Finite Group Databasebeta

← #2736566

Subgroups — #2736566

Order 512.

2396 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2821C212yesmaximal
2C2×#2546C211yesmaximal, characteristic, complemented
3C2×#2496C211yesmaximal, characteristic
4#17113C214yesmaximal, complemented
5#17108C214yesmaximal
6#16663C212yesmaximal, complemented
7#16656C212yesmaximal
8#16603C212yesmaximal, complemented
9#15158C211yesmaximal, characteristic, complemented
10#15146C212yesmaximal, complemented
11#15143C211yesmaximal, characteristic, complemented
12#15137C212yesmaximal, complemented
13#15135C212yesmaximal
14#15132C212yesmaximal
15#15125C211yesmaximal, characteristic
16#13921C211yesmaximal, characteristic, complemented
17C4×#393C211yesmaximal, characteristic
18C22×C4⋊Q8C2212yes—
19C22×C4⋊Q8C2211yescharacteristic
20C22×#126C2212yes—
21C22×C4×Q8C2211yescharacteristic
22#2821—216no—
23#2546—28no—
24#2496—28no—
25C2×#400C2214yes—
26C2×#400C2212yes—
27C2×#399C2214yes—
28C2×#398C2214yes—
29C2×#398C2214yes—
30C2×#397C2214yes—
31C2×#397C2212yes—
32C2×#395C2212yes—
33C2×#395C2212yes—
34C2×#395C2212yes—
35C2×#393C2212yes—
36C2×#393C2211yescharacteristic
37C2×#391C2212yes—
38C2×#389C2212yes—
39C2×#389C2212yes—
40C2×#389C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#389C2212yes—
43C2×#387C2212yes—
44C2×#384C2211yescharacteristic
45C2×#384C2212yes—
46C2×#383C2212yes—
47C2×#380C2211yescharacteristic
48C2×#380C2211yescharacteristic
49C2×#380C2212yes—
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C22⋊C4C2211yescharacteristic
52C2×C4×C22⋊C4C2212yes—
53#1446C2212yes—
54#1446C2211yescharacteristic
55#1445C2214yes—
56#1444C2214yes—
57#1444C2214yes—
58#1444C2214yes—
59#1443C2214yes—
60#1443C2214yes—
61#1442C2214yes—
62#1442C2212yes—
63#1442C2211yescharacteristic
64#1442C2214yes—
65#1442C2214yes—
66#1442C2214yes—
67#1442C2212yes—
68#1441C2212yes—
69#1441C2214yes—
70#1441C2214yes—
71#1441C2214yes—
72#1440C2214yes—
73#1439C2211yescharacteristic
74#1439C2214yes—
75#1439C2212yes—
76#1439C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82C4⋊#95C2212yes—
83#1434C2212yes—
84C4×#95C2212yes—
85C4×#95C2211yescharacteristic
86C23×Q8C2312yes—
87C23×Q8C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C2×C4⋊Q8—28no—
90C2×C4⋊Q8—24no—
91C2×C4⋊Q8—28no—
92C2×C4⋊Q8—44no—
93C2×C4⋊Q8—44no—
94C2×#126—28no—
95C2×#126—28no—
96C2×#126—44no—
97C2×C4×Q8—24no—
98C2×C4×Q8—44no—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2314yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129#400—216no—
130#400—28no—
131#399—216no—
132#398—216no—
133#398—216no—
134#397—216no—
135#397—28no—
136#395—28no—
137#395—28no—
138#395—44no—
139#393—28no—
140#393D418yes—
141#391—28no—
142#389—24no—
143#389—28no—
144#389—28no—
145#389—24no—
146#389—44no—
147#387—28no—
148#384—24no—
149#384—44no—
150#383—28no—
151#380—24no—
152#380—24no—
153#380—44no—
154C4×C4⋊C4D418yes—
155C4×C22⋊C4—24no—
156C4×C22⋊C4—44no—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2311yescharacteristic
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2311yescharacteristic
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2311yescharacteristic
176C2×#95C2314yes—
177C2×#95C2312yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C2×#95C2314yes—
186C2×#95C2311yescharacteristic
187C2×#95C2312yes—
188C2×#95C2312yes—
189C25C2411yescharacteristic
190C22×Q8—22no—
191C22×Q8—24no—
192C22×Q8—24no—
193C22×Q8—22no—
194C22×Q8—24no—
195C22×Q8—42no—
196C22×Q8—42no—
197C22×Q8—44no—
198C22×Q8—44no—
199C22×Q8C2×D414yes—
200C23×C4—22no—
201C23×C4—24no—
202C23×C4—22no—
203C23×C4—24no—
204C23×C4—22no—
205C23×C4C2412yes—
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2414yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C4⋊Q8—432no—
222C4⋊Q8—416no—
223#126—432no—
224C4×Q8—416no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—28no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
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⋊C4—44no—
273C2×C4⋊C4C2×D414yes—
274C2×C4⋊C4C2×D414yes—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—28no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—41no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—42no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4C2×D414yes—
295C2×C22⋊C4C2×D418yes—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—22no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42C2×D414yes—
308C2×C42C2×D418yes—
309#95—22no—
