Finite Group Databasebeta

← #4414531

Subgroups — #4414531

Order 512.

2585 conjugacy classes of subgroups.

The subgroup lattice has 745 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2821C211yesmaximal, characteristic, complemented
2C2×#2857C211yesmaximal, characteristic, complemented
3C2×#2852C211yesmaximal, characteristic
4C2×#2283C211yesmaximal, characteristic, complemented
5#16518C212yesmaximal, complemented
6#16517C212yesmaximal, complemented
7#16394C212yesmaximal, complemented
8#16392C212yesmaximal
9#16265C212yesmaximal
10#16264C212yesmaximal, complemented
11#14902C212yesmaximal, complemented
12#14901C212yesmaximal, complemented
13#14870C212yesmaximal, complemented
14#14869C212yesmaximal, complemented
15#14842C212yesmaximal
16#14840C212yesmaximal, complemented
17#14678C211yesmaximal, characteristic, complemented
18#14665C211yesmaximal, characteristic, complemented
19#14663C211yesmaximal, characteristic, complemented
20C22×C4⋊Q8C2211yescharacteristic
21C22×#126C2211yescharacteristic
22C22×C22⋊Q8C2212yes—
23C23×C42C2211yescharacteristic
24#2857—28no—
25#2852—28no—
26#2821—28no—
27C2×#400C2212yes—
28C2×#399C2212yes—
29C2×#398C2212yes—
30C2×#398C2214yes—
31C2×#397C2212yes—
32C2×#396C2214yes—
33C2×#395C2212yes—
34C2×#393C2212yes—
35C2×#391C2212yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#388C2211yescharacteristic
39C2×#388C2211yescharacteristic
40C2×#388C2211yescharacteristic
41C2×#387C2211yescharacteristic
42C2×#387C2211yescharacteristic
43C2×#387C2211yescharacteristic
44C2×#386C2211yescharacteristic
45C2×#386C2211yescharacteristic
46C2×#386C2211yescharacteristic
47C2×#385C2211yescharacteristic
48C2×#385C2211yescharacteristic
49C2×#385C2211yescharacteristic
50C2×#384C2211yescharacteristic
51#2283—28no—
52C2×#383C2211yescharacteristic
53C2×#383C2211yescharacteristic
54C2×#383C2211yescharacteristic
55C2×#382C2212yes—
56C2×#381C2212yes—
57C2×#380C2212yes—
58C2×#380C2212yes—
59C2×#380C2212yes—
60C2×C4×C4⋊C4C2212yes—
61C2×C4×C22⋊C4C2212yes—
62C2×#376C2212yes—
63#1446C2212yes—
64#1446C2212yes—
65#1445C2212yes—
66#1445C2212yes—
67#1445C2212yes—
68#1445C2212yes—
69#1444C2212yes—
70#1444C2212yes—
71#1444C2212yes—
72#1443C2214yes—
73#1442C2212yes—
74#1442C2214yes—
75#1442C2212yes—
76#1442C2212yes—
77#1441C2212yes—
78#1441C2214yes—
79#1441C2214yes—
80#1441C2214yes—
81#1441C2212yes—
82#1441C2212yes—
83#1440C2214yes—
84#1440C2214yes—
85#1439C2212yes—
86#1439C2214yes—
87#1439C2214yes—
88#1439C2212yes—
89#1439C2212yes—
90#1438C2214yes—
91#1437C2214yes—
92#1436C2214yes—
93C4⋊#95C2214yes—
94#1434C2214yes—
95C23×Q8C2312yes—
96C23×D4C2311yescharacteristic
97C24×C4C2311yescharacteristic
98C24×C4C2311yescharacteristic
99C24×C4C2311yescharacteristic
100C2×C4⋊Q8—24no—
101C2×C4⋊Q8—24no—
102C2×C4⋊Q8—24no—
103C2×#126—24no—
104C2×#126—24no—
105C2×#126—24no—
106C2×C22⋊Q8—28no—
107C2×C22⋊Q8—48no—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2312yes—
121C22×C4⋊C4C2311yescharacteristic
122C22×C4⋊C4C2314yes—
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2311yescharacteristic
129C22×C22⋊C4C2314yes—
130C22×C22⋊C4C2314yes—
131C22×C22⋊C4C2314yes—
132C22×C22⋊C4C2311yescharacteristic
133C22×C22⋊C4C2311yescharacteristic
134C22×C22⋊C4C2311yescharacteristic
