Finite Group Databasebeta

← #644952

Subgroups — #644952

Order 512.

2453 conjugacy classes of subgroups.

The subgroup lattice has 703 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2852C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic, complemented
3#17067C211yesmaximal, characteristic, complemented
4#17066C211yesmaximal, characteristic, complemented
5#17063C211yesmaximal, characteristic, complemented
6#17062C211yesmaximal, characteristic, complemented
7#16401C212yesmaximal
8#16399C212yesmaximal, complemented
9#15105C212yesmaximal, complemented
10#15103C212yesmaximal, complemented
11#14892C212yesmaximal, complemented
12#14888C212yesmaximal
13#14841C212yesmaximal, complemented
14#14839C212yesmaximal
15#14356C212yesmaximal, complemented
16#14291C212yesmaximal, complemented
17#14249C212yesmaximal, complemented
18#14244C212yesmaximal, complemented
19C4×#380C211yesmaximal, characteristic, complemented
20C22×#126C2211yescharacteristic
21C22×#117C2212yes—
22C23×C42C2211yescharacteristic
23#2852—28no—
24C2×#400C2212yes—
25C2×#399C2212yes—
26C2×#398C2212yes—
27C2×#397C2212yes—
28C2×#397C2214yes—
29C2×#394C2214yes—
30C2×#388C2211yescharacteristic
31C2×#388C2211yescharacteristic
32C2×#388C2211yescharacteristic
33C2×#387C2211yescharacteristic
34C2×#387C2211yescharacteristic
35C2×#387C2211yescharacteristic
36C2×#386C2211yescharacteristic
37C2×#386C2211yescharacteristic
38C2×#386C2211yescharacteristic
39C2×#385C2211yescharacteristic
40C2×#385C2211yescharacteristic
41C2×#385C2211yescharacteristic
42#2285—28no—
43C2×#384C2211yescharacteristic
44C2×#384C2211yescharacteristic
45C2×#383C2211yescharacteristic
46C2×#382C2212yes—
47C2×#380C2212yes—
48C2×#380C2212yes—
49C2×#380C2212yes—
50C2×C4×C4⋊C4C2212yes—
51C2×C4×C4⋊C4C2212yes—
52C2×C4×C4⋊C4C2212yes—
53C2×C4×C22⋊C4C2212yes—
54#1445C2212yes—
55#1445C2212yes—
56#1445C2212yes—
57#1445C2212yes—
58#1444C2212yes—
59#1444C2212yes—
60#1444C2212yes—
61#1444C2212yes—
62#1442C2212yes—
63#1442C2212yes—
64#1442C2212yes—
65#1442C2212yes—
66#1441C2214yes—
67#1441C2214yes—
68#1441C2214yes—
69#1441C2212yes—
70#1441C2212yes—
71#1441C2212yes—
72#1441C2212yes—
73#1440C2214yes—
74#1440C2214yes—
75#1439C2212yes—
76#1439C2214yes—
77#1439C2212yes—
78#1439C2212yes—
79#1439C2212yes—
80#1437C2214yes—
81#1437C2214yes—
82#1436C2214yes—
83#1436C2214yes—
84#1436C2214yes—
85#1436C2214yes—
86C4⋊#95C2212yes—
87C4⋊#95C2214yes—
88#1434C2214yes—
89#1434C2211yescharacteristic
90#1434C2211yescharacteristic
91C22⋊#95C2212yes—
92C4×#95C2212yes—
93C23×D4C2311yescharacteristic
94C24×C4C2311yescharacteristic
95C24×C4C2311yescharacteristic
96C24×C4C2311yescharacteristic
97C2×#126—24no—
98C2×#126—28no—
99C2×#117—28no—
100C2×#117—48no—
101C22×C4⋊C4C2312yes—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2312yes—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2314yes—
110C22×C4⋊C4C2314yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2312yes—
115C22×C4⋊C4C2314yes—
116C22×C22⋊C4C2314yes—
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2311yescharacteristic
124C22×C22⋊C4C2314yes—
125C22×C22⋊C4C2314yes—
126C22×C22⋊C4C2311yescharacteristic
127C22×C22⋊C4C2311yescharacteristic
128C22×C22⋊C4C2314yes—
129C22×C42—28no—
130C22×C42D418yes—
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2314yes—
135C22×C42C2314yes—
136C22×C42C2311yescharacteristic
137#400—44no—
138#399—44no—
139#398—28no—
140#397—28no—
141#397—48no—
142#394—48no—
143#388—24no—
144#388—24no—
145#388—24no—
146#387—24no—
147#387—24no—
148#387—24no—
149#386—24no—
150#386—24no—
151#386—24no—
152#385—24no—
153#385—24no—
154#385—24no—
155#384—24no—
156#384—24no—
157#383—24no—
158#382—28no—
159#380—28no—
160#380—28no—
161#380—28no—
162C4×C4⋊C4—28no—
163C4×C4⋊C4—28no—
164C4×C4⋊C4—28no—
165C4×C22⋊C4—44no—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2312yes—
169C2×#95C2312yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2312yes—
173C2×#95C2312yes—
