Finite Group Databasebeta

← #6530031

Subgroups — #6530031

Order 512.

1432 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34966C211yesmaximal, characteristic
2#34225C211yesmaximal, characteristic
3#29288C212yesmaximal, complemented
4#29287C212yesmaximal, complemented
5#29276C212yesmaximal, complemented
6#29275C212yesmaximal, complemented
7#26517C214yesmaximal, complemented
8#26449C214yesmaximal, complemented
9#26013C214yesmaximal
10#26009C214yesmaximal
11#21974C211yesmaximal, characteristic
12#21963C211yesmaximal, characteristic
13#21837C211yesmaximal, characteristic
14#21359C211yesmaximal, characteristic
15#21312C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#423C2211yescharacteristic
19#2606—21no—
20#2605—21no—
21#2570C2212yes—
22#2550C2212yes—
23#2442C2212yes—
24#2440C2212yes—
25#2438—22no—
26#2402—21no—
27#2399—21no—
28#2370C2214yes—
29#2362—22no—
30#2356—21no—
31#2351—21no—
32#2347C2212yes—
33#2346C2212yes—
34#2323—22no—
35#2320—21no—
36C4×C4⋊Q8C2212yes—
37C4×C4⋊Q8C2212yes—
38C4×C4⋊Q8C2212yes—
39C4×#125C2212yes—
40C4×C22⋊Q8—22no—
41#2285C2211yescharacteristic
42#2283C2211yescharacteristic
43#2276—21no—
44C4×#117C2211yescharacteristic
45C2×C4×C4⋊C4C2211yescharacteristic
46#2101C2214yes—
47#2100C2214yes—
48#2088C2214yes—
49#2088C2214yes—
50#2088C2214yes—
51#2087C2214yes—
52#2059C2214yes—
53#2058C2214yes—
54#1991C2214yes—
55#1989C2212yes—
56#1989C2212yes—
57#1989C2214yes—
58#1988C2212yes—
59#1988C2211yescharacteristic
60#1988C2211yescharacteristic
61#1988C2214yes—
62#1987C2212yes—
63#1987C2211yescharacteristic
64#1987C2211yescharacteristic
65#1949C2214yes—
66#1949C2214yes—
67#1948C2212yes—
68#1948C2212yes—
69#1948C2212yes—
70#1948C2214yes—
71#1947C2212yes—
72#1939C2214yes—
73#1920C2214yes—
74#1920C2214yes—
75#1920C2214yes—
76#1918C2214yes—
77#1853C2214yes—
78#1851C2212yes—
79#1851C2212yes—
80#1842C2212yes—
81#1831C2212yes—
82#1825C2214yes—
83#1809C2214yes—
84C4×#106C2212yes—
85C4×#106C2212yes—
86#1736C2214yes—
87C22×M16C2312yes—
88#533—24no—
89#533Q818yescomplemented
90C2×C4⋊Q8C2312yes—
91C2×C4⋊Q8C2311yescharacteristic
92C2×C4⋊Q8C2311yescharacteristic
93C2×#125C2312yes—
94C2×#125C2312yes—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—22no—
98C2×C4×Q8—21no—
99C2×C4×Q8—21no—
100C2×C4×Q8—22no—
101C2×C4×Q8—22no—
102C2×C4×Q8—22no—
103C2×#117—21no—
104C2×#117D412yes—
105C2×#117C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C42—21no—
108C22×C42C2311yescharacteristic
109#431—24no—
110C2×#107C2314yes—
111C2×#106C2314yes—
112C2×#106C2314yes—
113C2×#106C2314yes—
114#423—24no—
115C2×C4⋊C8C2311yescharacteristic
116C2×C4⋊C8C2312yes—
117C2×C4⋊C8C2311yescharacteristic
118C2×C4⋊C8C2312yes—
119C2×C4⋊C8C2314yes—
120C2×C4⋊C8C2314yes—
121C2×C22⋊C8C2312yes—
122C2×C4×C8C2312yes—
123#400—22no—
124#400—22no—
125#398—22no—
126#398—22no—
127#398—22no—
128#398—22no—
129#395—22no—
130#395—22no—
131#391—22no—
132#391—22no—
133#391—22no—
134#391—22no—
135#389—24no—
136#389—22no—
137#389—22no—
138#389—22no—
139#389—22no—
140#389C2312yes—
141#389C2312yes—
142#389C2314yes—
143#389C2312yes—
144#389C2314yes—
145#389C2312yes—
146#388—24no—
147#388—24no—