310#95—22no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—28no—
315#95—28no—
316#95—24no—
317#95—28no—
318#95—28no—
319#95—24no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—22no—
324#95—24no—
325#95—24no—
326#95—28no—
327#95—24no—
328#95—42no—
329#95—44no—
330#95—44no—
331#95—42no—
332#95—44no—
333#95—42no—
334#95—42no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95C2×D418yes—
339#95C2×D414yes—
340#95C2×D418yes—
341C24—21no—
342C24—21no—
343C24—22no—
344C24—21no—
345C24—22no—
346C24—21no—
347C24—21no—
348C24—22no—
349C24—22no—
350C24—21no—
351C24C22×D412yes—
352C24C2511yescharacteristic
353C2×Q8—28no—
354C2×Q8—48no—
355C2×Q8—44no—
356C2×Q8—44no—
357C2×Q8—48no—
358C2×Q8—48no—
359C2×Q8—48no—
360C2×Q8—48no—
361C2×Q8—88no—
362C2×Q8—88no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—48no—
447C22×C4—48no—
448C22×C4C22×D414yes—
449C22×C4C22×D412yes—
450C22×C4C22×D414yes—
451C22×C4C22×D412yes—
452C22×C4C22×D414yes—
453C22×C4C22×D414yes—
454C22×C4C22×D412yes—
455C22×C4C22×D412yes—
456C22×C4C22×D412yes—
457C22×C4H(5,𝔽2)12yes—
458C22×C4H(5,𝔽2)14yes—
459C22×C42−1+414yes—
460C22×C42−1+412yes—
461C22×C42−1+414yes—
462C22×C42−1+412yes—
463C22×C42−1+412yes—
464C4⋊C4—28no—
465C4⋊C4—28no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—48no—
469C4⋊C4—416no—
470C4⋊C4—416no—
471C4⋊C4—44no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—44no—
475C4⋊C4—48no—
476C4⋊C4—44no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C4⋊C4—88no—
480C4⋊C4—88no—
481C4⋊C4—88no—
482C22⋊C4—28no—
483C22⋊C4—48no—
484C22⋊C4—44no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—88no—
489C42—28no—
490C42—216no—
491C42—48no—
492C42—44no—
493C42—48no—
494C23—22no—
495C23—24no—
496C23—22no—
497C23—22no—
498C23—24no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—42no—
506C23—42no—
507C23—42no—
508C23—42no—
509C23—41no—
510C23—42no—
511C23—41no—
512C23—42no—
513C23—42no—
514C23—42no—
515C23—42no—
516C23—42no—
517C23C23×D411yescharacteristic
518C23C23×D411yescharacteristic
519C23C2×H(5,𝔽2)12yes—
520C23C2×H(5,𝔽2)11yescharacteristic
521C23C2×2−1+411yescharacteristic
522C23C2×2−1+411yescharacteristic
523C23C2×2−1+412yes—
524C23C2×2−1+411yescharacteristic
525C23C2×2−1+412yes—
526C23#58512yes—
527C23#58511yescharacteristic
528Q8—416no—
529Q8—816no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—22no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—24no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—44no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—44no—
584C2×C4—42no—
585C2×C4—41no—
586C2×C4—42no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—42no—
605C2×C4—44no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—88no—
621C2×C4#53514yes—
622C2×C4#53614yes—
623C2×C4#53614yes—
624C2×C4D4214yes—
625C2×C4#56318yes—
626C2×C4#56318yes—
627C22—44no—
628C22—42no—
629C22—44no—
630C22—42no—
631C22—42no—
632C22—81no—
633C22—81no—
634C22—82no—
635C22—82no—
636C22—81no—
637C22—81no—
638C22C2×#53511yescharacteristic
639C22C2×#53611yescharacteristic
640C22C2×#53611yescharacteristic
641C22C2×D4211yescharacteristic
642C22#348711yescharacteristic
643C22#348911yescharacteristic
644C22#349612yes—
645C22#349711yescharacteristic
646C22#349911yescharacteristic
647C22#349912yes—
648C22#349911yescharacteristic
649C22C2×#56312yes—
650C22C2×#56312yes—
651C22#354112yes—
652C22#354212yes—
653C22#354711yescharacteristic
654C22#355012yes—
655C22#355011yescharacteristic
656C22#355212yes—
657C22#355512yes—
658C22#356312yes—
659C22#356312yes—
660C22#356512yes—
661C4—24no—
662C4—44no—
663C4—44no—
664C4—42no—
665C4—42no—
666C4—44no—
667C4—44no—
668C4—44no—
669C4—42no—
670C4—81no—
671C4—82no—
672C4—82no—
673C4—84no—
674C4—84no—
675C4—82no—
676C4#282118yescomplemented
677C2—82no—
678C2C2×#282111yescharacteristic
679C2#3933911yescharacteristic
680C2#3934311yescharacteristic
681C2#3936011yescharacteristic
682C2#4287911yescharacteristic
683C2#4288811yescharacteristic
684C2#4291111yescharacteristic
685C2#4548512yes—
686C2#4614712yes—
687C2#5311912yes—
688C2#5312412yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2546C2semidirect
4. #17113C2semidirect
6. #16663C2semidirect
8. #16603C2semidirect
9. #15158C2semidirect
10. #15146C2semidirect
11. #15143C2semidirect
12. #15137C2semidirect
16. #13921C2semidirect
676. 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×#25462
C2×#24962
#171132
#171082
#166632
#166562
#166032
#151582
#151462
#151432
#151372
#151352
#151322
#151252
#139212
C4×#3932