174C2×#95C2311yescharacteristic
175C2×#95C2314yes—
176C2×#95C2311yescharacteristic
177C2×#95C2311yescharacteristic
178C2×#95C2311yescharacteristic
179C2×#95C2314yes—
180C2×#95C2311yescharacteristic
181C2×#95C2311yescharacteristic
182C2×#95C2311yescharacteristic
183C2×#95C2311yescharacteristic
184C2×#95C2311yescharacteristic
185C2×#95C2311yescharacteristic
186C2×#95C2311yescharacteristic
187C2×#95C2311yescharacteristic
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C2×#95C2314yes—
197C2×#95C2314yes—
198C2×#95C2312yes—
199C25C2411yescharacteristic
200C25C2411yescharacteristic
201C22×D4—22no—
202C22×D4—24no—
203C22×D4—42no—
204C22×D4—42no—
205C23×C4—24no—
206C23×C4—24no—
207C23×C4—24no—
208C23×C4—24no—
209C23×C4—24no—
210C23×C4—24no—
211C23×C4—22no—
212C23×C4—22no—
213C23×C4—42no—
214C23×C4—42no—
215C23×C4C2×D414yes—
216C23×C4C2×D414yes—
217C23×C4C2×D414yes—
218C23×C4C2411yescharacteristic
219C23×C4C2414yes—
220C23×C4C2414yes—
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2414yes—
227C23×C4C2411yescharacteristic
228C23×C4C2414yes—
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C23×C4C2411yescharacteristic
234C23×C4C2411yescharacteristic
235#126—416no—
236#117—432no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
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—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—28no—
253C2×C4⋊C4—28no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—44no—
265C2×C4⋊C4—44no—
266C2×C4⋊C4—44no—
267C2×C4⋊C4—44no—
268C2×C4⋊C4—44no—
269C2×C4⋊C4—44no—
270C2×C4⋊C4—44no—
271C2×C4⋊C4—44no—
272C2×C4⋊C4—44no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—28no—
278C2×C22⋊C4—28no—
279C2×C22⋊C4—28no—
280C2×C22⋊C4—24no—
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—24no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—28no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—28no—
296C2×C22⋊C4—42no—
297C2×C22⋊C4—42no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4—44no—
300C2×C22⋊C4—44no—
301C2×C22⋊C4—44no—
302C2×C22⋊C4—44no—
303C2×C22⋊C4—44no—
304C2×C42—24no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—232no—
308C2×C42—28no—
309C2×C42—28no—
310C2×C42—44no—
311C2×C42—44no—
312C2×C42—44no—
313C2×C42—44no—
314C2×C42C2×D414yes—
315C2×C42C2×D414yes—
316C2×C42C2×D414yes—
317C2×C42C2×D414yes—
318C2×C42#4118yes—
319#95—28no—
320#95—28no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—24no—
334#95—24no—
335#95—24no—
336#95—28no—
337#95—28no—
338#95—24no—
339#95—24no—
340#95—28no—
341#95—44no—
342#95—44no—
343#95—44no—
344#95—44no—
345#95—44no—
346#95—44no—
347#95—44no—
348#95—44no—
349#95—44no—
350#95—44no—
351#95#4118yes—
352C24—22no—
353C24—22no—
354C24—21no—
355C24—22no—
356C24—21no—
357C24—22no—
358C24—22no—
359C24—21no—
360C24—22no—
361C24—22no—
362C24—21no—
363C24—22no—
364C24—22no—
365C24—21no—
366C24—22no—
367C24—22no—
368C24C22×D412yes—
369C24H(5,𝔽2)14yes—
370C24C2511yescharacteristic
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—88no—
375C22×C4—232no—
376C22×C4—28no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—28no—
391C22×C4—28no—
392C22×C4—24no—
393C22×C4—24no—
394C22×C4—24no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—28no—
399C22×C4—28no—
400C22×C4—28no—
401C22×C4—28no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—24no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—24no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—21no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—21no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—24no—
447C22×C4—24no—
448C22×C4—24no—
449C22×C4—24no—
450C22×C4—24no—
451C22×C4—24no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—44no—
455C22×C4—44no—
456C22×C4—416no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4C22×D412yes—
460C22×C4C22×D412yes—
461C22×C4C22×D412yes—
462C22×C4C22×D412yes—
463C22×C4C22×D412yes—
464C22×C4C22×D412yes—
465C22×C4C2×#4118yes—
