Finite Group Databasebeta

← #4389959

Subgroups — #4389959

Order 512.

3020 conjugacy classes of subgroups.

The subgroup lattice has 705 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2820C211yesmaximal, characteristic, complemented
2C2×#2545C211yesmaximal, characteristic, complemented
3C2×#2562C211yesmaximal, characteristic, complemented
4C2×#2580C211yesmaximal, characteristic, complemented
5C2×#2634C211yesmaximal, characteristic, complemented
6C2×#2631C211yesmaximal, characteristic
7#16809C212yesmaximal, complemented
8#16511C212yesmaximal, complemented
9#15808C214yesmaximal, complemented
10#15776C214yesmaximal, complemented
11#15526C214yesmaximal, complemented
12#15505C214yesmaximal, complemented
13#15065C212yesmaximal, complemented
14#14928C212yesmaximal, complemented
15#13781C211yesmaximal, characteristic, complemented
16C22×C4⋊Q8C2211yescharacteristic
17C22×#125C2211yescharacteristic
18C22×#124C2211yescharacteristic
19C22×#124C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22#2820—28no—
23#2634—28no—
24#2631—28no—
25#2580—28no—
26#2562—28no—
27#2545—28no—
28C2×#400C2212yes—
29C2×#400C2212yes—
30C2×#400C2212yes—
31C2×#399C2212yes—
32C2×#399C2212yes—
33C2×#399C2214yes—
34C2×#398C2212yes—
35C2×#397C2212yes—
36C2×#397C2212yes—
37C2×#396C2214yes—
38C2×#396C2214yes—
39C2×#395C2212yes—
40C2×#395C2212yes—
41C2×#394C2212yes—
42C2×#394C2212yes—
43C2×#393C2212yes—
44C2×#393C2212yes—
45C2×#392C2212yes—
46C2×#392C2212yes—
47C2×#391C2211yescharacteristic
48C2×#390C2212yes—
49C2×#390C2211yescharacteristic
50C2×#390C2214yes—
51C2×#389C2212yes—
52C2×#388C2212yes—
53C2×#388C2212yes—
54C2×#388C2214yes—
55C2×#387C2214yes—
56C2×#386C2214yes—
57C2×#385C2214yes—
58C2×#383C2211yescharacteristic
59C2×#382C2211yescharacteristic
60C2×#380C2212yes—
61C2×#380C2212yes—
62C2×C4×C4⋊C4C2211yescharacteristic
63#1445C2214yes—
64#1445C2214yes—
65#1444C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1439C2214yes—
70#1439C2214yes—
71#1438C2214yes—
72#1438C2214yes—
73#1438C2214yes—
74#1437C2214yes—
75#1437C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78C4⋊#95C2214yes—
79#1434C2212yes—
80C4×#95C2212yes—
81C23×Q8C2311yescharacteristic
82C23×Q8C2311yescharacteristic
83C23×D4C2311yescharacteristic
84C23×D4C2311yescharacteristic
85C23×D4C2314yes—
86C24×C4C2312yes—
87C2×C4⋊Q8—24no—
88C2×C4⋊Q8—44no—
89C2×#125—24no—
90C2×#125—24no—
91C2×#125—24no—
92C2×#124—24no—
93C2×#124—24no—
94C2×#124—24no—
95C2×#124—24no—
96C2×#124—44no—
97C2×C4×Q8—24no—
98C2×C4×Q8—24no—
99C2×C4×Q8—24no—
100C2×C4×D4—24no—
101C2×C4×D4—24no—
102C2×C4×D4—24no—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C42C2312yes—
132C22×C42C2314yes—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136#400—28no—
137#400—28no—
138#400—28no—
139#399—28no—
140#399—28no—
141#399—48no—
142#398—28no—
143#397—28no—
144#397—28no—
145#396—216no—
146#396—216no—
147#395—28no—
148#395—28no—
149#394—28no—
150#394—28no—
151#393—28no—
152#393—28no—
153#392—28no—
154#392—28no—
155#391—24no—
156#390—28no—
157#390—24no—
158#390—48no—
159#389—28no—
160#388—28no—
161#388—28no—
162#388—216no—
163#387—216no—
164#386—216no—
165#385—216no—
166#383D418yes—
167#382D418yescomplemented
168#380—28no—
169#380—28no—
170C4×C4⋊C4D418yescomplemented
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C25C2412yes—
192C25C2412yes—
193C22×Q8—22no—
194C22×Q8—22no—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—24no—
200C22×Q8—42no—
201C22×Q8—42no—
202C22×Q8—42no—
203C22×D4—24no—
204C22×D4—28no—
205C22×D4—22no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4—28no—
212C22×D4—42no—
213C22×D4—42no—
214C22×D4—42no—
215C22×D4—44no—
216C22×D4—44no—
217C22×D4—44no—
218C22×D4—44no—
219C22×D4—44no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—44no—
224C23×C4—44no—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2414yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2414yes—
234C23×C4C2412yes—
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C4⋊Q8—416no—
239#125—416no—
240#124—416no—
241#124—416no—
242C4×Q8—416no—
243C4×D4—416no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
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⋊C4C2×D418yes—
280C2×C4⋊C4C2×D418yes—
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⋊C4—28no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—42no—
308C2×C22⋊C4—42no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—42no—
328C2×C42—42no—
329C2×C42—44no—
330C2×C42—44no—
331C2×C42—44no—
332C2×C42C2×D414yes—
333C2×C42C2×D414yes—
334C2×C42C2×D414yes—
335C2×C42C2×D414yes—
