Finite Group Databasebeta

← #815840

Subgroups — #815840

Order 512.

2462 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2514C211yesmaximal, characteristic, complemented
2C2×#2354C211yesmaximal, characteristic
3#16693C212yesmaximal
4#15842C214yesmaximal, complemented
5#15835C214yesmaximal, complemented
6#15196C211yesmaximal, characteristic, complemented
7#15185C211yesmaximal, characteristic, complemented
8#15177C211yesmaximal, characteristic
9#15102C212yesmaximal, complemented
10#14898C212yesmaximal
11#14138C212yesmaximal, complemented
12#14129C212yesmaximal, complemented
13#13997C212yesmaximal, complemented
14#13952C212yesmaximal, complemented
15#13875C212yesmaximal, complemented
16C4×#394C211yesmaximal, characteristic
17#13810C211yesmaximal, characteristic, complemented
18C22×C22⋊Q8C2212yes—
19C22×C4×D4C2211yescharacteristic
20C22×#117C2212yes—
21#2514—28no—
22C2×#401C2212yes—
23C2×#400C2212yes—
24C2×#394C2211yescharacteristic
25#2354—28no—
26C2×#391C2211yescharacteristic
27C2×#391C2211yescharacteristic
28C2×#391C2211yescharacteristic
29C2×#391C2212yes—
30C2×#390C2212yes—
31C2×#389C2211yescharacteristic
32C2×#389C2212yes—
33C2×#388C2214yes—
34C2×#387C2212yes—
35C2×#384C2211yescharacteristic
36C2×#383C2212yes—
37C2×#382C2211yescharacteristic
38C2×#380C2212yes—
39C2×#380C2212yes—
40C2×C4×C4⋊C4C2211yescharacteristic
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C22⋊C4C2212yes—
45C2×C4×C22⋊C4C2212yes—
46C2×#376C2212yes—
47#1445C2214yes—
48#1445C2214yes—
49#1444C2212yes—
50#1444C2212yes—
51#1444C2212yes—
52#1443C2214yes—
53#1442C2211yescharacteristic
54#1442C2212yes—
55#1442C2212yes—
56#1441C2211yescharacteristic
57#1441C2212yes—
58#1441C2212yes—
59#1440C2212yes—
60#1440C2214yes—
61#1440C2214yes—
62#1440C2214yes—
63#1439C2212yes—
64#1439C2214yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1438C2211yescharacteristic
69#1438C2212yes—
70#1437C2214yes—
71#1437C2214yes—
72#1436C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2211yescharacteristic
80C4⋊#95C2214yes—
81#1434C2214yes—
82#1434C2212yes—
83#1434C2214yes—
84C4×#95C2211yescharacteristic
85C23×Q8C2311yescharacteristic
86C23×D4C2311yescharacteristic
87C24×C4C2312yes—
88C2×C22⋊Q8—28no—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—44no—
91C2×C4×D4—28no—
92C2×C4×D4—24no—
93C2×#117—28no—
94C2×#117—44no—
95C2×#117—44no—
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2311yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2314yes—
115C22×C22⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C42C2312yes—
119C22×C42C2311yescharacteristic
120C22×C42C2311yescharacteristic
121C22×C42C2314yes—
122C22×C42C2311yescharacteristic
123C22×C42C2312yes—
124C22×C42C2314yes—
125#401—28no—
126#400—28no—
127#394—24no—
128#391—24no—
129#391—24no—
130#391—24no—
131#391—44no—
132#390—28no—
133#389—24no—
134#389—44no—
135#388—216no—
136#387—28no—
137#384—24no—
138#383—28no—
139#382—24no—
140#380—28no—
141#380—28no—
142C4×C4⋊C4—28no—
143C4×C4⋊C4—28no—
144C4×C4⋊C4—24no—
145C4×C4⋊C4D418yes—
146C4×C22⋊C4—28no—
147C4×C22⋊C4—44no—
148#376—28no—
149C2×#95C2312yes—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2311yescharacteristic
154C2×#95C2312yes—
155C2×#95C2312yes—
156C2×#95C2312yes—
157C2×#95C2312yes—
158C2×#95C2311yescharacteristic
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2312yes—
163C2×#95C2314yes—
164C2×#95C2314yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2311yescharacteristic
168C2×#95C2312yes—
169C2×#95C2314yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2311yescharacteristic
182C25C2412yes—
183C22×Q8—22no—
184C22×Q8—24no—
185C22×Q8—24no—
186C22×Q8—22no—
187C22×Q8—22no—
188C22×D4—24no—
189C22×D4—22no—
190C22×D4—22no—
191C22×D4—22no—
192C22×D4—42no—
193C23×C4—24no—
194C23×C4—24no—
195C23×C4—24no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—24no—
199C23×C4#4118yes—
200C23×C4C2411yescharacteristic
201C23×C4C2414yes—
202C23×C4C2412yes—
203C23×C4C2412yes—
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2414yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C22⋊Q8—432no—
217C4×D4—232no—
218#117—432no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—28no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—44no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4C2×D414yes—
251C2×C4⋊C4C2×D414yes—
252C2×C4⋊C4#4118yes—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—28no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—28no—
265C2×C22⋊C4—28no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—42no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4—42no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—28no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—28no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—28no—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—24no—
293C2×C42—22no—
294C2×C42—44no—
295C2×C42—44no—
296C2×C42—44no—
297C2×C42C2×D414yes—
298C2×C42C2×D418yes—
