Finite Group Databasebeta

← #722909

Subgroups — #722909

Order 512.

3338 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2583C211yesmaximal, characteristic
2C2×#2434C211yesmaximal, characteristic
3C22×#395C211yesmaximal, characteristic
4C2×#2345C211yesmaximal, characteristic
5#16806C212yesmaximal, complemented
6#16507C212yesmaximal, complemented
7#15623C214yesmaximal, complemented
8#15360C214yesmaximal, complemented
9#15128C212yesmaximal, complemented
10#15097C212yesmaximal, complemented
11#14920C212yesmaximal
12#14728C212yesmaximal
13#14281C211yesmaximal, characteristic
14#14044C212yesmaximal
15#14043C212yesmaximal
16#13922C211yesmaximal, characteristic
17#13782C211yesmaximal, characteristic
18C24×Q8C2211yescharacteristic
19C22×C4⋊Q8C2212yes—
20C22×C22⋊Q8C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2212yes—
23#2583—28no—
24#2434—28no—
25C2×#398C2211yescharacteristic
26C2×#398C2211yescharacteristic
27C2×#398C2214yes—
28C2×#395—28no—
29C2×#395C2212yes—
30C2×#395C2214yes—
31C2×#395C2212yes—
32C2×#395C2211yescharacteristic
33C2×#395C2211yescharacteristic
34C2×#395C2212yes—
35C2×#395C2212yes—
36#2345—28no—
37C2×#391C2212yes—
38C2×#391C2214yes—
39C2×#389C2212yes—
40C2×#389C2212yes—
41C2×#389C2212yes—
42C2×#389C2214yes—
43C2×#387C2212yes—
44C2×#387C2212yes—
45C2×#385C2212yes—
46C2×#385C2212yes—
47C2×#385C2212yes—
48C2×#385C2212yes—
49C2×#385C2212yes—
50C2×#380C2211yescharacteristic
51C2×#380C2211yescharacteristic
52C2×#380C2211yescharacteristic
53C2×C4×C4⋊C4C2212yes—
54C2×C4×C22⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2211yescharacteristic
56C2×C4×C22⋊C4C2211yescharacteristic
57C2×C4×C22⋊C4C2211yescharacteristic
58C22×#95C2212yes—
59C22×#95C2211yescharacteristic
60#1446C2212yes—
61#1446C2212yes—
62#1446C2212yes—
63#1445C2214yes—
64#1445C2212yes—
65#1444C2214yes—
66#1444C2214yes—
67#1442C2214yes—
68#1442C2214yes—
69#1442C2214yes—
70#1441C2214yes—
71#1441C2214yes—
72#1441C2214yes—
73#1441C2214yes—
74#1441C2214yes—
75#1441C2214yes—
76#1436C2214yes—
77#1436C2214yes—
78#1436C2214yes—
79C4⋊#95C2214yes—
80C4⋊#95C2212yes—
81#1434C2212yes—
82C22⋊#95C2212yes—
83C22⋊#95C2212yes—
84C22⋊#95C2212yes—
85C22⋊#95C2212yes—
86C22⋊#95C2212yes—
87C23×Q8—24no—
88C23×Q8—24no—
89C23×Q8—24no—
90C23×Q8—48no—
91C23×Q8C2312yes—
92C23×Q8C2312yes—
93C23×Q8C2311yescharacteristic
94C24×C4C2312yes—
95C24×C4C2311yescharacteristic
96C24×C4C2311yescharacteristic
97C24×C4C2312yes—
98C24×C4C2311yescharacteristic
99C2×C4⋊Q8—28no—
100C2×C4⋊Q8—48no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8—24no—
103C2×C22⋊Q8—24no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—24no—
106C22×C4⋊C4—28no—
107C22×C4⋊C4—48no—
108C22×C4⋊C4D418yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2314yes—
116C22×C4⋊C4C2312yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2311yescharacteristic
121C22×C4⋊C4C2312yes—
122C22×C4⋊C4C2312yes—
123C22×C4⋊C4C2312yes—
124C22×C4⋊C4C2312yes—
125C22×C4⋊C4C2312yes—
126C22×C4⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2311yescharacteristic
129C22×C22⋊C4C2311yescharacteristic
130C22×C22⋊C4C2311yescharacteristic
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2311yescharacteristic
133C22×C42C2312yes—
134C22×C42C2312yes—
135C22×C42C2312yes—
136C22×C42C2312yes—
137#398—24no—
138#398—24no—
139#398—48no—
140#395—24no—
141#395—24no—
142#395—28no—
143#395—28no—
144#395—28no—
145#395—28no—
146#395—216no—
147#391—28no—
148#391—48no—
149#389—28no—
150#389—28no—
151#389—28no—
152#389—48no—
153#387—28no—
154#387—28no—
155#385—28no—
156#385—28no—
157#385—28no—
158#385—28no—
159#385Q8116yescomplemented
160#380—24no—
