Finite Group Databasebeta

← #6497124

Subgroups — #6497124

Order 512.

1493 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
1#35284C211yesmaximal, characteristic, complemented
2#34123C211yesmaximal, characteristic
3#33197C211yesmaximal, characteristic, complemented
4#33068C211yesmaximal, characteristic, complemented
5#32958C212yesmaximal
6#29999C211yesmaximal, characteristic, complemented
7#29942C212yesmaximal, complemented
8#29869C211yesmaximal, characteristic, complemented
9#27544C214yesmaximal
10#27480C214yesmaximal, complemented
11#27384C214yesmaximal, complemented
12#27320C214yesmaximal, complemented
13#22119C212yesmaximal, complemented
14#21926C211yesmaximal, characteristic, complemented
15#21614C212yesmaximal, complemented
16C2×C4×#41C2211yescharacteristic
17C2×#477C2212yes—
18C2×#476C2212yes—
19#2868C2211yescharacteristic
20#2859C2211yescharacteristic
21#2857C2212yes—
22#2647—21no—
23#2629—21no—
24#2507C2214yes—
25#2497C2214yes—
26#2446—22no—
27#2429—22no—
28#2356C2212yes—
29#2353—22no—
30#2349C2212yes—
31#2346C2212yes—
32#2343—22no—
33#2339C2212yes—
34#2330—21no—
35#2314—21no—
36C4×C4⋊Q8C2212yes—
37#2295C2211yescharacteristic
38#2295C2211yescharacteristic
39#2290—21no—
40#2290—21no—
41#2283—21no—
42#2283C2211yescharacteristic
43C42×D4C2211yescharacteristic
44C42×D4C2211yescharacteristic
45C4×#117C2211yescharacteristic
46C2×#376—21no—
47#2101C2214yes—
48#2087C2214yes—
49#2085C2214yes—
50#2083C2214yes—
51#2077C2214yes—
52#2075C2214yes—
53#2072C2214yes—
54#2068C2214yes—
55#1987C2212yes—
56#1987C2212yes—
57#1948C2212yes—
58#1948C2214yes—
59#1947C2212yes—
60#1943C2214yes—
61#1935C2214yes—
62#1927C2214yes—
63#1924C2214yes—
64#1920C2214yes—
65#1919C2214yes—
66#1918C2214yes—
67#1917C2214yes—
68#1865C2214yes—
69#1864C2214yes—
70#1862C2214yes—
71#1862C2214yes—
72#1854C2214yes—
73#1842C2211yescharacteristic
74#1831C2211yescharacteristic
75#1803C2212yes—
76#1802C2212yes—
77#1802C2214yes—
78#1797C2212yes—
79#1788C2214yes—
80#1776C2214yes—
81#1767C2212yes—
82#1762C2214yes—
83C22×#41C2311yescharacteristic
84C2×C4⋊Q8—22no—
85C2×C4⋊Q8C2312yes—
86C2×#125—21no—
87C2×C22⋊Q8—22no—
88C4×#41—24no—
89C4×#41—24no—
90C2×C4×Q8—22no—
91C2×C4×Q8C2311yescharacteristic
92C2×C4×Q8C2311yescharacteristic
93C2×C4×D4—21no—
94C2×C4×D4—21no—
95C2×C4×D4C2311yescharacteristic
96C2×C4×D4C2312yes—
97C2×C4×D4C2312yes—
98C2×C4×D4C2311yescharacteristic
99C2×#117—21no—
100C2×#117D411yescharacteristic
101C2×#117D411yescharacteristic
102C2×#117C2311yescharacteristic
103C22×C42—21no—
104C22×C42—21no—
105C22×C42—21no—
106C22×C42C2311yescharacteristic
107#477—28no—
108#476—28no—
109C2×#107C2314yes—
110C2×#107C2314yes—
111C2×#106C2314yes—
112C2×#106C2314yes—
113C2×C4⋊C8C2312yes—
114C2×C4⋊C8C2312yes—
115C2×C4⋊C8C2314yes—
116C2×#103C2314yes—
117C2×#103C2314yes—
118C2×#102C2314yes—
119C2×#102C2314yes—
120C2×C22⋊C8C2314yes—
121C2×#97C2314yes—
122#400—24no—
123#398—24no—
124#398—24no—
125#396—24no—
126#391—22no—
127#391—22no—
128#391C2312yes—
129#391C2312yes—
130#389—22no—
131#389—22no—
132#389—22no—
133#389—24no—
134#389C2314yes—
135#389C2314yes—
136#389C2312yes—
137#389C2314yes—
138#389C2312yes—
139#388—24no—
140#388—22no—
141#388—22no—
142#387—24no—
143#387—22no—
144#387—22no—
145#387—22no—
