Finite Group Databasebeta

← #790712

Subgroups — #790712

Order 512.

2468 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2568C211yesmaximal, characteristic, complemented
2C2×#2362C211yesmaximal, characteristic
3#16901C212yesmaximal
4#15890C214yesmaximal, complemented
5#15742C214yesmaximal
6#15164C212yesmaximal, complemented
7#15156C211yesmaximal, characteristic, complemented
8#15143C211yesmaximal, characteristic, complemented
9#15125C211yesmaximal, characteristic
10#15065C212yesmaximal
11#14241C212yesmaximal, complemented
12#13969C212yesmaximal
13#13926C212yesmaximal, complemented
14#13885C212yesmaximal, complemented
15#13879C212yesmaximal
16C4×#399C211yesmaximal, characteristic
17Q8×#95C211yesmaximal, characteristic, complemented
18C22×C4⋊Q8C2211yescharacteristic
19C22×C22⋊Q8C2211yescharacteristic
20C22×C4×Q8C2212yes—
21C22×C4×Q8C2211yescharacteristic
22C22×#117C2211yescharacteristic
23#2568—28no—
24C2×#400C2214yes—
25C2×#400C2212yes—
26C2×#399C2211yescharacteristic
27C2×#395C2212yes—
28C2×#395C2212yes—
29C2×#393C2212yes—
30#2362—28no—
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#391C2211yescharacteristic
34C2×#391C2212yes—
35C2×#389C2211yescharacteristic
36C2×#388C2212yes—
37C2×#388C2212yes—
38C2×#387C2212yes—
39C2×#384C2212yes—
40C2×#382C2212yes—
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—
48#1446C2212yes—
49#1445C2212yes—
50#1445C2214yes—
51#1445C2211yescharacteristic
52#1444C2214yes—
53#1444C2211yescharacteristic
54#1443C2214yes—
55#1443C2214yes—
56#1442C2212yes—
57#1441C2214yes—
58#1441C2212yes—
59#1441C2214yes—
60#1441C2214yes—
61#1439C2211yescharacteristic
62#1439C2214yes—
63#1439C2214yes—
64#1439C2212yes—
65#1439C2212yes—
66#1439C2214yes—
67#1437C2214yes—
68#1436C2214yes—
69#1436C2214yes—
70#1436C2214yes—
71#1436C2214yes—
72C4⋊#95C2212yes—
73C4⋊#95C2214yes—
74C4⋊#95C2214yes—
75C4⋊#95C2211yescharacteristic
76C4⋊#95C2212yes—
77C4⋊#95C2212yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2214yes—
82#1434C2214yes—
83C4×#95C2211yescharacteristic
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C23×Q8C2312yes—
87C24×C4C2311yescharacteristic
88C2×C4⋊Q8—24no—
89C2×C4⋊Q8—44no—
90C2×C22⋊Q8—24no—
91C2×C22⋊Q8—24no—
92C2×C22⋊Q8Q818yescomplemented
93C2×C4×Q8—28no—
94C2×C4×Q8—28no—
95C2×C4×Q8—44no—
96C2×C4×Q8—44no—
97C2×C4×Q8D418yes—
98C2×#117—24no—
99C2×#117—44no—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2312yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2312yes—
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2314yes—
123C22×C42C2311yescharacteristic
124C22×C42C2312yes—
125C22×C42C2314yes—
126C22×C42C2312yes—
127C22×C42C2311yescharacteristic
128C22×C42C2312yes—
129C22×C42C2312yes—
130C22×C42C2312yes—
131#400—216no—
132#400—44no—
133#399—24no—
134#395—28no—
135#395—44no—
136#393—44no—
137#391—28no—
138#391—24no—
139#391—28no—
140#391—28no—
141#389—24no—
142#388—28no—
143#388—28no—
144#387—28no—
145#384—44no—
146#382—28no—
147#380—44no—
148C4×C4⋊C4—24no—
149C4×C4⋊C4—24no—
150C4×C4⋊C4—28no—
151C4×C4⋊C4—28no—
152C4×C22⋊C4—24no—
153C4×C22⋊C4—28no—
154C2×#95C2312yes—
155C2×#95C2311yescharacteristic
156C2×#95C2314yes—
157C2×#95C2312yes—
158C2×#95C2314yes—
159C2×#95C2314yes—
160C2×#95C2314yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2311yescharacteristic