336C2×C42C2×D414yes—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—28no—
342#95—28no—
343#95—28no—
344#95—28no—
345#95—28no—
346#95—28no—
347#95—28no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95C2×D418yes—
352#95C2×D418yes—
353#95C2×D418yes—
354#95C2×D418yes—
355#95C2×D418yes—
356#95C2×D418yes—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—24no—
361C24—22no—
362C24—22no—
363C24—24no—
364C24—24no—
365C24—24no—
366C24—22no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—24no—
371C24—24no—
372C24—24no—
373C24—24no—
374C24—24no—
375C24—24no—
376C242−1+414yes—
377C24C2511yescharacteristic
378C2×Q8—28no—
379C2×Q8—48no—
380C2×Q8—48no—
381C2×Q8—44no—
382C2×Q8—44no—
383C2×Q8—44no—
384C2×Q8—44no—
385C2×Q8—44no—
386C2×Q8—84no—
387C2×Q8—84no—
388C2×D4—28no—
389C2×D4—48no—
390C2×D4—48no—
391C2×D4—44no—
392C2×D4—416no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—416no—
396C2×D4—44no—
397C2×D4—44no—
398C2×D4—84no—
399C2×D4—84no—
400C2×D4—88no—
401C2×D4—88no—
402C2×D4—88no—
403C2×D4—88no—
404C2×D4—88no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
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—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—48no—
464C22×C4—48no—
465C22×C4—48no—
466C22×C4—88no—
467C22×C4—88no—
468C22×C4C22×D418yes—
469C22×C4C22×D414yes—
470C22×C4C22×D418yes—
471C22×C4C22×D414yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4H(5,𝔽2)14yes—
482C4⋊C4—48no—
483C4⋊C4—48no—
484C4⋊C4—416no—
485C4⋊C4—416no—
486C4⋊C4—44no—
487C4⋊C4—48no—
488C4⋊C4—44no—
489C4⋊C4—48no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—48no—
493C4⋊C4—88no—
494C4⋊C4—88no—
495C22⋊C4—416no—
496C22⋊C4—416no—
497C22⋊C4—48no—
498C22⋊C4—48no—
499C22⋊C4—416no—
500C22⋊C4—48no—
501C22⋊C4—48no—
502C22⋊C4—48no—
503C22⋊C4—84no—
504C22⋊C4—84no—
505C22⋊C4—88no—
506C22⋊C4—88no—
507C22⋊C4—88no—
508C22⋊C4—88no—
509C22⋊C4—88no—
510C42—44no—
511C42—44no—
512C42—48no—
513C42—88no—
514C42#127116yes—
515C23—24no—
516C23—24no—
517C23—24no—
518C23—28no—
519C23—28no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—44no—
527C23—44no—
528C23—42no—
529C23—42no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—42no—
539C23—44no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—44no—
548C23—42no—
549C23—42no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23—44no—
555C23—44no—
556C23—44no—
557C23—44no—
558C23—44no—
559C23C23×D411yescharacteristic
560C23C23×D411yescharacteristic
561C23C23×D411yescharacteristic
562C23C2×H(5,𝔽2)12yes—
563C23C2×2−1+412yes—
564C23#58514yes—
565C23#58512yes—
566C23#58512yes—
567Q8—416no—
568Q8—88no—
569D4—416no—
570D4—88no—
571D4—832no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—24no—
588C2×C4—28no—
589C2×C4—24no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—28no—
596C2×C4—28no—
597C2×C4—28no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—42no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—816no—
642C2×C4C2×#12714yes—
643C2×C4C2×#12714yes—
644C2×C4C2×#12714yes—
645C22—44no—
646C22—48no—
647C22—48no—
648C22—44no—
649C22—44no—
650C22—82no—
651C22—84no—
652C22—84no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—84no—
658C22—82no—
659C22—84no—
660C22—84no—
661C22—84no—
662C22—84no—
663C22—84no—
664C22—82no—
665C22C22×#12711yescharacteristic
666C22#344912yes—
667C22#348411yescharacteristic
668C22#348511yescharacteristic
669C22#348611yescharacteristic
670C22#348711yescharacteristic
671C22#348812yes—
672C22#348812yes—
673C22#348912yes—
674C22#348912yes—
675C22#350012yes—
676C22#350012yes—
677C22#354114yes—
678C22#356214yes—
679C22#356414yes—
680C22#356614yes—
681C4—24no—
682C4—24no—
683C4—24no—
684C4—48no—
685C4—48no—
686C4—44no—
687C4—44no—
688C4—44no—
689C4—82no—
690C4—82no—
691C4—82no—
692C4—82no—
693C4—82no—
694C2—84no—
695C2—162no—
696C2#3532911yescharacteristic
697C2#3556511yescharacteristic
698C2#3563411yescharacteristic
699C2#4060312yes—
700C2#4217512yes—
701C2#4772414yes—
702C2#5069412yes—
703C2#5107112yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#2631C2
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×#2820C2semidirect
2. C2×#2545C2semidirect
3. C2×#2562C2semidirect
4. C2×#2580C2semidirect
5. C2×#2634C2semidirect
7. #16809C2semidirect
8. #16511C2semidirect
9. #15808C2semidirect
10. #15776C2semidirect
11. #15526C2semidirect
12. #15505C2semidirect
13. #15065C2semidirect
14. #14928C2semidirect
15. #13781C2semidirect
167. #382D4semidirect
170. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28202
C2×#25452
C2×#25622
C2×#25802
C2×#26342
C2×#26312
#168092
#165112
#158082
#157762
#155262
#155052
#150652
#149282
#137812