466C22×C4C2×#4114yes—
467C22×C4C2×#4114yes—
468C22×C4C2×#4114yes—
469C22×C4C2×#4114yes—
470C22×C4H(5,𝔽2)12yes—
471C22×C42−1+414yes—
472C22×C42−1+412yes—
473C4⋊C4—416no—
474C4⋊C4—48no—
475C4⋊C4—48no—
476C4⋊C4—48no—
477C4⋊C4—48no—
478C4⋊C4—44no—
479C4⋊C4—44no—
480C4⋊C4—44no—
481C4⋊C4—48no—
482C4⋊C4—44no—
483C4⋊C4—44no—
484C4⋊C4—48no—
485C4⋊C4—44no—
486C4⋊C4—88no—
487C4⋊C4—88no—
488C22⋊C4—416no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—44no—
492C22⋊C4—416no—
493C22⋊C4—44no—
494C22⋊C4—44no—
495C22⋊C4—44no—
496C22⋊C4—44no—
497C22⋊C4—44no—
498C22⋊C4—44no—
499C22⋊C4—88no—
500C22⋊C4—88no—
501C42—28no—
502C42—28no—
503C42—28no—
504C42—416no—
505C42—88no—
506C42#123116yes—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—28no—
514C23—24no—
515C23—24no—
516C23—28no—
517C23—44no—
518C23—44no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23—44no—
531C23—44no—
532C23—44no—
533C23—42no—
534C23—42no—
535C23—42no—
536C23—42no—
537C23—42no—
538C23—42no—
539C23—44no—
540C23#53514yes—
541C23C23×D411yescharacteristic
542C23C22×#4112yes—
543C23C22×#4112yes—
544C23C2×H(5,𝔽2)12yes—
545C23C2×H(5,𝔽2)11yescharacteristic
546C23C2×2−1+412yes—
547C23C2×2−1+411yescharacteristic
548C23#58512yes—
549C23#58512yes—
550D4—88no—
551C2×C4—232no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—28no—
565C2×C4—28no—
566C2×C4—28no—
567C2×C4—28no—
568C2×C4—24no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—28no—
574C2×C4—28no—
575C2×C4—416no—
576C2×C4—416no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—44no—
587C2×C4—41no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—44no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—44no—
605C2×C4—44no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—44no—
609C2×C4—44no—
610C2×C4—44no—
611C2×C4—44no—
612C2×C4—44no—
613C2×C4—44no—
614C2×C4—44no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—42no—
620C2×C4—44no—
621C2×C4—44no—
622C2×C4—44no—
623C2×C4—42no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—44no—
630C2×C4—44no—
631C2×C4—44no—
632C2×C4—44no—
633C2×C4C2×#12314yes—
634C2×C4C2×#12314yes—
635C2×C4C2×#12314yes—
636C2×C4#53214yes—
637C2×C4#53614yes—
638C2×C4#53714yes—
639C22—24no—
640C22—24no—
641C22—24no—
642C22—42no—
643C22—48no—
644C22—48no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—81no—
649C22—84no—
650C22—84no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—81no—
655C22—81no—
656C22—81no—
657C22C22×#12311yescharacteristic
658C22C2×#53211yescharacteristic
659C22C2×#53511yescharacteristic
660C22C2×#53611yescharacteristic
661C22C2×#53711yescharacteristic
662C22#344911yescharacteristic
663C22#344911yescharacteristic
664C22#346114yes—
665C22#347514yes—
666C22#349214yes—
667C22#349412yes—
668C22#349512yes—
669C22#351912yes—
670C22#352212yes—
671C22#355412yes—
672C22#355512yes—
673C22#356014yes—
674C4—24no—
675C4—24no—
676C4—24no—
677C4—24no—
678C4—24no—
679C4—24no—
680C4—48no—
681C4—81no—
682C4—84no—
683C4—81no—
684C4—84no—
685C4—81no—
686C4—84no—
687C4—81no—
688C4—81no—
689C4—81no—
690C4—81no—
691C2—44no—
692C2—161no—
693C2#3463111yescharacteristic
694C2#3481011yescharacteristic
695C2#3522711yescharacteristic
696C2#3523811yescharacteristic
697C2#3526311yescharacteristic
698C2#3754711yescharacteristic
699C2#3762211yescharacteristic
700C2#3799614yes—
701C2#3799814yes—
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. C2×#2285C2semidirect
3. #17067C2semidirect
4. #17066C2semidirect
5. #17063C2semidirect
6. #17062C2semidirect
8. #16399C2semidirect
9. #15105C2semidirect
10. #15103C2semidirect
11. #14892C2semidirect
13. #14841C2semidirect
15. #14356C2semidirect
16. #14291C2semidirect
17. #14249C2semidirect
18. #14244C2semidirect
19. C4×#380C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28522
C2×#22852
#170672
#170662
#170632
#170622
#164012
#163992
#151052
#151032
#148922
#148882
#148412
#148392
#143562
#142912
#142492
#142442
C4×#3802