146#387—22no—
147#387—24no—
148#386—22no—
149#386—22no—
150#385—22no—
151#385—22no—
152#384—21no—
153#384—21no—
154#384C2312yes—
155#384C2314yes—
156#383—21no—
157#383—22no—
158#382—21no—
159#382—22no—
160#382C2312yes—
161#381—21no—
162#381—22no—
163#380—22no—
164#380—22no—
165#380C2312yes—
166#380C2314yes—
167C4×C4⋊C4—22no—
168C4×C4⋊C4—21no—
169C4×C4⋊C4—21no—
170C4×C4⋊C4D412yes—
171C4×C4⋊C4C2314yes—
172C4×C4⋊C4C2312yes—
173C4×C4⋊C4C2312yes—
174C4×C4⋊C4C2311yescharacteristic
175C4×C4⋊C4C2311yescharacteristic
176C4×C4⋊C4C2312yes—
177C4×C22⋊C4—21no—
178C4×C22⋊C4—22no—
179C4×C22⋊C4—22no—
180C4×C22⋊C4—21no—
181C4×C22⋊C4—21no—
182C4×C22⋊C4D412yes—
183C4×C22⋊C4C2312yes—
184#376—21no—
185#376—21no—
186#376—21no—
187#376—22no—
188#376—22no—
189#376D411yescharacteristic
190#376D411yescharacteristic
191C2×#95—22no—
192C2×#95—22no—
193C43C2311yescharacteristic
194C43C2311yescharacteristic
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#336C2314yes—
204#336C2314yes—
205#336C2314yes—
206C2×#41—22no—
207C2×#41—22no—
208C2×#41—24no—
209C2×#41—42no—
210C22×Q8—22no—
211C22×Q8C2411yescharacteristic
212C22×D4—21no—
213C22×D4C2411yescharacteristic
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4C2×D411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4C2411yescharacteristic
222C22×C8C2414yes—
223C22×C8C2414yes—
224C4⋊Q8—24no—
225C4⋊Q8—24no—
226C4⋊Q8—84no—
227#125—44no—
228C22⋊Q8—84no—
229C4×Q8—22no—
230C4×Q8—22no—
231C4×Q8—22no—
232C4×Q8—84no—
233C4×Q8#4114yes—
234C4×D4—28no—
235C4×D4—28no—
236C4×D4—22no—
237C4×D4—22no—
238C4×D4—24no—
239C4×D4—24no—
240C4×D4—22no—
241C4×D4—42no—
242C4×D4—42no—
243C4×D4#4114yes—
244#117—28no—
245#117—22no—
246#117—22no—
247#117—24no—
248#117—42no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2412yes—
264C2×C4⋊C4C2412yes—
265C2×C4⋊C4C2414yes—
266C2×C4⋊C4C2414yes—
267C2×C4⋊C4C2411yescharacteristic
268C2×C4⋊C4C2411yescharacteristic
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4C2×D412yes—
277C2×C22⋊C4C2×D412yes—
278C2×C42—28no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—22no—
287C2×C42—22no—
288C2×C42—24no—
289C2×C42—44no—
290C2×C42—44no—
291C2×C42C2×D411yescharacteristic
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2×D411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D412yes—
297C2×C42C2×D412yes—
298C2×C42C2×D411yescharacteristic
299C2×C42C2411yescharacteristic
300C2×C42C2412yes—
301C2×C42C2412yes—
302C2×C42C2411yescharacteristic
303#107—28no—
304#107—44no—
305#106—28no—
306#106—44no—
307C4⋊C8—24no—
308C4⋊C8—24no—
309C4⋊C8—44no—
310#103—28no—
311#103—44no—
312#102—28no—
313#102—28no—
314C22⋊C8—44no—
315#97—44no—
316#95—22no—
317#95—22no—
318#95—22no—
319#95—22no—
320#95—22no—
321#95—22no—
322#95—21no—
323#95—21no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95C2×D412yes—
335#95C2×D412yes—
336#95C2×D412yes—
337#95C2×D412yes—
338C24—21no—
339C24C22×D411yescharacteristic
340#41—48no—
341#41—44no—
342#41—44no—
343C2×Q8—21no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—44no—
347C2×Q8—44no—
348C2×Q8—44no—
349C2×Q8C2×#4112yes—
350C2×D4—24no—
351C2×D4—21no—
352C2×D4—22no—
353C2×D4—44no—
354C2×D4C2×#4112yes—
355C2×D42−1+414yes—
