Finite Group Databasebeta

← #4389956

Subgroups — #4389956

Order 512.

2926 conjugacy classes of subgroups.

The subgroup lattice has 693 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2817C211yesmaximal, characteristic, complemented
2C2×#2561C211yesmaximal, characteristic
3C2×#2482C211yesmaximal, characteristic
4C2×#2499C211yesmaximal, characteristic, complemented
5C2×#2625C211yesmaximal, characteristic, complemented
6C2×#2626C211yesmaximal, characteristic
7#17064C212yesmaximal, complemented
8#16812C212yesmaximal, complemented
9#15825C214yesmaximal, complemented
10#15808C214yesmaximal, complemented
11#15539C214yesmaximal, complemented
12#15480C214yesmaximal
13#15065C212yesmaximal, complemented
14#15020C212yesmaximal, complemented
15#13780C211yesmaximal, characteristic, complemented
16C22×C22⋊Q8C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C23×C22⋊C4C2211yescharacteristic
20#2817—28no—
21#2626—28no—
22#2625—28no—
23#2561—28no—
24#2499—28no—
25#2482—28no—
26C2×#400C2212yescomplemented
27C2×#400C2212yescomplemented
28C2×#400C2212yescomplemented
29C2×#399C2212yes—
30C2×#399C2212yes—
31C2×#398C2212yescomplemented
32C2×#397C2212yes—
33C2×#397C2214yes—
34C2×#397C2214yes—
35C2×#397C2214yes—
36C2×#397C2212yes—
37C2×#396C2214yes—
38C2×#395C2212yescomplemented
39C2×#395C2212yescomplemented
40C2×#393C2212yes—
41C2×#393C2212yes—
42C2×#391C2212yescomplemented
43C2×#391C2211yescharacteristic, complemented
44C2×#391C2211yescharacteristic, complemented
45C2×#389C2212yescomplemented
46C2×#388C2212yes—
47C2×#388C2212yes—
48C2×#388C2214yes—
49C2×#387C2214yescomplemented
50C2×#385C2212yes—
51C2×#385C2212yes—
52C2×#384C2211yescharacteristic, complemented
53C2×#382C2211yescharacteristic, complemented
54C2×#381C2212yes—
55C2×#381C2211yescharacteristic
56C2×#380C2211yescharacteristic
57C2×#379C2212yes—
58C2×#379C2212yes—
59C2×#379C2212yes—
60C2×C4×C22⋊C4C2211yescharacteristic
61#1445C2214yescomplemented
62#1445C2214yescomplemented
63#1445C2214yescomplemented
64#1445C2214yescomplemented
65#1444C2214yescomplemented
66#1444C2214yescomplemented
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1439C2214yes—
72#1439C2214yes—
73#1437C2214yes—
74#1437C2214yes—
75#1437C2214yes—
76#1436C2214yescomplemented
77C4⋊#95C2214yescomplemented
78C4⋊#95C2214yescomplemented
79#1434C2212yescomplemented
80C4×#95C2212yescomplemented
81C26C2311yescharacteristic
82C23×Q8C2311yescharacteristic
83C23×Q8C2311yescharacteristic
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⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C22⋊C4—24no—
108C22×C22⋊C4—24no—
109C22×C22⋊C4—24no—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2312yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C42C2312yes—
127C22×C42C2314yes—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130#400—28no—
131#400—28no—
132#400—28no—
133#399—28no—
134#399—28no—
135#398—28no—
136#397—216no—
137#397—216no—
138#397—216no—
139#397—28no—
140#397—28no—
141#396—216no—
142#395—28no—
143#395—28no—
144#393—28no—
145#393—28no—
146#391—28no—
147#391—24no—
148#391—24no—
149#389—28no—
150#388—28no—
151#388—28no—
152#388—216no—
153#387—216no—
154#385—28no—
155#385—28no—
156#384—24no—
157#382—24no—
158#381—28no—
159#381D418yes—
160#380D418yes—
161#379—28no—
162#379—28no—
163#379—28no—
164C4×C22⋊C4D418yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C25—22no—
188C25—24no—
189C25—22no—
190C25—22no—
191C25—42no—
192C25—44no—
193C25—42no—
194C25—42no—
195C25C2412yes—
196C25C2411yescharacteristic
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—22no—
200C22×Q8—22no—
201C22×Q8—22no—
202C22×Q8—22no—
203C22×Q8—42no—
204C22×Q8—42no—
205C22×Q8—42no—
206C22×Q8—42no—
207C23×C4—22no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—42no—
211C23×C4—42no—
212C23×C4—44no—
213C23×C4C2×D414yes—
214C23×C4C2×D414yes—
215C23×C4C2×D414yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2414yes—
224C23×C4C2412yes—
225C23×C4C2414yes—
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C22⋊Q8—416no—
230C22⋊Q8—416no—
231#117—416no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—28no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4C2×D418yes—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—28no—
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—28no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—28no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—416no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—44no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4C2×D418yes—
311C2×C22⋊C4C2×D418yes—
312C2×C22⋊C4C2×D418yes—
313C2×C42—24no—
314C2×C42—24no—
315C2×C42—24no—
316C2×C42—42no—
317C2×C42—42no—
318C2×C42—44no—
319C2×C42—44no—
320C2×C42—44no—
321C2×C42C2×D414yes—
322C2×C42C2×D414yes—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