135C22×C22⋊C4C2314yes—
136C22×C42—24no—
137C22×C42—24no—
138C22×C42D418yes—
139C22×C42C2314yes—
140C22×C42C2311yescharacteristic
141C22×C42C2311yescharacteristic
142C22×C42C2311yescharacteristic
143C22×C42C2311yescharacteristic
144#400—28no—
145#399—28no—
146#398—28no—
147#398—48no—
148#397—28no—
149#396—48no—
150#395—28no—
151#393—28no—
152#391—28no—
153#389—28no—
154#389—28no—
155#388—24no—
156#388—24no—
157#388—24no—
158#387—24no—
159#387—24no—
160#387—24no—
161#386—24no—
162#386—24no—
163#386—24no—
164#385—24no—
165#385—24no—
166#385—24no—
167#384—24no—
168#383—24no—
169#383—24no—
170#383D418yescomplemented
171#382—28no—
172#381—28no—
173#380—28no—
174#380—28no—
175#380—28no—
176C4×C4⋊C4—28no—
177C4×C22⋊C4—28no—
178#376—28no—
179C2×#95C2314yes—
180C2×#95C2314yes—
181C2×#95C2311yescharacteristic
182C2×#95C2314yes—
183C2×#95C2311yescharacteristic
184C2×#95C2312yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2311yescharacteristic
190C2×#95C2311yescharacteristic
191C2×#95C2311yescharacteristic
192C2×#95C2311yescharacteristic
193C2×#95C2311yescharacteristic
194C2×#95C2311yescharacteristic
195C2×#95C2311yescharacteristic
196C2×#95C2311yescharacteristic
197C2×#95C2311yescharacteristic
198C2×#95C2311yescharacteristic
199C2×#95C2314yes—
200C2×#95C2314yes—
201C2×#95C2314yes—
202C2×#95C2314yes—
203C2×#95C2314yes—
204C2×#95C2314yes—
205C2×#95C2314yes—
206C2×#95C2314yes—
207C2×#95C2314yes—
208C2×#95C2314yes—
209C2×#95C2312yes—
210C2×#95C2312yes—
211C2×#95C2312yes—
212C25C2411yescharacteristic
213C25C2411yescharacteristic
214C22×Q8—24no—
215C22×Q8—24no—
216C22×Q8—24no—
217C22×Q8—44no—
218C22×Q8—44no—
219C22×D4—22no—
220C22×D4—24no—
221C22×D4—42no—
222C22×D4—42no—
223C23×C4—24no—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4—22no—
230C23×C4—24no—
231C23×C4—24no—
232C23×C4—42no—
233C23×C4—42no—
234C23×C4—42no—
235C23×C4C2×D414yes—
236C23×C4C2×D414yes—
237C23×C4C2×D414yes—
238C23×C4C2411yescharacteristic
239C23×C4C2411yescharacteristic
240C23×C4C2414yes—
241C23×C4C2414yes—
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2411yescharacteristic
245C23×C4C2411yescharacteristic
246C23×C4C2414yes—
247C23×C4C2411yescharacteristic
248C23×C4C2414yes—
249C23×C4C2411yescharacteristic
250C23×C4C2411yescharacteristic
251C23×C4C2411yescharacteristic
252C23×C4C2411yescharacteristic
253C23×C4C2411yescharacteristic
254C23×C4C2411yescharacteristic
255C4⋊Q8—416no—
256#126—416no—
257C22⋊Q8—432no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—28no—
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—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—28no—
283C2×C4⋊C4—28no—
284C2×C4⋊C4—24no—
285C2×C4⋊C4—24no—
286C2×C4⋊C4—28no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4—44no—
290C2×C4⋊C4—42no—
291C2×C4⋊C4—44no—
292C2×C4⋊C4—44no—
293C2×C4⋊C4—44no—
294C2×C4⋊C4—44no—
295C2×C4⋊C4—44no—
296C2×C4⋊C4—44no—
297C2×C22⋊C4—28no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—28no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—28no—
310C2×C22⋊C4—22no—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—22no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—24no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—28no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—42no—
