Finite Group Databasebeta

← #2736618

Subgroups — #2736618

Order 512.

2552 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×#2821C212yesmaximal, complemented
2C2×#2551C211yesmaximal, characteristic, complemented
3C2×#2512C211yesmaximal, characteristic
4#17106C214yesmaximal, complemented
5#17104C214yesmaximal, complemented
6#16676C212yesmaximal, complemented
7#16674C212yesmaximal, complemented
8#16511C212yesmaximal, complemented
9#15155C211yesmaximal, characteristic
10#15152C211yesmaximal, characteristic, complemented
11#15148C211yesmaximal, characteristic
12#15144C212yesmaximal
13#15143C212yesmaximal, complemented
14#15138C212yesmaximal
15#15137C212yesmaximal, complemented
16#13921C211yesmaximal, characteristic, complemented
17C4×#400C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2212yes—
19C22×C4⋊Q8C2212yes—
20C22×#126C2212yes—
21#2821—216no—
22#2551—28no—
23#2512—28no—
24C2×#400C2212yes—
25C2×#400C2211yescharacteristic
26C2×#399C2212yes—
27C2×#399C2212yes—
28C2×#399C2212yes—
29C2×#399C2214yes—
30C2×#398C2214yes—
31C2×#397C2214yes—
32C2×#397C2212yes—
33C2×#396C2212yes—
34C2×#395C2214yes—
35C2×#395C2214yes—
36C2×#395C2214yes—
37C2×#393C2214yes—
38C2×#393C2214yes—
39C2×#390C2211yescharacteristic
40C2×#390C2211yescharacteristic
41C2×#390C2211yescharacteristic
42C2×#390C2212yes—
43C2×#390C2212yes—
44C2×#389C2211yescharacteristic
45C2×#389C2212yes—
46C2×#389C2212yes—
47C2×#389C2212yes—
48C2×#388C2212yes—
49C2×#384C2211yescharacteristic
50C2×#384C2211yescharacteristic
51C2×#384C2212yes—
52C2×#383C2212yes—
53C2×C4×C4⋊C4C2211yescharacteristic
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C4⋊C4C2211yescharacteristic
56#1445C2212yes—
57#1445C2211yescharacteristic
58#1444C2214yes—
59#1444C2214yes—
60#1444C2214yes—
61#1443C2212yes—
62#1443C2214yes—
63#1442C2214yes—
64#1442C2211yescharacteristic
65#1442C2212yes—
66#1442C2212yes—
67#1441C2214yes—
68#1441C2211yescharacteristic
69#1441C2212yes—
70#1440C2214yes—
71#1440C2214yes—
72#1439C2214yes—
73#1439C2212yes—
74#1439C2214yes—
75#1439C2214yes—
76#1439C2214yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2211yescharacteristic
82#1434C2212yes—
83C4×#95C2212yes—
84C4×#95C2212yes—
85C4×#95C2211yescharacteristic
86C23×Q8C2314yes—
87C23×D4C2311yescharacteristic
88C2×C4⋊Q8—28no—
89C2×C4⋊Q8—28no—
90C2×C4⋊Q8—28no—
91C2×C4⋊Q8—44no—
92C2×C4⋊Q8—44no—
93C2×C4⋊Q8—44no—
94C2×#126—28no—
95C2×#126—28no—
96C2×#126—44no—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C42C2312yes—
127C22×C42C2312yes—
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2312yes—
131C22×C42C2311yescharacteristic
132#400—28no—
133#400D418yescomplemented
134#399—28no—
135#399—28no—
136#399—44no—
137#399—48no—
138#398—216no—
139#397—216no—
140#397—28no—
141#396—28no—
142#395—216no—
143#395—216no—
144#395—48no—
145#393—216no—
146#393—48no—
147#390—24no—
148#390—24no—
149#390—24no—
150#390—44no—
151#390—44no—
152#389—24no—
153#389—28no—
154#389—28no—
155#389—44no—
156#388—28no—
157#384—24no—
158#384—24no—
159#384—44no—
160#383—28no—
161C4×C4⋊C4—24no—
162C4×C4⋊C4—24no—
163C4×C4⋊C4D418yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2311yescharacteristic
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2311yescharacteristic
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2311yescharacteristic
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2312yes—
189C2×#95C2312yes—
190C2×#95C2312yes—
191C2×#95C2311yescharacteristic
192C25C2412yes—
193C22×Q8—28no—
194C22×Q8—28no—
195C22×Q8—44no—
196C22×Q8—44no—
197C22×Q8—44no—
198C22×Q8—44no—
199C22×Q8—44no—
200C22×D4—22no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—22no—
204C22×D4—22no—
205C23×C4C2412yes—
206C23×C4C2414yes—
207C23×C4C2414yes—
208C23×C4C2411yescharacteristic
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2412yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C4⋊Q8—432no—
222C4⋊Q8—432no—
223#126—432no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
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—28no—
240C2×C4⋊C4—28no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—28no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
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⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4C2×D414yes—
273C2×C4⋊C4C2×D414yes—
274C2×C4⋊C4C2×D414yes—
275C2×C4⋊C4C2×D414yes—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—28no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—44no—
296C2×C22⋊C4—44no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—24no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—24no—
314C2×C42—22no—
315C2×C42—22no—
316C2×C42—24no—
317C2×C42—24no—
318C2×C42C2×D418yes—
319C2×C42C2×D414yes—
