Finite Group Databasebeta

← #647610

Subgroups — #647610

Order 512.

2721 conjugacy classes of subgroups.

The subgroup lattice has 724 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2853C211yesmaximal, characteristic
2C22×#389C211yesmaximal, characteristic, complemented
3#17043C212yesmaximal, complemented
4#17039C212yesmaximal, complemented
5#16983C212yesmaximal, complemented
6#16982C212yesmaximal, complemented
7#15092C211yesmaximal, characteristic, complemented
8#15091C211yesmaximal, characteristic
9#15012C212yesmaximal, complemented
10#14972C212yesmaximal, complemented
11#14919C212yesmaximal, complemented
12#14912C212yesmaximal
13#14843C211yesmaximal, characteristic, complemented
14#14777C211yesmaximal, characteristic
15#14326C212yesmaximal, complemented
16#14322C212yesmaximal, complemented
17#14043C212yesmaximal, complemented
18#14019C212yesmaximal
19C4×#381C211yesmaximal, characteristic, complemented
20C22×#123C2211yescharacteristic
21C23×C4⋊C4C2211yescharacteristic
22C23×C4⋊C4C2212yes—
23C23×C4⋊C4C2211yescharacteristic
24C23×C42C2211yescharacteristic
25#2853—28no—
26C2×#400C2212yes—
27C2×#399C2212yes—
28C2×#389—28no—
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#389C2212yes—
33C2×#389C2214yes—
34C2×#388C2211yescharacteristic
35C2×#388C2211yescharacteristic
36C2×#388C2211yescharacteristic
37C2×#388C2211yescharacteristic
38C2×#388C2214yes—
39C2×#387C2211yescharacteristic
40C2×#387C2211yescharacteristic
41C2×#387C2211yescharacteristic
42C2×#387C2211yescharacteristic
43C2×#386C2211yescharacteristic
44C2×#386C2211yescharacteristic
45C2×#386C2212yes—
46C2×#386C2212yes—
47C2×#385C2211yescharacteristic
48C2×#385C2211yescharacteristic
49C2×#385C2212yes—
50C2×#385C2212yes—
51C2×#381C2211yescharacteristic
52C2×#381C2212yes—
53C2×C4×C22⋊C4C2212yes—
54C22×#95C2212yes—
55#1445C2214yes—
56#1445C2212yes—
57#1445C2212yes—
58#1444C2214yes—
59#1444C2212yes—
60#1444C2212yes—
61#1444C2212yes—
62#1443C2214yes—
63#1443C2214yes—
64#1442C2214yes—
65#1442C2214yes—
66#1442C2212yes—
67#1442C2214yes—
68#1442C2212yes—
69#1442C2212yes—
70#1441C2214yes—
71#1441C2212yes—
72#1441C2212yes—
73#1441C2214yes—
74#1441C2212yes—
75#1440C2214yes—
76#1439C2214yes—
77#1439C2212yes—
78#1439C2212yes—
79#1439C2214yes—
80#1439C2212yes—
81#1436C2214yes—
82#1436C2214yes—
83#1434C2212yes—
84#1434C2212yes—
85C22⋊#95C2212yes—
86C22⋊#95C2212yes—
87C22⋊#95C2212yes—
88C22⋊#95C2212yes—
89C22⋊#95C2211yescharacteristic
90C22⋊#95C2211yescharacteristic
91C4×#95C2212yes—
92C4×#95C2212yes—
93C23×D4C2311yescharacteristic
94C24×C4C2311yescharacteristic
95C24×C4C2311yescharacteristic
96C24×C4C2312yes—
97C24×C4C2311yescharacteristic
98C24×C4C2312yes—
99C2×#123—24no—
100C2×#123—44no—
101C22×C4⋊C4—24no—
102C22×C4⋊C4—28no—
103C22×C4⋊C4—28no—
104C22×C4⋊C4—24no—
105C22×C4⋊C4—48no—
106C22×C4⋊C4—44no—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2314yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2312yes—
116C22×C4⋊C4C2314yes—