148#387—24no—
149#387—22no—
150#387—22no—
151#387—22no—
152#387—22no—
153#387—24no—
154#385—22no—
155#385—22no—
156#384C2312yes—
157#384C2312yes—
158#383—21no—
159#383—22no—
160#382—21no—
161#382—22no—
162#382C2312yes—
163#382C2312yes—
164#381—21no—
165#380—21no—
166#380—22no—
167#380C2312yes—
168#380C2312yes—
169C4×C4⋊C4—22no—
170C4×C4⋊C4—22no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4—22no—
174C4×C4⋊C4—21no—
175C4×C4⋊C4—21no—
176C4×C4⋊C4—21no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—24no—
179C4×C4⋊C4D411yescharacteristic
180C4×C4⋊C4D411yescharacteristic
181C4×C4⋊C4C2311yescharacteristic
182C4×C4⋊C4C2312yes—
183C4×C4⋊C4C2311yescharacteristic
184C4×C4⋊C4C2314yes—
185C4×C4⋊C4C2314yes—
186C4×C4⋊C4C2311yescharacteristic
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2311yescharacteristic
189C4×C4⋊C4C2312yes—
190C4×C22⋊C4—21no—
191C4×C22⋊C4D412yes—
192C4×C22⋊C4C2311yescharacteristic
193C4×C22⋊C4C2311yescharacteristic
194#376—21no—
195#376—22no—
196#376D411yescharacteristic
197#376D411yescharacteristic
198C2×#95—21no—
199C43C2312yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#336C2314yes—
213#336C2314yes—
214C22×Q8—21no—
215C22×Q8—21no—
216C22×Q8—22no—
217C23×C4—22no—
218C23×C4C2×D411yescharacteristic
219C23×C4C2×D411yescharacteristic
220C23×C4C2411yescharacteristic
221C2×M16—24no—
222C2×M16—44no—
223C22×C8C2414yes—
224C22×C8C2412yes—
225C22×C8C2412yes—
226C4⋊Q8—24no—
227C4⋊Q8—22no—
228C4⋊Q8—22no—
229C4⋊Q8—42no—
230C4⋊Q8C2×Q814yes—
231C4⋊Q8C2×Q814yes—
232#125—24no—
233#125—24no—
234#125—42no—
235#125C2×Q818yes—
236C22⋊Q8—28no—
237C22⋊Q8—28no—
238C22⋊Q8—48no—
239C4×Q8—28no—
240C4×Q8—28no—
241C4×Q8—48no—
242C4×Q8—84no—
243C4×Q8—84no—
244#117—24no—
245#117—44no—
246#117—82no—
247#117C2×Q818yes—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4C2×D412yes—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2412yes—
271C2×C4⋊C4C2412yes—
272C2×C4⋊C4C2412yes—
273C2×C4⋊C4C2414yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4C2×D412yes—
281C2×C22⋊C4C2×D412yes—
282C2×C42—22no—
283C2×C42—21no—
284C2×C42—24no—
285C2×C42—22no—
286C2×C42—21no—
287C2×C42—22no—
288C2×C42—24no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—82no—
292C2×C42C2×D412yes—
293C2×C42C2×D412yes—
294C2×C42C2×D412yes—
295C2×C42C2×D412yes—
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×Q814yes—
299C2×C42#4114yes—
300C2×C42C2412yes—
301C2×C42C2412yes—
302C2×C42C2411yescharacteristic
303C2×C42C2411yescharacteristic
304#107—44no—
305#106—28no—
306#106—44no—
307#106—44no—
308C4⋊C8—24no—
309C4⋊C8—24no—
310C4⋊C8—22no—
311C4⋊C8—22no—
312C4⋊C8—44no—
313C4⋊C8—44no—
314C22⋊C8—24no—
315C4×C8—24no—
316#95—22no—
317#95—21no—
318#95—21no—
319#95—22no—
320#95—24no—
321#95—22no—
322#95—22no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—24no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—21no—
333#95—21no—
334#95—24no—
335#95—24no—
336#95C2×D411yescharacteristic
337#95C2×D412yes—
338#95C2×D412yes—
339#95C2×D411yescharacteristic
340#95C2×D411yescharacteristic
341#95C2×D411yescharacteristic
342C24C22×D411yescharacteristic
343C2×Q8—24no—
344C2×Q8—24no—