321C2×C22⋊C4—42no—
322C2×C22⋊C4—42no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4—44no—
327C2×C22⋊C4—44no—
328C2×C22⋊C4—44no—
329C2×C22⋊C4—44no—
330C2×C42—22no—
331C2×C42—22no—
332C2×C42—22no—
333C2×C42—22no—
334C2×C42—22no—
335C2×C42—22no—
336C2×C42—232no—
337C2×C42—28no—
338C2×C42—28no—
339C2×C42—44no—
340C2×C42C2×D414yes—
341C2×C42C2×D414yes—
342C2×C42C2×D414yes—
343C2×C42C2×D414yes—
344C2×C42C2×D414yes—
345C2×C42#4114yes—
346#95—24no—
347#95—28no—
348#95—24no—
349#95—24no—
350#95—28no—
351#95—28no—
352#95—22no—
353#95—22no—
354#95—22no—
355#95—22no—
356#95—22no—
357#95—22no—
358#95—22no—
359#95—22no—
360#95—22no—
361#95—22no—
362#95—22no—
363#95—22no—
364#95—28no—
365#95—28no—
366#95—44no—
367#95—44no—
368#95—44no—
369#95—44no—
370#95—44no—
371#95—44no—
372#95—44no—
373#95—44no—
374#95—44no—
375#95—44no—
376#95C2×D418yes—
377#95C2×D418yes—
378#95C2×D418yes—
379C24—21no—
380C24—21no—
381C24—22no—
382C24—22no—
383C24—22no—
384C24—22no—
385C24—22no—
386C24—21no—
387C24—21no—
388C24—21no—
389C24—22no—
390C24—22no—
391C24—22no—
392C24—22no—
393C24—22no—
394C24—22no—
395C24C22×D412yes—
396C242−1+414yes—
397C24C2511yescharacteristic
398C2×Q8—48no—
399C2×Q8—48no—
400C2×Q8—48no—
401C2×Q8—88no—
402C2×Q8—88no—
403C2×D4—44no—
404C2×D4—44no—
405C2×D4—44no—
406C2×D4—88no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—28no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—24no—
423C22×C4—24no—
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—28no—
433C22×C4—28no—
434C22×C4—28no—
435C22×C4—21no—
436C22×C4—24no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—24no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—24no—
453C22×C4—22no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—24no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—24no—
466C22×C4—22no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—22no—
477C22×C4—22no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—22no—
483C22×C4—22no—
484C22×C4—24no—
485C22×C4—24no—
486C22×C4—22no—
487C22×C4—44no—
488C22×C4—44no—
489C22×C4—44no—
490C22×C4—48no—
491C22×C4—48no—
492C22×C4—48no—
493C22×C4—48no—
494C22×C4—48no—
495C22×C4—48no—
496C22×C4C22×D412yes—
497C22×C4C22×D418yes—
498C22×C4C22×D412yes—
499C22×C4C22×D412yes—
500C22×C4C22×D412yes—
501C22×C4C22×D412yes—
502C22×C4C22×D412yes—
503C22×C4C22×D412yes—
504C22×C4C22×D412yes—
505C22×C4C22×D412yes—
506C22×C4C2×#4112yes—
507C22×C4C2×#4112yes—
508C22×C4C2×#4112yes—
509C22×C4H(5,𝔽2)14yes—
510C22×C42−1+414yes—
511C4⋊C4—416no—
512C4⋊C4—48no—
513C4⋊C4—48no—
514C4⋊C4—44no—
515C4⋊C4—48no—
516C4⋊C4—44no—
517C4⋊C4—44no—
518C4⋊C4—44no—
519C4⋊C4—48no—
520C4⋊C4—48no—
521C4⋊C4—44no—
522C4⋊C4—416no—
523C4⋊C4—44no—
524C4⋊C4—88no—
525C4⋊C4—88no—
526C22⋊C4—416no—
527C22⋊C4—44no—
528C22⋊C4—44no—
529C22⋊C4—44no—
530C22⋊C4—416no—
531C22⋊C4—44no—
532C22⋊C4—44no—
533C22⋊C4—44no—
534C22⋊C4—44no—
535C22⋊C4—44no—
536C22⋊C4—44no—
537C22⋊C4—88no—
538C22⋊C4—88no—
539C42—28no—
540C42—28no—
541C42—28no—
542C42—416no—
543C42#121116yes—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—22no—
548C23—22no—
549C23—22no—
550C23—24no—
551C23—24no—
552C23—28no—
553C23—28no—
554C23—42no—
555C23—42no—
556C23—42no—
557C23—42no—
558C23—44no—
