Finite Group Databasebeta

← #646825

Subgroups — #646825

Order 512.

2464 conjugacy classes of subgroups.

The subgroup lattice has 733 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2859C211yesmaximal, characteristic, complemented
2C2×#2664C212yesmaximal
3C2×C4×#126C211yesmaximal, characteristic
4#16821C212yesmaximal
5#16820C212yesmaximal, complemented
6#16705C212yesmaximal, complemented
7#16701C212yesmaximal
8#15152C211yesmaximal, characteristic, complemented
9#15151C211yesmaximal, characteristic, complemented
10#15150C211yesmaximal, characteristic
11#15135C211yesmaximal, characteristic
12#15133C212yesmaximal, complemented
13#15131C212yesmaximal
14#14818C212yesmaximal, complemented
15#14808C212yesmaximal
16#14289C212yesmaximal, complemented
17#14046C212yesmaximal, complemented
18#13974C212yesmaximal
19C4×#384C211yesmaximal, characteristic, complemented
20C22×#126C2211yescharacteristic
21C22×#125C2212yes—
22#2859—28no—
23#2664—216no—
24C2×#400C2212yes—
25C2×#399C2212yes—
26C2×#398C2214yes—
27C2×#398C2212yes—
28C2×#398C2212yes—
29C2×#397C2212yes—
30C2×#397C2212yes—
31C2×#396C2214yes—
32C4×#126—28no—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#389C2212yes—
37C2×#389C2212yes—
38C2×#389C2214yes—
39C2×#389C2212yes—
40C2×#388C2214yes—
41C2×#388C2211yescharacteristic
42C2×#388C2211yescharacteristic
43C2×#388C2211yescharacteristic
44C2×#387C2211yescharacteristic
45C2×#387C2211yescharacteristic
46C2×#387C2211yescharacteristic
47C2×#384C2211yescharacteristic
48C2×#384C2211yescharacteristic
49C2×#384C2212yes—
50C2×#383C2211yescharacteristic
51C2×#382C2212yes—
52C2×#380C2212yes—
53C2×#380C2212yes—
54C2×C4×C4⋊C4C2211yescharacteristic
55C2×C4×C4⋊C4C2212yes—
56C2×C4×C4⋊C4C2211yescharacteristic
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×C4×C4⋊C4C2212yes—
59C2×C4×C4⋊C4C2212yes—
60C2×C4×C22⋊C4C2211yescharacteristic
61C2×C4×C22⋊C4C2211yescharacteristic
62C2×C4×C22⋊C4C2211yescharacteristic
63C2×C43C2211yescharacteristic
64#1445C2212yes—
65#1445C2212yes—
66#1445C2212yes—
67#1445C2212yes—
68#1445C2212yes—
69#1445C2212yes—
70#1444C2212yes—
71#1443C2214yes—
72#1442C2212yes—
73#1442C2214yes—
74#1441C2214yes—
75#1441C2214yes—
76#1441C2214yes—
77#1441C2214yes—
78#1441C2214yes—
79#1441C2212yes—
80#1441C2214yes—
81#1440C2214yes—
82#1439C2214yes—
83#1439C2212yes—
84#1436C2214yes—
85#1436C2214yes—
86C4⋊#95C2212yes—
87C4⋊#95C2212yes—
88C4⋊#95C2212yes—
89C4⋊#95C2212yes—
90C4⋊#95C2211yescharacteristic
91C4⋊#95C2211yescharacteristic
92C4⋊#95C2212yes—
93C4⋊#95C2212yes—
94C24×C4C2311yescharacteristic
95C2×#126—24no—
96C2×#126—24no—
97C2×#126D418yes—
98C2×#125—28no—
99C2×#125—28no—
100C2×#125—28no—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2314yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2311yescharacteristic
116C22×C4⋊C4C2311yescharacteristic
117C22×C4⋊C4C2312yes—
118C22×C4⋊C4C2314yes—
119C22×C4⋊C4C2312yes—
120C22×C4⋊C4C2311yescharacteristic
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2314yes—
128C22×C22⋊C4C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2311yescharacteristic
136C22×C42C2311yescharacteristic
137C22×C42C2314yes—
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140#400—28no—
141#399—28no—
142#398—28no—
143#398—216no—
144#398—44no—
145#397—28no—
146#397—44no—
147#396—216no—
148#389—216no—
149#389—28no—
150#389—28no—
151#389—28no—
152#389—28no—
153#389—28no—
154#389—44no—
155#388—216no—
156#388—24no—
157#388—24no—
158#388—24no—
159#387—24no—
160#387—24no—
161#387—24no—
162#384—28no—
163#384—24no—
164#384—24no—
165#383—24no—
166#382—28no—
167#380—28no—
168#380—44no—
169C4×C4⋊C4—24no—
170C4×C4⋊C4—24no—
171C4×C4⋊C4—24no—
172C4×C4⋊C4—28no—
173C4×C4⋊C4—44no—
174C4×C4⋊C4—44no—
175C4×C22⋊C4—24no—
176C4×C22⋊C4—24no—
177C4×C22⋊C4—24no—
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2312yes—
181C2×#95C2312yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2311yescharacteristic
