Finite Group Databasebeta

← #647567

Subgroups — #647567

Order 512.

2208 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2859C211yesmaximal, characteristic, complemented
2C2×C4×#126C211yesmaximal, characteristic
3#16965C212yesmaximal, complemented
4#16683C212yesmaximal, complemented
5#16682C212yesmaximal
6#16673C212yesmaximal
7#15221C212yesmaximal, complemented
8#15220C212yesmaximal
9#15152C211yesmaximal, characteristic, complemented
10#15151C211yesmaximal, characteristic, complemented
11#15150C211yesmaximal, characteristic
12#15135C211yesmaximal, characteristic
13#15024C212yesmaximal, complemented
14#15020C212yesmaximal
15#14328C212yesmaximal, complemented
16#14324C212yesmaximal
17#14045C212yesmaximal, complemented
18#13973C212yesmaximal
19C4×#382C211yesmaximal, characteristic, complemented
20C22×#126C2211yescharacteristic
21C22×#125C2212yes—
22#2859—28no—
23C2×#400C2212yes—
24C2×#399C2212yes—
25C2×#398C2212yes—
26C2×#398C2212yes—
27C2×#397C2212yes—
28C2×#397C2212yes—
29C4×#126—28no—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#389C2212yes—
34C2×#389C2212yes—
35C2×#389C2212yes—
36C2×#388C2211yescharacteristic
37C2×#388C2211yescharacteristic
38C2×#388C2211yescharacteristic
39C2×#387C2211yescharacteristic
40C2×#387C2211yescharacteristic
41C2×#387C2211yescharacteristic
42C2×#384C2211yescharacteristic
43C2×#384C2211yescharacteristic
44C2×#383C2211yescharacteristic
45C2×#382C2212yes—
46C2×#382C2212yes—
47C2×#380C2212yes—
48C2×#380C2212yes—
49C2×C4×C4⋊C4C2211yescharacteristic
50C2×C4×C4⋊C4C2211yescharacteristic
51C2×C4×C4⋊C4C2211yescharacteristic
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C22⋊C4C2211yescharacteristic
54C2×C4×C22⋊C4C2211yescharacteristic
55C2×C4×C22⋊C4C2211yescharacteristic
56C2×C43C2211yescharacteristic
57#1445C2212yes—
58#1445C2212yes—
59#1445C2212yes—
60#1445C2212yes—
61#1445C2212yes—
62#1445C2212yes—
63#1445C2214yes—
64#1444C2212yes—
65#1444C2214yes—
66#1444C2214yes—
67#1443C2214yes—
68#1442C2214yes—
69#1442C2214yes—
70#1442C2214yes—
71#1442C2212yes—
72#1441C2212yes—
73#1441C2214yes—
74#1441C2214yes—
75#1439C2212yes—
76#1439C2214yes—
77#1439C2214yes—
78#1439C2214yes—
79#1437C2214yes—
80#1436C2214yes—
81C4⋊#95C2211yescharacteristic
82C4⋊#95C2211yescharacteristic
83C4⋊#95C2212yes—
84C4⋊#95C2212yes—
85C4⋊#95C2212yes—
86C4⋊#95C2212yes—
87C4⋊#95C2214yes—
88#1434C2212yes—
89#1434C2212yes—
90#1434C2214yes—
91C4×#95C2212yes—
92C4×#95C2212yes—
93C24×C4C2311yescharacteristic
94C2×#126—24no—
95C2×#126—24no—
96C2×#126—24no—
97C2×#125—28no—
98C2×#125—44no—
99C2×#125—44no—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C4⋊C4C2311yescharacteristic
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×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2314yes—
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135C22×C42C2314yes—
136C22×C42C2311yescharacteristic
137#400—44no—
138#399—44no—
139#398—28no—
140#398—44no—
141#397—28no—
142#397—44no—
143#389—28no—
144#389—28no—
145#389—28no—
146#389—28no—
147#389—28no—
148#389—44no—
149#388—24no—
150#388—24no—
151#388—24no—
152#387—24no—
153#387—24no—
154#387—24no—
155#384—24no—
156#384—24no—
157#383—24no—
158#382—28no—
159#382—28no—
160#380—28no—
161#380—44no—
162C4×C4⋊C4—24no—
163C4×C4⋊C4—24no—
164C4×C4⋊C4—24no—
165C4×C4⋊C4—44no—
166C4×C22⋊C4—24no—
167C4×C22⋊C4—24no—
168C4×C22⋊C4—24no—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2314yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2311yescharacteristic
178C2×#95C2314yes—
179C2×#95C2314yes—
180C2×#95C2311yescharacteristic
