Finite Group Databasebeta

← #826746

Subgroups — #826746

Order 512.

2630 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2365C211yesmaximal, characteristic, complemented
2C2×#2360C211yesmaximal, characteristic
3#16890C212yesmaximal, complemented
4#16056C214yesmaximal, complemented
5#15975C214yesmaximal
6#15202C212yesmaximal, complemented
7#15193C211yesmaximal, characteristic, complemented
8#15191C211yesmaximal, characteristic, complemented
9#15161C212yesmaximal
10#15125C211yesmaximal, characteristic
11#14172C212yesmaximal
12#14130C212yesmaximal
13#14046C212yesmaximal, complemented
14#14017C212yesmaximal, complemented
15#13997C212yesmaximal
16#13933C211yesmaximal, characteristic, complemented
17C4×#399C211yesmaximal, characteristic
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×Q8C2211yescharacteristic
21C22×C4×Q8C2212yes—
22C23×C4⋊C4C2211yescharacteristic
23C2×#399C2211yescharacteristic
24C2×#393C2212yes—
25#2365—28no—
26#2360—28no—
27C2×#391C2212yes—
28C2×#391C2211yescharacteristic
29C2×#391C2212yes—
30C2×#391C2212yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#389C2211yescharacteristic
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2214yes—
37C2×#387C2212yes—
38C2×#387C2214yes—
39C2×#385C2212yes—
40C2×#385C2214yes—
41C2×#380C2212yes—
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2212yes—
48C2×C4×C22⋊C4C2212yes—
49#1446C2212yes—
50#1446C2212yes—
51#1446C2212yes—
52#1445C2212yes—
53#1445C2211yescharacteristic
54#1445C2212yes—
55#1445C2214yes—
56#1445C2214yes—
57#1444C2211yescharacteristic
58#1444C2214yes—
59#1444C2212yes—
60#1444C2214yes—
61#1442C2212yes—
62#1442C2214yes—
63#1442C2214yes—
64#1442C2214yes—
65#1442C2214yes—
66#1442C2214yes—
67#1441C2214yes—
68#1439C2212yes—
69#1439C2211yescharacteristic
70#1439C2212yes—
71#1439C2214yes—
72#1437C2214yes—
73#1436C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76C4⋊#95C2211yescharacteristic
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2214yes—
82#1434C2212yes—
83C22⋊#95C2214yes—
84C4×#95C2211yescharacteristic
85C23×Q8C2311yescharacteristic
86C23×Q8C2312yes—
87C24×C4C2311yescharacteristic
88C24×C4C2312yes—
89C2×C4⋊Q8—28no—
90C2×C4⋊Q8—24no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8—24no—
93C2×C22⋊Q8Q818yescomplemented
94C2×C4×Q8—24no—
95C2×C4×Q8—28no—
96C2×C4×Q8—48no—
97C2×C4×Q8—44no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4—44no—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2311yescharacteristic
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2311yescharacteristic
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2312yes—
126C22×C42C2314yes—
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129C22×C42C2312yes—
130#399—24no—
131#393—28no—
132#391—28no—
133#391—24no—
134#391—28no—
135#391—44no—
136#389—24no—
137#389—28no—
138#389—28no—
139#389—28no—
140#389—48no—
141#389—44no—
142#387—28no—
143#387—216no—
144#385—28no—
145#385—48no—
146#380—28no—
147C4×C4⋊C4—28no—
148C4×C4⋊C4—24no—
149C4×C4⋊C4—44no—
150C4×C4⋊C4D418yes—
151C4×C22⋊C4—24no—
152C4×C22⋊C4—28no—
153C4×C22⋊C4—44no—
154C2×#95C2312yes—
155C2×#95C2314yes—
156C2×#95C2311yescharacteristic
157C2×#95C2311yescharacteristic
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2311yescharacteristic
168C2×#95C2312yes—
169C2×#95C2311yescharacteristic
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2312yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2312yes—
183C2×#95C2312yes—
184C2×#95C2312yes—
