Finite Group Databasebeta

← #4418510

Subgroups — #4418510

Order 512.

2849 conjugacy classes of subgroups.

The subgroup lattice has 707 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2562C211yesmaximal, characteristic, complemented
2C2×#2664C211yesmaximal, characteristic, complemented
3C2×#2636C211yesmaximal, characteristic
4C2×#2349C211yesmaximal, characteristic, complemented
5#17074C212yesmaximal, complemented
6#16899C212yesmaximal, complemented
7#16401C212yesmaximal, complemented
8#16400C212yesmaximal, complemented
9#15436C214yesmaximal, complemented
10#15302C214yesmaximal, complemented
11#15129C212yesmaximal, complemented
12#14836C212yesmaximal, complemented
13#14806C212yesmaximal
14#14766C212yesmaximal, complemented
15#14664C211yesmaximal, characteristic, complemented
16#14291C211yesmaximal, characteristic, complemented
17#13954C211yesmaximal, characteristic, complemented
18C22×#126C2211yescharacteristic
19C22×#125C2211yescharacteristic
20C22×C4×D4C2211yescharacteristic
21C22×C4×D4C2212yes—
22#2664—28no—
23#2636—28no—
24#2562—28no—
25C2×#400C2212yes—
26C2×#400C2211yescharacteristic
27C2×#400C2211yescharacteristic
28C2×#399C2211yescharacteristic
29C2×#399C2211yescharacteristic
30C2×#398C2212yes—
31C2×#398C2211yescharacteristic
32C2×#397C2211yescharacteristic
33C2×#396C2212yes—
34C2×#396C2212yes—
35C2×#394C2211yescharacteristic
36C2×#394C2212yes—
37C2×#394C2212yes—
38C2×#394C2214yes—
39C2×#392C2211yescharacteristic
40#2349—28no—
41C2×#389C2212yes—
42C2×#389C2212yes—
43C2×#389C2214yes—
44C2×#388C2212yes—
45C2×#388C2214yes—
46C2×#386C2212yes—
47C2×#386C2214yes—
48C2×#385C2212yes—
49C2×#385C2214yes—
50C2×#382C2211yescharacteristic
51C2×#382C2211yescharacteristic
52C2×#380C2212yes—
53C2×#380C2212yes—
54C2×#380C2212yes—
55C2×#380C2214yes—
56C2×#380C2214yes—
57C2×C4×C4⋊C4C2211yescharacteristic
58C2×C4×C4⋊C4C2211yescharacteristic
59C22×#95C2212yes—
60#1445C2212yes—
61#1445C2212yes—
62#1445C2214yes—
63#1444C2212yes—
64#1444C2212yes—
65#1442C2214yes—
66#1441C2214yes—
67#1441C2212yes—
68#1441C2214yes—
69#1441C2212yes—
70#1441C2214yes—
71#1441C2214yes—
72#1441C2212yes—
73#1441C2214yes—
74#1441C2212yes—
75#1440C2214yes—
76#1440C2214yes—
77#1439C2214yes—
78#1439C2214yes—
79#1439C2212yes—
80#1439C2214yes—
81#1439C2212yes—
82#1437C2212yes—
83#1436C2212yes—
84C4⋊#95C2212yes—
85C4⋊#95C2212yes—
86C22⋊#95C2214yes—
87C23×D4C2311yescharacteristic
88C23×D4C2312yes—
89C24×C4C2314yes—
90C24×C4C2312yes—
91C2×#126—24no—
92C2×#126—24no—
93C2×#126—24no—
94C2×#125—24no—
95C2×#125—24no—
96C2×#125—24no—
97C2×C4×D4—24no—
98C2×C4×D4—28no—
99C2×C4×D4—24no—
100C2×C4×D4—48no—
101C2×C4×D4D418yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2314yes—
118C22×C4⋊C4C2312yes—
119C22×C4⋊C4C2314yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2314yes—
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2312yes—
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2314yes—
127C22×C22⋊C4C2312yes—
128C22×C22⋊C4C2312yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2312yes—
131C22×C22⋊C4C2312yes—
132C22×C22⋊C4C2314yes—
133C22×C22⋊C4C2314yes—
134C22×C42C2312yes—
135C22×C42C2312yes—
136C22×C42C2311yescharacteristic
137#400—28no—
138#400—24no—
139#400—24no—
140#399—24no—
141#399—24no—
142#398—28no—
143#398—24no—
144#397—24no—
145#396—28no—
146#396—28no—
147#394—28no—
148#394—24no—
149#394—28no—
150#394—48no—
151#392—24no—
152#389—28no—
153#389—28no—
154#389—48no—
155#388—216no—
156#388—28no—
157#386—28no—
158#386—48no—
159#385—28no—
160#385—216no—
161#382—24no—
162#382—24no—
163#380—28no—
164#380—28no—
165#380—28no—
166#380—48no—
167#380—48no—
168C4×C4⋊C4—24no—
169C4×C4⋊C4D418yescomplemented
170C2×#95—28no—
171C2×#95C2311yescharacteristic
172C2×#95C2311yescharacteristic
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2312yes—
176C2×#95C2312yes—
