Finite Group Databasebeta

← #695792

Subgroups — #695792

Order 512.

3274 conjugacy classes of subgroups.

The subgroup lattice has 741 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2503C211yesmaximal, characteristic, complemented
2C2×#2616C211yesmaximal, characteristic
3C2×#2345C211yesmaximal, characteristic, complemented
4C2×#2311C211yesmaximal, characteristic
5#16397C214yesmaximal, complemented
6#15858C214yesmaximal, complemented
7#15815C214yesmaximal, complemented
8#15220C212yesmaximal, complemented
9#15179C212yesmaximal, complemented
10#14838C214yesmaximal, complemented
11#14138C211yesmaximal, characteristic, complemented
12#14129C211yesmaximal, characteristic, complemented
13#13997C212yesmaximal, complemented
14#13875C212yesmaximal, complemented
15#13851C211yesmaximal, characteristic
16C23×#41C2211yescharacteristic
17C22×C22⋊Q8C2211yescharacteristic
18C22×C22⋊Q8C2212yes—
19C22×#117C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21#2616—28no—
22#2503—28no—
23C2×#400C2214yes—
24C2×#395C2212yes—
25C2×#394C2214yes—
26C2×#394C2212yes—
27#2345—28no—
28#2311—28no—
29C2×#391C2211yescharacteristic
30C2×#391C2211yescharacteristic
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#390C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#387C2214yes—
37C2×#386C2214yes—
38C2×#382C2211yescharacteristic
39C2×#382C2211yescharacteristic
40C2×#381C2214yes—
41C2×#380C2212yes—
42C2×#380C2212yes—
43C2×#380C2212yes—
44C2×#380C2212yes—
45C2×#380C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2212yes—
52#1446C2212yes—
53#1445C2214yes—
54#1443C2214yes—
55#1443C2214yes—
56#1442C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1440C2212yes—
60#1440C2214yes—
61#1440C2212yes—
62#1440C2214yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1438C2212yes—
69#1438C2212yes—
70#1438C2212yes—
71#1437C2214yes—
72#1437C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2214yes—
78#1434C2214yes—
79C22×#41—24no—
80C22×#41—24no—
81C22×#41—28no—
82C22×#41—24no—
83C22×#41—44no—
84C23×Q8C2311yescharacteristic
85C23×Q8C2311yescharacteristic
86C23×D4C2312yes—
87C23×D4C2311yescharacteristic
88C24×C4C2312yes—
89C24×C4C2311yescharacteristic
90C24×C4C2312yes—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8—44no—
97C2×#117—24no—
98C2×#117—44no—
99C22×C4⋊C4—24no—
100C22×C4⋊C4—24no—
101C22×C4⋊C4D418yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C22⋊C4C2312yes—
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2314yes—
131#400—216no—
132#395—28no—
133#394—216no—
134#394—28no—
135#391—28no—
136#391—24no—
137#391—24no—
138#391—44no—
139#390—28no—
140#389—28no—
141#389—44no—
142#387—216no—
143#386—216no—
144#382—24no—
145#382—24no—
146#381—216no—
147#380—28no—
148#380—28no—
149#380—28no—
150#380—28no—
151#380D418yes—
152C4×C4⋊C4—28no—
153C4×C22⋊C4—24no—
154C4×C22⋊C4—24no—
155C4×C22⋊C4—28no—
156C4×C22⋊C4—28no—
157C4×C22⋊C4—44no—
158C2×#95C2314yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C25C2412yes—
182C25C2411yescharacteristic
183C2×#41—232no—
184C2×#41—432no—
185C2×#41—432no—
186C2×#41—416no—
187C2×#41—416no—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—22no—
192C22×Q8—24no—
193C22×Q8—22no—
194C22×Q8—24no—
195C22×Q8—22no—
196C22×Q8—24no—
197C22×Q8—42no—
198C22×D4—24no—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—24no—
206C22×D4—22no—
207C22×D4—42no—
208C22×D4—42no—
209C22×D4—42no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—24no—
217C23×C4—24no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—44no—
221C23×C4—44no—
222C23×C4C2×D414yes—
223C23×C4C2×D414yes—
224C23×C4C2×D418yes—
225C23×C4#4118yes—
226C23×C4#4114yes—
227C23×C4C2412yes—
228C23×C4C2412yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2414yes—
232C23×C4C2412yes—
233C23×C4C2414yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2412yes—
237C23×C4C2412yes—
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C22⋊Q8—432no—
241C22⋊Q8—416no—
242#117—416no—
243C2×C4⋊C4—232no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4C2×D414yes—
271C2×C4⋊C4C2×D414yes—
272C2×C4⋊C4C2×D418yes—
273C2×C4⋊C4C2×D418yes—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—42no—
299C2×C22⋊C4—42no—
300C2×C22⋊C4—42no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—44no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4C2×D418yes—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—28no—
321C2×C42—44no—
322C2×C42—44no—
323C2×C42—44no—
324#95—24no—
325#95—28no—
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—24no—
336#95—24no—
337#95—28no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—42no—
342#95—42no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95C2×D418yes—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C24—22no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—22no—
367C24C22×D412yes—
368C24C22×D412yes—
369C24C2×#4114yes—
370C24C2×#4112yes—
