Finite Group Databasebeta

← #722748

Subgroups — #722748

Order 512.

2432 conjugacy classes of subgroups.

The subgroup lattice has 700 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
2C2×#2323C211yesmaximal, characteristic
3#17067C212yesmaximal, complemented
4#17064C212yesmaximal
5#15353C214yesmaximal
6#15349C214yesmaximal, complemented
7#15139C212yesmaximal, complemented
8#15135C212yesmaximal
9#15102C212yesmaximal, complemented
10#15100C212yesmaximal
11#14340C211yesmaximal, characteristic
12#14150C211yesmaximal, characteristic, complemented
13#13976C212yesmaximal, complemented
14#13969C212yesmaximal
15#13942C211yesmaximal, characteristic, complemented
16#13910C211yesmaximal, characteristic, complemented
17#13852C211yesmaximal, characteristic
18C22×C22⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic
20C22×C4×Q8C2212yes—
21C22×#117C2211yescharacteristic
22C2×#400C2214yes—
23C2×#399C2214yes—
24Q8×C4⋊C4—28no—
25#2323—28no—
26C2×#391C2211yescharacteristic
27C2×#391C2212yes—
28C2×#389C2211yescharacteristic
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#389C2214yes—
32C2×#389C2212yes—
33C2×#389C2214yes—
34C2×#388C2212yes—
35C2×#388C2212yes—
36C2×#387C2212yes—
37C2×#384C2211yescharacteristic
38C2×#384C2212yes—
39C2×#384C2212yes—
40C2×#382C2211yescharacteristic
41C2×#380C2211yescharacteristic
42C2×#380C2212yes—
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47C2×C4×C22⋊C4C2212yes—
48#1445C2214yes—
49#1444C2214yes—
50#1444C2212yes—
51#1444C2212yes—
52#1444C2212yes—
53#1444C2214yes—
54#1444C2214yes—
55#1443C2214yes—
56#1442C2214yes—
57#1442C2214yes—
58#1442C2212yes—
59#1441C2212yes—
60#1441C2212yes—
61#1441C2211yescharacteristic
62#1441C2212yes—
63#1441C2211yescharacteristic
64#1441C2214yes—
65#1441C2214yes—
66#1439C2212yes—
67#1439C2212yes—
68#1439C2211yescharacteristic
69#1439C2212yes—
70#1439C2211yescharacteristic
71#1439C2214yes—
72#1439C2214yes—
73#1437C2214yes—
74#1436C2214yes—
75#1436C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78C4⋊#95C2212yes—
79C4⋊#95C2212yes—
80C4⋊#95C2212yes—
81C4⋊#95C2212yes—
82#1434C2212yes—
83#1434C2212yes—
84C4×#95C2212yes—
85C23×Q8C2311yescharacteristic
86C24×C4C2311yescharacteristic
87C2×C22⋊Q8—24no—
88C2×C22⋊Q8—24no—
89C2×C22⋊Q8Q818yes—
90C2×C4×Q8—28no—
91C2×C4×Q8—24no—
92C2×C4×Q8—24no—
93C2×C4×Q8—44no—
94C2×C4×Q8—44no—
95C2×C4×Q8D418yes—
96C2×#117—24no—
97C2×#117—24no—
98C2×#117—24no—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2314yes—
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2312yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C42C2311yescharacteristic
127C22×C42C2312yes—
128C22×C42C2312yes—
129C22×C42C2312yes—
130C22×C42C2314yes—
131C22×C42C2312yes—
132C22×C42C2311yescharacteristic
133#400—48no—
134#399—48no—
135#391—24no—
136#391—28no—
137#389—28no—
138#389—24no—
139#389—28no—
140#389—216no—
141#389—48no—
142#389—44no—
143#388—28no—
144#388—28no—
145#387—28no—
146#384—28no—
147#384—24no—
148#384—44no—
149#382—24no—
150#380—24no—
151#380—44no—
152C4×C4⋊C4—24no—
153C4×C4⋊C4—28no—
154C4×C4⋊C4—24no—
155C4×C4⋊C4—24no—
156C4×C22⋊C4—44no—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2312yes—
163C2×#95C2311yescharacteristic
164C2×#95C2311yescharacteristic
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C25C2411yescharacteristic
