Finite Group Databasebeta

← #799290

Subgroups — #799290

Order 512.

2734 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C211yesmaximal, characteristic
2C2×D4×C4⋊C4C211yesmaximal, characteristic
3#17005C212yesmaximal, complemented
4#15579C214yesmaximal, complemented
5#15487C214yesmaximal, complemented
6#15188C211yesmaximal, characteristic, complemented
7#15180C211yesmaximal, characteristic, complemented
8#15156C211yesmaximal, characteristic, complemented
9#14841C212yesmaximal, complemented
10#14822C212yesmaximal, complemented
11#14142C212yesmaximal, complemented
12#14027C212yesmaximal
13#14027C212yesmaximal
14#14000C212yesmaximal, complemented
15#13923C212yesmaximal, complemented
16C4×#400C211yesmaximal, characteristic, complemented
17#13820C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×#124C2212yes—
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22C22×C4×D4C2211yescharacteristic
23C23×C4⋊C4C2212yes—
24#2547—28no—
25C2×#400C2211yescharacteristic
26C2×#400C2212yes—
27C2×#397C2214yes—
28D4×C4⋊C4—28no—
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#390C2212yes—
35C2×#389C2211yescharacteristic
36C2×#389C2211yescharacteristic
37C2×#385C2212yes—
38C2×#385C2212yes—
39C2×#384C2211yescharacteristic
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×#380C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C22⋊C4C2212yes—
47#1445C2211yescharacteristic
48#1443C2212yes—
49#1443C2214yes—
50#1443C2214yes—
51#1442C2211yescharacteristic
52#1442C2212yes—
53#1442C2214yes—
54#1442C2212yes—
55#1442C2212yes—
56#1442C2214yes—
57#1442C2214yes—
58#1442C2214yes—
59#1441C2211yescharacteristic
60#1441C2212yes—
61#1441C2212yes—
62#1441C2212yes—
63#1441C2212yes—
64#1441C2212yes—
65#1440C2214yes—
66#1440C2214yes—
67#1440C2214yes—
68#1440C2214yes—
69#1437C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73C4⋊#95C2212yes—
74C4⋊#95C2211yescharacteristic
75C4⋊#95C2212yes—
76#1434C2214yes—
77C22⋊#95C2214yes—
78C22⋊#95C2214yes—
79C22⋊#95C2212yes—
80C22⋊#95C2214yes—
81C22⋊#95C2214yes—
82C4×#95C2212yes—
83C4×#95C2211yescharacteristic
84C4×#95C2212yes—
85C23×Q8C2312yes—
86C23×D4C2311yescharacteristic
87C24×C4C2312yes—
88C24×C4C2312yes—
89C24×C4C2312yes—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—44no—
92C2×#124—28no—
93C2×#124—28no—
94C2×#124—28no—
95C2×C4×D4—24no—
96C2×C4×D4—24no—
97C2×C4×D4—24no—
98C2×C4×D4—24no—
99C2×C4×D4—44no—
100C2×C4×D4—44no—
101C2×C4×D4D418yes—
102C22×C4⋊C4—28no—
103C22×C4⋊C4—28no—
104C22×C4⋊C4—44no—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2311yescharacteristic
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2312yes—
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2314yes—
132#400—24no—
133#400—44no—
134#397—216no—
135#391—28no—
136#391—28no—
137#391—28no—
138#391—44no—
139#391—44no—
140#390—28no—
141#389—24no—
142#389—24no—
143#385—28no—
144#385—28no—
145#384—24no—
146#380—28no—
147#380—28no—
148#380—28no—
149C4×C4⋊C4—24no—
150C4×C4⋊C4—24no—
151C4×C4⋊C4—44no—
152C4×C22⋊C4—44no—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2311yescharacteristic
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2311yescharacteristic
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2311yescharacteristic
171C2×#95C2311yescharacteristic
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C25C2412yes—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8—24no—
193C22×Q8—24no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—24no—
197C22×D4C2×D414yes—
198C22×D4#4118yes—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—42no—
209C23×C4—42no—
210C23×C4—42no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4C2×D418yes—
216C23×C4C2414yes—
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2414yes—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2411yescharacteristic
226C23×C4C2412yes—
227C23×C4C2411yescharacteristic
228C23×C4C2412yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C4⋊Q8—416no—
233#124—432no—
234C4×D4—232no—
235C4×D4—416no—
236C4×D4—416no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—432no—
264C2×C4⋊C4—42no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—41no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4C2×D414yes—
272C2×C4⋊C4#4114yes—
273C2×C4⋊C4#4114yes—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—28no—
282C2×C22⋊C4—28no—
283C2×C22⋊C4—28no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4C2×D418yes—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—24no—
296C2×C42—22no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—44no—
305C2×C42—44no—
306C2×C42—44no—
307C2×C42C2×D414yes—
308C2×C42#4114yes—