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—44no—
339#95—44no—
340#95—44no—
341#95C2×D418yes—
342#95C2×D418yes—
343#95C2×D418yes—
344#95C2×D418yes—
345C24—22no—
346C24—22no—
347C24—24no—
348C24—22no—
349C24—22no—
350C24—28no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—24no—
358C24—24no—
359C24—24no—
360C24—24no—
361C24—48no—
362C24—48no—
363C24—44no—
364C24—48no—
365C24—48no—
366C24—44no—
367C24—84no—
368C24—82no—
369C24—82no—
370C24—84no—
371C24—84no—
372C24—88no—
373C24—88no—
374C24—88no—
375C24—88no—
376C24C22×D412yes—
377C24C22×D412yes—
378C24C22×D412yes—
379C242−1+414yes—
380C24C2511yescharacteristic
381C2×Q8—44no—
382C2×Q8—44no—
383C2×Q8—48no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—84no—
387C2×Q8—82no—
388C2×Q8—82no—
389C2×Q8—84no—
390C2×Q8—84no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
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—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—44no—
453C22×C4—44no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—44no—
458C22×C4—84no—
459C22×C4—84no—
460C22×C4#127116yes—
461C22×C4C22×D414yes—
462C22×C4C22×D418yes—
463C22×C4C22×D414yes—
464C22×C4C22×D418yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C4⋊C4—48no—
472C4⋊C4—416no—
473C4⋊C4—44no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—44no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C4⋊C4—88no—
480C4⋊C4—88no—
481C4⋊C4—88no—
482C4⋊C4—88no—
483C22⋊C4—48no—
484C22⋊C4—48no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—416no—
488C22⋊C4—48no—
489C22⋊C4—48no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—84no—
495C22⋊C4—84no—
496C22⋊C4—88no—
497C22⋊C4—88no—
498C22⋊C4—88no—
499C42—44no—
500C42—44no—
501C42—48no—
502C42—88no—
503C23—24no—
504C23—22no—
505C23—24no—
506C23—24no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—24no—
511C23—24no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—28no—
518C23—28no—
519C23—416no—
520C23—44no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—44no—
530C23—42no—
531C23—44no—
532C23—42no—
533C23—42no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—44no—
539C23—44no—
540C23—44no—
541C23—44no—
542C23—88no—
543C23—88no—
544C23—816no—
545C23—816no—
546C23—832no—
547C23—816no—
548C23—816no—
549C23C2×#12714yes—
550C23C23×D411yescharacteristic
551C23C23×D411yescharacteristic
552C23C23×D411yescharacteristic
553C23C2×2−1+412yes—
554C23#58514yes—
555C23#58512yes—
556C23#58512yes—
557C23#58512yes—
558Q8—88no—
559Q8—88no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—24no—
576C2×C4—28no—
577C2×C4—24no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—416no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—88no—
631C2×C4C2×#12714yes—
632C2×C4C2×#12714yes—
633C22—24no—
634C22—24no—
635C22—24no—
636C22—44no—
637C22—44no—
638C22—48no—
639C22—44no—
640C22—44no—
641C22—44no—
642C22—44no—
643C22—48no—
644C22—44no—
645C22—48no—
646C22—832no—
647C22—82no—
648C22—84no—
649C22—82no—
650C22—82no—
651C22—84no—
652C22—82no—
653C22C22×#12711yescharacteristic
654C22#344912yes—
655C22#348511yescharacteristic
656C22#348512yes—
657C22#348511yescharacteristic
658C22#348512yes—
659C22#348511yescharacteristic
660C22#348611yescharacteristic
661C22#348912yes—
662C22#348912yes—
663C22#350012yes—
664C22#350012yes—
665C22#356414yes—
666C22#356414yes—
667C22#356614yes—
668C22#356814yes—
669C4—24no—
670C4—24no—
671C4—48no—
672C4—44no—
673C4—48no—
674C4—44no—
675C4—82no—
676C4—82no—
677C4—82no—
678C4—82no—
679C4—82no—
680C4—82no—
681C4—82no—
682C2—44no—
683C2—84no—
684C2#3532911yescharacteristic
685C2#3546311yescharacteristic
686C2#3563411yescharacteristic
687C2#4141812yes—
688C2#4217312yes—
689C2#4681512yes—
690C2#4721812yes—
691C2#4883414yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
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×#2817C2semidirect
4. C2×#2499C2semidirect
5. C2×#2625C2semidirect
7. #17064C2semidirect
8. #16812C2semidirect
9. #15825C2semidirect
10. #15808C2semidirect
11. #15539C2semidirect
13. #15065C2semidirect
14. #15020C2semidirect
15. #13780C2semidirect
26. C2×#400C22semidirect
27. C2×#400C22semidirect
28. C2×#400C22semidirect
31. C2×#398C22semidirect
38. C2×#395C22semidirect
39. C2×#395C22semidirect
42. C2×#391C22semidirect
43. C2×#391C22semidirect
44. C2×#391C22semidirect
45. C2×#389C22semidirect
49. C2×#387C22semidirect
52. C2×#384C22semidirect
53. C2×#382C22semidirect
61. #1445C22semidirect
62. #1445C22semidirect
63. #1445C22semidirect
64. #1445C22semidirect
65. #1444C22semidirect
66. #1444C22semidirect
76. #1436C22semidirect
77. C4⋊#95C22semidirect
78. C4⋊#95C22semidirect
79. #1434C22semidirect
80. C4×#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28172
C2×#25612
C2×#24822
C2×#24992
C2×#26252
C2×#26262
#170642
#168122
#158252
#158082
#155392
#154802
#150652
#150202
#137802