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—22no—
190C22×Q8—22no—
191C22×Q8C2×D414yes—
192C22×Q8C2×Q814yes—
193C22×Q8#4114yes—
194C23×C4—22no—
195C23×C4—22no—
196C23×C4—22no—
197C23×C4—22no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4C2×Q814yes—
201C23×C4C2412yes—
202C23×C4C2412yes—
203C23×C4C2414yes—
204C23×C4C2411yescharacteristic
205C23×C4C2414yes—
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2412yes—
209C23×C4C2411yescharacteristic
210C23×C4C2412yes—
211C23×C4C2412yes—
212C23×C4C2412yes—
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C22⋊Q8—232no—
218C4×Q8—232no—
219C4×Q8—432no—
220#117—416no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
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—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
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—22no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
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×D418yes—
274C2×C4⋊C4C2×Q814yes—
275C2×C4⋊C4C2×Q818yes—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—42no—
289C2×C22⋊C4—42no—
290C2×C22⋊C4—42no—
291C2×C22⋊C4—44no—
292C2×C22⋊C4—44no—
293C2×C22⋊C4—44no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4C2×Q818yes—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—22no—
299C2×C42—28no—
300C2×C42—24no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42—24no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—24no—
309C2×C42—44no—
310C2×C42—44no—
311C2×C42—44no—
312C2×C42C2×D418yes—
313C2×C42C2×D414yes—
314C2×C42#4114yes—
315C2×C42#4114yes—
316#95—24no—
317#95—28no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—22no—
324#95—22no—
325#95—24no—
326#95—24no—
327#95—28no—
328#95—28no—
329#95—28no—
330#95—28no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—28no—
335#95—28no—
336#95—42no—
337#95—44no—
338#95—42no—
339#95—42no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345C24—21no—
346C24—21no—
347C24—21no—
348C24—21no—
349C24—22no—
350C24—21no—
351C24—21no—
352C24—22no—
353C24—22no—
354C24—22no—
355C24C22×Q812yes—
356C24C2511yescharacteristic
357C2×Q8—28no—
358C2×Q8—28no—
359C2×Q8—28no—
360C2×Q8—28no—
361C2×Q8—28no—
362C2×Q8—28no—
363C2×Q8C22⋊Q8116yes—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
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—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—28no—
405C22×C4—28no—
406C22×C4—28no—
407C22×C4—21no—
408C22×C4—22no—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—24no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—44no—
446C22×C4—44no—
447C22×C4C22×D414yes—
448C22×C4C22×D414yes—
449C22×C4C22×D412yes—
450C22×C4C22×D412yes—
451C22×C4C22×D412yes—
452C22×C4C22×Q814yes—
453C22×C4C22×Q812yes—
454C22×C4C22×Q814yes—
455C22×C4C22×Q812yes—
456C22×C4C2×#4114yes—
457C22×C4C2×#4112yes—
458C22×C4C2×#4112yes—
459C22×C4C2×#4112yes—
460C22×C4C2×#4112yes—
461C22×C4C2×#4112yes—
462C22×C4C2×#4112yes—
463C22×C4C2×#4112yes—
464C22×C4H(5,𝔽2)14yes—
465C22×C4H(5,𝔽2)12yes—
466C22×C4H(5,𝔽2)12yes—
467C22×C42−1+414yes—
468C22×C42−1+414yes—
469C4⋊C4—28no—
470C4⋊C4—216no—
471C4⋊C4—28no—
472C4⋊C4—216no—
473C4⋊C4—216no—
474C4⋊C4—28no—
475C4⋊C4—44no—
476C4⋊C4—48no—
477C4⋊C4—416no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—48no—
482C4⋊C4—48no—
483C4⋊C4—84no—
484C4⋊C4—84no—
485C4⋊C4—88no—
