Finite Group Databasebeta

← #762063

Subgroups — #762063

Order 512.

3210 conjugacy classes of subgroups.

The subgroup lattice has 742 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Q8C211yesmaximal, characteristic, complemented
2C22×#388C211yesmaximal, characteristic
3#17020C214yesmaximal, complemented
4#15176C212yesmaximal, complemented
5#15175C212yesmaximal, complemented
6#15065C212yesmaximal, complemented
7#14972C212yesmaximal, complemented
8#14822C211yesmaximal, characteristic, complemented
9#14734C211yesmaximal, characteristic
10#14720C211yesmaximal, characteristic, complemented
11#14717C211yesmaximal, characteristic
12#14079C214yesmaximal, complemented
13#14044C212yesmaximal, complemented
14#14040C212yesmaximal, complemented
15#13991C212yesmaximal
16#13971C212yesmaximal
17#13753C211yesmaximal, characteristic, complemented
18C22×C22⋊Q8C2211yescharacteristic
19C22×C4×Q8C2211yescharacteristic, complemented
20C23×C4⋊C4C2211yescharacteristic
21C23×C22⋊C4C2211yescharacteristic
22C23×C22⋊C4C2211yescharacteristic
23C23×C22⋊C4C2211yescharacteristic
24C23×C42C2211yescharacteristic
25C2×#400C2212yescomplemented
26C2×#399C2212yes—
27C2×#395C2212yescomplemented
28C2×#393C2212yes—
29C4×C22⋊Q8—28no—
30C2×#391C2211yescharacteristic, complemented
31C2×#389C2211yescharacteristic, complemented
32C2×#388—28no—
33C2×#388C2212yes—
34C2×#388C2214yes—
35C2×#388C2212yes—
36C2×#388C2212yes—
37C2×#388C2212yes—
38C2×#388C2214yes—
39C2×#387C2212yescomplemented
40C2×#385C2212yes—
41C2×#385C2212yes—
42C2×#385C2212yes—
43C2×#380C2211yescharacteristic
44C2×#379C2212yes—
45C2×#379C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic, complemented
47C2×C4×C4⋊C4C2212yescomplemented
48C2×C4×C22⋊C4C2212yes—
49C2×C4×C22⋊C4C2212yes—
50C2×#376C2212yescomplemented
51C22×#95C2211yescharacteristic
52C22×#95C2211yescharacteristic
53#1445C2214yescomplemented
54#1444C2214yescomplemented
55#1443C2214yescomplemented
56#1442C2214yescomplemented
57#1442C2214yescomplemented
58#1441C2212yescomplemented
59#1441C2212yescomplemented
60#1441C2214yescomplemented
61#1440C2214yes—
62#1440C2214yes—
63#1439C2212yes—
64#1439C2212yes—
65#1439C2214yes—
66#1439C2214yes—
67#1439C2214yes—
68#1439C2214yes—
69#1437C2214yes—
70#1437C2214yes—
71#1436C2214yescomplemented
72#1436C2214yescomplemented
73C4⋊#95C2214yescomplemented
74#1434C2212yescomplemented
75#1434C2214yescomplemented
76#1434C2212yescomplemented
77C22⋊#95C2211yescharacteristic
78C22⋊#95C2212yes—
79C22⋊#95C2211yescharacteristic
80C22⋊#95C2212yes—
81C22⋊#95C2212yes—
82C22⋊#95C2212yes—
83C22⋊#95C2212yes—
84C4×#95C2212yescomplemented
85C26C2311yescharacteristic
86C23×Q8C2311yescharacteristic
87C24×C4C2311yescharacteristic
88C24×C4C2311yescharacteristic
89C24×C4C2311yescharacteristic
90C24×C4C2311yescharacteristic
91C24×C4C2311yescharacteristic
92C24×C4C2311yescharacteristic
93C2×C22⋊Q8—24no—
94C2×C22⋊Q8—28no—
95C2×C4×Q8—24no—
96C2×C4×Q8—44no—
97C22×C4⋊C4—24no—
98C22×C4⋊C4—44no—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2314yes—
105C22×C4⋊C4C2314yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2312yes—
109C22×C22⋊C4—24no—
110C22×C22⋊C4—28no—