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2311yescharacteristic
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2311yescharacteristic
185C25C2411yescharacteristic
186C22×Q8—24no—
187C22×Q8—24no—
188C22×Q8—24no—
189C22×Q8—24no—
190C22×Q8—24no—
191C22×Q8—44no—
192C22×Q8—44no—
193C22×Q8C2×D414yes—
194C22×Q8C2×Q814yes—
195C22×Q8#4114yes—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4C2×Q814yes—
201C23×C4C2412yes—
202C23×C4C2414yes—
203C23×C4C2411yescharacteristic
204C23×C4C2412yes—
205C23×C4C2412yes—
206C23×C4C2414yes—
207C23×C4C2411yescharacteristic
208C23×C4C2412yes—
209C23×C4C2412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2412yes—
212C23×C4C2411yescharacteristic
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2412yes—
216C23×C4C2411yescharacteristic
217C4⋊Q8—416no—
218C22⋊Q8—232no—
219C4×Q8—416no—
220C4×Q8—432no—
221#117—416no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—24no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—28no—
234C2×C4⋊C4—28no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—28no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—44no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4C2×D414yes—
262C2×C4⋊C4C2×D418yes—
263C2×C4⋊C4C2×Q814yes—
264C2×C4⋊C4C2×Q818yes—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—41no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4C2×Q818yes—
285C2×C42—24no—
286C2×C42—24no—
287C2×C42—24no—
288C2×C42—24no—
289C2×C42—22no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—28no—
293C2×C42—28no—
294C2×C42—24no—
295C2×C42—24no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—44no—
299C2×C42—44no—
300C2×C42—44no—
301C2×C42—42no—
302C2×C42C2×D414yes—
303C2×C42C2×D418yes—
304C2×C42#4118yes—
305#95—22no—
306#95—22no—
307#95—24no—
308#95—24no—
309#95—24no—
310#95—28no—
311#95—28no—
312#95—24no—
313#95—24no—
314#95—28no—
315#95—28no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—28no—
321#95—24no—
322#95—22no—
323#95—24no—
324#95—28no—
325#95—44no—
326#95—44no—
327#95—42no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
334#95—44no—
335#95#4114yes—
336C24—21no—
337C24—21no—
338C24—21no—
339C24—22no—
340C24—21no—
341C24—22no—
342C24—22no—
343C24—21no—
344C24—21no—
345C24—22no—
346C24C22×Q812yes—
347C24C2511yescharacteristic
348C2×Q8—28no—
349C2×Q8—28no—
350C2×Q8—216no—
351C2×Q8—216no—
352C2×Q8—48no—
353C2×Q8—48no—
354C2×Q8—48no—
355C2×Q8—88no—
356C2×Q8—88no—
357C2×Q8C22⋊Q8116yes—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—22no—
394C22×C4—28no—
395C22×C4—216no—
396C22×C4—21no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—21no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—48no—
437C22×C4C22×D414yes—
438C22×C4C22×D412yes—
439C22×C4C22×D414yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×Q814yes—
443C22×C4C22×Q812yes—
444C22×C4C22×Q814yes—
445C22×C4C22×Q812yes—
446C22×C4C2×#4114yes—
447C22×C4C2×#4114yes—
448C22×C4C2×#4112yes—
449C22×C4C2×#4114yes—
450C22×C4C2×#4114yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4114yes—
453C22×C4H(5,𝔽2)12yes—
454C22×C4H(5,𝔽2)14yes—
455C22×C42−1+412yes—
456C22×C42−1+414yes—
457C4⋊C4—28no—
458C4⋊C4—216no—
459C4⋊C4—28no—
460C4⋊C4—48no—
461C4⋊C4—48no—
462C4⋊C4—416no—
463C4⋊C4—44no—