117C22×C4⋊C4C2311yescharacteristic
118C22×C4⋊C4C2311yescharacteristic
119C22×C4⋊C4C2311yescharacteristic
120C22×C4⋊C4C2314yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2312yes—
124C22×C22⋊C4C2311yescharacteristic
125C22×C22⋊C4C2311yescharacteristic
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2314yes—
129C22×C22⋊C4C2312yes—
130C22×C22⋊C4C2311yescharacteristic
131C22×C22⋊C4C2311yescharacteristic
132C22×C22⋊C4C2311yescharacteristic
133C22×C22⋊C4C2314yes—
134C22×C42—44no—
135C22×C42D418yes—
136C22×C42C2314yes—
137C22×C42C2311yescharacteristic
138C22×C42C2311yescharacteristic
139C22×C42C2311yescharacteristic
140C22×C42C2311yescharacteristic
141#400—28no—
142#399—28no—
143#389—28no—
144#389—28no—
145#389—28no—
146#389—28no—
147#389—48no—
148#388—24no—
149#388—24no—
150#388—24no—
151#388—24no—
152#388—48no—
153#387—24no—
154#387—24no—
155#387—24no—
156#387—24no—
157#386—24no—
158#386—24no—
159#386—28no—
160#386—44no—
161#385—24no—
162#385—24no—
163#385—28no—
164#385—44no—
165#381—24no—
166#381—28no—
167C4×C22⋊C4—44no—
168C2×#95—28no—
169C2×#95C2312yes—
170C2×#95C2312yes—
171C2×#95C2314yes—
172C2×#95C2314yes—
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2314yes—
177C2×#95C2311yescharacteristic
178C2×#95C2311yescharacteristic
179C2×#95C2314yes—
180C2×#95C2314yes—
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—
202C25C2411yescharacteristic
203C25C2411yescharacteristic
204C22×D4—22no—
205C22×D4—24no—
206C22×D4—24no—
207C22×D4—22no—
208C22×D4—22no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—24no—
212C23×C4—24no—
213C23×C4—24no—
214C23×C4—24no—
215C23×C4—24no—
216C23×C4—22no—
217C23×C4—22no—
218C23×C4—28no—
219C23×C4—24no—
220C23×C4—24no—
221C23×C4—24no—
222C23×C4—24no—
223C23×C4—24no—
224C23×C4—24no—
225C23×C4—44no—
226C23×C4—42no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4C2×D414yes—
230C23×C4C2×D414yes—
231C23×C4C2×D414yes—
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235C23×C4C2411yescharacteristic
236C23×C4C2411yescharacteristic
237C23×C4C2411yescharacteristic
238C23×C4C2414yes—
239C23×C4C2414yes—
240C23×C4C2411yescharacteristic
241C23×C4C2411yescharacteristic
242C23×C4C2411yescharacteristic
243C23×C4C2411yescharacteristic
244C23×C4C2414yes—
245C23×C4C2414yes—
246C23×C4C2411yescharacteristic
247C23×C4C2411yescharacteristic
248C23×C4C2411yescharacteristic
249#123—416no—
250C2×C4⋊C4—232no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—28no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—28no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—28no—
266C2×C4⋊C4—28no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—28no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—44no—
273C2×C4⋊C4—44no—
274C2×C4⋊C4—44no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—42no—
278C2×C4⋊C4—42no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—416no—
281C2×C4⋊C4—432no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4#4118yes—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—28no—