111C22×C22⋊C4—24no—
112C22×C22⋊C4—24no—
113C22×C22⋊C4—44no—
114C22×C22⋊C4—44no—
115C22×C22⋊C4C2312yes—
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2312yes—
118C22×C22⋊C4C2312yes—
119C22×C22⋊C4C2312yes—
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2314yes—
122C22×C22⋊C4C2312yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C42—44no—
130C22×C42D418yes—
131C22×C42C2312yes—
132C22×C42C2314yes—
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2312yes—
136C22×C42C2311yescharacteristic
137C22×C42C2311yescharacteristic
138#400—44no—
139#399—44no—
140#395—44no—
141#393—44no—
142#391—24no—
143#389—24no—
144#388—216no—
145#388—28no—
146#388—28no—
147#388—28no—
148#388—28no—
149#388—48no—
150#387—28no—
151#385—28no—
152#385—28no—
153#385—28no—
154#380—24no—
155#379—28no—
156#379—28no—
157C4×C4⋊C4—28no—
158C4×C4⋊C4—24no—
159C4×C22⋊C4—28no—
160C4×C22⋊C4—28no—
161#376—44no—
162C2×#95—24no—
163C2×#95—24no—
164C2×#95C2312yes—
165C2×#95C2312yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2312yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C25—22no—
191C25—24no—
192C25—24no—
193C25—24no—
194C25—22no—
195C25—22no—
196C25—42no—
197C25—42no—
198C25—42no—
199C25C2412yes—
200C25C2411yescharacteristic
201C22×Q8—22no—
202C22×Q8—24no—
203C22×Q8—24no—
204C22×Q8—42no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—42no—
221C23×C4—42no—
222C23×C4—42no—
223C23×C4—42no—
224C23×C4—42no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4—42no—
228C23×C4C2×D414yes—
229C23×C4C2×D414yes—
230C23×C4C2×D414yes—
231C23×C4C2412yes—
232C23×C4C2412yes—
233C23×C4C2414yes—
234C23×C4C2412yes—
235C23×C4C2412yes—
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2411yescharacteristic
239C23×C4C2412yes—
240C23×C4C2412yes—
241C23×C4C2414yes—
242C23×C4C2412yes—
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C22⋊Q8—232no—
247C4×Q8—416no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—28no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—416no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C22⋊C4—232no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—28no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—28no—
303C2×C22⋊C4—44no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—42no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—44no—
308C2×C22⋊C4—416no—
309C2×C22⋊C4—416no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4#4118yes—
315C2×C22⋊C4#4118yes—
316C2×C22⋊C4#4118yes—
317C2×C42—24no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—24no—
321C2×C42—28no—
322C2×C42—28no—
323C2×C42—28no—
324C2×C42—416no—
325C2×C42—44no—
326C2×C42—42no—
327C2×C42—42no—
328C2×C42—44no—
329C2×C42C2×D414yes—
330C2×C42C2×D414yes—
331C2×C42C2×D414yes—
332C2×C42C2×D414yes—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—24no—
341#95—24no—
342#95—28no—
343#95—28no—
344#95—28no—
345#95—28no—
346#95—28no—
347#95—28no—
348#95—24no—
349#95—24no—
350#95—44no—
351#95—44no—
352#95—42no—
353#95—42no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359C24—22no—
360C24—22no—
361C24—22no—
362C24—22no—
363C24—28no—
364C24—22no—
365C24—22no—
366C24—24no—
367C24—24no—
