Finite Group Databasebeta

← #828546

Subgroups — #828546

Order 512.

2502 conjugacy classes of subgroups.

The subgroup lattice has 710 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2506C211yesmaximal, characteristic, complemented
2C2×#2354C211yesmaximal, characteristic
3#16633C212yesmaximal
4#15841C214yesmaximal, complemented
5#15781C214yesmaximal, complemented
6#15197C211yesmaximal, characteristic, complemented
7#15152C211yesmaximal, characteristic, complemented
8#15129C211yesmaximal, characteristic
9#15088C212yesmaximal, complemented
10#14813C212yesmaximal
11#14160C212yesmaximal, complemented
12#13993C212yesmaximal, complemented
13#13951C212yesmaximal, complemented
14#13951C212yesmaximal, complemented
15#13876C212yesmaximal, complemented
16C4×#394C211yesmaximal, characteristic
17#13821C211yesmaximal, characteristic, complemented
18C22×#125C2212yes—
19C22×C4×D4C2211yescharacteristic
20C22×#117C2212yes—
21#2506—28no—
22C2×#400C2212yes—
23C2×#400C2212yes—
24C2×#398C2212yes—
25C2×#394C2211yescharacteristic
26C2×#394C2212yes—
27#2354—28no—
28C2×#390C2212yes—
29C2×#389C2211yescharacteristic
30C2×#389C2211yescharacteristic
31C2×#389C2211yescharacteristic
32C2×#389C2212yes—
33C2×#389C2211yescharacteristic
34C2×#388C2214yes—
35C2×#388C2214yes—
36C2×#387C2212yes—
37C2×#384C2211yescharacteristic
38C2×#384C2211yescharacteristic
39C2×#380C2212yes—
40C2×#380C2212yes—
41C2×#380C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2212yes—
48C2×C4×C22⋊C4C2212yes—
49#1445C2212yes—
50#1443C2214yes—
51#1442C2211yescharacteristic
52#1442C2212yes—
53#1442C2214yes—
54#1442C2214yes—
55#1441C2212yes—
56#1441C2211yescharacteristic
57#1441C2212yes—
58#1441C2212yes—
59#1441C2212yes—
60#1440C2214yes—
61#1440C2212yes—
62#1439C2212yes—
63#1439C2214yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1438C2211yescharacteristic
70#1437C2214yes—
71#1436C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74C4⋊#95C2211yescharacteristic
75C4⋊#95C2212yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2214yes—
79C4⋊#95C2214yes—
80C4⋊#95C2212yes—
81#1434C2214yes—
82#1434C2214yes—
83#1434C2212yes—
84C4×#95C2211yescharacteristic
85C23×D4C2311yescharacteristic
86C24×C4C2312yes—
87C2×#125—28no—
88C2×#125—28no—
89C2×#125—44no—
90C2×C4×D4—28no—
91C2×C4×D4—24no—
92C2×#117—28no—
93C2×#117—28no—
94C2×#117—28no—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C42C2314yes—
123C22×C42C2312yes—
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2314yes—
129#400—28no—
130#400—44no—
131#398—28no—
132#394—24no—
133#394—44no—
134#390—28no—
135#389—24no—
136#389—24no—
137#389—24no—
138#389—28no—
139#389—24no—
140#388—216no—
141#388—48no—
142#387—28no—
143#384—24no—
144#384—24no—
145#380—28no—
146#380—28no—
147#380—44no—
148C4×C4⋊C4—28no—
149C4×C4⋊C4—24no—
150C4×C4⋊C4—44no—
151C4×C4⋊C4—44no—
152C4×C4⋊C4D418yes—
153C4×C22⋊C4—28no—
154C4×C22⋊C4—44no—
155C2×#95C2311yescharacteristic
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2312yes—
159C2×#95C2311yescharacteristic
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2311yescharacteristic
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2311yescharacteristic
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C25C2412yes—
185C22×D4—24no—
186C22×D4—22no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—42no—
190C23×C4—24no—
191C23×C4—24no—
192C23×C4—24no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4#4118yes—
197C23×C4C2412yes—
198C23×C4C2412yes—
199C23×C4C2414yes—
200C23×C4C2414yes—
201C23×C4C2411yescharacteristic
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213#125—432no—
214C4×D4—232no—
215#117—432no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—41no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4C2×D414yes—
263C2×C4⋊C4C2×D414yes—
264C2×C4⋊C4#4114yes—
265C2×C4⋊C4#4118yes—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—42no—
283C2×C22⋊C4—42no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4—44no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—28no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—28no—
300C2×C42—28no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—22no—
305C2×C42—24no—
306C2×C42—22no—
307C2×C42—44no—
308C2×C42—44no—
309C2×C42—44no—
310C2×C42C2×D414yes—
311C2×C42C2×D418yes—
312#95—28no—