187C2×#95C2311yescharacteristic
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2311yescharacteristic
193C2×#95C2311yescharacteristic
194C2×#95C2311yescharacteristic
195C2×#95C2311yescharacteristic
196C2×#95C2311yescharacteristic
197C2×#95C2311yescharacteristic
198C2×#95C2311yescharacteristic
199C2×#95C2311yescharacteristic
200C2×#95C2311yescharacteristic
201C2×#95C2311yescharacteristic
202C2×#95C2314yes—
203C2×#95C2314yes—
204C2×#95C2314yes—
205C2×#95C2314yes—
206C2×#95C2314yes—
207C2×#95C2314yes—
208C2×#95C2314yes—
209C43D418yes—
210C25C2411yescharacteristic
211C23×C4—22no—
212C23×C4—24no—
213C23×C4—22no—
214C23×C4—24no—
215C23×C4—22no—
216C23×C4C2414yes—
217C23×C4C2414yes—
218C23×C4C2414yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2414yes—
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234#126—232no—
235#125—432no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—24no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—28no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—28no—
261C2×C4⋊C4—28no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—28no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—28no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—24no—
281C2×C4⋊C4—28no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—42no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4—44no—
288C2×C4⋊C4—44no—
289C2×C4⋊C4C2×D414yes—
290C2×C4⋊C4C2×D414yes—
291C2×C4⋊C4C2×D414yes—
292C2×C4⋊C4#4118yes—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—28no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—28no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—28no—
308C2×C22⋊C4—44no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—44no—
311C2×C22⋊C4—44no—
312C2×C22⋊C4C2×D414yes—
313C2×C22⋊C4C2×D414yes—
314C2×C22⋊C4C2×D414yes—
315C2×C42—24no—
316C2×C42—22no—
317C2×C42—22no—
318C2×C42—24no—
319C2×C42—24no—
320C2×C42—22no—
321C2×C42—22no—
322C2×C42—22no—
323C2×C42—22no—
324C2×C42—22no—
325C2×C42—24no—
326C2×C42—22no—
327C2×C42—24no—
328C2×C42—22no—
329C2×C42—28no—
330C2×C42—42no—
331C2×C42—44no—
332C2×C42—44no—
333C2×C42C2×D414yes—
334C2×C42C2×D414yes—
335C2×C42C2×D414yes—
336C2×C42C2×D414yes—
337C2×C42C2×D414yes—
338C2×C42C2×D414yes—
339C2×C42C2×D414yes—
340C2×C42C2×D414yes—
341C2×C42#4114yes—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95—22no—
353#95—22no—
354#95—24no—
355#95—24no—
356#95—28no—
357#95—28no—
358#95—28no—
359#95—28no—
360#95—28no—
361#95—28no—
362#95—28no—
363#95—28no—
364#95—24no—
365#95—24no—
366#95—44no—
367#95—44no—
368#95—44no—
369#95—44no—
370#95—44no—
371#95—44no—
372#95—44no—
373C24—21no—
374C24—21no—
375C24—22no—
376C24—21no—
377C24—21no—
378C24—22no—
379C24—21no—
380C24—22no—
381C24—21no—
382C24—22no—
383C24C22×D412yes—
384C24C2511yescharacteristic
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—21no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—24no—
407C22×C4—21no—
408C22×C4—24no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—24no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—24no—
419C22×C4—22no—
420C22×C4—24no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—24no—
444C22×C4—24no—
445C22×C4—24no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—21no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—24no—
461C22×C4—24no—
462C22×C4—24no—
463C22×C4—44no—
464C22×C4—44no—
465C22×C4—44no—
466C22×C4—48no—
467C22×C4—48no—
468C22×C4C22×D412yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D412yes—
476C22×C4C22×D412yes—
477C22×C4C22×D412yes—
478C22×C4C22×D412yes—
479C22×C4C22×D412yes—
480C22×C4C22×D412yes—
481C22×C4C2×#4118yes—
482C22×C4C2×#4112yes—
483C22×C4C2×#4112yes—
484C22×C4C2×#4112yes—
485C22×C4C2×#4114yes—
486C22×C4H(5,𝔽2)12yes—
487C22×C4H(5,𝔽2)12yes—
488C22×C42−1+414yes—
489C22×C42−1+412yes—
490C22×C42−1+414yes—
491C22×C42−1+412yes—
492C4⋊C4—28no—