305C2×C22⋊C4—28no—
306C2×C22⋊C4—24no—
307C2×C22⋊C4—24no—
308C2×C22⋊C4—24no—
309C2×C22⋊C4—42no—
310C2×C22⋊C4—42no—
311C2×C22⋊C4—42no—
312C2×C22⋊C4—44no—
313C2×C22⋊C4—44no—
314C2×C22⋊C4—44no—
315C2×C22⋊C4—44no—
316C2×C22⋊C4#4118yes—
317C2×C42—28no—
318C2×C42—416no—
319C2×C42—42no—
320C2×C42—42no—
321C2×C42—42no—
322C2×C42—42no—
323C2×C42—44no—
324C2×C42—44no—
325C2×C42C2×D414yes—
326C2×C42C2×D414yes—
327C2×C42C2×D414yes—
328C2×C42C2×D414yes—
329#95—28no—
330#95—28no—
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—22no—
342#95—22no—
343#95—28no—
344#95—28no—
345#95—28no—
346#95—28no—
347#95—24no—
348#95—24no—
349#95—28no—
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—22no—
363C24—22no—
364C24—22no—
365C24—22no—
366C24—21no—
367C24—22no—
368C24—22no—
369C24—22no—
370C24—22no—
371C24—22no—
372C24—21no—
373C24—21no—
374C24—22no—
375C24C22×D412yes—
376C24C2×#4114yes—
377C24H(5,𝔽2)14yes—
378C242−1+412yes—
379C242−1+412yes—
380C24C2511yescharacteristic
381C2×D4—44no—
382C2×D4—44no—
383C2×D4—44no—
384C2×D4—48no—
385C2×D4—48no—
386C22×C4—232no—
387C22×C4—28no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—28no—
395C22×C4—28no—
396C22×C4—216no—
397C22×C4—28no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—24no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—24no—
423C22×C4—22no—
424C22×C4—28no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—24no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—22no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—24no—
460C22×C4—22no—
461C22×C4—22no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—24no—
469C22×C4—22no—
470C22×C4—44no—
471C22×C4—416no—
472C22×C4—416no—
473C22×C4—416no—
474C22×C4—48no—
475C22×C4—48no—
476C22×C4—48no—
477C22×C4—48no—
478C22×C4—48no—
479C22×C4—48no—
480C22×C4—48no—
481C22×C4—88no—
482C22×C4—88no—
483C22×C4C22×D412yes—
484C22×C4C22×D412yes—
485C22×C4C22×D412yes—
486C22×C4C22×D412yes—
487C22×C4C22×D412yes—
488C22×C4C22×D412yes—
489C22×C4C2×#4118yes—
490C22×C4C2×#4118yes—
491C22×C4C2×#4114yes—
492C22×C4H(5,𝔽2)14yes—
493C22×C42−1+414yes—
494C4⋊C4—216no—
495C4⋊C4—216no—
496C4⋊C4—416no—
497C4⋊C4—44no—
498C4⋊C4—48no—
499C4⋊C4—48no—
500C4⋊C4—416no—
501C4⋊C4—44no—
502C4⋊C4—44no—
503C4⋊C4—44no—
504C4⋊C4—44no—
505C4⋊C4—416no—
506C4⋊C4—44no—
507C4⋊C4—88no—
508C22⋊C4—216no—
509C22⋊C4—416no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—44no—
513C22⋊C4—44no—
514C22⋊C4—44no—
515C22⋊C4—44no—
516C22⋊C4—48no—
517C22⋊C4—44no—
518C22⋊C4—44no—
519C22⋊C4—44no—
520C22⋊C4—88no—
521C42—44no—
522C42—44no—
523C42—44no—
524C42—44no—
525C42—88no—
526C23—22no—
527C23—22no—
528C23—22no—
529C23—22no—
530C23—28no—
531C23—28no—
532C23—28no—
533C23—24no—
534C23—24no—
535C23—28no—
536C23—24no—
537C23—24no—
538C23—41no—
539C23—41no—
540C23—41no—
541C23—41no—
542C23—41no—
543C23—41no—
544C23—41no—
545C23—42no—
546C23—42no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—42no—