368C24—24no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—22no—
373C24—24no—
374C24—416no—
375C24—416no—
376C24—48no—
377C24—48no—
378C24—44no—
379C24—44no—
380C24—44no—
381C24—44no—
382C24—44no—
383C24—44no—
384C24—48no—
385C24—48no—
386C24—88no—
387C24—88no—
388C24—84no—
389C24—84no—
390C24C22×D412yes—
391C24C2×#4114yes—
392C24C2×#4114yes—
393C24C2×#4114yes—
394C242−1+412yes—
395C242−1+412yes—
396C24C2511yescharacteristic
397C2×Q8—28no—
398C2×Q8—416no—
399C2×Q8—44no—
400C2×Q8—44no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—28no—
409C22×C4—28no—
410C22×C4—28no—
411C22×C4—28no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—22no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—28no—
443C22×C4—28no—
444C22×C4—28no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—22no—
462C22×C4—24no—
463C22×C4—22no—
464C22×C4—24no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—24no—
469C22×C4—24no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—44no—
473C22×C4—44no—
474C22×C4—44no—
475C22×C4—416no—
476C22×C4—416no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4—44no—
482C22×C4—44no—
483C22×C4—44no—
484C22×C4—44no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—88no—
490C22×C4—88no—
491C22×C4C22×D412yes—
492C22×C4C22×D412yes—
493C22×C4C22×D412yes—
494C22×C4C22×D412yes—
495C22×C4C22×D412yes—
496C22×C4C22×D412yes—
497C22×C4C2×#4114yes—
498C22×C4C2×#4118yes—
499C22×C4C2×#4114yes—
500C22×C4C2×#4114yes—
501C22×C4C2×#4114yes—
502C22×C4H(5,𝔽2)14yes—
503C22×C4H(5,𝔽2)14yes—
504C4⋊C4—28no—
505C4⋊C4—216no—
506C4⋊C4—48no—
507C4⋊C4—44no—
508C4⋊C4—48no—
509C4⋊C4—48no—
510C4⋊C4—44no—
511C4⋊C4—48no—
512C4⋊C4—88no—
513C4⋊C4—88no—
514C22⋊C4—216no—
515C22⋊C4—216no—
516C22⋊C4—216no—
517C22⋊C4—216no—
518C22⋊C4—416no—
519C22⋊C4—416no—
520C22⋊C4—48no—
521C22⋊C4—48no—
522C22⋊C4—48no—
523C22⋊C4—48no—
524C22⋊C4—48no—
525C22⋊C4—48no—
526C22⋊C4—88no—
527C22⋊C4—88no—
528C42—28no—
529C42—28no—
530C42—44no—
531C42—44no—
532C42—416no—
533C42—48no—
534C42—48no—
535C23—22no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—28no—
540C23—28no—
541C23—28no—
542C23—28no—
543C23—24no—
544C23—24no—
545C23—24no—
546C23—24no—
547C23—24no—
548C23—28no—
549C23—28no—
550C23—24no—
551C23—24no—
552C23—416no—
553C23—416no—
554C23—416no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—44no—
562C23—44no—
563C23—44no—
564C23—44no—
565C23—44no—
566C23—44no—
567C23—44no—
568C23—44no—
569C23—42no—
570C23—42no—
571C23—44no—
572C23—44no—
573C23—832no—
574C23—832no—
575C23—88no—
576C23—88no—
577C23—88no—
578C23—88no—
579C23#52514yes—
580C23#53214yes—
581C23#53614yes—
582C23#55218yes—
583C23#55618yes—
584C23C23×D411yescharacteristic
585C23C22×#4112yes—
586C23C22×#4112yes—
587C23C22×#4112yes—
588C23C2×H(5,𝔽2)12yes—
589C23C2×H(5,𝔽2)12yes—
590C23C2×2−1+411yescharacteristic
591C23C2×2−1+411yescharacteristic
592C23#58512yes—
593Q8—416no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—28no—
601C2×C4—28no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—22no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—22no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—28no—
617C2×C4—28no—