493C4⋊C4—28no—
494C4⋊C4—28no—
495C4⋊C4—216no—
496C4⋊C4—416no—
497C4⋊C4—416no—
498C4⋊C4—416no—
499C4⋊C4—48no—
500C4⋊C4—44no—
501C4⋊C4—48no—
502C4⋊C4—48no—
503C4⋊C4—416no—
504C4⋊C4—416no—
505C4⋊C4—48no—
506C4⋊C4—44no—
507C4⋊C4—44no—
508C4⋊C4—48no—
509C4⋊C4—48no—
510C4⋊C4—48no—
511C4⋊C4—88no—
512C22⋊C4—28no—
513C22⋊C4—28no—
514C22⋊C4—28no—
515C22⋊C4—416no—
516C22⋊C4—44no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—88no—
520C42—28no—
521C42—28no—
522C42—28no—
523C42—28no—
524C42—28no—
525C42—44no—
526C42—44no—
527C42—44no—
528C42—44no—
529C42—88no—
530C42#121116yes—
531C23—22no—
532C23—24no—
533C23—22no—
534C23—22no—
535C23—24no—
536C23—41no—
537C23—41no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—42no—
543C23—42no—
544C23—42no—
545C23—42no—
546C23—41no—
547C23—41no—
548C23—42no—
549C23—42no—
550C23—42no—
551C23—42no—
552C23—42no—
553C23—42no—
554C23C23×D411yescharacteristic
555C23C23×D411yescharacteristic
556C23C22×#4111yescharacteristic
557C23C22×#4112yes—
558C23C2×H(5,𝔽2)11yescharacteristic
559C23C2×H(5,𝔽2)11yescharacteristic
560C23C2×2−1+412yes—
561C23C2×2−1+412yes—
562C23C2×2−1+411yescharacteristic
563C23C2×2−1+411yescharacteristic
564C23#58512yes—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—24no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—28no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—42no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—44no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—41no—
619C2×C4—44no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—44no—
633C2×C4—41no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—42no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—44no—
645C2×C4—44no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—42no—
650C2×C4—42no—
651C2×C4—44no—
652C2×C4—42no—
653C2×C4—42no—
654C2×C4—42no—
655C2×C4—44no—
656C2×C4—42no—
657C2×C4—88no—
658C2×C4C2×#12114yes—
659C2×C4C2×#12114yes—
660C2×C4C2×#12114yes—
661C2×C4#53214yes—
662C2×C4#53614yes—
663C2×C4#53614yes—
664C2×C4#53714yes—
665C2×C4#53714yes—
666C2×C4#53918yes—
667C2×C4#54118yes—
668C22—42no—
669C22—42no—
670C22—44no—
671C22—44no—
672C22—42no—
673C22—81no—
674C22—82no—
675C22—82no—
676C22—81no—
677C22—81no—
678C22—81no—
679C22C22×#12111yescharacteristic
680C22C2×#53211yescharacteristic
681C22C2×#53611yescharacteristic
682C22C2×#53611yescharacteristic
683C22C2×#53711yescharacteristic
684C22C2×#53711yescharacteristic
685C22#344911yescharacteristic
686C22C2×#53912yes—
687C22C2×#54112yes—
688C22D4×#4112yes—
689C22#348012yes—
690C22#349811yescharacteristic
691C22#349911yescharacteristic
692C22#349912yes—
693C22#350912yes—
694C22#351111yescharacteristic
695C22#351111yescharacteristic
696C22#352012yes—
697C22#352312yes—
698C22#355412yes—
699C22#355512yes—
700C22#355912yes—
701C22#356312yes—
702C4—24no—
703C4—24no—
704C4—24no—
705C4—24no—
706C4—24no—
707C4—42no—
708C4—42no—
709C4—48no—
710C4—42no—
711C4—42no—
712C4—81no—
713C4—84no—
714C4—81no—
715C4—81no—
716C4—84no—
717C4—84no—
718C4—81no—
719C4#266418yescomplemented
720C2—82no—
721C2C2×#266411yescharacteristic
722C2#3493711yescharacteristic
723C2#3524111yescharacteristic
724C2#3527311yescharacteristic
725C2#3531611yescharacteristic
726C2#3754911yescharacteristic
727C2#3762211yescharacteristic
728C2#3940412yes—
729C2#3940712yes—
730C2#3978812yes—
731C2#3979112yes—
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×#2859C2semidirect
5. #16820C2semidirect
6. #16705C2semidirect
8. #15152C2semidirect
9. #15151C2semidirect
12. #15133C2semidirect
14. #14818C2semidirect
16. #14289C2semidirect
17. #14046C2semidirect
19. C4×#384C2semidirect
719. C4#2664semidirect

Maximal subgroups

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

subgroupindex
C2×#28592
C2×#26642
C2×C4×#1262
#168212
#168202
#167052
#167012
#151522
#151512
#151502
#151352
#151332
#151312
#148182
#148082
#142892
#140462
#139742
C4×#3842