Finite Group Databasebeta

← #2737608

Subgroups — #2737608

Order 512.

2370 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2538C211yesmaximal, characteristic
2C2×#2453C211yesmaximal, characteristic, complemented
3#17188C214yesmaximal, complemented
4#17119C214yesmaximal, complemented
5#16992C212yesmaximal, complemented
6#16811C212yesmaximal
7#16800C212yesmaximal
8#15216C212yesmaximal, complemented
9#15182C212yesmaximal, complemented
10#15179C212yesmaximal, complemented
11#15151C211yesmaximal, characteristic, complemented
12#15145C211yesmaximal, characteristic, complemented
13#15135C212yesmaximal, complemented
14#15129C211yesmaximal, characteristic
15#14631C212yesmaximal, complemented
16#13931C211yesmaximal, characteristic, complemented
17C4×#394C211yesmaximal, characteristic
18C22×#125C2211yescharacteristic
19C22×C22⋊Q8C2212yes—
20C22×C4×Q8C2211yescharacteristic
21C22×C4×D4C2211yescharacteristic
22#2538—28no—
23#2453—28no—
24C2×#400C2212yes—
25C2×#399C2214yes—
26C2×#398C2212yes—
27C2×#398C2212yes—
28C2×#398C2212yes—
29C2×#398C2212yes—
30C2×#398C2212yes—
31C2×#396C2214yes—
32C2×#394C2211yescharacteristic
33C2×#394C2212yes—
34C2×#391C2211yescharacteristic
35C2×#391C2211yescharacteristic
36C2×#391C2212yes—
37C2×#390C2212yes—
38C2×#389C2211yescharacteristic
39C2×#389C2211yescharacteristic
40C2×#389C2212yes—
41C2×#389C2212yes—
42C2×#389C2212yes—
43C2×#387C2212yes—
44C2×#384C2211yescharacteristic
45C2×#384C2212yes—
46C2×#380C2212yes—
47C2×#380C2212yes—
48C2×#380C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C22⋊C4C2212yes—
51C2×C4×C22⋊C4C2212yes—
52#1445C2214yes—
53#1445C2212yes—
54#1445C2214yes—
55#1445C2212yes—
56#1444C2214yes—
57#1444C2214yes—
58#1442C2211yescharacteristic
59#1442C2214yes—
60#1442C2214yes—
61#1442C2214yes—
62#1441C2212yes—
63#1441C2211yescharacteristic
64#1441C2214yes—
65#1441C2214yes—
66#1441C2214yes—
67#1440C2214yes—
68#1440C2212yes—
69#1440C2214yes—
70#1439C2214yes—
71#1439C2212yes—
72#1439C2214yes—
73#1439C2214yes—
74#1439C2214yes—
75#1439C2214yes—
76#1439C2214yes—
77#1438C2211yescharacteristic
78C4⋊#95C2211yescharacteristic
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82#1434C2212yes—
83#1434C2212yes—
84C4×#95C2211yescharacteristic
85C23×Q8C2311yescharacteristic
86C23×D4C2311yescharacteristic
87C24×C4C2312yes—
88C2×#125—24no—
89C2×#125—44no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—44no—
92C2×C22⋊Q8—44no—
93C2×C4×Q8—44no—
94C2×C4×Q8D418yes—
95C2×C4×D4—24no—
96C2×C4×D4—44no—
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2314yes—
125C22×C42C2311yescharacteristic
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2312yes—
130#400—28no—
131#399—48no—
132#398—28no—
133#398—28no—
134#398—44no—
135#398—44no—
136#398—44no—
137#396—216no—
138#394—24no—
139#394—44no—
140#391—24no—
141#391—24no—
142#391—44no—
143#390—28no—
144#389—24no—
145#389—28no—
146#389—24no—
147#389—28no—
148#389—44no—
149#387—28no—
150#384—24no—
151#384—44no—
152#380—28no—
153#380—44no—
154#380—44no—
155C4×C4⋊C4—24no—
156C4×C22⋊C4—44no—
157C4×C22⋊C4—44no—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2311yescharacteristic
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2311yescharacteristic
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2311yescharacteristic
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2311yescharacteristic
187C25C2412yes—
188C22×Q8—24no—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8—24no—
192C22×Q8C2×D414yes—
193C22×D4—24no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—22no—
197C22×D4—42no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—24no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4C2412yes—
206C23×C4C2414yes—
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C23×C4C2414yes—
211C23×C4C2411yescharacteristic
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2412yes—
217C23×C4C2411yescharacteristic
218C23×C4C2412yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221#125—416no—
222C22⋊Q8—432no—
223C4×Q8—416no—
224C4×D4—416no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—28no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
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—28no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—41no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4—44no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—42no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4C2×D414yes—
275C2×C4⋊C4C2×D418yes—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—28no—
286C2×C22⋊C4—28no—
287C2×C22⋊C4—42no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—44no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—42no—
293C2×C22⋊C4—42no—
294C2×C22⋊C4—42no—
295C2×C22⋊C4—42no—