309#95—22no—
310#95—22no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—28no—
320#95—22no—
321#95—24no—
322#95—24no—
323#95—22no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95—42no—
330#95—44no—
331#95—42no—
332#95—42no—
333#95—42no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—24no—
347C24—24no—
348C24—22no—
349C24—22no—
350C24—24no—
351C24C22×D414yes—
352C24C2×#4118yes—
353C24C2511yescharacteristic
354C2×Q8—48no—
355C2×Q8—48no—
356C2×Q8—48no—
357C2×Q8—48no—
358C2×Q8—48no—
359C2×Q8—48no—
360C2×Q8—48no—
361C2×D4—28no—
362C2×D4—28no—
363C2×D4—216no—
364C2×D4—216no—
365C2×D4#124116yes—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—216no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—216no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—24no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—48no—
448C22×C4—48no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4—88no—
458C22×C4—88no—
459C22×C4—88no—
460C22×C4—84no—
461C22×C4—88no—
462C22×C4C22×D412yes—
463C22×C4C22×D414yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C2×#4112yes—
467C22×C4C2×#4114yes—
468C22×C4C2×#4112yes—
469C22×C4C2×#4112yes—
470C22×C4C2×#4112yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4112yes—
474C22×C4C2×#4114yes—
475C22×C4H(5,𝔽2)12yes—
476C22×C4H(5,𝔽2)14yes—
477C22×C42−1+414yes—
478C22×C42−1+412yes—
479C4⋊C4—28no—
480C4⋊C4—28no—
481C4⋊C4—28no—
482C4⋊C4—44no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—44no—
486C4⋊C4—48no—
487C4⋊C4—48no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—48no—
491C4⋊C4—48no—
492C4⋊C4—84no—
493C22⋊C4—216no—
494C22⋊C4—48no—
495C22⋊C4—416no—
496C22⋊C4—48no—
497C22⋊C4—416no—
498C22⋊C4—416no—
499C22⋊C4—88no—
500C42—28no—
501C42—28no—
502C42—48no—
503C42—44no—
504C42—48no—
505C42—88no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—28no—
510C23—28no—
511C23—28no—
512C23—24no—
513C23—24no—
514C23—24no—
515C23—28no—
516C23—28no—
517C23—28no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—44no—
522C23—44no—
523C23—44no—
524C23—42no—
525C23—44no—
526C23—44no—
527C23—44no—
528C23C2×#12418yes—
529C23C23×D411yescharacteristic
530C23C22×#4111yescharacteristic
531C23C22×#4111yescharacteristic
532C23C22×#4112yes—
533C23C22×#4111yescharacteristic
534C23C2×H(5,𝔽2)12yes—
535C23C2×H(5,𝔽2)11yescharacteristic
536C23C2×2−1+411yescharacteristic
537C23C2×2−1+412yes—
538C23#58511yescharacteristic
539C23#58512yes—
540Q8—816no—
541D4—232no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—432no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—44no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—41no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—41no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—816no—
633C2×C4—816no—
634C2×C4#52514yes—
635C2×C4C2×#12414yes—
636C2×C4#53414yes—
637C2×C4#54614yes—
638C2×C4#55318yes—
639C2×C4#55318yes—
640C2×C4#55614yes—
641C22—28no—
642C22—28no—
643C22—28no—
644C22—44no—
645C22—48no—
646C22—44no—
647C22—48no—
648C22—48no—
649C22—44no—
650C22—48no—
651C22—44no—
652C22C2×#52511yescharacteristic
653C22C22×#12411yescharacteristic
654C22C2×#53411yescharacteristic
655C22C2×#54611yescharacteristic
656C22C2×#55312yes—
657C22C2×#55312yes—
658C22C2×#55611yescharacteristic
659C22#347511yescharacteristic
660C22#348212yes—
661C22#348511yescharacteristic
662C22#348911yescharacteristic
663C22#349911yescharacteristic
664C22#350611yescharacteristic
665C22#350712yes—
666C22#351412yes—
667C22#351512yes—
668C22#351612yes—
669C22#352011yescharacteristic
670C22#352012yes—
671C22#352112yes—
672C22#355212yes—
673C22#355412yes—
674C22#355812yes—
675C4—24no—
676C4—24no—
677C4—42no—
678C4—44no—
679C4—44no—
680C4—42no—
681C4—44no—
682C4—42no—
683C4—84no—
684C4—84no—
685C4—81no—
686C4—82no—
687C4—82no—
688C4—82no—
689C4—82no—
690C4#259118yes—
691C2—48no—
692C2C2×#259111yescharacteristic
693C2#3584611yescharacteristic
694C2#3604211yescharacteristic
695C2#3616111yescharacteristic
696C2#3630311yescharacteristic
697C2#3890611yescharacteristic
698C2#4291711yescharacteristic
699C2#4515512yes—
700C2#4522012yes—
701C2#5943712yes—
702C2#5957512yes—
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
3. #17005C2semidirect
4. #15579C2semidirect
5. #15487C2semidirect
6. #15188C2semidirect
7. #15180C2semidirect
8. #15156C2semidirect
9. #14841C2semidirect
10. #14822C2semidirect
11. #14142C2semidirect
14. #14000C2semidirect
15. #13923C2semidirect
16. C4×#400C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25472
C2×D4×C4⋊C42
#170052
#155792
#154872
#151882
#151802
#151562
#148412
#148222
#141422
#140272
#140002
#139232
C4×#4002
#138202