161#380—24no—
162#380—24no—
163C4×C4⋊C4—28no—
164C4×C22⋊C4—24no—
165C4×C22⋊C4—24no—
166C4×C22⋊C4—24no—
167C4×C22⋊C4D418yes—
168C2×#95—28no—
169C2×#95—24no—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2314yes—
174C2×#95C2311yescharacteristic
175C2×#95C2311yescharacteristic
176C2×#95C2312yes—
177C2×#95C2314yes—
178C2×#95C2314yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2312yes—
182C2×#95C2312yes—
183C2×#95C2314yes—
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2312yes—
188C2×#95C2312yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2312yes—
198C2×#95C2314yes—
199C25C2411yescharacteristic
200C22×Q8—24no—
201C22×Q8—22no—
202C22×Q8—24no—
203C22×Q8—24no—
204C22×Q8—24no—
205C22×Q8—22no—
206C22×Q8—24no—
207C22×Q8—24no—
208C22×Q8—22no—
209C22×Q8—28no—
210C22×Q8—28no—
211C22×Q8—44no—
212C22×Q8—44no—
213C22×Q8—44no—
214C22×Q8—432no—
215C22×Q8—432no—
216C22×Q8—432no—
217C22×Q8—416no—
218C23×C4—24no—
219C23×C4—24no—
220C23×C4—22no—
221C23×C4—22no—
222C23×C4—24no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4—28no—
226C23×C4—24no—
227C23×C4—24no—
228C23×C4—24no—
229C23×C4—44no—
230C23×C4—42no—
231C23×C4—44no—
232C23×C4—44no—
233C23×C4—44no—
234C23×C4—44no—
235C23×C4C2×D414yes—
236C23×C4C2×D414yes—
237C23×C4C2×D414yes—
238C23×C4C2×D414yes—
239C23×C4C2×Q818yes—
240C23×C4#4114yes—
241C23×C4C2412yes—
242C23×C4C2412yes—
243C23×C4C2412yes—
244C23×C4C2412yes—
245C23×C4C2412yes—
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249C23×C4C2412yes—
250C23×C4C2412yes—
251C23×C4C2414yes—
252C23×C4C2414yes—
253C23×C4C2411yescharacteristic
254C23×C4C2411yescharacteristic
255C23×C4C2412yes—
256C23×C4C2411yescharacteristic
257C4⋊Q8—432no—
258C22⋊Q8—416no—
259C2×C4⋊C4—232no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—24no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—28no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—28no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—24no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—24no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—42no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C4⋊C4—44no—
298C2×C4⋊C4—44no—
299C2×C4⋊C4—432no—
300C2×C4⋊C4—44no—
301C2×C4⋊C4—44no—
302C2×C4⋊C4—44no—
303C2×C4⋊C4—44no—
304C2×C4⋊C4C2×D414yes—
305C2×C4⋊C4C2×D414yes—
306C2×C4⋊C4C2×D414yes—
307C2×C4⋊C4C2×D414yes—
308C2×C4⋊C4C2×Q8116yescomplemented
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—24no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—24no—
319C2×C22⋊C4C2×D414yes—
320C2×C22⋊C4C2×D414yes—
321C2×C22⋊C4C2×Q818yes—
322C2×C22⋊C4C2×Q818yes—
323C2×C42—24no—
324C2×C42—24no—
325C2×C42—24no—
326C2×C42—24no—
327C2×C42—44no—
328C2×C42—42no—
329C2×C42—44no—
330C2×C42—44no—
331C2×C42C2×D418yes—
332#95—24no—
333#95—24no—
334#95—28no—
335#95—28no—
336#95—24no—
337#95—24no—
338#95—22no—
339#95—22no—
340#95—24no—
341#95—24no—
342#95—24no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—28no—
348#95—28no—
349#95—28no—
350#95—28no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95C2×D418yes—
360#95C2×Q8116yescomplemented
361C24—24no—
362C24—22no—
363C24—22no—
364C24C22×D412yes—
365C24C22×D412yes—
366C24C22×Q814yes—
367C24C2×#4112yes—
368C242−1+414yes—
369C24C2511yescharacteristic
370C2×Q8—216no—
371C2×Q8—216no—
372C2×Q8—416no—
373C2×Q8—416no—
374C2×Q8—416no—
375C2×Q8—416no—
376C2×Q8—48no—
377C2×Q8—48no—
378C2×Q8—48no—
379C2×Q8—4128no—
380C2×Q8—48no—
381C2×Q8—48no—
382C2×Q8—44no—
383C2×Q8—84no—
384C2×Q8—84no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—216no—
394C22×C4—28no—
395C22×C4—216no—
396C22×C4—21no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—28no—