551C23—42no—
552C23—44no—
553C23—42no—
554C23—42no—
555C23#53214yes—
556C23#53414yes—
557C23#53614yes—
558C23#53918yes—
559C23#54218yes—
560C23C23×D411yescharacteristic
561C23C22×#4112yes—
562C23C22×#4112yes—
563C23C2×H(5,𝔽2)12yes—
564C23C2×H(5,𝔽2)12yes—
565C23C2×2−1+411yescharacteristic
566C23C2×2−1+411yescharacteristic
567C23C2×2−1+412yes—
568C23#58512yes—
569D4—88no—
570C2×C4—232no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—28no—
578C2×C4—28no—
579C2×C4—28no—
580C2×C4—28no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—28no—
593C2×C4—28no—
594C2×C4—28no—
595C2×C4—416no—
596C2×C4—432no—
597C2×C4—416no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—44no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—44no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—41no—
633C2×C4—41no—
634C2×C4—44no—
635C2×C4—42no—
636C2×C4—44no—
637C2×C4—42no—
638C2×C4—44no—
639C2×C4—44no—
640C2×C4—44no—
641C2×C4—44no—
642C2×C4—44no—
643C2×C4—44no—
644C2×C4—42no—
645C2×C4—42no—
646C2×C4—44no—
647C2×C4—44no—
648C2×C4—44no—
649C2×C4—44no—
650C2×C4—44no—
651C2×C4—44no—
652C2×C4—44no—
653C2×C4—44no—
654C2×C4—42no—
655C2×C4—42no—
656C2×C4—816no—
657C2×C4#52514yes—
658C2×C4#53514yes—
659C2×C4#53614yes—
660C22—24no—
661C22—24no—
662C22—24no—
663C22—24no—
664C22—216no—
665C22—216no—
666C22—24no—
667C22—24no—
668C22—81no—
669C22—81no—
670C22—81no—
671C22—81no—
672C22—81no—
673C22—84no—
674C22—84no—
675C22—81no—
676C22—81no—
677C22#266318yes—
678C22C2×#52511yescharacteristic
679C22C2×#53211yescharacteristic
680C22C2×#53411yescharacteristic
681C22C2×#53511yescharacteristic
682C22C2×#53611yescharacteristic
683C22C2×#53611yescharacteristic
684C22#344911yescharacteristic
685C22C2×#53912yes—
686C22C2×#54212yes—
687C22#348114yes—
688C22#349214yes—
689C22#351514yes—
690C22#351612yes—
691C22#351612yes—
692C22#353414yes—
693C22#355212yes—
694C22#355212yes—
695C4—24no—
696C4—24no—
697C4—24no—
698C4—42no—
699C4—42no—
700C4—42no—
701C4—48no—
702C4—48no—
703C4—81no—
704C4—81no—
705C4—81no—
706C4—84no—
707C4—81no—
708C4—81no—
709C4—84no—
710C4—81no—
711C4—81no—
712C2—28no—
713C2—161no—
714C2C2×#266311yescharacteristic
715C2#3496511yescharacteristic
716C2#3505711yescharacteristic
717C2#3513911yescharacteristic
718C2#3524511yescharacteristic
719C2#3744611yescharacteristic
720C2#3761711yescharacteristic
721C2#3857314yes—
722C2#3857514yes—
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
2. C22×#389C2semidirect
3. #17043C2semidirect
4. #17039C2semidirect
5. #16983C2semidirect
6. #16982C2semidirect
7. #15092C2semidirect
9. #15012C2semidirect
10. #14972C2semidirect
11. #14919C2semidirect
13. #14843C2semidirect
15. #14326C2semidirect
16. #14322C2semidirect
17. #14043C2semidirect
19. C4×#381C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28532
C22×#3892
#170432
#170392
#169832
#169822
#150922
#150912
#150122
#149722
#149192
#149122
#148432
#147772
#143262
#143222
#140432
#140192
C4×#3812