618C2×C4—28no—
619C2×C4—28no—
620C2×C4—28no—
621C2×C4—24no—
622C2×C4—24no—
623C2×C4—416no—
624C2×C4—416no—
625C2×C4—416no—
626C2×C4—416no—
627C2×C4—42no—
628C2×C4—42no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—42no—
632C2×C4—42no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—42no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—42no—
647C2×C4—42no—
648C2×C4—42no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—42no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—42no—
657C2×C4—44no—
658C2×C4—44no—
659C2×C4—44no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—44no—
663C2×C4—44no—
664C2×C4—44no—
665C2×C4—44no—
666C2×C4—44no—
667C2×C4—44no—
668C2×C4—44no—
669C2×C4—44no—
670C2×C4—44no—
671C2×C4—44no—
672C2×C4—88no—
673C2×C4—88no—
674C2×C4#52514yes—
675C2×C4#53414yes—
676C2×C4#53514yes—
677C22—24no—
678C22—216no—
679C22—24no—
680C22—24no—
681C22—24no—
682C22—216no—
683C22—24no—
684C22—24no—
685C22—48no—
686C22—44no—
687C22—48no—
688C22—44no—
689C22—44no—
690C22—832no—
691C22—82no—
692C22—82no—
693C22—84no—
694C22—82no—
695C22—82no—
696C22—84no—
697C22#260418yes—
698C22C2×#52511yescharacteristic
699C22C2×#52511yescharacteristic
700C22C2×#53211yescharacteristic
701C22C2×#53411yescharacteristic
702C22C2×#53511yescharacteristic
703C22C2×#53611yescharacteristic
704C22#344911yescharacteristic
705C22C2×#55212yes—
706C22C2×#55612yes—
707C22#347514yes—
708C22#348114yes—
709C22#351414yes—
710C22#351914yes—
711C22#352012yes—
712C22#352012yes—
713C22#353512yes—
714C22#353512yes—
715C4—24no—
716C4—24no—
717C4—24no—
718C4—42no—
719C4—44no—
720C4—44no—
721C4—42no—
722C4—48no—
723C4—42no—
724C4—44no—
725C4—82no—
726C4—84no—
727C4—82no—
728C4—82no—
729C4—82no—
730C2—28no—
731C2—84no—
732C2C2×#260411yescharacteristic
733C2#3487511yescharacteristic
734C2#3495711yescharacteristic
735C2#3507611yescharacteristic
736C2#3513111yescharacteristic
737C2#3524011yescharacteristic
738C2#3744111yescharacteristic
739C2#5949214yes—
740C2#5970514yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC26C23
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×C4×C22⋊Q8C2semidirect
3. #17020C2semidirect
4. #15176C2semidirect
5. #15175C2semidirect
6. #15065C2semidirect
7. #14972C2semidirect
8. #14822C2semidirect
10. #14720C2semidirect
12. #14079C2semidirect
13. #14044C2semidirect
14. #14040C2semidirect
17. #13753C2semidirect
19. C22×C4×Q8C22semidirect
25. C2×#400C22semidirect
27. C2×#395C22semidirect
30. C2×#391C22semidirect
31. C2×#389C22semidirect
39. C2×#387C22semidirect
46. C2×C4×C4⋊C4C22semidirect
47. C2×C4×C4⋊C4C22semidirect
50. C2×#376C22semidirect
53. #1445C22semidirect
54. #1444C22semidirect
55. #1443C22semidirect
56. #1442C22semidirect
57. #1442C22semidirect
58. #1441C22semidirect
59. #1441C22semidirect
60. #1441C22semidirect
71. #1436C22semidirect
72. #1436C22semidirect
73. C4⋊#95C22semidirect
74. #1434C22semidirect
75. #1434C22semidirect
76. #1434C22semidirect
84. C4×#95C22semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Q82
C22×#3882
#170202
#151762
#151752
#150652
#149722
#148222
#147342
#147202
#147172
#140792
#140442
#140402
#139912
#139712
#137532