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2311yescharacteristic
184C2×#95C2311yescharacteristic
185C2×#95C2311yescharacteristic
186C2×#95C2311yescharacteristic
187C2×#95C2311yescharacteristic
188C2×#95C2311yescharacteristic
189C2×#95C2311yescharacteristic
190C2×#95C2311yescharacteristic
191C2×#95C2311yescharacteristic
192C2×#95C2311yescharacteristic
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2314yes—
199C2×#95C2314yes—
200C2×#95C2314yes—
201C2×#95C2314yes—
202C43D418yes—
203C25C2411yescharacteristic
204C23×C4—22no—
205C23×C4—24no—
206C23×C4—22no—
207C23×C4—24no—
208C23×C4—22no—
209C23×C4C2414yes—
210C23×C4C2414yes—
211C23×C4C2411yescharacteristic
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2414yes—
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2414yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227#126—232no—
228#125—432no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—28no—
242C2×C4⋊C4—28no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—28no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—42no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4#4114yes—
279C2×C4⋊C4#4118yes—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—22no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—28no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—44no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4#4114yes—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—24no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—24no—
309C2×C42—24no—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—22no—
313C2×C42—22no—
314C2×C42—24no—
315C2×C42—22no—
316C2×C42—24no—
317C2×C42—22no—
318C2×C42—28no—
319C2×C42—42no—
320C2×C42—44no—
321C2×C42—44no—
322C2×C42C2×D414yes—
323C2×C42C2×D414yes—
324C2×C42C2×D414yes—
325C2×C42C2×D414yes—
326C2×C42C2×D414yes—
327C2×C42C2×D414yes—
328C2×C42C2×D414yes—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—28no—
342#95—28no—
343#95—28no—
344#95—28no—
345#95—28no—
346#95—28no—
347#95—24no—
348#95—28no—
349#95—24no—
350#95—28no—
351#95—24no—
352#95—24no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95—44no—
359#95—44no—
360#95—44no—
361#95—44no—
362C24—21no—
363C24—21no—
364C24—22no—
365C24—21no—
366C24—21no—
367C24—21no—
368C24—22no—
369C24—21no—
370C24—22no—
371C24—22no—
372C24C2×#4112yes—
373C24C2511yescharacteristic
374C22×C4—21no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—24no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—24no—
399C22×C4—21no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—22no—
406C22×C4—21no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—24no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—24no—
420C22×C4—24no—
421C22×C4—24no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—24no—
436C22×C4—24no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—21no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—21no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—22no—
450C22×C4—21no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—22no—
456C22×C4—44no—
457C22×C4—44no—
458C22×C4—44no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C22×D412yes—
468C22×C4C2×#4112yes—
469C22×C4C2×#4118yes—
470C22×C4C2×#4112yes—
471C22×C4C2×#4112yes—
472C22×C4C2×#4112yes—
473C22×C4C2×#4112yes—
474C22×C4C2×#4114yes—
475C22×C4H(5,𝔽2)12yes—
476C22×C4H(5,𝔽2)12yes—
477C22×C4H(5,𝔽2)12yes—
478C22×C42−1+414yes—
479C22×C42−1+414yes—
480C22×C42−1+412yes—
481C4⋊C4—28no—
482C4⋊C4—28no—
483C4⋊C4—28no—
484C4⋊C4—216no—
