Finite Group Databasebeta

← #646058

Subgroups — #646058

Order 512.

2288 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊Q8C211yesmaximal, characteristic, complemented
2C2×#2298C211yesmaximal, characteristic
3#17067C211yesmaximal, characteristic, complemented
4#17066C211yesmaximal, characteristic
5#17042C212yesmaximal, complemented
6#17040C212yesmaximal
7#16507C211yesmaximal, characteristic, complemented
8#16506C211yesmaximal, characteristic
9#15119C212yesmaximal, complemented
10#15098C212yesmaximal, complemented
11#15088C212yesmaximal, complemented
12#15079C212yesmaximal
13#15045C212yesmaximal
14#14887C212yesmaximal
15#14690C211yesmaximal, characteristic, complemented
16#14302C212yesmaximal, complemented
17#14301C212yesmaximal
18#14225C212yesmaximal
19#14223C212yesmaximal, complemented
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4×Q8C2212yes—
22C22×#117C2211yescharacteristic
23C2×#398C2212yes—
24C2×#397C2212yes—
25C2×#395C2212yes—
26C2×#393C2212yes—
27C4×C4⋊Q8—28no—
28#2298—28no—
29C2×#391C2212yes—
30C2×#389C2212yes—
31C2×#389C2212yes—
32C2×#388C2211yescharacteristic
33C2×#388C2211yescharacteristic
34C2×#388C2211yescharacteristic
35C2×#388C2211yescharacteristic
36C2×#388C2214yes—
37C2×#387C2211yescharacteristic
38C2×#387C2211yescharacteristic
39C2×#387C2211yescharacteristic
40C2×#387C2211yescharacteristic
41C2×#387C2214yes—
42C2×#384C2211yescharacteristic
43C2×C4×C4⋊C4C2211yescharacteristic
44C2×C4×C4⋊C4C2212yes—
45C2×C4×C4⋊C4C2212yes—
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49C2×#376C2211yescharacteristic
50C2×C43C2211yescharacteristic
51#1446C2212yes—
52#1446C2212yes—
53#1445C2214yes—
54#1445C2212yes—
55#1445C2212yes—
56#1444C2212yes—
57#1444C2212yes—
58#1444C2212yes—
59#1444C2214yes—
60#1444C2212yes—
61#1443C2214yes—
62#1442C2212yes—
63#1442C2214yes—
64#1442C2212yes—
65#1442C2212yes—
66#1442C2214yes—
67#1442C2212yes—
68#1441C2212yes—
69#1441C2212yes—
70#1441C2212yes—
71#1441C2214yes—
72#1441C2212yes—
73#1439C2214yes—
74#1439C2212yes—
75#1439C2212yes—
76#1439C2212yes—
77#1439C2214yes—
78#1439C2212yes—
79#1436C2214yes—
80#1436C2214yes—
81#1434C2212yes—
82#1434C2214yes—
83#1434C2211yescharacteristic
84#1434C2211yescharacteristic
85#1434C2212yes—
86#1434C2212yes—
87#1434C2212yes—
88#1434C2212yes—
89#1434C2212yes—
90#1434C2214yes—
91#1434C2214yes—
92C22⋊#95C2214yes—
93C23×Q8C2312yes—
94C24×C4C2311yescharacteristic
95C2×C4⋊Q8—28no—
96C2×C4⋊Q8—24no—
97C2×C4×Q8—28no—
98C2×C4×Q8—48no—
99C2×#117—24no—
100C2×#117—24no—
101C2×#117—24no—
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2314yes—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2314yes—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2312yes—
114C22×C4⋊C4C2311yescharacteristic
115C22×C4⋊C4C2314yes—
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2311yescharacteristic
120C22×C22⋊C4C2314yes—
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2311yescharacteristic
123C22×C22⋊C4C2314yes—
124C22×C42C2311yescharacteristic
125C22×C42C2314yes—
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133C22×C42C2311yescharacteristic
134C22×C42C2311yescharacteristic
135#398—28no—
136#397—28no—
137#395—28no—
138#393—28no—
139#391—28no—
140#389—28no—
141#389—28no—
142#388—24no—
143#388—24no—
144#388—24no—
145#388—24no—
146#388—48no—
147#387—24no—
148#387—24no—
149#387—24no—
150#387—24no—
151#387—48no—
152#384—24no—
153C4×C4⋊C4—28no—
154C4×C4⋊C4—28no—
155C4×C4⋊C4—24no—
156C4×C4⋊C4—24no—
157C4×C22⋊C4—24no—
158C4×C22⋊C4—24no—
159#376—24no—
160C2×#95C2312yes—
161C2×#95C2314yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2311yescharacteristic