185C25C2411yescharacteristic
186C22×Q8—22no—
187C22×Q8—24no—
188C22×Q8—22no—
189C22×Q8—28no—
190C22×Q8—24no—
191C22×Q8—24no—
192C22×Q8—24no—
193C22×Q8—24no—
194C22×Q8—44no—
195C22×Q8C2×Q814yes—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—44no—
203C23×C4—42no—
204C23×C4—44no—
205C23×C4C2×Q814yes—
206C23×C4C2411yescharacteristic
207C23×C4C2414yes—
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2414yes—
216C23×C4C2411yescharacteristic
217C23×C4C2412yes—
218C23×C4C2412yes—
219C23×C4C2412yes—
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C4⋊Q8—232no—
223C22⋊Q8—232no—
224C4×Q8—432no—
225C4×Q8—416no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—24no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—28no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—28no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—416no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
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⋊C4C2×D414yes—
266C2×C4⋊C4C2×D414yes—
267C2×C4⋊C4C2×Q818yes—
268C2×C4⋊C4C2×Q814yes—
269C2×C4⋊C4#4118yes—
270C2×C4⋊C4#4114yes—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—28no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—28no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—41no—
284C2×C22⋊C4—42no—
285C2×C22⋊C4—42no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4C2×Q818yes—
288C2×C22⋊C4#4114yes—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—22no—
292C2×C42—28no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—28no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—24no—
301C2×C42—44no—
302C2×C42—44no—
303C2×C42—44no—
304C2×C42C2×D414yes—
305C2×C42C2×D418yes—
306#95—22no—
307#95—24no—
308#95—22no—
309#95—24no—
310#95—24no—
311#95—28no—
312#95—24no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—28no—
317#95—28no—
318#95—28no—
319#95—24no—
320#95—22no—
321#95—24no—
322#95—22no—
323#95—42no—
324#95—44no—
325#95—42no—
326#95—42no—
327#95—42no—
328#95—42no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95—44no—
336#95C2×D418yes—
337C24—21no—
338C24—21no—
339C24—22no—
340C24—22no—
341C24—22no—
342C24—21no—
343C24—22no—
344C24—21no—
345C24C22×Q812yes—
346C24C2×#4112yes—
347C242−1+412yes—
348C24C2511yescharacteristic
349C2×Q8—216no—
350C2×Q8—28no—
351C2×Q8—216no—
352C2×Q8—416no—
353C2×Q8—416no—
354C2×Q8—48no—
355C2×Q8—44no—
356C2×Q8—44no—
357C2×Q8—48no—
358C2×Q8—48no—
359C22×C4—22no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—24no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—24no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—28no—
402C22×C4—216no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—24no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—48no—
435C22×C4—48no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—48no—
439C22×C4—84no—
440C22×C4—88no—
441C22×C4—88no—
442C22×C4C22×D412yes—
443C22×C4C22×D414yes—
444C22×C4C22×D414yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×Q814yes—
448C22×C4C22×Q812yes—
449C22×C4C22×Q814yes—
450C22×C4C22×Q812yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4118yes—
453C22×C4C2×#4114yes—
454C22×C4C2×#4112yes—
455C22×C4C2×#4112yes—
456C22×C4C2×#4114yes—
457C22×C4H(5,𝔽2)14yes—
458C22×C4H(5,𝔽2)14yes—
459C22×C42−1+414yes—
460C22×C42−1+412yes—
461C4⋊C4—28no—
462C4⋊C4—216no—
463C4⋊C4—216no—
464C4⋊C4—216no—
465C4⋊C4—28no—
466C4⋊C4—416no—
467C4⋊C4—48no—
468C4⋊C4—416no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—48no—
