Finite Group Databasebeta

← #4416072

Subgroups — #4416072

Order 512.

2092 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2862C211yesmaximal, characteristic, complemented
2C2×#2298C211yesmaximal, characteristic
3#17048C212yesmaximal, complemented
4#17046C212yesmaximal
5#16889C212yesmaximal, complemented
6#16812C212yesmaximal
7#16603C211yesmaximal, characteristic, complemented
8#16602C211yesmaximal, characteristic
9#16507C211yesmaximal, characteristic, complemented
10#16506C211yesmaximal, characteristic
11#15119C212yesmaximal, complemented
12#15097C212yesmaximal, complemented
13#15086C212yesmaximal, complemented
14#15050C212yesmaximal
15#15017C212yesmaximal
16#14887C212yesmaximal
17#14691C211yesmaximal, characteristic, complemented
18#14626C212yesmaximal, complemented
19#14625C212yesmaximal
20C22×C4⋊Q8C2211yescharacteristic
21C22×C4⋊Q8C2211yescharacteristic
22C22×#117C2211yescharacteristic
23#2862—28no—
24C2×#398C2212yes—
25C2×#397C2212yes—
26C2×#395C2212yes—
27C2×#395C2212yes—
28C2×#393C2212yes—
29C2×#393C2212yes—
30#2298—28no—
31C2×#391C2212yes—
32C2×#391C2212yes—
33C2×#391C2212yes—
34C2×#391C2212yes—
35C2×#388C2211yescharacteristic
36C2×#388C2211yescharacteristic
37C2×#388C2211yescharacteristic
38C2×#388C2211yescharacteristic
39C2×#387C2211yescharacteristic
40C2×#387C2211yescharacteristic
41C2×#387C2211yescharacteristic
42C2×#387C2211yescharacteristic
43C2×#382C2212yes—
44C2×#382C2212yes—
45C2×C4×C4⋊C4C2211yescharacteristic
46C2×C4×C4⋊C4C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48C2×C4×C22⋊C4C2211yescharacteristic
49C2×#376C2211yescharacteristic
50C2×C43C2211yescharacteristic
51#1446C2212yes—
52#1446C2212yes—
53#1446C2212yes—
54#1446C2212yes—
55#1445C2214yes—
56#1445C2214yes—
57#1445C2212yes—
58#1445C2212yes—
59#1444C2212yes—
60#1444C2212yes—
61#1444C2212yes—
62#1443C2214yes—
63#1443C2214yes—
64#1443C2214yes—
65#1442C2212yes—
66#1442C2214yes—
67#1442C2214yes—
68#1442C2214yes—
69#1442C2212yes—
70#1442C2212yes—
71#1441C2212yes—
72#1441C2214yes—
73#1441C2212yes—
74#1441C2212yes—
75#1440C2214yes—
76#1439C2212yes—
77#1439C2214yes—
78#1439C2214yes—
79#1439C2212yes—
80#1439C2212yes—
81#1436C2214yes—
82#1436C2214yes—
83#1434C2212yes—
84#1434C2214yes—
85#1434C2212yes—
86#1434C2211yescharacteristic
87#1434C2211yescharacteristic
88#1434C2212yes—
89#1434C2212yes—
90#1434C2212yes—
91#1434C2212yes—
92C22⋊#95C2214yes—
93C23×Q8C2312yes—
94C23×Q8C2312yes—
95C24×C4C2311yescharacteristic
96C2×C4⋊Q8—24no—
97C2×C4⋊Q8—24no—
98C2×C4⋊Q8—44no—
99C2×C4⋊Q8—44no—
100C2×#117—24no—
101C2×#117—24no—
102C2×#117—24no—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C4⋊C4C2311yescharacteristic
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2312yes—
112C22×C4⋊C4C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2314yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2311yescharacteristic
117C22×C22⋊C4C2311yescharacteristic
118C22×C22⋊C4C2311yescharacteristic
119C22×C22⋊C4C2314yes—
120C22×C22⋊C4C2311yescharacteristic
121C22×C22⋊C4C2311yescharacteristic
122C22×C22⋊C4C2314yes—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2311yescharacteristic
126C22×C42C2311yescharacteristic
127C22×C42C2311yescharacteristic
128C22×C42C2311yescharacteristic
129C22×C42C2311yescharacteristic
130C22×C42C2311yescharacteristic
131C22×C42C2311yescharacteristic
132C22×C42C2311yescharacteristic
133#398—44no—
134#397—44no—
