Finite Group Databasebeta

← #6498603

Subgroups — #6498603

Order 512.

1890 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3252C212yesmaximal
2#36645C211yesmaximal, characteristic, complemented
3C4×#529C211yesmaximal, characteristic
4#33027C212yesmaximal, complemented
5#33026C212yesmaximal, complemented
6#29869C211yesmaximal, characteristic, complemented
7#29868C211yesmaximal, characteristic, complemented
8#29799C211yesmaximal, characteristic, complemented
9#29798C211yesmaximal, characteristic, complemented
10#28581C214yesmaximal, complemented
11#28577C214yesmaximal
12#28209C214yesmaximal, complemented
13#28164C214yesmaximal, complemented
14#21892C212yesmaximal, complemented
15#20752C211yesmaximal, characteristic, complemented
16C2×#533C2212yes—
17C2×#529C2211yescharacteristic
18#3252—216no—
19C2×#479C2214yes—
20C2×#477C2212yes—
21C2×#476C2212yes—
22C2×#470C2214yes—
23C2×#468C2214yes—
24C2×#464C2214yes—
25C2×#462C2212yes—
26C2×#461C2212yes—
27C2×#423C2212yes—
28#2868C2211yescharacteristic
29#2868C2211yescharacteristic
30#2862C2212yes—
31#2859—21no—
32#2859—21no—
33#2857—22no—
34#2857C2214yes—
35C4×C4⋊Q8—22no—
36C4×C4⋊Q8C2214yes—
37C4×#125C2212yes—
38C4×#126—22no—
39C4×#124C2212yes—
40C4×C22⋊Q8C2212yes—
41C4×#123—22no—
42C4×#121C2212yes—
43#2298—21no—
44#2295C2211yescharacteristic
45#2293C2211yescharacteristic
46#2290—21no—
47#2285—21no—
48#2285—21no—
49#2283C2211yescharacteristic
50C42×Q8C2211yescharacteristic
51C42×D4—21no—
52C42×D4C2211yescharacteristic
53C4×#117—21no—
54C2×C43C2211yescharacteristic
55#2084C2214yes—
56#2083C2214yes—
57#2072C2214yes—
58#2071C2214yes—
59#2043C2214yes—
60#2043C2214yes—
61#2016C2214yes—
62#2013C2214yes—
63#1987C2212yes—
64#1987C2212yes—
65#1981C2212yes—
66#1981C2212yes—
67#1928C2218yes—
68#1927C2218yes—
69#1894C2218yes—
70#1878C2218yes—
71#1877C2218yes—
72#1831C2212yes—
73#1831C2214yes—
74#1831C2211yescharacteristic
75#1831C2211yescharacteristic
76#1805C2218yes—
77#1728C2214yes—
78C22×Q16C2314yes—
79C22×SD16C2314yes—
80C22×M16C2314yes—
81#533—28no—
82#533—28no—
83C2×C4⋊Q8—22no—
84C2×C4⋊Q8C2312yes—
85C2×C4⋊Q8C2314yes—
86C2×C4⋊Q8C2312yes—
87#529—24no—
88#529D418yes—
89C2×#126—21no—
90C2×#125—22no—
91C2×#125C2311yescharacteristic
92C2×#124C2311yescharacteristic
93C2×#123—21no—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8C2314yes—
96C2×C22⋊Q8C2311yescharacteristic
97C2×#121C2311yescharacteristic
98C2×C4×Q8—22no—
99C2×C4×Q8C2314yes—
100C2×C4×Q8C2311yescharacteristic
101C2×C4×Q8C2312yes—
102C2×C4×D4—21no—
103C2×C4×D4—22no—
104C2×C4×D4C2311yescharacteristic
105C2×C4×D4C2312yes—
106C2×#117—22no—
107C2×#117—21no—
108C2×#117—21no—
109C22×C42—21no—
110C22×C42D411yescharacteristic
111C22×C42D411yescharacteristic
112C22×C42D412yes—
113C22×C42C2312yes—
114C22×C42C2311yescharacteristic
115#479—216no—
116#477—28no—
117#476—28no—
118#470—216no—
119#468—216no—
120#464—216no—
121#462—28no—
122#461—28no—
123C2×#107C2314yes—
124C2×#106C2314yes—
125#423—28no—
126C2×C4⋊C8C2312yes—
127C2×C4⋊C8C2314yes—
128C2×C4⋊C8C2312yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2318yes—
133C2×#103C2318yes—
134C2×#102C2314yes—
135#391—24no—
136#391C2318yes—
137#391C2312yes—
138#390C2312yes—
139#389—24no—
140#389—24no—
141#389—24no—
142#389—24no—
143#389C2312yes—
144#389C2312yes—
145#388—22no—
146#388—22no—
147#388—22no—
148#388—22no—
149#387—22no—
150#387—22no—
151#387—22no—