431C22×C4—28no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—24no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—22no—
452C22×C4—44no—
453C22×C4—48no—
454C22×C4—48no—
455C22×C4—416no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—416no—
461C22×C4—416no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—416no—
465C22×C4—48no—
466C22×C4—48no—
467C22×C4—84no—
468C22×C4—84no—
469C22×C4#121116yes—
470C22×C4C22×D412yes—
471C22×C4C22×D414yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D414yes—
479C22×C4C22×D412yes—
480C22×C4C22×Q814yes—
481C22×C4C22×Q818yes—
482C22×C4C22×Q814yes—
483C22×C4C22×Q818yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4112yes—
486C22×C4H(5,𝔽2)18yes—
487C22×C42−1+414yes—
488C4⋊C4—28no—
489C4⋊C4—28no—
490C4⋊C4—28no—
491C4⋊C4—28no—
492C4⋊C4—416no—
493C4⋊C4—48no—
494C4⋊C4—416no—
495C4⋊C4—416no—
496C4⋊C4—416no—
497C4⋊C4—416no—
498C4⋊C4—48no—
499C4⋊C4—44no—
500C4⋊C4—48no—
501C4⋊C4—48no—
502C4⋊C4—84no—
503C4⋊C4—84no—
504C4⋊C4—84no—
505C4⋊C4—84no—
506C22⋊C4—28no—
507C22⋊C4—216no—
508C22⋊C4—48no—
509C22⋊C4—44no—
510C22⋊C4—44no—
511C22⋊C4C4⋊Q8116yes—
512C42—48no—
513C42—48no—
514C42—84no—
515C42—84no—
516C23—22no—
517C23—22no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—28no—
524C23—28no—
525C23—28no—
526C23—28no—
527C23C2×#12114yes—
528C23C2×C4⋊Q814yes—
529C23#53614yes—
530C23D4×Q818yes—
531C23#55718yes—
532C23C23×D411yescharacteristic
533C23C23×D411yescharacteristic
534C23C23×Q812yes—
535C23C22×#4111yescharacteristic
536C23C2×H(5,𝔽2)14yes—
537C23C2×2−1+412yes—
538C23C2×2−1+412yes—
539C23#58512yes—
540Q8—432no—
541Q8—432no—
542Q8—88no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—28no—
556C2×C4—28no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—28no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—28no—
570C2×C4—28no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—416no—
578C2×C4—416no—
579C2×C4—432no—
580C2×C4—432no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—44no—
589C2×C4—42no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—44no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—42no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—88no—
624C2×C4C2×#12114yes—
625C2×C4C2×#12114yes—
626C2×C4C2×C4⋊Q818yes—
627C2×C4#53614yes—
628C22—24no—
629C22—28no—
630C22—216no—
631C22—28no—
632C22—28no—
633C22—28no—
634C22—24no—
635C22#254618yes—
636C22C22×#12111yescharacteristic
637C22C22×C4⋊Q811yescharacteristic
638C22C2×#53611yescharacteristic
639C22C2×#53611yescharacteristic
640C22#344911yescharacteristic
641C22C2×D4×Q812yes—
642C22C2×#55712yes—
643C22#348412yes—
644C22#348412yes—
645C22#348912yes—
646C22#349012yes—
647C22#349512yes—
648C22#350114yes—
649C22#350314yes—
650C22#354114yes—
651C22#355214yes—
652C4—24no—
653C4—24no—
654C4—28no—
655C4—24no—
656C4—48no—
657C4—42no—
658C4—42no—
659C4—44no—
660C4—44no—
661C4—48no—
662C4—44no—
663C4—42no—
664C4—82no—
665C4—82no—
666C4—82no—
667C4—82no—
668C2—28no—
669C2C2×#254611yescharacteristic
670C2#3530611yescharacteristic
671C2#3534811yescharacteristic
672C2#3592812yes—
673C2#3958812yes—
674C2#3959212yes—
675C2#4059112yes—
676C2#4312514yes—
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
5. #16806C2semidirect
6. #16507C2semidirect
7. #15623C2semidirect
8. #15360C2semidirect
9. #15128C2semidirect
10. #15097C2semidirect
159. #385Q8semidirect
308. C2×C4⋊C4C2×Q8semidirect
360. #95C2×Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25832
C2×#24342
C22×#3952
C2×#23452
#168062
#165072
#156232
#153602
#151282
#150972
#149202
#147282
#142812
#140442
#140432
#139222
#137822