356C22×C4—24no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—42no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—44no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—44no—
382C22×C4—82no—
383C22×C4—82no—
384C22×C4—82no—
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D412yes—
388C22×C4C22×D412yes—
389C22×C4C22×D411yescharacteristic
390C22×C4C22×D412yes—
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D412yes—
393C22×C4C22×D411yescharacteristic
394C22×C4C2511yescharacteristic
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—24no—
407C4⋊C4—24no—
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—42no—
415C4⋊C4—44no—
416C4⋊C4—48no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4—82no—
420C4⋊C4—82no—
421C4⋊C4—84no—
422C4⋊C4—84no—
423C4⋊C4—84no—
424C4⋊C4—84no—
425C4⋊C4C2×#4112yes—
426C4⋊C4C2×#4112yes—
427C22⋊C4—24no—
428C22⋊C4—24no—
429C22⋊C4—24no—
430C22⋊C4—24no—
431C22⋊C4—42no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—82no—
435C22⋊C4—84no—
436C42—24no—
437C42—24no—
438C42—22no—
439C42—24no—
440C42—22no—
441C42—22no—
442C42—22no—
443C42—22no—
444C42—21no—
445C42—21no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—42no—
451C42—42no—
452C42—44no—
453C42—44no—
454C42—42no—
455C42—42no—
456C42—84no—
457C42#13714yes—
458C42C2×#4112yes—
459C42C2×#4112yes—
460C23—22no—
461C23—24no—
462C23—22no—
463C23—21no—
464C23—22no—
465C23—21no—
466C23—21no—
467C23—42no—
468C23—41no—
469C23—41no—
470C23C23×D411yescharacteristic
471Q8—24no—
472Q8—41no—
473Q8—41no—
474Q8—88no—
475D4—22no—
476D4—24no—
477D4—48no—
478D4#54114yes—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—48no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—44no—
507C2×C4—48no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—82no—
524C2×C4—88no—
525C2×C4—84no—
526C2×C4—88no—
527C2×C4#52512yes—
528C2×C4#52512yes—
529C2×C4C2×#13714yes—
530C2×C4C2×#13712yes—
531C2×C4#57512yes—
532C2×C4C22×#4111yescharacteristic
533C2×C4C22×#4111yescharacteristic
534C2×C4C2×2−1+412yes—
535C2×C4#58512yes—
536C8—44no—
537C8—44no—
538C22—24no—
539C22—42no—
540C22—42no—
541C22—44no—
542C22—44no—
543C22—41no—
544C22—41no—
545C22—42no—
546C22—42no—
547C22—41no—
548C22C2×#52511yescharacteristic
549C22#348912yes—
550C22C22×#13711yescharacteristic
551C22C2×#57511yescharacteristic
552C22#358512yes—
553C4—22no—
554C4—24no—
555C4—41no—
556C4—42no—
557C4—41no—
558C4—42no—
559C4—42no—
560C4—42no—
561C4—42no—
562C4—41no—
563C4—42no—
564C4—84no—
565C4—82no—
566C4—84no—
567C4—82no—
568C4#311614yes—
569C4C2×#54111yescharacteristic
570C4#345511yescharacteristic
571C4#349512yes—
572C2—44no—
573C2—81no—
574C2#3531011yescharacteristic
575C2C2×#311611yescharacteristic
576C2#6056811yescharacteristic
577C2#6210812yes—
578C2#6211512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35284C2semidirect
3. #33197C2semidirect
4. #33068C2semidirect
6. #29999C2semidirect
7. #29942C2semidirect
8. #29869C2semidirect
10. #27480C2semidirect
11. #27384C2semidirect
12. #27320C2semidirect
13. #22119C2semidirect
14. #21926C2semidirect
15. #21614C2semidirect

Maximal subgroups

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

subgroupindex
#352842
#341232
#331972
#330682
#329582
#299992
#299422
#298692
#275442
#274802
#273842
#273202
#221192
#219262
#216142