177C2×#95C2312yes—
178C2×#95C2312yes—
179C2×#95C2312yes—
180C2×#95C2314yes—
181C2×#95C2314yes—
182C2×#95C2314yes—
183C2×#95C2312yes—
184C2×#95C2314yes—
185C2×#95C2312yes—
186C2×#95C2312yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2311yescharacteristic
196C2×#95C2311yescharacteristic
197C25C2411yescharacteristic
198C25C2412yes—
199C22×D4—24no—
200C22×D4—24no—
201C22×D4—24no—
202C22×D4—24no—
203C22×D4—24no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—44no—
207C22×D4—44no—
208C22×D4C2×D414yes—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—28no—
212C23×C4—28no—
213C23×C4—28no—
214C23×C4—28no—
215C23×C4—44no—
216C23×C4—44no—
217C23×C4—44no—
218C23×C4—44no—
219C23×C4—44no—
220C23×C4C2×D418yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2414yes—
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2412yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2412yes—
236#126—416no—
237#125—416no—
238C4×D4—232no—
239C4×D4—432no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—28no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—28no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—44no—
281C2×C4⋊C4—44no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—44no—
287C2×C4⋊C4C2×D414yes—
288C2×C4⋊C4C2×D418yes—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—28no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—28no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—28no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—28no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4—22no—
309C2×C22⋊C4—24no—
310C2×C22⋊C4—24no—
311C2×C22⋊C4—28no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—24no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—24no—
316C2×C22⋊C4—44no—
317C2×C22⋊C4—44no—
318C2×C22⋊C4—44no—
319C2×C22⋊C4—44no—
320C2×C22⋊C4—44no—
321C2×C22⋊C4—44no—
322C2×C22⋊C4—44no—
323C2×C22⋊C4—44no—
324C2×C22⋊C4—44no—
325C2×C22⋊C4—44no—
326C2×C22⋊C4C2×D418yes—
327C2×C42—24no—
328C2×C42—44no—
329C2×C42—44no—
330C2×C42C2×D418yes—
331C2×C42C2×D414yes—
332C2×C42C2×D414yes—
333C2×C42#4114yes—
334#95—24no—
335#95—28no—
336#95—24no—
337#95—28no—
338#95—28no—
339#95—24no—
340#95—24no—
341#95—22no—
342#95—22no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95—24no—
349#95—28no—
350#95—44no—
351#95—44no—
352#95—44no—
353#95—44no—
354#95—44no—
355#95—44no—
356#95—44no—
357#95—44no—
358#95C2×D414yes—
359#95C2×D414yes—
360C24—22no—
361C24—22no—
362C24—24no—
363C24—22no—
364C24—24no—
365C24—22no—
366C24—24no—
367C24—24no—
368C24—21no—
369C24—21no—
370C24—21no—
371C24—22no—
372C24—22no—
373C24—22no—
374C24—22no—
375C24—24no—
376C24C22×D414yes—
377C24H(5,𝔽2)14yes—
378C242−1+414yes—
379C24C2511yescharacteristic
380C2×D4—28no—
381C2×D4—28no—
382C2×D4—216no—
383C2×D4—216no—
384C2×D4—28no—
385C2×D4—416no—
386C2×D4—48no—
387C2×D4—84no—
388C2×D4—84no—
389C2×D4—88no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—22no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—216no—
406C22×C4—21no—
407C22×C4—22no—
408C22×C4—24no—
409C22×C4—24no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—22no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—24no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—24no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—24no—
448C22×C4—22no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—24no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—416no—
459C22×C4—416no—
460C22×C4—416no—
461C22×C4—416no—
462C22×C4—88no—
463C22×C4—88no—
464C22×C4—88no—
465C22×C4—88no—
466C22×C4—88no—
467C22×C4C22×D414yes—
468C22×C4C22×D414yes—
469C22×C4C22×D412yes—
470C22×C4C22×D412yes—
471C22×C4C22×D412yes—
472C22×C4C22×D412yes—
473C22×C4C22×D412yes—
474C22×C4C22×D412yes—
475C22×C4C22×D414yes—
476C22×C4C2×#4112yes—
477C22×C4C2×#4112yes—
478C22×C4C2×#4112yes—