486C4⋊C4—88no—
487C4⋊C4—88no—
488C4⋊C4—88no—
489C22⋊C4—216no—
490C22⋊C4—48no—
491C22⋊C4—48no—
492C22⋊C4—44no—
493C22⋊C4—44no—
494C22⋊C4—84no—
495C22⋊C4—88no—
496C42—216no—
497C42—416no—
498C42—44no—
499C42—48no—
500C42—48no—
501C42—84no—
502C42#123116yes—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—24no—
507C23—24no—
508C23—41no—
509C23—41no—
510C23—42no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—42no—
516C23—41no—
517C23—41no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23C23×D411yescharacteristic
527C23C23×Q811yescharacteristic
528C23C22×#4111yescharacteristic
529C23C22×#4111yescharacteristic
530C23C22×#4111yescharacteristic
531C23C2×H(5,𝔽2)11yescharacteristic
532C23C2×H(5,𝔽2)11yescharacteristic
533C23C2×H(5,𝔽2)12yes—
534C23C2×2−1+412yes—
535C23C2×2−1+412yes—
536C23#58512yes—
537Q8—232no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—24no—
541C2×C4—24no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—28no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—41no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—42no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—42no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—416no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—41no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—41no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—42no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—44no—
626C2×C4—44no—
627C2×C4C2×C22⋊Q818yes—
628C2×C4C2×C22⋊Q814yes—
629C2×C4C2×#12314yes—
630C2×C4C2×#12314yes—
631C2×C4C2×#12314yes—
632C2×C4#53514yes—
633C2×C4#55314yes—
634C2×C4#55314yes—
635C2×C4#55414yes—
636C2×C4#55414yes—
637C22—42no—
638C22—44no—
639C22—44no—
640C22—42no—
641C22—42no—
642C22—81no—
643C22—81no—
644C22—82no—
645C22—81no—
646C22—82no—
647C22—81no—
648C22C22×C22⋊Q811yescharacteristic
649C22C22×#12311yescharacteristic
650C22C2×#53511yescharacteristic
651C22#346112yes—
652C22C2×#55311yescharacteristic
653C22C2×#55311yescharacteristic
654C22C2×#55411yescharacteristic
655C22C2×#55411yescharacteristic
656C22#348611yescharacteristic
657C22#348711yescharacteristic
658C22#349212yes—
659C22#349811yescharacteristic
660C22#349911yescharacteristic
661C22#350212yes—
662C22#350312yes—
663C22#350712yes—
664C22#351112yes—
665C22#351612yes—
666C22#352212yes—
667C22#353812yes—
668C22#354212yes—
669C22#354612yes—
670C22#355212yes—
671C4—24no—
672C4—24no—
673C4—28no—
674C4—44no—
675C4—44no—
676C4—44no—
677C4—44no—
678C4—44no—
679C4—42no—
680C4—81no—
681C4—84no—
682C4—82no—
683C4—84no—
684C4—82no—
685C4—81no—
686C4#248718yes—
687C2—82no—
688C2C2×#248711yescharacteristic
689C2#3518211yescharacteristic
690C2#3553011yescharacteristic
691C2#3574811yescharacteristic
692C2#3584511yescharacteristic
693C2#3928112yes—
694C2#3968912yes—
695C2#4279211yescharacteristic
696C2#4287611yescharacteristic
697C2#4492312yes—
698C2#4492612yes—
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×Q8×C4⋊C4C2semidirect
3. #17067C2semidirect
6. #15349C2semidirect
7. #15139C2semidirect
9. #15102C2semidirect
12. #14150C2semidirect
13. #13976C2semidirect
15. #13942C2semidirect
16. #13910C2semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C4⋊C42
C2×#23232
#170672
#170642
#153532
#153492
#151392
#151352
#151022
#151002
#143402
#141502
#139762
#139692
#139422
#139102
#138522