320#95—22no—
321#95—24no—
322#95—22no—
323#95—28no—
324#95—28no—
325#95—24no—
326#95—24no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—24no—
331#95—28no—
332#95—24no—
333#95—24no—
334#95—22no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—44no—
339#95—42no—
340#95—42no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95C2×D418yes—
345#95C2×D414yes—
346#95C2×D418yes—
347#95C2×D418yes—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—24no—
352C24—24no—
353C24—24no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—24no—
358C24—22no—
359C24C2511yescharacteristic
360C2×Q8—416no—
361C2×Q8—416no—
362C2×Q8—88no—
363C2×Q8—88no—
364C2×Q8—88no—
365C2×Q8—88no—
366C2×Q8—88no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4—48no—
371C2×D4—48no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—21no—
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—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—21no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4C22×D414yes—
451C22×C4C22×D412yes—
452C22×C4C22×D412yes—
453C22×C4C22×D412yes—
454C22×C4C22×D414yes—
455C22×C4C22×D414yes—
456C22×C4C22×D414yes—
457C22×C4C22×D412yes—
458C22×C4C22×D412yes—
459C22×C4C22×D412yes—
460C22×C4H(5,𝔽2)14yes—
461C22×C4H(5,𝔽2)12yes—
462C22×C4H(5,𝔽2)14yes—
463C22×C42−1+414yes—
464C22×C42−1+412yes—
465C22×C42−1+414yes—
466C22×C42−1+412yes—
467C22×C42−1+412yes—
468C4⋊C4—28no—
469C4⋊C4—28no—
470C4⋊C4—28no—
471C4⋊C4—28no—
472C4⋊C4—28no—
473C4⋊C4—44no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—416no—
477C4⋊C4—416no—
478C4⋊C4—44no—
479C4⋊C4—48no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C4⋊C4—88no—
483C4⋊C4—84no—
484C4⋊C4—84no—
485C4⋊C4—88no—
486C4⋊C4—88no—
487C22⋊C4—48no—
488C22⋊C4—48no—
489C22⋊C4—48no—
490C22⋊C4—48no—
491C22⋊C4—84no—
492C22⋊C4—88no—
493C22⋊C4—88no—
494C22⋊C4—84no—
495C22⋊C4—88no—
496C22⋊C4—88no—
497C42—28no—
498C42—216no—
499C42—48no—
500C42—44no—
501C42—44no—
502C42—48no—
503C23—42no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—44no—
512C23—44no—
513C23—42no—
514C23—44no—
515C23—44no—
516C23—44no—
517C23—44no—
518C23—44no—
519C23—42no—
520C23—44no—
521C23—42no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23C23×D411yescharacteristic
527C23C23×D411yescharacteristic
528C23C2×H(5,𝔽2)12yes—
529C23C2×H(5,𝔽2)11yescharacteristic
530C23C2×H(5,𝔽2)12yes—
531C23C2×2−1+411yescharacteristic
532C23C2×2−1+411yescharacteristic
533C23C2×2−1+412yes—
534C23C2×2−1+412yes—
535C23C2×2−1+411yescharacteristic
536C23#58511yescharacteristic
537Q8—832no—
538D4—88no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—41no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—41no—
608C2×C4—42no—
609C2×C4—44no—
610C2×C4—41no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4#53514yes—
630C2×C4#53614yes—
631C2×C4#53614yes—
632C2×C4D4214yes—
633C2×C4#56318yes—
634C2×C4#56318yes—
635C22—82no—
636C22—84no—
637C22—82no—
638C22—82no—
639C22—84no—
640C22—82no—
641C22—82no—
642C22—84no—
643C22—84no—
644C22—82no—
645C22—82no—
646C22C2×#53511yescharacteristic
647C22C2×#53611yescharacteristic
648C22C2×#53611yescharacteristic
649C22C2×D4211yescharacteristic
650C22#348611yescharacteristic
651C22#348911yescharacteristic
652C22#349711yescharacteristic
653C22#349811yescharacteristic
654C22#349812yes—
655C22#349912yes—
656C22#349911yescharacteristic
657C22C2×#56312yes—
658C22C2×#56312yes—
659C22#354312yes—
660C22#354412yes—
661C22#354412yes—
662C22#354711yescharacteristic
663C22#355012yes—
664C22#355011yescharacteristic
665C22#355212yes—
666C22#355212yes—
667C22#355912yes—
668C22#356112yes—
669C4—24no—
670C4—44no—
671C4—44no—
672C4—42no—
673C4—42no—
674C4—42no—
675C4—44no—
676C4—44no—
677C4—42no—
678C4—44no—
679C4—44no—
680C4—81no—
681C4—84no—
682C4—82no—
683C4—84no—
684C4#282118yescomplemented
685C2—162no—
686C2C2×#282111yescharacteristic
687C2#3933711yescharacteristic
688C2#3934311yescharacteristic
689C2#3936111yescharacteristic
690C2#4278911yescharacteristic
691C2#4281211yescharacteristic
692C2#4291411yescharacteristic
693C2#4599312yes—
694C2#4599812yes—
695C2#5303912yes—
696C2#5311812yes—
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×#2821C2semidirect
2. C2×#2551C2semidirect
4. #17106C2semidirect
5. #17104C2semidirect
6. #16676C2semidirect
7. #16674C2semidirect
8. #16511C2semidirect
10. #15152C2semidirect
13. #15143C2semidirect
15. #15137C2semidirect
16. #13921C2semidirect
17. C4×#400C2semidirect
133. #400D4semidirect
684. C4#2821semidirect

Maximal subgroups

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

subgroupindex
C2×#28212
C2×#25512
C2×#25122
#171062
#171042
#166762
#166742
#165112
#151552
#151522
#151482
#151442
#151432
#151382
#151372
#139212
C4×#4002