135#395—28no—
136#395—44no—
137#393—28no—
138#393—44no—
139#391—28no—
140#391—28no—
141#391—28no—
142#391—28no—
143#388—24no—
144#388—24no—
145#388—24no—
146#388—24no—
147#387—24no—
148#387—24no—
149#387—24no—
150#387—24no—
151#382—28no—
152#382—28no—
153C4×C4⋊C4—24no—
154C4×C4⋊C4—24no—
155C4×C22⋊C4—24no—
156C4×C22⋊C4—24no—
157#376—24no—
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C2×#95C2312yes—
167C2×#95C2312yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2311yescharacteristic
172C2×#95C2311yescharacteristic
173C2×#95C2311yescharacteristic
174C2×#95C2311yescharacteristic
175C2×#95C2311yescharacteristic
176C2×#95C2311yescharacteristic
177C2×#95C2311yescharacteristic
178C2×#95C2311yescharacteristic
179C2×#95C2311yescharacteristic
180C2×#95C2311yescharacteristic
181C2×#95C2311yescharacteristic
182C2×#95C2311yescharacteristic
183C2×#95C2314yes—
184C2×#95C2314yes—
185C2×#95C2314yes—
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—24no—
200C22×Q8—24no—
201C22×Q8—24no—
202C22×Q8—24no—
203C22×Q8—24no—
204C22×Q8—44no—
205C22×Q8—44no—
206C22×Q8—44no—
207C22×Q8—44no—
208C23×C4—22no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—24no—
212C23×C4—42no—
213C23×C4C2411yescharacteristic
214C23×C4C2414yes—
215C23×C4C2414yes—
216C23×C4C2414yes—
217C23×C4C2414yes—
218C23×C4C2411yescharacteristic
219C23×C4C2411yescharacteristic
220C23×C4C2411yescharacteristic
221C23×C4C2411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C23×C4C2411yescharacteristic
227C23×C4C2411yescharacteristic
228C23×C4C2411yescharacteristic
229C23×C4C2411yescharacteristic
230C23×C4C2411yescharacteristic
231C4⋊Q8—416no—
232C4⋊Q8—416no—
233#117—232no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
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—28no—
252C2×C4⋊C4—42no—
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—44no—
262C2×C4⋊C4#4114yes—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—42no—
276C2×C22⋊C4—42no—
277C2×C22⋊C4—44no—
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—22no—
292C2×C42—22no—
293C2×C42—24no—
294C2×C42—42no—
295C2×C42—42no—
296C2×C42—42no—
297C2×C42—42no—
298C2×C42C2×D414yes—
299C2×C42C2×D414yes—
300C2×C42C2×D414yes—
301C2×C42C2×D414yes—
302C2×C42C2×D414yes—
303C2×C42C2×D414yes—
304C2×C42C2×D414yes—
305#95—22no—
306#95—22no—
307#95—22no—
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—28no—
318#95—28no—
319#95—28no—
320#95—28no—
321#95—24no—
322#95—28no—
323#95—24no—
324#95—28no—
325#95—24no—
326#95—24no—
327#95—28no—
328#95—24no—
329#95—24no—
330#95—44no—
331#95—44no—
332#95—44no—
333#95—44no—
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—
343C24—21no—
344C24—21no—
345C24—22no—
346C24—22no—
347C24—22no—
348C24—22no—
349C24—22no—
350C24—22no—
351C24C2×#4112yes—
352C24C2511yescharacteristic
353C2×Q8—48no—
354C2×Q8—48no—
355C2×Q8—48no—
356C2×Q8—48no—
357C2×Q8—48no—
358C2×Q8—48no—
359C2×Q8—88no—
360C2×Q8—88no—
361C2×Q8—88no—
362C2×Q8—88no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—28no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
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—21no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—24no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—24no—
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—22no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—24no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—48no—