559C23—44no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—42no—
568C23—42no—
569C23—42no—
570C23—42no—
571C23—42no—
572C23—42no—
573C23—44no—
574C23—44no—
575C23—44no—
576C23—44no—
577C23#53614yes—
578C23C23×D411yescharacteristic
579C23C23×D411yescharacteristic
580C23C22×#4111yescharacteristic
581C23C2×H(5,𝔽2)12yes—
582C23C2×2−1+412yes—
583C23C2×2−1+412yes—
584C23#58512yes—
585C23#58512yes—
586C23#58512yes—
587Q8—816no—
588D4—88no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—24no—
603C2×C4—24no—
604C2×C4—24no—
605C2×C4—24no—
606C2×C4—24no—
607C2×C4—24no—
608C2×C4—24no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—28no—
613C2×C4—28no—
614C2×C4—28no—
615C2×C4—416no—
616C2×C4—416no—
617C2×C4—416no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—41no—
635C2×C4—41no—
636C2×C4—41no—
637C2×C4—41no—
638C2×C4—41no—
639C2×C4—41no—
640C2×C4—41no—
641C2×C4—41no—
642C2×C4—41no—
643C2×C4—41no—
644C2×C4—41no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—41no—
648C2×C4—41no—
649C2×C4—41no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—42no—
654C2×C4—44no—
655C2×C4—44no—
656C2×C4—44no—
657C2×C4—42no—
658C2×C4—42no—
659C2×C4—42no—
660C2×C4—44no—
661C2×C4—44no—
662C2×C4—42no—
663C2×C4—42no—
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—44no—
673C2×C4—44no—
674C2×C4—44no—
675C2×C4—44no—
676C2×C4C2×#12114yes—
677C2×C4C2×#12114yes—
678C2×C4C2×#12114yes—
679C2×C4#53514yes—
680C2×C4#53614yes—
681C22—24no—
682C22—24no—
683C22—24no—
684C22—48no—
685C22—48no—
686C22—42no—
687C22—42no—
688C22—42no—
689C22—42no—
690C22—84no—
691C22—84no—
692C22—81no—
693C22—81no—
694C22—81no—
695C22—81no—
696C22—81no—
697C22—81no—
698C22—81no—
699C22C22×#12111yescharacteristic
700C22C2×#53511yescharacteristic
701C22C2×#53611yescharacteristic
702C22C2×#53611yescharacteristic
703C22#344911yescharacteristic
704C22#344911yescharacteristic
705C22#344911yescharacteristic
706C22#348812yes—
707C22#348912yes—
708C22#348912yes—
709C22#349412yes—
710C22#349512yes—
711C22#349512yes—
712C22#353114yes—
713C22#355114yes—
714C22#355214yes—
715C22#356414yes—
716C4—24no—
717C4—24no—
718C4—24no—
719C4—24no—
720C4—24no—
721C4—48no—
722C4—42no—
723C4—84no—
724C4—81no—
725C4—81no—
726C4—84no—
727C4—81no—
728C4—81no—
729C4—84no—
730C4—81no—
731C4—81no—
732C4—81no—
733C2—44no—
734C2—161no—
735C2#3525211yescharacteristic
736C2#3530611yescharacteristic
737C2#3530611yescharacteristic
738C2#3669911yescharacteristic
739C2#3754811yescharacteristic
740C2#3755011yescharacteristic
741C2#3761511yescharacteristic
742C2#4613114yes—
743C2#4613914yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
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×#2821C2semidirect
2. C2×#2857C2semidirect
4. C2×#2283C2semidirect
5. #16518C2semidirect
6. #16517C2semidirect
7. #16394C2semidirect
10. #16264C2semidirect
11. #14902C2semidirect
12. #14901C2semidirect
13. #14870C2semidirect
14. #14869C2semidirect
16. #14840C2semidirect
17. #14678C2semidirect
18. #14665C2semidirect
19. #14663C2semidirect
170. #383D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28212
C2×#28572
C2×#28522
C2×#22832
#165182
#165172
#163942
#163922
#162652
#162642
#149022
#149012
#148702
#148692
#148422
#148402
#146782
#146652
#146632