345C2×Q8—44no—
346C2×Q8—44no—
347C2×Q8—44no—
348C2×Q8—42no—
349C2×Q8—42no—
350C2×Q8—82no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—42no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—82no—
366C22×C4C22⋊Q814yes—
367C22×C4C22⋊Q814yes—
368C22×C4C22×D412yes—
369C22×C4C22×D412yes—
370C22×C4C22×D412yes—
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D412yes—
376C22×C4C22×D411yescharacteristic
377C22×C4C22×Q812yes—
378C22×C4C2×#4112yes—
379C22×C4C2511yescharacteristic
380M16—84no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—24no—
388C4⋊C4—28no—
389C4⋊C4—28no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—24no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—24no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—22no—
404C4⋊C4—42no—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—44no—
410C4⋊C4—48no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C4⋊C4—82no—
416C4⋊C4—82no—
417C4⋊C4—84no—
418C4⋊C4—82no—
419C4⋊C4C22×Q814yes—
420C4⋊C4C22×Q814yes—
421C4⋊C42−1+414yes—
422C22⋊C4—28no—
423C22⋊C4—24no—
424C22⋊C4—42no—
425C22⋊C4—44no—
426C22⋊C4—82no—
427C42—28no—
428C42—24no—
429C42—24no—
430C42—24no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—42no—
436C42—48no—
437C42—42no—
438C42—44no—
439C42—42no—
440C42—81no—
441C42—81no—
442C42—82no—
443C42—84no—
444C42C22⋊Q814yes—
445C42C22⋊Q814yes—
446C42C22×Q812yes—
447C42C22×Q812yes—
448C42C2×#4112yes—
449C42C2×#4112yes—
450C23—22no—
451C23—22no—
452C23—21no—
453C23—21no—
454C23C2×C22⋊Q812yes—
455C23C23×D411yescharacteristic
456Q8—48no—
457Q8—48no—
458Q8—88no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—28no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—88no—
501C2×C4C2×C22⋊Q812yes—
502C2×C4C2×C22⋊Q812yes—
503C2×C4C2×C22⋊Q812yes—
504C2×C4C2×C22⋊Q812yes—
505C2×C4C2×C22⋊Q812yes—
506C2×C4#57514yes—
507C2×C4#57712yes—
508C2×C4#57712yes—
509C2×C4C23×Q811yescharacteristic
510C2×C4C22×#4111yescharacteristic
511C2×C4C2×2−1+412yes—
512C2×C4#58512yes—
513C8—42no—
514C8—42no—
515C8—44no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—42no—
520C22—42no—
521C22C22×C22⋊Q811yescharacteristic
522C22#348912yes—
523C22C2×#57512yes—
524C22C2×#57711yescharacteristic
525C22C2×#57711yescharacteristic
526C4—22no—
527C4—22no—
528C4—22no—
529C4—22no—
530C4—24no—
531C4—44no—
532C4—44no—
533C4—42no—
534C4—44no—
535C4—42no—
536C4—42no—
537C4—82no—
538C4—82no—
539C4—82no—
540C4—84no—
541C4#308414yes—
542C4#350212yes—
543C4#350911yescharacteristic
544C4#351011yescharacteristic
545C2—42no—
546C2#3566211yescharacteristic
547C2#6048411yescharacteristic
548C2C2×#308411yescharacteristic
549C2#6212612yes—
550C2#6214512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
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
3. #29288C2semidirect
4. #29287C2semidirect
5. #29276C2semidirect
6. #29275C2semidirect
7. #26517C2semidirect
8. #26449C2semidirect
89. #533Q8semidirect

Maximal subgroups

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

subgroupindex
#349662
#342252
#292882
#292872
#292762
#292752
#265172
#264492
#260132
#260092
#219742
#219632
#218372
#213592
#213122