371C24C2511yescharacteristic
372#41—4128no—
373C2×Q8—28no—
374C2×Q8—48no—
375C2×Q8—48no—
376C2×Q8—44no—
377C2×Q8—44no—
378C2×Q8—44no—
379C2×Q8—44no—
380C2×Q8—44no—
381C2×Q8—48no—
382C2×Q8—48no—
383C2×D4—28no—
384C2×D4—216no—
385C2×D4—48no—
386C2×D4—48no—
387C2×D4—48no—
388C2×D4—48no—
389C2×D4—44no—
390C2×D4—48no—
391C2×D4—48no—
392C2×D4—48no—
393C2×D4—48no—
394C2×D4—44no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—216no—
400C22×C4—216no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—216no—
428C22×C4—216no—
429C22×C4—216no—
430C22×C4—216no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—22no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—416no—
470C22×C4—48no—
471C22×C4—48no—
472C22×C4—48no—
473C22×C4—48no—
474C22×C4—88no—
475C22×C4—84no—
476C22×C4—84no—
477C22×C4#121116yes—
478C22×C4C22×D414yes—
479C22×C4C22×D414yes—
480C22×C4C22×D418yes—
481C22×C4C22×D412yes—
482C22×C4C22×D412yes—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C2×#4114yes—
486C22×C4C2×#4112yes—
487C22×C4C2×#4112yes—
488C22×C4C2×#4114yes—
489C22×C4H(5,𝔽2)14yes—
490C22×C42−1+414yes—
491C4⋊C4—216no—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—216no—
495C4⋊C4—416no—
496C4⋊C4—44no—
497C4⋊C4—48no—
498C4⋊C4—48no—
499C4⋊C4—88no—
500C4⋊C4—88no—
501C22⋊C4—28no—
502C22⋊C4—48no—
503C22⋊C4—48no—
504C22⋊C4—416no—
505C22⋊C4—48no—
506C22⋊C4—44no—
507C22⋊C4—48no—
508C22⋊C4—48no—
509C22⋊C4—48no—
510C22⋊C4—48no—
511C22⋊C4—48no—
512C22⋊C4—88no—
513C22⋊C4—88no—
514C22⋊C4—88no—
515C42—48no—
516C42—44no—
517C42—44no—
518C42—48no—
519C42—88no—
520C23—22no—
521C23—22no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—22no—
526C23—24no—
527C23—22no—
528C23—24no—
529C23—24no—
530C23—22no—
531C23—22no—
532C23—24no—
533C23—24no—
534C23—24no—
535C23—24no—
536C23—24no—
537C23—24no—
538C23—24no—
539C23—42no—
540C23—42no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—42no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23—42no—
563C23—42no—
564C23—42no—
565C23—42no—
566C23—42no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—42no—
574C23—42no—
575C23C2×#12114yes—
576C23C23×D411yescharacteristic
577C23C23×D411yescharacteristic
578C23C22×#4111yescharacteristic
579C23C22×#4112yes—
580C23C2×H(5,𝔽2)12yes—
581C23C2×2−1+412yes—
582C23#58512yes—
583C23#58512yes—
584C23#58512yes—
585Q8—416no—
586Q8—88no—
587D4—416no—
588D4—432no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—232no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—28no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—28no—
601C2×C4—28no—
602C2×C4—28no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—28no—
606C2×C4—24no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—432no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—42no—
639C2×C4—42no—
640C2×C4—42no—
641C2×C4—44no—
642C2×C4—42no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—42no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—44no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—42no—
662C2×C4—44no—
663C2×C4—42no—
664C2×C4—816no—
665C2×C4C2×#12114yes—
666C2×C4C2×#12114yes—
667C2×C4#52514yes—
668C2×C4#53514yes—
669C2×C4#54618yes—
670C2×C4#55118yes—
671C22—24no—
672C22—24no—
673C22—24no—
674C22—44no—
675C22—44no—
676C22—44no—
677C22—44no—
678C22—44no—
679C22—44no—
680C22—44no—
681C22—44no—
682C22—44no—
683C22—44no—
684C22—44no—
685C22—44no—
686C22—44no—
687C22—82no—
688C22—82no—
689C22—82no—
690C22—82no—
691C22—82no—
692C22—82no—
693C22—82no—
694C22—82no—
695C22C22×#12111yescharacteristic
696C22C2×#52511yescharacteristic
697C22C2×#53511yescharacteristic
698C22#346112yes—
699C22C2×#54612yes—
700C22C2×#55112yes—
701C22#348512yes—
702C22#348511yescharacteristic
703C22#348611yescharacteristic
704C22#349312yes—
705C22#349512yes—
706C22#349712yes—
707C22#350012yes—
708C22#350712yes—
709C22#351011yescharacteristic
710C22#351011yescharacteristic
711C22#351212yes—
712C22#353912yes—
713C22#354112yes—
714C22#354712yes—
715C22#355112yes—
716C4—24no—
717C4—44no—
718C4—44no—
719C4—48no—
720C4—44no—
721C4—44no—
722C4—82no—
723C4—82no—
724C4—84no—
725C4—82no—
726C4—82no—
727C4—82no—
728C4#241118yes—
729C2—44no—
730C2—84no—
731C2C2×#241111yescharacteristic
732C2#3515311yescharacteristic
733C2#3561711yescharacteristic
734C2#3575812yes—
735C2#3797412yes—
736C2#3956612yes—
737C2#4264312yes—
738C2#4509212yes—
739C2#4511412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×#41C22
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×#2503C2semidirect
3. C2×#2345C2semidirect
5. #16397C2semidirect
6. #15858C2semidirect
7. #15815C2semidirect
8. #15220C2semidirect
9. #15179C2semidirect
10. #14838C2semidirect
11. #14138C2semidirect
12. #14129C2semidirect
13. #13997C2semidirect
14. #13875C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25032
C2×#26162
C2×#23452
C2×#23112
#163972
#158582
#158152
#152202
#151792
#148382
#141382
#141292
#139972
#138752
#138512