152#387—22no—
153#386—22no—
154#385—22no—
155#384—22no—
156#384—21no—
157#384—21no—
158#384—21no—
159#384—21no—
160#384C2314yes—
161#383—21no—
162#382—22no—
163#382—22no—
164#382C2312yes—
165#381—21no—
166#380—24no—
167#380—24no—
168#380C2312yes—
169C4×C4⋊C4—22no—
170C4×C4⋊C4—22no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4—24no—
174C4×C4⋊C4—24no—
175C4×C4⋊C4—21no—
176C4×C4⋊C4—21no—
177C4×C4⋊C4C2312yes—
178C4×C4⋊C4C2312yes—
179C4×C4⋊C4C2311yescharacteristic
180C4×C4⋊C4C2311yescharacteristic
181C4×C22⋊C4—21no—
182C4×C22⋊C4—22no—
183C4×C22⋊C4—22no—
184C4×C22⋊C4—22no—
185C4×C22⋊C4—22no—
186C4×C22⋊C4—22no—
187C4×C22⋊C4—21no—
188C4×C22⋊C4—21no—
189#376—21no—
190#376—21no—
191C43—21no—
192C43D412yes—
193C43D411yescharacteristic
194C43D411yescharacteristic
195C43C2312yes—
196C43C2311yescharacteristic
197C43C2311yescharacteristic
198#340C2318yes—
199#340C2318yes—
200#336C2314yes—
201#336C2314yes—
202#336C2318yes—
203C22×Q8—22no—
204C22×Q8C2414yes—
205C22×Q8C2411yescharacteristic
206C22×D4—21no—
207C22×D4C2411yescharacteristic
208C23×C4—21no—
209C23×C4—22no—
210C23×C4C2×D412yes—
211C23×C4C2×D411yescharacteristic
212C23×C4C2×D412yes—
213C23×C4C2×D411yescharacteristic
214C23×C4C2411yescharacteristic
215C2×Q16—28no—
216C2×Q16—48no—
217C2×SD16—28no—
218C2×SD16—48no—
219C2×M16—28no—
220C2×M16—48no—
221C22×C8C2414yes—
222C22×C8C2414yes—
223C4⋊Q8—24no—
224C4⋊Q8—24no—
225C4⋊Q8—24no—
226C4⋊Q8—24no—
227C4⋊Q8—48no—
228C4⋊Q8—48no—
229#126—28no—
230#125—22no—
231#125—48no—
232#125C2×D414yes—
233#124—22no—
234#124C2×D414yes—
235#123—28no—
236C22⋊Q8—22no—
237C22⋊Q8—28no—
238C22⋊Q8—28no—
239C22⋊Q8—48no—
240C22⋊Q8C2×D414yes—
241#121—22no—
242#121C2×D414yes—
243C4×Q8—22no—
244C4×Q8—28no—
245C4×Q8—28no—
246C4×Q8—44no—
247C4×Q8—48no—
248C4×Q8C2×D414yes—
249C4×D4—28no—
250C4×D4—22no—
251C4×D4—44no—
252C4×D4—48no—
253C4×D4C2×D414yes—
254#117—28no—
255#117—48no—
256#117—44no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2414yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2411yescharacteristic
277C2×C4⋊C4C2411yescharacteristic
278C2×C22⋊C4—21no—
279C2×C22⋊C4—21no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C42—21no—
289C2×C42—24no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—21no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—21no—
296C2×C42—22no—
297C2×C42—44no—
298C2×C42—44no—
299C2×C42—44no—
300C2×C42C2×D412yes—
301C2×C42C2×D412yes—
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42C2×D412yes—
305C2×C42C2×D412yes—
306C2×C42C2×D412yes—
307C2×C42C2×D412yes—
308C2×C42C2×D412yes—
309C2×C42C2×D412yes—
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D414yes—
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42#4114yes—
316C2×C42C2412yes—
317C2×C42C2412yes—
318C2×C42C2411yescharacteristic
319C2×C42C2411yescharacteristic
320#107—28no—
321#106—28no—
322C4⋊C8—24no—
323C4⋊C8—24no—
324C4⋊C8—28no—
325#103—28no—
326#103—28no—
327#103—28no—
328#103—48no—
329#103—48no—
330#102—28no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—21no—
339#95—22no—
340#95—21no—
341#95—22no—
342#95—28no—
343#95—28no—
344C24—21no—
345C24C22×D411yescharacteristic
346C2×Q8—21no—
347C2×Q8—24no—
348C2×Q8—24no—
349C2×Q8—24no—
350C2×Q8—28no—
351C2×Q8—28no—
352C2×Q8—48no—
353C2×Q8—44no—
354C2×Q8C22×D412yes—
355C2×D4—24no—
356C2×D4—21no—
357C2×D4—24no—
358C2×D4—44no—
359C2×D4C22×D412yes—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—44no—