485C4⋊C4—48no—
486C4⋊C4—44no—
487C4⋊C4—416no—
488C4⋊C4—416no—
489C4⋊C4—48no—
490C4⋊C4—48no—
491C4⋊C4—44no—
492C4⋊C4—48no—
493C4⋊C4—48no—
494C4⋊C4—44no—
495C4⋊C4—48no—
496C4⋊C4—48no—
497C4⋊C4—88no—
498C4⋊C4—88no—
499C22⋊C4—28no—
500C22⋊C4—28no—
501C22⋊C4—28no—
502C22⋊C4—44no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—88no—
506C22⋊C4—88no—
507C42—28no—
508C42—28no—
509C42—28no—
510C42—28no—
511C42—28no—
512C42—28no—
513C42—44no—
514C42—44no—
515C42—44no—
516C42—44no—
517C42—88no—
518C23—22no—
519C23—24no—
520C23—22no—
521C23—22no—
522C23—24no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—42no—
530C23—42no—
531C23—42no—
532C23—41no—
533C23—41no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—42no—
540C23—42no—
541C23C23×D411yescharacteristic
542C23C22×#4111yescharacteristic
543C23C22×#4111yescharacteristic
544C23C22×#4112yes—
545C23C2×H(5,𝔽2)11yescharacteristic
546C23C2×H(5,𝔽2)11yescharacteristic
547C23C2×H(5,𝔽2)11yescharacteristic
548C23C2×2−1+412yes—
549C23C2×2−1+412yes—
550C23C2×2−1+411yescharacteristic
551C23#58512yes—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
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—22no—
576C2×C4—28no—
577C2×C4—28no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—28no—
582C2×C4—24no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—41no—
607C2×C4—44no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—44no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—44no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—42no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—42no—
639C2×C4—44no—
640C2×C4—42no—
641C2×C4—42no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—88no—
647C2×C4#52514yes—
648C2×C4#53214yes—
649C2×C4#53514yes—
650C2×C4#53614yes—
651C2×C4#53614yes—
652C2×C4#53714yes—
653C2×C4#53918yes—
654C2×C4#54118yes—
655C22—42no—
656C22—44no—
657C22—42no—
658C22—44no—
659C22—42no—
660C22—81no—
661C22—81no—
662C22—81no—
663C22—82no—
664C22—82no—
665C22—81no—
666C22C2×#52511yescharacteristic
667C22C2×#53211yescharacteristic
668C22C2×#53511yescharacteristic
669C22C2×#53611yescharacteristic
670C22C2×#53611yescharacteristic
671C22C2×#53711yescharacteristic
672C22#344911yescharacteristic
673C22C2×#53912yes—
674C22C2×#54112yes—
675C22#348212yes—
676C22#348212yes—
677C22#350811yescharacteristic
678C22#350811yescharacteristic
679C22#351111yescharacteristic
680C22#351111yescharacteristic
681C22#351412yes—
682C22#351512yes—
683C22#351512yes—
684C22#351712yes—
685C22#354412yes—
686C22#354412yes—
687C22#356112yes—
688C22#356312yes—
689C4—24no—
690C4—24no—
691C4—24no—
692C4—42no—
693C4—42no—
694C4—48no—
695C4—42no—
696C4—42no—
697C4—42no—
698C4—42no—
699C4—81no—
700C4—81no—
701C4—81no—
702C4—84no—
703C4—84no—
704C4—81no—
705C4—84no—
706C4#266418yes—
707C2—82no—
708C2C2×#266411yescharacteristic
709C2#3497911yescharacteristic
710C2#3513811yescharacteristic
711C2#3519111yescharacteristic
712C2#3529211yescharacteristic
713C2#3755511yescharacteristic
714C2#3763211yescharacteristic
715C2#4007212yes—
716C2#4007512yes—
717C2#4009612yes—
718C2#4009912yes—
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
3. #16965C2semidirect
4. #16683C2semidirect
7. #15221C2semidirect
9. #15152C2semidirect
10. #15151C2semidirect
13. #15024C2semidirect
15. #14328C2semidirect
17. #14045C2semidirect
19. C4×#382C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28592
C2×C4×#1262
#169652
#166832
#166822
#166732
#152212
#152202
#151522
#151512
#151502
#151352
#150242
#150202
#143282
#143242
#140452
#139732
C4×#3822