472C4⋊C4—44no—
473C4⋊C4—44no—
474C4⋊C4—48no—
475C4⋊C4—84no—
476C4⋊C4—88no—
477C4⋊C4C22⋊Q8116yes—
478C22⋊C4—216no—
479C22⋊C4—28no—
480C22⋊C4—416no—
481C22⋊C4—44no—
482C22⋊C4—48no—
483C22⋊C4—48no—
484C42—28no—
485C42—48no—
486C42—416no—
487C42—48no—
488C42—48no—
489C42—44no—
490C42—48no—
491C23—22no—
492C23—22no—
493C23—24no—
494C23—22no—
495C23—24no—
496C23—22no—
497C23—22no—
498C23—24no—
499C23—24no—
500C23—24no—
501C23—22no—
502C23—24no—
503C23—41no—
504C23—41no—
505C23—42no—
506C23—41no—
507C23—41no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23—42no—
512C23—42no—
513C23#55714yes—
514C23C23×D411yescharacteristic
515C23C23×Q811yescharacteristic
516C23C22×#4111yescharacteristic
517C23C22×#4112yes—
518C23C22×#4111yescharacteristic
519C23C2×H(5,𝔽2)12yes—
520C23C2×H(5,𝔽2)12yes—
521C23C2×2−1+411yescharacteristic
522C23C2×2−1+411yescharacteristic
523C23C2×2−1+412yes—
524C23#58511yescharacteristic
525Q8—432no—
526Q8—416no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—28no—
533C2×C4—28no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—28no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—28no—
543C2×C4—28no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—28no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—41no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—41no—
562C2×C4—44no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—44no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—44no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—44no—
580C2×C4—42no—
581C2×C4—416no—
582C2×C4—41no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—816no—
614C2×C4C2×C22⋊Q818yes—
615C2×C4C2×C22⋊Q814yes—
616C2×C4#52514yes—
617C2×C4#53514yes—
618C2×C4#54814yes—
619C2×C4#55418yes—
620C2×C4#55918yes—
621C22—28no—
622C22—24no—
623C22—44no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22—42no—
628C22—44no—
629C22—42no—
630C22—44no—
631C22—42no—
632C22C22×C22⋊Q811yescharacteristic
633C22C2×#52511yescharacteristic
634C22C2×#53511yescharacteristic
635C22C2×#54811yescharacteristic
636C22C2×#55412yes—
637C22C2×#55711yescharacteristic
638C22C2×#55912yes—
639C22#348011yescharacteristic
640C22#348112yes—
641C22#348411yescharacteristic
642C22#348911yescharacteristic
643C22#349911yescharacteristic
644C22#350211yescharacteristic
645C22#350312yes—
646C22#351612yes—
647C22#352012yes—
648C22#352012yes—
649C22#352012yes—
650C22#352312yes—
651C22#352311yescharacteristic
652C22#353612yes—
653C22#354212yes—
654C22#355012yes—
655C4—28no—
656C4—24no—
657C4—42no—
658C4—44no—
659C4—42no—
660C4—48no—
661C4—44no—
662C4—44no—
663C4—44no—
664C4—42no—
665C4—82no—
666C4—81no—
667C4—82no—
668C4—84no—
669C4—82no—
670C4#258018yes—
671C2—44no—
672C2C2×#258011yescharacteristic
673C2#3573911yescharacteristic
674C2#3599711yescharacteristic
675C2#3652811yescharacteristic
676C2#3657711yescharacteristic
677C2#3916711yescharacteristic
678C2#4247811yescharacteristic
679C2#4499712yes—
680C2#4534512yes—
681C2#5963412yes—
682C2#5966312yes—
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
1. C2×#2365C2semidirect
3. #16890C2semidirect
4. #16056C2semidirect
6. #15202C2semidirect
7. #15193C2semidirect
8. #15191C2semidirect
13. #14046C2semidirect
14. #14017C2semidirect
16. #13933C2semidirect
93. C2×C22⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23652
C2×#23602
#168902
#160562
#159752
#152022
#151932
#151912
#151612
#151252
#141722
#141302
#140462
#140172
#139972
#139332
C4×#3992