479C22×C4H(5,𝔽2)14yes—
480C22×C42−1+414yes—
481C4⋊C4—28no—
482C4⋊C4—216no—
483C4⋊C4—48no—
484C4⋊C4—48no—
485C4⋊C4—48no—
486C4⋊C4—48no—
487C4⋊C4—44no—
488C4⋊C4—44no—
489C4⋊C4—44no—
490C4⋊C4—44no—
491C4⋊C4—44no—
492C4⋊C4—48no—
493C4⋊C4—84no—
494C4⋊C4—84no—
495C4⋊C4—88no—
496C4⋊C4—84no—
497C4⋊C4—88no—
498C4⋊C4—88no—
499C22⋊C4—216no—
500C22⋊C4—416no—
501C22⋊C4—416no—
502C22⋊C4—48no—
503C22⋊C4—44no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—48no—
507C22⋊C4—84no—
508C22⋊C4—84no—
509C22⋊C4—84no—
510C22⋊C4—88no—
511C22⋊C4—84no—
512C22⋊C4—88no—
513C42—48no—
514C42—84no—
515C42#121116yes—
516C23—24no—
517C23—24no—
518C23—24no—
519C23—24no—
520C23—24no—
521C23—24no—
522C23—24no—
523C23—24no—
524C23—24no—
525C23—24no—
526C23—28no—
527C23—28no—
528C23—28no—
529C23—28no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—44no—
534C23—44no—
535C23—44no—
536C23—44no—
537C23—44no—
538C23—41no—
539C23—42no—
540C23—42no—
541C23—41no—
542C23—42no—
543C23—41no—
544C23—42no—
545C23—41no—
546C23—41no—
547C23—42no—
548C23—42no—
549C23—42no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23—44no—
554C23#53718yes—
555C23C23×D411yescharacteristic
556C23C23×D411yescharacteristic
557C23C22×#4111yescharacteristic
558C23C2×H(5,𝔽2)12yes—
559C23C2×H(5,𝔽2)12yes—
560C23C2×2−1+412yes—
561C23C2×2−1+412yes—
562C23#58514yes—
563D4—232no—
564D4—816no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—22no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—22no—
571C2×C4—24no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—24no—
575C2×C4—24no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—44no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—41no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—44no—
615C2×C4—44no—
616C2×C4—41no—
617C2×C4—42no—
618C2×C4—42no—
619C2×C4—42no—
620C2×C4—41no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—42no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4—44no—
634C2×C4—44no—
635C2×C4—44no—
636C2×C4—44no—
637C2×C4—44no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—816no—
641C2×C4—832no—
642C2×C4C2×#12114yes—
643C2×C4C2×#12114yes—
644C2×C4C2×#12114yes—
645C2×C4#53718yes—
646C22—28no—
647C22—28no—
648C22—28no—
649C22—44no—
650C22—48no—
651C22—48no—
652C22—44no—
653C22—44no—
654C22—48no—
655C22—48no—
656C22—44no—
657C22—81no—
658C22—82no—
659C22—82no—
660C22—81no—
661C22—82no—
662C22—82no—
663C22—84no—
664C22—81no—
665C22C22×#12111yescharacteristic
666C22C2×#53712yes—
667C22C2×#53712yes—
668C22#348411yescharacteristic
669C22#348512yes—
670C22#348611yescharacteristic
671C22#349212yes—
672C22#349811yescharacteristic
673C22#349911yescharacteristic
674C22#350012yes—
675C22#351012yes—
676C22#351112yes—
677C22#353414yes—
678C22#353514yes—
679C22#354114yes—
680C22#356314yes—
681C4—24no—
682C4—28no—
683C4—24no—
684C4—24no—
685C4—44no—
686C4—44no—
687C4—81no—
688C4—84no—
689C4—81no—
690C4—82no—
691C4—82no—
692C4—84no—
693C4—82no—
694C4—81no—
695C4—82no—
696C2—48no—
697C2—161no—
698C2#3522311yescharacteristic
699C2#3527311yescharacteristic
700C2#3561711yescharacteristic
701C2#4049212yes—
702C2#4136812yes—
703C2#4137312yes—
704C2#4279012yes—
705C2#5110614yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#392C22
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×#2562C2semidirect
2. C2×#2664C2semidirect
4. C2×#2349C2semidirect
5. #17074C2semidirect
6. #16899C2semidirect
7. #16401C2semidirect
8. #16400C2semidirect
9. #15436C2semidirect
10. #15302C2semidirect
11. #15129C2semidirect
12. #14836C2semidirect
14. #14766C2semidirect
15. #14664C2semidirect
16. #14291C2semidirect
17. #13954C2semidirect
169. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25622
C2×#26642
C2×#26362
C2×#23492
#170742
#168992
#164012
#164002
#154362
#153022
#151292
#148362
#148062
#147662
#146642
#142912
#139542