299#95—28no—
300#95—22no—
301#95—22no—
302#95—24no—
303#95—24no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—24no—
311#95—28no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—28no—
316#95—28no—
317#95—22no—
318#95—42no—
319#95—44no—
320#95—44no—
321#95—42no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—42no—
326#95—42no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95C2×D418yes—
331#95#4114yes—
332C24—22no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—22no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—22no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24C2×#4114yes—
347C24C2511yescharacteristic
348C2×Q8—44no—
349C2×Q8—44no—
350C2×Q8—44no—
351C2×Q8—48no—
352C2×Q8—48no—
353C2×D4—28no—
354C2×D4—48no—
355C2×D4—48no—
356C2×D4—44no—
357C2×D4—44no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—21no—
362C22×C4—22no—
363C22×C4—24no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—22no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—216no—
399C22×C4—216no—
400C22×C4—216no—
401C22×C4—21no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—21no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—24no—
439C22×C4—22no—
440C22×C4—48no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—48no—
444C22×C4C22×D414yes—
445C22×C4C22×D412yes—
446C22×C4C22×D414yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C2×#4114yes—
450C22×C4C2×#4118yes—
451C22×C4C2×#4114yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4114yes—
454C22×C4C2×#4114yes—
455C22×C4H(5,𝔽2)12yes—
456C22×C4H(5,𝔽2)12yes—
457C22×C42−1+412yes—
458C22×C42−1+412yes—
459C4⋊C4—28no—
460C4⋊C4—28no—
461C4⋊C4—28no—
462C4⋊C4—216no—
463C4⋊C4—44no—
464C4⋊C4—48no—
465C4⋊C4—416no—
466C4⋊C4—48no—
467C4⋊C4—44no—
468C4⋊C4—48no—
469C4⋊C4—44no—
470C4⋊C4—84no—
471C4⋊C4—88no—
472C22⋊C4—216no—
473C22⋊C4—48no—
474C22⋊C4—48no—
475C22⋊C4—416no—
476C22⋊C4—416no—
477C22⋊C4—48no—
478C22⋊C4—48no—
479C22⋊C4—88no—
480C22⋊C4—88no—
481C42—28no—
482C42—216no—
483C42—416no—
484C42—416no—
485C42—44no—
486C42—44no—
487C42—48no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—24no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—42no—
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—
523C23C23×D411yescharacteristic
524C23C22×#4112yes—
525C23C22×#4112yes—
526C23C22×#4111yescharacteristic
527C23C2×H(5,𝔽2)11yescharacteristic
528C23C2×H(5,𝔽2)11yescharacteristic
529C23C2×2−1+411yescharacteristic
530C23C2×2−1+411yescharacteristic
531C23#58512yes—
532C23#58512yes—
533C23#58511yescharacteristic
534Q8—88no—
535D4—416no—
536C2×C4—22no—
537C2×C4—24no—
538C2×C4—22no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—28no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—28no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—28no—
560C2×C4—22no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—41no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—432no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—41no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—44no—
625C2×C4#52514yes—
626C2×C4#53214yes—
627C2×C4#53514yes—
628C2×C4#54714yes—
629C2×C4#55118yes—
630C2×C4#55318yes—
631C22—44no—
632C22—44no—
633C22—44no—
634C22—44no—
635C22—44no—
636C22—44no—
637C22—44no—
638C22—82no—
639C22—82no—
640C22—82no—
641C22—82no—
642C22—82no—
643C22—82no—
644C22—82no—
645C22—82no—
646C22C2×#52511yescharacteristic
647C22C2×#53211yescharacteristic
648C22C2×#53511yescharacteristic
649C22C2×#54711yescharacteristic
650C22C2×#55112yes—
651C22C2×#55312yes—
652C22#347512yes—
653C22#347512yes—
654C22#348711yescharacteristic
655C22#348911yescharacteristic
656C22#349312yes—
657C22#349512yes—
658C22#350011yescharacteristic
659C22#350712yes—
660C22#351711yescharacteristic
661C22#351812yes—
662C22#351812yes—
663C22#352011yescharacteristic
664C22#352112yes—
665C22#353912yes—
666C22#354711yescharacteristic
667C22#355211yescharacteristic
668C22#356212yes—
669C4—24no—
670C4—44no—
671C4—42no—
672C4—44no—
673C4—44no—
674C4—48no—
675C4—44no—
676C4—42no—
677C4—82no—
678C4—81no—
679C4—82no—
680C4—84no—
681C4—81no—
682C4—81no—
683C4—82no—
684C4#260318yes—
685C2—84no—
686C2C2×#260311yescharacteristic
687C2#3575911yescharacteristic
688C2#3604811yescharacteristic
689C2#3618511yescharacteristic
690C2#3642711yescharacteristic
691C2#3892011yescharacteristic
692C2#4288011yescharacteristic
693C2#4529012yes—
694C2#4539612yes—
695C2#5928012yes—
696C2#5956612yes—
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×#2514C2semidirect
4. #15842C2semidirect
5. #15835C2semidirect
6. #15196C2semidirect
7. #15185C2semidirect
9. #15102C2semidirect
11. #14138C2semidirect
12. #14129C2semidirect
13. #13997C2semidirect
14. #13952C2semidirect
15. #13875C2semidirect
17. #13810C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25142
C2×#23542
#166932
#158422
#158352
#151962
#151852
#151772
#151022
#148982
#141382
#141292
#139972
#139522
#138752
C4×#3942
#138102