172C2×#95C2311yescharacteristic
173C2×#95C2312yes—
174C2×#95C2312yes—
175C2×#95C2314yes—
176C2×#95C2311yescharacteristic
177C2×#95C2311yescharacteristic
178C2×#95C2311yescharacteristic
179C2×#95C2311yescharacteristic
180C2×#95C2311yescharacteristic
181C2×#95C2311yescharacteristic
182C2×#95C2311yescharacteristic
183C2×#95C2311yescharacteristic
184C2×#95C2311yescharacteristic
185C2×#95C2311yescharacteristic
186C2×#95C2314yes—
187C2×#95C2314yes—
188C2×#95C2314yes—
189C2×#95C2314yes—
190C2×#95C2314yes—
191C2×#95C2314yes—
192C2×#95C2314yes—
193C2×#95C2314yes—
194C2×#95C2314yes—
195C2×#95C2314yes—
196C43D418yes—
197C25C2411yescharacteristic
198C22×Q8—24no—
199C22×Q8—28no—
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8—44no—
203C23×C4—22no—
204C23×C4—22no—
205C23×C4—22no—
206C23×C4—24no—
207C23×C4—42no—
208C23×C4C2411yescharacteristic
209C23×C4C2414yes—
210C23×C4C2414yes—
211C23×C4C2414yes—
212C23×C4C2411yescharacteristic
213C23×C4C2411yescharacteristic
214C23×C4C2411yescharacteristic
215C23×C4C2411yescharacteristic
216C23×C4C2411yescharacteristic
217C23×C4C2411yescharacteristic
218C23×C4C2414yes—
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C4⋊Q8—232no—
227C4×Q8—432no—
228#117—232no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—28no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—28no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—28no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4—44no—
261C2×C4⋊C4#4114yes—
262C2×C4⋊C4#4118yes—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—28no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—42no—
278C2×C22⋊C4—44no—
279C2×C22⋊C4—44no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4—44no—
282C2×C22⋊C4#4114yes—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—24no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—28no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—28no—
296C2×C42—28no—
297C2×C42—24no—
298C2×C42—42no—
299C2×C42—42no—
300C2×C42—42no—
301C2×C42C2×D414yes—
302C2×C42C2×D414yes—
303C2×C42C2×D414yes—
304C2×C42C2×D414yes—
305C2×C42C2×D414yes—
306C2×C42C2×D414yes—
307C2×C42C2×D414yes—
308#95—22no—
309#95—22no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—28no—
321#95—28no—
322#95—28no—
323#95—24no—
324#95—28no—
325#95—24no—
326#95—28no—
327#95—24no—
328#95—28no—
329#95—24no—
330#95—28no—
331#95—28no—
332#95—24no—
333#95—24no—
334#95—44no—
335#95—44no—
336#95—44no—
337#95—44no—
338#95—44no—
339#95—44no—
340#95—44no—
341#95—44no—
342#95—44no—
343#95—44no—
344C24—21no—
345C24—22no—
346C24—21no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24—22no—
352C24C2×#4112yes—
353C24C2511yescharacteristic
354C2×Q8—216no—
355C2×Q8—416no—
356C2×Q8—416no—
357C2×Q8—48no—
358C2×Q8—48no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—28no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—24no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—24no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—24no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—24no—
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—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—24no—
426C22×C4—24no—
427C22×C4—24no—
428C22×C4—24no—
429C22×C4—22no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—22no—
433C22×C4—48no—
434C22×C4—48no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4C22×D412yes—
438C22×C4C22×D412yes—
439C22×C4C22×D412yes—
440C22×C4C22×D412yes—
441C22×C4C22×D412yes—
442C22×C4C22×D412yes—
443C22×C4C22×D412yes—
444C22×C4C2×#4112yes—
445C22×C4C2×#4112yes—
446C22×C4C2×#4118yes—
