Finite Group Databasebeta

← #6459007

Subgroups — #6459007

Order 512.

1705 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34268C211yesmaximal, characteristic
2#34188C211yesmaximal, characteristic
3#27680C212yesmaximal, complemented
4#27652C212yesmaximal, complemented
5#27642C214yesmaximal, complemented
6#27615C212yesmaximal, complemented
7#27609C212yesmaximal
8#27583C212yesmaximal, complemented
9#24016C212yesmaximal
10#23763C212yesmaximal, complemented
11#23756C212yesmaximal, complemented
12#21591C214yesmaximal, complemented
13#21369C212yesmaximal, complemented
14#20745C212yesmaximal, complemented
15#20720C211yesmaximal, characteristic
16C2×#536C2211yescharacteristic
17C2×#519C2211yescharacteristic
18C2×C4×#41C2211yescharacteristic
19C2×#485C2214yes—
20C2×#433C2212yes—
21#2367—22no—
22#2365—22no—
23#2359—22no—
24#2359—22no—
25#2355—22no—
26#2354—22no—
27#2352—22no—
28#2352—22no—
29#2323C2212yes—
30#2323C2212yes—
31#2314C2212yes—
32#2314C2212yes—
33#2311C2212yes—
34#2311C2211yescharacteristic
35C4×C22⋊Q8C2212yes—
36C4×C22⋊Q8C2212yes—
37#2296C2211yescharacteristic
38C2×#391C2211yescharacteristic
39#2285C2212yes—
40#2285C2212yes—
41#2285C2211yescharacteristic
42C2×#384C2211yescharacteristic
43#2274C2212yes—
44#2274C2211yescharacteristic
45#2099C2212yes—
46#2099C2212yes—
47#2096C2212yes—
48#2096C2212yes—
49#2084C2212yes—
50#2083C2212yes—
51#2077C2214yes—
52#2077C2214yes—
53#2072C2212yes—
54#2072C2212yes—
55#2036C2214yes—
56#2035C2214yes—
57#1949C2214yes—
58#1948C2214yes—
59#1927C2214yes—
60#1927C2214yes—
61#1921C2214yes—
62#1919C2214yes—
63#1872C2214yes—
64#1854C2214yes—
65#1853C2214yes—
66#1853C2214yes—
67#1825C2214yes—
68#1825C2214yes—
69#1817C2212yes—
70#1805C2214yes—
71#1803C2214yes—
72#1798C2212yes—
73#1797C2212yes—
74#1787C2214yes—
75#1780C2212yes—
76#1776C2214yes—
77C4×#106C2214yes—
78#1763C2214yes—
79C4×#103C2214yes—
80#1750C2212yes—
81#1749C2211yescharacteristic
82C4×C22⋊C8C2211yescharacteristic
83C22×#41C2311yescharacteristic
84C23×Q8C2311yescharacteristic
85#536—28no—
86C2×C4⋊Q8—21no—
87C2×#124—21no—
88C2×#123—21no—
89C2×#123—21no—
90C2×C22⋊Q8C2312yes—
91C2×C22⋊Q8C2312yes—
92#519—28no—
93C4×#41—28no—
94C2×C4×Q8—22no—
95C2×C4×Q8D412yes—
96C2×C4×Q8C2312yes—
97C2×C4×Q8C2312yes—
98C2×C4×D4D412yes—
99C2×C4×D4D412yes—
100C2×C4×D4D412yes—
101C2×#117—22no—
102C2×#117—22no—
103C2×#117—22no—
104C2×#117C2312yes—
105C2×#117C2311yescharacteristic
106C2×#117C2311yescharacteristic
107C2×#117C2311yescharacteristic
108C2×#117C2311yescharacteristic
109C2×#117C2311yescharacteristic
110C22×C4⋊C4—22no—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2312yes—
113C22×C42C2311yescharacteristic
114C22×C42C2311yescharacteristic
115C22×C42C2311yescharacteristic
116#485—216no—
117#433—28no—
118C2×#107C2314yes—
119C2×#106C2314yes—
120C2×#106C2314yes—
121C2×#106C2314yes—
122C2×C4⋊C8C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×C22⋊C8C2314yes—
128C2×C22⋊C8C2314yes—
129C2×C22⋊C8C2312yes—
130C2×C22⋊C8C2312yes—
131C2×#97C2312yes—
132C2×C4×C8C2312yes—
133#391—22no—
134#391—21no—
135#391—22no—
136#391C2312yes—
137#391C2312yes—
138#390—22no—
139#390—21no—
140#389—22no—
141#389—22no—
142#389C2312yes—
143#389C2312yes—