313#95—22no—
314#95—22no—
315#95—24no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—28no—
322#95—28no—
323#95—22no—
324#95—22no—
325#95—24no—
326#95—28no—
327#95—24no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—42no—
334#95—42no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—42no—
340#95C2×D418yes—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24C2×#4114yes—
356C24C2511yescharacteristic
357C2×D4—28no—
358C2×D4—48no—
359C2×D4—48no—
360C2×D4—44no—
361C2×D4—44no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—216no—
404C22×C4—216no—
405C22×C4—216no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—48no—
445C22×C4—48no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4C22×D412yes—
449C22×C4C22×D414yes—
450C22×C4C22×D414yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C2×#4114yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4118yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4114yes—
459C22×C4C2×#4114yes—
460C22×C4H(5,𝔽2)12yes—
461C22×C4H(5,𝔽2)12yes—
462C22×C4H(5,𝔽2)12yes—
463C22×C42−1+414yes—
464C22×C42−1+412yes—
465C4⋊C4—28no—
466C4⋊C4—28no—
467C4⋊C4—28no—
468C4⋊C4—416no—
469C4⋊C4—48no—
470C4⋊C4—44no—
471C4⋊C4—48no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—44no—
477C4⋊C4—48no—
478C4⋊C4—44no—
479C4⋊C4—48no—
480C4⋊C4—48no—
481C4⋊C4—84no—
482C4⋊C4#124116yes—
483C22⋊C4—216no—
484C22⋊C4—216no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—416no—
489C22⋊C4—88no—
490C22⋊C4—88no—
491C22⋊C4—88no—
492C42—28no—
493C42—216no—
494C42—44no—
495C42—416no—
496C42—44no—
497C42—48no—
498C42—88no—
499C23—24no—
500C23—24no—
501C23—24no—
502C23—24no—
503C23—24no—
504C23—24no—
505C23—24no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—42no—
533C23—42no—
534C23C23×D411yescharacteristic
535C23C22×#4112yes—
536C23C22×#4112yes—
537C23C22×#4111yescharacteristic
538C23C2×H(5,𝔽2)11yescharacteristic
539C23C2×H(5,𝔽2)11yescharacteristic
540C23C2×H(5,𝔽2)11yescharacteristic
541C23C2×2−1+411yescharacteristic
542C23C2×2−1+412yes—
543C23#58512yes—
544C23#58511yescharacteristic
545D4—416no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—28no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—432no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—41no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—41no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—42no—
636C2×C4#52514yes—
637C2×C4C2×#12418yes—
638C2×C4C2×#12414yes—
639C2×C4#53714yes—
640C2×C4#54614yes—
641C2×C4#55218yes—
642C2×C4#55318yes—
643C22—44no—
644C22—44no—
645C22—44no—
646C22—44no—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—82no—
651C22—82no—
652C22—82no—
653C22—82no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22C2×#52511yescharacteristic
659C22C22×#12411yescharacteristic
660C22C2×#53711yescharacteristic
661C22C2×#54611yescharacteristic
662C22C2×#55212yes—
663C22C2×#55312yes—
664C22#347512yes—
665C22#348212yes—
666C22#348711yescharacteristic
667C22#348811yescharacteristic
668C22#349212yes—
669C22#349412yes—
670C22#349911yescharacteristic
671C22#350611yescharacteristic
672C22#350712yes—
673C22#351111yescharacteristic
674C22#351712yes—
675C22#352012yes—
676C22#352412yes—
677C22#354112yes—
678C22#354711yescharacteristic
679C22#354911yescharacteristic
680C22#355512yes—
681C4—24no—
682C4—28no—
683C4—44no—
684C4—42no—
685C4—42no—
686C4—44no—
687C4—42no—
688C4—42no—
689C4—44no—
690C4—82no—
691C4—82no—
692C4—81no—
693C4—84no—
694C4—84no—
695C4—82no—
696C4#256218yes—
697C2—84no—
698C2C2×#256211yescharacteristic
699C2#3568611yescharacteristic
700C2#3604911yescharacteristic
701C2#3630311yescharacteristic
702C2#3642611yescharacteristic
703C2#3946911yescharacteristic
704C2#4287811yescharacteristic
705C2#4497612yes—
706C2#4499012yes—
707C2#5929312yes—
708C2#5985712yes—
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. C2×#2506C2semidirect
4. #15841C2semidirect
5. #15781C2semidirect
6. #15197C2semidirect
7. #15152C2semidirect
9. #15088C2semidirect
11. #14160C2semidirect
12. #13993C2semidirect
13. #13951C2semidirect
14. #13951C2semidirect
15. #13876C2semidirect
17. #13821C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25062
C2×#23542
#166332
#158412
#157812
#151972
#151522
#151292
#150882
#148132
#141602
#139932
#139512
#138762
C4×#3942
#138212