Finite Group Databasebeta

← #5956429

Subgroups — #5956429

Order 512.

3940 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
1C22×#541C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊Q8C212yesmaximal, complemented
3C2×#2290C211yesmaximal, characteristic
4#15218C211yesmaximal, characteristic, complemented
5#15216C211yesmaximal, characteristic, complemented
6#15180C211yesmaximal, characteristic, complemented
7#15179C212yesmaximal, complemented
8#15178C211yesmaximal, characteristic, complemented
9#15177C212yesmaximal
10#15072C214yesmaximal, complemented
11#15071C214yesmaximal, complemented
12#14917C214yesmaximal
13#14916C214yesmaximal, complemented
14#13786C212yesmaximal, complemented
15#13777C211yesmaximal, characteristic, complemented
16C2×#541—216no—
17C2×#541—28no—
18C22×C4⋊Q8C2211yescharacteristic
19C22×#126C2214yes—
20C22×#125C2214yes—
21C22×#125C2211yescharacteristic
22C22×C22⋊Q8C2212yes—
23C22×C22⋊Q8C2212yes—
24C22×C4×Q8C2212yes—
25C22×C4×D4C2211yescharacteristic
26C22×#117C2211yescharacteristic
27C23×C42C2211yescharacteristic
28C4×C22⋊Q8—216no—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#389C2212yes—
33#2290—28no—
34C2×#388C2212yes—
35C2×#387C2212yes—
36C2×#387C2214yes—
37C2×#386C2212yes—
38C2×#385C2212yes—
39C2×#385C2214yes—
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2214yes—
45C2×C4×C22⋊C4C2214yes—
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×#376C2211yescharacteristic
49#1444C2214yes—
50#1444C2214yes—
51#1444C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1442C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1440C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1439C2214yes—
62#1439C2214yes—
63#1439C2214yes—
64#1438C2214yes—
65#1437C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68#1436C2214yes—
69C4⋊#95C2212yes—
70C4⋊#95C2211yescharacteristic
71C4⋊#95C2212yes—
72C4⋊#95C2211yescharacteristic
73#1434C2214yes—
74#1434C2214yes—
75C23×Q8C2312yes—
76C23×D4C2311yescharacteristic
77C24×C4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2312yes—
80#541—2128no—
81C2×C4⋊Q8—28no—
82C2×C4⋊Q8—24no—
83C2×#126—216no—
84C2×#126—216no—
85C2×#126—216no—
86C2×#125—28no—
87C2×#125—216no—
88C2×#125—216no—
89C2×#125—24no—
90C2×#125—216no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8—28no—
93C2×C22⋊Q8—28no—
94C2×C22⋊Q8—28no—
95C2×C22⋊Q8—28no—
96C2×C22⋊Q8—28no—
97C2×C4×Q8—28no—
98C2×C4×Q8—28no—
99C2×C4×Q8D4116yescomplemented
100C2×C4×D4—24no—
101C2×C4×D4—24no—
102C2×C4×D4—24no—
103C2×#117—28no—
104C2×#117D418yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C42—24no—
127C22×C42—28no—
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2314yes—
133C22×C42C2311yescharacteristic
134C22×C42C2314yes—
135C22×C42C2311yescharacteristic
136#391—28no—
137#391—28no—
138#391—28no—
139#389—28no—
140#388—28no—
141#387—28no—
142#387—216no—
143#386—28no—
144#385—28no—
145#385—216no—
146#380—28no—
147#380—28no—
148C4×C4⋊C4—24no—
149C4×C4⋊C4—216no—
150C4×C4⋊C4—28no—
151C4×C22⋊C4—24no—
152C4×C22⋊C4—24no—
153C4×C22⋊C4—216no—
154#376—24no—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2314yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C25C2411yescharacteristic
177C25C2411yescharacteristic
178C22×Q8—24no—
179C22×Q8—24no—
180C22×Q8—24no—
181C22×Q8—24no—
182C22×Q8—24no—
183C22×Q8—24no—
184C22×Q8C2×D418yes—
185C22×D4—22no—
186C22×D4—22no—
187C22×D4—24no—
188C22×D4—22no—
189C22×D4—42no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—22no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4C2×D414yes—
206C23×C4#4114yes—
207C23×C4C2414yes—
208C23×C4C2412yes—
209C23×C4C2414yes—
210C23×C4C2412yes—
211C23×C4C2414yes—
212C23×C4C2411yescharacteristic
213C23×C4C2414yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C4⋊Q8—232no—
221#126—2128no—
222#125—232no—
223#125—2128no—
224C22⋊Q8—264no—
225C22⋊Q8—264no—
226C4×Q8—264no—
227C4×D4—232no—
228#117—232no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—28no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—28no—
244C2×C4⋊C4—28no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—28no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4C2×D418yes—
262C2×C4⋊C4C2×D418yes—
263C2×C4⋊C4C2×D4116yes—
264C2×C4⋊C4#4118yes—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—28no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—42no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4C2×D418yes—
292C2×C42—24no—
293C2×C42—22no—
294C2×C42—28no—
295C2×C42—28no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—28no—
299C2×C42—22no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—44no—
306C2×C42—44no—
307C2×C42—416no—
308C2×C42C2×D414yes—
309C2×C42C2×D4116yes—
310C2×C42C2×D418yes—
311C2×C42C2×D414yes—
312C2×C42#4114yes—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—28no—
322#95—24no—
323#95—28no—
324#95—28no—
325#95—28no—
326#95—28no—
327#95—28no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334C24—22no—
