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—28no—
302C2×C42—42no—
303C2×C42—42no—
304C2×C42—44no—
305C2×C42—44no—
306C2×C42C2×D414yes—
307C2×C42C2×D414yes—
308#95—28no—
309#95—28no—
310#95—28no—
311#95—28no—
312#95—28no—
313#95—28no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—44no—
329#95—44no—
330#95C2×D418yes—
331#95C2×D418yes—
332#95C2×D418yes—
333#95C2×D418yes—
334C24—24no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24C22×D412yes—
340C24C22×D412yes—
341C24C22×D412yes—
342C24C2511yescharacteristic
343C2×Q8—44no—
344C2×Q8—44no—
345C2×Q8—48no—
346C2×Q8—44no—
347C2×Q8—44no—
348C2×Q8—84no—
349C2×Q8—82no—
350C2×Q8—84no—
351C2×Q8—84no—
352C2×Q8—82no—
353C22×C4—22no—
354C22×C4—24no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—28no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
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—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—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—44no—
419C22×C4—44no—
420C22×C4—48no—
421C22×C4—48no—
422C22×C4—48no—
423C22×C4—48no—
424C22×C4—44no—
425C22×C4—44no—
426C22×C4—48no—
427C22×C4—44no—
428C22×C4—84no—
429C22×C4—82no—
430C22×C4—82no—
431C22×C4#127116yes—
432C22×C4C22×D418yes—
433C22×C4C22×D414yes—
434C22×C4C22×D418yes—
435C22×C4C22×D412yes—
436C22×C4C22×D412yes—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D414yes—
442C22×C42−1+414yes—
443C22×C42−1+418yes—
444C4⋊C4—48no—
445C4⋊C4—48no—
446C4⋊C4—416no—
447C4⋊C4—48no—
448C4⋊C4—48no—
449C4⋊C4—48no—
450C4⋊C4—44no—
451C4⋊C4—44no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—48no—
455C4⋊C4—48no—
456C4⋊C4—416no—
457C4⋊C4—84no—
458C4⋊C4—84no—
459C4⋊C4—88no—
460C4⋊C4—88no—
461C4⋊C4—88no—
462C22⋊C4—216no—
463C22⋊C4—48no—
464C22⋊C4—48no—
465C22⋊C4—48no—
466C22⋊C4—48no—
467C22⋊C4—48no—
468C42—44no—
469C42—44no—
470C42—48no—
471C42—88no—
472C23—24no—
473C23—22no—
474C23—24no—
475C23—24no—
476C23—22no—
477C23—22no—
478C23—22no—
479C23—24no—
480C23—24no—
481C23—24no—
482C23—24no—
483C23—44no—
484C23—44no—
485C23—44no—
486C23—44no—
487C23C2×#12714yes—
488C23C23×D411yescharacteristic
489C23C23×D411yescharacteristic
490C23C23×D411yescharacteristic
491C23C2×2−1+414yes—
492C23C2×2−1+412yes—
493C23#58512yes—
494C23#58512yes—
495C23#58512yes—
496Q8—88no—
497Q8—88no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—24no—
508C2×C4—28no—
509C2×C4—28no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—28no—
513C2×C4—28no—
514C2×C4—28no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—28no—
520C2×C4—28no—
521C2×C4—28no—
522C2×C4—416no—
523C2×C4—416no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—44no—
546C2×C4—44no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—44no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—44no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—44no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—44no—
573C2×C4—88no—
574C2×C4C2×#12714yes—
575C2×C4C2×#12714yes—
576C2×C4#53618yes—
577C22—24no—
578C22—24no—
579C22—24no—
580C22—44no—
581C22—44no—
582C22—48no—
583C22—44no—
584C22—44no—
585C22C22×#12711yescharacteristic
586C22C2×#53612yes—
587C22#348511yescharacteristic
588C22#348512yes—
589C22#348511yescharacteristic
590C22#348512yes—
591C22#348511yescharacteristic
592C22#348611yescharacteristic
593C22#348912yes—
594C22#348912yes—
595C22#349912yes—
596C22#349912yes—
597C22#355514yes—
598C22#355714yes—
599C22#356314yes—
600C22#356414yes—
601C4—24no—
602C4—24no—
603C4—28no—
604C4—44no—
605C4—44no—
606C4—48no—
607C4—48no—
608C4—82no—
609C4—82no—
610C4—82no—
611C4—82no—
612C4—82no—
613C4—82no—
614C4—82no—
615C2—44no—
616C2#3531711yescharacteristic
617C2#3546311yescharacteristic
618C2#3563411yescharacteristic
619C2#4141112yes—
620C2#4267612yes—
621C2#4681812yes—
622C2#4721112yes—
623C2#4882914yes—
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
7. #16964C2semidirect
8. #16813C2semidirect
9. #15856C2semidirect
10. #15804C2semidirect
13. #15173C2semidirect
14. #15153C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28172
C2×#25832
C2×#24842
C2×#24992
C2×#26252
C2×#26272
#169642
#168132
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