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×C42—24no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—24no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—22no—
311C2×C42—24no—
312C2×C42—44no—
313C2×C42C2×D414yes—
314C2×C42C2×D418yes—
315#95—22no—
316#95—22no—
317#95—28no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—28no—
325#95—24no—
326#95—22no—
327#95—24no—
328#95—28no—
329#95—42no—
330#95—42no—
331#95—42no—
332#95—42no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—42no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95#4114yes—
344C24—22no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24—22no—
356C24—22no—
357C24—22no—
358C242−1+414yes—
359C24C2511yescharacteristic
360C2×Q8—28no—
361C2×Q8—28no—
362C2×Q8—28no—
363C2×Q8—48no—
364C2×Q8—48no—
365C2×D4—28no—
366C2×D4—48no—
367C2×D4—48no—
368C2×D4—44no—
369C2×D4—44no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—22no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—48no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—48no—
456C22×C4—48no—
457C22×C4C22×D414yes—
458C22×C4C22×D412yes—
459C22×C4C22×D414yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C2×#4114yes—
463C22×C4C2×#4112yes—
464C22×C4H(5,𝔽2)12yes—
465C22×C4H(5,𝔽2)12yes—
466C22×C4H(5,𝔽2)14yes—
467C22×C4H(5,𝔽2)12yes—
468C22×C4H(5,𝔽2)12yes—
469C22×C42−1+414yes—
470C22×C42−1+414yes—
471C4⋊C4—28no—
472C4⋊C4—48no—
473C4⋊C4—48no—
474C4⋊C4—44no—
475C4⋊C4—44no—
476C4⋊C4—416no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—48no—
482C4⋊C4—44no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—84no—
487C4⋊C4—88no—
488C4⋊C4—88no—
489C4⋊C4—88no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C22⋊C4—48no—
493C22⋊C4—48no—
494C22⋊C4—48no—
495C22⋊C4—416no—
496C22⋊C4—88no—
497C22⋊C4—88no—
498C22⋊C4—88no—
499C42—28no—
500C42—48no—
501C42—44no—
502C42—44no—
503C42—48no—
504C23—24no—
505C23—24no—
506C23—24no—
507C23—24no—
508C23—24no—
509C23—24no—
510C23—24no—
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—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23C23×D411yescharacteristic
540C23C22×#4111yescharacteristic
541C23C2×H(5,𝔽2)12yes—
542C23C2×H(5,𝔽2)11yescharacteristic
543C23C2×H(5,𝔽2)11yescharacteristic
544C23C2×H(5,𝔽2)11yescharacteristic
545C23C2×H(5,𝔽2)11yescharacteristic
546C23C2×2−1+412yes—
547C23C2×2−1+412yes—
548C23C2×2−1+412yes—
549C23#58511yescharacteristic
550Q8—416no—
551D4—416no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—24no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—42no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—41no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—41no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—44no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—42no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—44no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—42no—
635C2×C4—44no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—816no—
641C2×C4#53414yes—
642C2×C4#53514yes—
643C2×C4#53614yes—
644C2×C4#55014yes—
645C2×C4#56218yes—
646C2×C4#56218yes—
647C22—44no—
648C22—44no—
649C22—44no—
650C22—44no—
651C22—44no—
652C22—44no—
653C22—44no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22—82no—
659C22—82no—
660C22—82no—
661C22—82no—
662C22C2×#53411yescharacteristic
663C22C2×#53511yescharacteristic
664C22C2×#53611yescharacteristic
665C22C2×#55011yescharacteristic
666C22#348611yescharacteristic
667C22#348811yescharacteristic
668C22#349911yescharacteristic
669C22#350912yes—
670C22#351711yescharacteristic
671C22#351911yescharacteristic
672C22#352012yes—
673C22C2×#56212yes—
674C22C2×#56212yes—
675C22#353612yes—
676C22#353812yes—
677C22#354112yes—
678C22#354311yescharacteristic
679C22#354611yescharacteristic
680C22#354812yes—
681C22#355212yes—
682C22#355212yes—
683C22#355212yes—
684C22#356312yes—
685C4—24no—
686C4—44no—
687C4—44no—
688C4—42no—
689C4—44no—
690C4—42no—
691C4—44no—
692C4—82no—
693C4—81no—
694C4—81no—
695C4—81no—
696C4—82no—
697C4—84no—
698C4—84no—
699C4—82no—
700C4#281618yes—
701C2—84no—
702C2C2×#281611yescharacteristic
703C2#3913311yescharacteristic
704C2#3913811yescharacteristic
705C2#3974411yescharacteristic
706C2#4279211yescharacteristic
707C2#4281411yescharacteristic
708C2#4291611yescharacteristic
709C2#4565612yes—
710C2#4567112yes—
711C2#5295912yes—
712C2#5298212yes—
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
2. C2×#2453C2semidirect
3. #17188C2semidirect
4. #17119C2semidirect
5. #16992C2semidirect
8. #15216C2semidirect
9. #15182C2semidirect
10. #15179C2semidirect
11. #15151C2semidirect
12. #15145C2semidirect
13. #15135C2semidirect
15. #14631C2semidirect
16. #13931C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25382
C2×#24532
#171882
#171192
#169922
#168112
#168002
#152162
#151822
#151792
#151512
#151452
#151352
#151292
#146312
#139312
C4×#3942