144#388—24no—
145#388—24no—
146#388—22no—
147#388—22no—
148#388—22no—
149#388—22no—
150#387—24no—
151#387—24no—
152#387—22no—
153#387—22no—
154#387—22no—
155#387—22no—
156#386—22no—
157#386—22no—
158#386—21no—
159#386—21no—
160#385—22no—
161#385—22no—
162#385—22no—
163#384—22no—
164#384C2312yes—
165#384C2312yes—
166#384C2312yes—
167#382—22no—
168#382—22no—
169#382C2312yes—
170#380—22no—
171#380—22no—
172#380—22no—
173#380C2314yes—
174#380C2314yes—
175#380C2312yes—
176#380C2312yes—
177C4×C4⋊C4—24no—
178C4×C4⋊C4—22no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4—22no—
185C4×C4⋊C4C2314yes—
186C4×C4⋊C4C2314yes—
187C4×C4⋊C4C2312yes—
188C4×C4⋊C4C2312yes—
189C4×C22⋊C4—22no—
190C4×C22⋊C4—22no—
191C4×C22⋊C4—24no—
192C4×C22⋊C4—22no—
193C2×#95—22no—
194#340C2314yes—
195#340C2314yes—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#336C2312yes—
203#336C2312yes—
204#336C2314yes—
205C2×#41—24no—
206C2×#41—24no—
207C2×#41—42no—
208C22×Q8—24no—
209C22×Q8—21no—
210C22×Q8—24no—
211C22×Q8—42no—
212C22×Q8C2×D411yescharacteristic
213C22×Q8C2412yes—
214C22×D4C2×D411yescharacteristic
215C22×D4C2×D411yescharacteristic
216C22×D4C2×D411yescharacteristic
217C23×C4—21no—
218C23×C4—22no—
219C23×C4—21no—
220C23×C4C2×D412yes—
221C23×C4C2×D412yes—
222C23×C4C2×D412yes—
223C23×C4C2411yescharacteristic
224C23×C4C2411yescharacteristic
225C23×C4C2411yescharacteristic
226C22×C8C2414yes—
227C22×C8C2414yes—
228C4⋊Q8—28no—
229#124—28no—
230#123—28no—
231#123—28no—
232C22⋊Q8—24no—
233C22⋊Q8—24no—
234C22⋊Q8—24no—
235C22⋊Q8#4118yes—
236C4×Q8—28no—
237C4×Q8—28no—
238C4×Q8—28no—
239C4×Q8—48no—
240C4×D4—28no—
241C4×D4—28no—
242C4×D4—28no—
243#117—24no—
244#117—24no—
245#117—24no—
246#117—24no—
247#117—28no—
248#117—48no—
249#117—84no—
250#117—84no—
251#117#4118yes—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—42no—
266C2×C4⋊C4—42no—
267C2×C4⋊C4—42no—
268C2×C4⋊C4—84no—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2×D412yes—
272C2×C4⋊C4C2414yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2414yes—
276C2×C4⋊C4C2412yes—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—21no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4C2×D412yes—
287C2×C22⋊C4C2×D412yes—
288C2×C22⋊C4C2×D412yes—
289C2×C42—21no—
290C2×C42—24no—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—22no—
294C2×C42—22no—
295C2×C42—21no—
296C2×C42—22no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42C2×D412yes—
302C2×C42C2×D412yes—
303C2×C42C2×D412yes—
304C2×C42C2411yescharacteristic
305C2×C42C2411yescharacteristic
306C2×C42C2411yescharacteristic
307C2×C42C2411yescharacteristic
308#107—28no—
309#106—28no—
310#106—44no—
311#106—44no—
312C4⋊C8—28no—
313#103—28no—
314#103—28no—
315#103—44no—
316#103—44no—
317C22⋊C8—28no—
318C22⋊C8—28no—
319C22⋊C8—42no—
320C22⋊C8—42no—
321#97—24no—
322C4×C8—24no—
323#95—22no—
324#95—22no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—24no—
329#95—24no—
330#95—22no—
331#95—22no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95—22no—
336#95—22no—
337#95—22no—
338C24C22×D411yescharacteristic
339C24C22×D411yescharacteristic
340C24C22×D411yescharacteristic
341#41—416no—