440C22×C4—48no—
441C22×C4—44no—
442C22×C4—44no—
443C22×C4C22×D412yes—
444C22×C4C22×D412yes—
445C22×C4C22×D412yes—
446C22×C4C22×D412yes—
447C22×C4C22×D412yes—
448C22×C4C22×D412yes—
449C22×C4C22×D412yes—
450C22×C4C2×#4112yes—
451C22×C4C2×#4112yes—
452C22×C4C2×#4112yes—
453C22×C4C2×#4112yes—
454C22×C4C2×#4112yes—
455C22×C4H(5,𝔽2)14yes—
456C22×C42−1+414yes—
457C22×C42−1+414yes—
458C4⋊C4—28no—
459C4⋊C4—28no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—48no—
465C4⋊C4—44no—
466C4⋊C4—48no—
467C4⋊C4—44no—
468C4⋊C4—88no—
469C4⋊C4—88no—
470C22⋊C4—28no—
471C22⋊C4—28no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—44no—
476C22⋊C4—88no—
477C22⋊C4—88no—
478C42—28no—
479C42—28no—
480C42—28no—
481C42—28no—
482C42—28no—
483C42—44no—
484C42—44no—
485C42—44no—
486C42—44no—
487C42—44no—
488C23—22no—
489C23—24no—
490C23—24no—
491C23—24no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23—41no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23—44no—
505C23C23×D411yescharacteristic
506C23C22×#4111yescharacteristic
507C23C22×#4111yescharacteristic
508C23C2×H(5,𝔽2)12yes—
509C23C2×2−1+412yes—
510C23C2×2−1+412yes—
511C23#58512yes—
512C23#58512yes—
513C23#58512yes—
514Q8—816no—
515Q8—816no—
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—24no—
525C2×C4—24no—
526C2×C4—24no—
527C2×C4—24no—
528C2×C4—28no—
529C2×C4—28no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—24no—
543C2×C4—24no—
544C2×C4—24no—
545C2×C4—416no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—44no—
560C2×C4—41no—
561C2×C4—44no—
562C2×C4—44no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
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—44no—
575C2×C4—44no—
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—44no—
601C2×C4—44no—
602C2×C4—44no—
603C2×C4#52514yes—
604C2×C4#53514yes—
605C2×C4#53614yes—
606C2×C4#53614yes—
607C22—48no—
608C22—42no—
609C22—42no—
610C22—42no—
611C22—84no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—81no—
616C22C2×#52511yescharacteristic
617C22C2×#53511yescharacteristic
618C22C2×#53611yescharacteristic
619C22C2×#53611yescharacteristic
620C22#344911yescharacteristic
621C22#344911yescharacteristic
622C22#344911yescharacteristic
623C22#349512yes—
624C22#349512yes—
625C22#351412yes—
626C22#351412yes—
627C22#351512yes—
628C22#351812yes—
629C22#353114yes—
630C22#354514yes—
631C22#355214yes—
632C22#356514yes—
633C4—24no—
634C4—24no—
635C4—24no—
636C4—24no—
637C4—42no—
638C4—42no—
639C4—42no—
640C4—42no—
641C4—42no—
642C4—42no—
643C4—84no—
644C4—84no—
645C4—84no—
646C4—81no—
647C4—81no—
648C4—81no—
649C4—84no—
650C4—81no—
651C2—82no—
652C2#3512911yescharacteristic
653C2#3520011yescharacteristic
654C2#3532511yescharacteristic
655C2#3669911yescharacteristic
656C2#3754511yescharacteristic
657C2#3760611yescharacteristic
658C2#3761711yescharacteristic
659C2#4583714yes—
660C2#4583914yes—
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×#2862C2semidirect
3. #17048C2semidirect
5. #16889C2semidirect
7. #16603C2semidirect
9. #16507C2semidirect
11. #15119C2semidirect
12. #15097C2semidirect
13. #15086C2semidirect
17. #14691C2semidirect
18. #14626C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28622
C2×#22982
#170482
#170462
#168892
#168122
#166032
#166022
#165072
#165062
#151192
#150972
#150862
#150502
#150172
#148872
#146912
#146262
#146252