447C22×C4C2×#4112yes—
448C22×C4C2×#4114yes—
449C22×C4C2×#4112yes—
450C22×C4C2×#4112yes—
451C22×C4H(5,𝔽2)14yes—
452C22×C42−1+414yes—
453C22×C42−1+414yes—
454C4⋊C4—28no—
455C4⋊C4—216no—
456C4⋊C4—28no—
457C4⋊C4—216no—
458C4⋊C4—416no—
459C4⋊C4—48no—
460C4⋊C4—48no—
461C4⋊C4—44no—
462C4⋊C4—48no—
463C4⋊C4—416no—
464C4⋊C4—44no—
465C4⋊C4—44no—
466C4⋊C4—44no—
467C4⋊C4—88no—
468C22⋊C4—28no—
469C22⋊C4—28no—
470C22⋊C4—416no—
471C22⋊C4—44no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—88no—
476C42—28no—
477C42—28no—
478C42—28no—
479C42—28no—
480C42—28no—
481C42—28no—
482C42—44no—
483C42—44no—
484C42—44no—
485C42—416no—
486C42—44no—
487C23—22no—
488C23—24no—
489C23—24no—
490C23—24no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—44no—
498C23—42no—
499C23—42no—
500C23—44no—
501C23—42no—
502C23—44no—
503C23—44no—
504C23C23×D411yescharacteristic
505C23C22×#4112yes—
506C23C22×#4111yescharacteristic
507C23C22×#4111yescharacteristic
508C23C2×H(5,𝔽2)12yes—
509C23C2×2−1+412yes—
510C23C2×2−1+412yes—
511C23#58512yes—
512C23#58512yes—
513Q8—432no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—28no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—28no—
530C2×C4—28no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—28no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—28no—
542C2×C4—24no—
543C2×C4—28no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—416no—
547C2×C4—41no—
548C2×C4—44no—
549C2×C4—44no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—42no—
556C2×C4—44no—
557C2×C4—44no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—44no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—44no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—44no—
581C2×C4—44no—
582C2×C4—44no—
583C2×C4—44no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—44no—
587C2×C4—44no—
588C2×C4—44no—
589C2×C4—44no—
590C2×C4—44no—
591C2×C4—44no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—44no—
596C2×C4—44no—
597C2×C4—44no—
598C2×C4—44no—
599C2×C4—44no—
600C2×C4#52514yes—
601C2×C4#52514yes—
602C2×C4#53514yes—
603C2×C4#53614yes—
604C2×C4#53614yes—
605C22—42no—
606C22—42no—
607C22—48no—
608C22—42no—
609C22—81no—
610C22—81no—
611C22—81no—
612C22—81no—
613C22—84no—
614C22C2×#52511yescharacteristic
615C22C2×#52511yescharacteristic
616C22C2×#53511yescharacteristic
617C22C2×#53611yescharacteristic
618C22C2×#53611yescharacteristic
619C22#344911yescharacteristic
620C22#344911yescharacteristic
621C22#346114yes—
622C22#348112yes—
623C22#348112yes—
624C22#349512yes—
625C22#349512yes—
626C22#351512yes—
627C22#351614yes—
628C22#351812yes—
629C22#354414yes—
630C22#355114yes—
631C4—24no—
632C4—24no—
633C4—24no—
634C4—24no—
635C4—24no—
636C4—42no—
637C4—42no—
638C4—42no—
639C4—48no—
640C4—42no—
641C4—42no—
642C4—81no—
643C4—81no—
644C4—81no—
645C4—84no—
646C4—81no—
647C4—84no—
648C4—84no—
649C2—82no—
650C2#3460211yescharacteristic
651C2#3496311yescharacteristic
652C2#3520011yescharacteristic
653C2#3524611yescharacteristic
654C2#3532511yescharacteristic
655C2#3755011yescharacteristic
656C2#3763211yescharacteristic
657C2#3885014yes—
658C2#3885114yes—
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×C4×C4⋊Q8C2semidirect
3. #17067C2semidirect
5. #17042C2semidirect
7. #16507C2semidirect
9. #15119C2semidirect
10. #15098C2semidirect
11. #15088C2semidirect
15. #14690C2semidirect
16. #14302C2semidirect
19. #14223C2semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊Q82
C2×#22982
#170672
#170662
#170422
#170402
#165072
#165062
#151192
#150982
#150882
#150792
#150452
#148872
#146902
#143022
#143012
#142252
#142232