335C24—21no—
336C24—22no—
337C24—21no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—21no—
344C24—21no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24C22×D412yes—
350C24C2×#4112yes—
351C242−1+414yes—
352C24C2511yescharacteristic
353C2×Q8—216no—
354C2×Q8—216no—
355C2×Q8—216no—
356C2×Q8—48no—
357C2×Q8—48no—
358C2×Q8—48no—
359C2×Q8—48no—
360C2×D4—28no—
361C2×D4—48no—
362C2×D4—48no—
363C2×D4—44no—
364C2×D4—44no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—216no—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—24no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—28no—
417C22×C4—28no—
418C22×C4—28no—
419C22×C4—216no—
420C22×C4—28no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—48no—
436C22×C4—48no—
437C22×C4—48no—
438C22×C4—48no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—44no—
442C22×C4—48no—
443C22×C4—44no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4C22×D418yes—
447C22×C4C22×D418yes—
448C22×C4C22×D414yes—
449C22×C4C22×D412yes—
450C22×C4C22×D414yes—
451C22×C4C22×D414yes—
452C22×C4C22×D414yes—
453C22×C4C22×D412yes—
454C22×C4C22×D414yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4112yes—
457C22×C4C2×#4118yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4114yes—
461C22×C4C2×#4112yes—
462C22×C42−1+414yes—
463C4⋊C4—28no—
464C4⋊C4—232no—
465C4⋊C4—232no—
466C4⋊C4—232no—
467C4⋊C4—232no—
468C4⋊C4—216no—
469C4⋊C4—216no—
470C4⋊C4—232no—
471C4⋊C4—216no—
472C4⋊C4—48no—
473C4⋊C4—416no—
474C4⋊C4—48no—
475C22⋊C4—28no—
476C22⋊C4—28no—
477C22⋊C4—232no—
478C22⋊C4—232no—
479C22⋊C4—216no—
480C22⋊C4—48no—
481C22⋊C4—416no—
482C22⋊C4—48no—
483C22⋊C4—88no—
484C42—28no—
485C42—232no—
486C42—28no—
487C42—416no—
488C42—48no—
489C42—44no—
490C42—44no—
491C42#120116yes—
492C23—24no—
493C23—22no—
494C23—22no—
495C23—24no—
496C23—24no—
497C23—24no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—22no—
503C23—22no—
504C23—22no—
505C23—24no—
506C23—24no—
507C23—41no—
508C23—41no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—41no—
514C23—41no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—41no—
520C23—41no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—41no—
528C23—41no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23#54114yescomplemented
534C23C23×D412yes—
535C23C23×D411yescharacteristic
536C23C22×#4111yescharacteristic
537C23C22×#4111yescharacteristic
538C23C22×#4112yes—
539C23C2×2−1+412yes—
540C23C2×2−1+412yes—
541C23#58512yes—
542C23#58512yes—
543Q8—432no—
544D4—416no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—28no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—28no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—28no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—28no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—432no—
577C2×C4—42no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—44no—
589C2×C4—41no—
590C2×C4—44no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—42no—
598C2×C4—44no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—416no—
608C2×C4—416no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4C2×#12018yes—
624C2×C4C2×#12014yes—
625C2×C4#52514yes—
626C2×C4#52518yes—
627C2×C4#53214yes—
628C2×C4#53218yes—
629C22—28no—
630C22—24no—
631C22—44no—
632C22—44no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—42no—
637C22—42no—
638C22—44no—
639C22—44no—
640C22—42no—
641C22—42no—
642C22—42no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22—82no—
647C22C22×#12011yescharacteristic
648C22C2×#52512yes—
649C22C2×#52511yescharacteristic
650C22C2×#53211yescharacteristic
651C22C2×#53212yes—
652C22C2×#54111yescharacteristic
653C22#345511yescharacteristic
654C22#346112yes—
655C22#348612yes—
656C22#348712yes—
657C22#348912yes—
658C22#348912yes—
659C22#348912yes—
660C22#348912yes—
661C22#349512yes—
662C22#350911yescharacteristic
663C22#351211yescharacteristic
664C22#351512yes—
665C22#351612yes—
666C22#352112yes—
667C22#352312yes—
668C4—28no—
669C4—44no—
670C4—48no—
671C4—44no—
672C4—48no—
673C4—42no—
674C4—42no—
675C4—48no—
676C4—42no—
677C4—42no—
678C4—82no—
679C4—84no—
680C4#238818yes—
681C2—44no—
682C2—82no—
683C2C2×#238811yescharacteristic
684C2#3520512yes—
685C2#3524412yes—
686C2#3530911yescharacteristic
687C2#3531011yescharacteristic
688C2#3563512yes—
689C2#3563712yes—
690C2#3563812yes—
691C2#3565912yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C22×#541C2semidirect
2. C2×C4×C22⋊Q8C2semidirect
4. #15218C2semidirect
5. #15216C2semidirect
6. #15180C2semidirect
7. #15179C2semidirect
8. #15178C2semidirect
10. #15072C2semidirect
11. #15071C2semidirect
13. #14916C2semidirect
14. #13786C2semidirect
15. #13777C2semidirect
99. C2×C4×Q8D4semidirect
533. C23#541semidirect

Maximal subgroups

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

subgroupindex
C22×#5412
C2×C4×C22⋊Q82
C2×#22902
#152182
#152162
#151802
#151792
#151782
#151772
#150722
#150712
#149172
#149162
#137862
#137772