379C22×C4—44no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—44no—
383C22×C4#12114yes—
384C22×C4#12114yes—
385C22×C4C22×D412yes—
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D412yes—
391C22×C4C22×D412yes—
392C22×C4C22×D412yes—
393C22×C4C22×D412yes—
394C22×C4C22×D411yescharacteristic
395C22×C4C2×#4112yes—
396C22×C4C2511yescharacteristic
397Q16—88no—
398SD16—88no—
399M16—88no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—28no—
403C2×C8—28no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—44no—
415C4⋊C4—48no—
416C4⋊C4—48no—
417C4⋊C4—48no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—48no—
421C4⋊C4—48no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—48no—
426C4⋊C4C22×D412yes—
427C4⋊C4C22×D412yes—
428C22⋊C4—24no—
429C22⋊C4—24no—
430C22⋊C4—24no—
431C22⋊C4—24no—
432C22⋊C4—24no—
433C22⋊C4—44no—
434C22⋊C4—44no—
435C22⋊C4—48no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C42—24no—
439C42—24no—
440C42—22no—
441C42—24no—
442C42—22no—
443C42—24no—
444C42—24no—
445C42—24no—
446C42—24no—
447C42—42no—
448C42—42no—
449C42—42no—
450C42—42no—
451C42—44no—
452C42—44no—
453C42—48no—
454C42—44no—
455C42—42no—
456C42—42no—
457C42#12114yes—
458C42#12114yes—
459C42#13714yes—
460C42#13714yes—
461C42C22×D412yes—
462C42C22×D412yes—
463C42C2×#4112yes—
464C42C2×#4112yes—
465C23—22no—
466C23—21no—
467C23—21no—
468C23—22no—
469C23—22no—
470C23—42no—
471C23—41no—
472C23—41no—
473C23—42no—
474C23C2×#12112yes—
475C23C23×D411yescharacteristic
476Q8—24no—
477Q8—42no—
478Q8—44no—
479Q8—44no—
480Q8—48no—
481Q8—88no—
482D4—24no—
483D4—48no—
484D4—42no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—28no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—44no—
506C2×C4—48no—
507C2×C4—48no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—48no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—48no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—48no—
523C2×C4—88no—
524C2×C4C2×#12112yes—
525C2×C4C2×#12112yes—
526C2×C4C2×#12112yes—
527C2×C4C2×#12112yes—
528C2×C4C2×#12112yes—
529C2×C4C2×#13714yes—
530C2×C4C2×#13714yes—
531C2×C4C2×#13712yes—
532C2×C4C2×#13712yes—
533C2×C4#57514yes—
534C2×C4C23×D411yescharacteristic
535C2×C4C22×#4111yescharacteristic
536C2×C4#58514yes—
537C8—44no—
538C8—44no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—82no—
548C22—82no—
549C22C22×#12111yescharacteristic
550C22#344912yes—
551C22C22×#13711yescharacteristic
552C22C22×#13711yescharacteristic
553C22C2×#57512yes—
554C4—22no—
555C4—24no—
556C4—24no—
557C4—24no—
558C4—22no—
559C4—22no—
560C4—22no—
561C4—42no—
562C4—42no—
563C4—42no—
564C4—42no—
565C4—84no—
566C4—82no—
567C4—82no—
568C4—84no—
569C4#303914yes—
570C4#350012yes—
571C4#351212yes—
572C2—42no—
573C2—82no—
574C2#3532811yescharacteristic
575C2C2×#303911yescharacteristic
576C2#6041511yescharacteristic
577C2#6095012yes—
578C2#6096612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #36645C2semidirect
4. #33027C2semidirect
5. #33026C2semidirect
6. #29869C2semidirect
7. #29868C2semidirect
8. #29799C2semidirect
9. #29798C2semidirect
10. #28581C2semidirect
12. #28209C2semidirect
13. #28164C2semidirect
14. #21892C2semidirect
15. #20752C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32522
#366452
C4×#5292
#330272
#330262
#298692
#298682
#297992
#297982
#285812
#285772
#282092
#281642
#218922
#207522