464C4⋊C4—48no—
465C4⋊C4—44no—
466C4⋊C4—48no—
467C4⋊C4—48no—
468C4⋊C4—44no—
469C4⋊C4—48no—
470C4⋊C4—48no—
471C4⋊C4—88no—
472C4⋊C4—88no—
473C22⋊C4—216no—
474C22⋊C4—44no—
475C22⋊C4—48no—
476C22⋊C4—48no—
477C22⋊C4—44no—
478C22⋊C4—48no—
479C22⋊C4—88no—
480C42—28no—
481C42—216no—
482C42—48no—
483C42—48no—
484C42—416no—
485C42—48no—
486C42—44no—
487C42—48no—
488C23—22no—
489C23—24no—
490C23—22no—
491C23—24no—
492C23—22no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—42no—
497C23—42no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—42no—
502C23—42no—
503C23—41no—
504C23—42no—
505C23—42no—
506C23—42no—
507C23—41no—
508C23—42no—
509C23—42no—
510C23—42no—
511C23C23×D411yescharacteristic
512C23C23×Q811yescharacteristic
513C23C22×#4111yescharacteristic
514C23C22×#4112yes—
515C23C22×#4111yescharacteristic
516C23C2×H(5,𝔽2)12yes—
517C23C2×H(5,𝔽2)11yescharacteristic
518C23C2×2−1+412yes—
519C23C2×2−1+411yescharacteristic
520C23#58511yescharacteristic
521C23#58512yes—
522Q8—232no—
523Q8—816no—
524C2×C4—24no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—24no—
531C2×C4—22no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—24no—
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—44no—
568C2×C4—42no—
569C2×C4—44no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—416no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—41no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4C2×C22⋊Q818yes—
612C2×C4C2×C22⋊Q814yes—
613C2×C4#53214yes—
614C2×C4#53414yes—
615C2×C4#55014yes—
616C2×C4#55118yes—
617C2×C4#55418yes—
618C2×C4#55618yes—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—44no—
623C22—44no—
624C22—82no—
625C22—81no—
626C22—82no—
627C22—81no—
628C22—81no—
629C22—81no—
630C22C22×C22⋊Q811yescharacteristic
631C22C2×#53211yescharacteristic
632C22C2×#53411yescharacteristic
633C22C2×#55011yescharacteristic
634C22C2×#55112yes—
635C22C2×#55412yes—
636C22C2×#55612yes—
637C22Q8×#4111yescharacteristic
638C22#348211yescharacteristic
639C22#348711yescharacteristic
640C22#348911yescharacteristic
641C22#349312yes—
642C22#349512yes—
643C22#349911yescharacteristic
644C22#350212yes—
645C22#350311yescharacteristic
646C22#351112yes—
647C22#351412yes—
648C22#351612yes—
649C22#351711yescharacteristic
650C22#355212yes—
651C22#355412yes—
652C22#355812yes—
653C4—28no—
654C4—28no—
655C4—24no—
656C4—42no—
657C4—44no—
658C4—42no—
659C4—44no—
660C4—44no—
661C4—42no—
662C4—84no—
663C4—82no—
664C4—84no—
665C4—82no—
666C4—81no—
667C4—82no—
668C4#256418yes—
669C2—82no—
670C2C2×#256411yescharacteristic
671C2#3584511yescharacteristic
672C2#3606511yescharacteristic
673C2#3644511yescharacteristic
674C2#3659411yescharacteristic
675C2#3913511yescharacteristic
676C2#4291711yescharacteristic
677C2#4525912yes—
678C2#4527512yes—
679C2#5923912yes—
680C2#5924512yes—
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×#2568C2semidirect
4. #15890C2semidirect
6. #15164C2semidirect
7. #15156C2semidirect
8. #15143C2semidirect
11. #14241C2semidirect
13. #13926C2semidirect
14. #13885C2semidirect
17. Q8×#95C2semidirect
92. C2×C22⋊Q8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25682
C2×#23622
#169012
#158902
#157422
#151642
#151562
#151432
#151252
#150652
#142412
#139692
#139262
#138852
#138792
C4×#3992
Q8×#952