342C2×Q8—24no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—22no—
346C2×Q8—24no—
347C2×Q8—24no—
348C2×Q8—41no—
349C2×Q8—416no—
350C2×Q8—42no—
351C2×Q8—42no—
352C2×Q8C2×#4114yes—
353C2×D4—22no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—41no—
360C2×D4—41no—
361C2×D4—41no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—44no—
388C22×C4—42no—
389C22×C4—84no—
390C22×C4—82no—
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D412yes—
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D412yes—
395C22×C4C22×D412yes—
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D412yes—
398C22×C4C2×#4114yes—
399C22×C4C2×#4114yes—
400C22×C4C2511yescharacteristic
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C4⋊C4—28no—
408C4⋊C4—24no—
409C4⋊C4—24no—
410C4⋊C4—28no—
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—42no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—82no—
428C4⋊C4—84no—
429C4⋊C4#12318yes—
430C4⋊C4C2×#4114yes—
431C22⋊C4—24no—
432C22⋊C4—28no—
433C22⋊C4—28no—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—42no—
437C22⋊C4—44no—
438C22⋊C4—82no—
439C22⋊C4—82no—
440C22⋊C4—84no—
441C22⋊C4—84no—
442C22⋊C4#12318yes—
443C42—24no—
444C42—24no—
445C42—24no—
446C42—24no—
447C42—22no—
448C42—22no—
449C42—42no—
450C42—44no—
451C42—44no—
452C42—44no—
453C42—82no—
454C42—82no—
455C42—84no—
456C42—84no—
457C42C2×#4114yes—
458C42C2×#4114yes—
459C23—22no—
460C23—22no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—22no—
472C23C2×#12312yes—
473C23#52512yes—
474C23C23×D411yescharacteristic
475Q8—28no—
476Q8—44no—
477Q8—48no—
478Q8—44no—
479D4—44no—
480D4—44no—
481D4—44no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—28no—
493C2×C4—28no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—48no—
503C2×C4—48no—
504C2×C4—48no—
505C2×C4—48no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—164no—
524C2×C4—162no—
525C2×C4C2×#12312yes—
526C2×C4C2×#12314yes—
527C2×C4C2×#12314yes—
528C2×C4#52512yes—
529C2×C4#57714yes—
530C2×C4C22×#4112yes—
531C2×C4C22×#4112yes—
532C2×C4#58512yes—
533C8—44no—
534C8—44no—
535C22—24no—
536C22—24no—
537C22—22no—
538C22—22no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—42no—
544C22—42no—
545C22—41no—
546C22—41no—
547C22—42no—
548C22—42no—
549C22—41no—
550C22C22×#12311yescharacteristic
551C22C2×#52511yescharacteristic
552C22#344911yescharacteristic
553C22C2×#57712yes—
554C22#358612yes—
555C4—24no—
556C4—24no—
557C4—24no—
558C4—22no—
559C4—22no—
560C4—42no—
561C4—44no—
562C4—41no—
563C4—44no—
564C4—84no—
565C4—82no—
566C4—82no—
567C4—84no—
568C4#347514yes—
569C2—42no—
570C2—42no—
571C2—81no—
572C2#3480911yescharacteristic
573C2#6052212yes—
574C2#6059912yes—
575C2#6101211yescharacteristic
576C2#6101811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27680C2semidirect
4. #27652C2semidirect
5. #27642C2semidirect
6. #27615C2semidirect
8. #27583C2semidirect
10. #23763C2semidirect
11. #23756C2semidirect
12. #21591C2semidirect
13. #21369C2semidirect
14. #20745C2semidirect

Maximal subgroups

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

subgroupindex
#342682
#341882
#276802
#276522
#276422
#276152
#276092
#275832
#240162
#237632
#237562
#215912
#213692
#207452
#207202