Finite Group Databasebeta

← #6480287

Subgroups — #6480287

Order 512.

1674 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35320C211yesmaximal, characteristic, complemented
2#34280C211yesmaximal, characteristic
3#33040C214yesmaximal, complemented
4#30067C212yesmaximal, complemented
5#30039C212yesmaximal, complemented
6#29560C212yesmaximal, complemented
7#29554C212yesmaximal, complemented
8#29361C212yesmaximal, complemented
9#28388C212yesmaximal, complemented
10#27592C212yesmaximal, complemented
11#24304C212yesmaximal, complemented
12#24209C212yesmaximal, complemented
13#23940C212yesmaximal
14#22058C214yesmaximal, complemented
15#20785C211yesmaximal, characteristic, complemented
16C2×#540C2211yescharacteristic
17C2×#533C2211yescharacteristic
18C22×C4⋊Q8C2211yescharacteristic
19C2×#478C2214yes—
20C2×#461C2214yes—
21C2×#448C2214yes—
22#2862C2212yes—
23#2862C2212yes—
24#2642C2212yes—
25#2642C2212yes—
26#2568—22no—
27#2568—22no—
28#2503—22no—
29#2503—22no—
30#2374—22no—
31#2364—22no—
32#2363—22no—
33#2359—22no—
34#2330C2211yescharacteristic
35#2328C2211yescharacteristic
36#2324C2212yes—
37#2314C2211yescharacteristic
38#2310C2212yes—
39#2310C2212yes—
40C4×#127C2211yescharacteristic
41C4×#125C2211yescharacteristic
42C4×#121C2212yes—
43#2293C2212yes—
44#2275C2212yes—
45C4×#117C2211yescharacteristic
46#2097C2214yes—
47#2072C2214yes—
48#2058C2212yes—
49#2057C2212yes—
50#2043C2212yes—
51#2043C2212yes—
52#2026C2214yes—
53#2023C2212yes—
54#2023C2212yes—
55#2014C2214yes—
56#2013C2212yes—
57#2013C2212yes—
58#1989C2214yes—
59#1981C2214yes—
60#1973C2214yes—
61#1968C2214yes—
62#1935C2214yes—
63#1933C2214yes—
64#1928C2214yes—
65#1928C2214yes—
66#1927C2214yes—
67#1927C2214yes—
68#1877C2214yes—
69#1877C2214yes—
70#1877C2214yes—
71#1877C2214yes—
72#1875C2214yes—
73#1854C2214yes—
74#1839C2212yes—
75#1832C2212yes—
76#1818C2212yes—
77#1797C2212yes—
78#1787C2214yes—
79#1768C2212yes—
80#1750C2212yes—
81#1745C2211yescharacteristic
82#1744C2211yescharacteristic
83C23×Q8C2312yes—
84C22×SD16C2314yes—
85C22×SD16C2314yes—
86#540—28no—
87#533—44no—
88C2×C4⋊Q8—22no—
89C2×C4⋊Q8—44no—
90C2×C4⋊Q8C2312yes—
91C2×C4⋊Q8C2312yes—
92C2×C4⋊Q8C2312yes—
93C2×#127C2311yescharacteristic
94C2×#125C2312yes—
95C2×#125C2311yescharacteristic
96C2×#123—22no—
97C2×C22⋊Q8—22no—
98C2×#121—22no—
99C2×#121C2312yes—
100C2×C4×Q8—22no—
101C2×C4×D4—22no—
102C2×C4×D4—22no—
103C2×C4×D4C2312yes—
104C2×C4×D4C2312yes—
105C2×C4×D4C2312yes—
106C2×#117—22no—
107C2×#117C2312yes—
108C2×#117C2311yescharacteristic
109C2×#117C2311yescharacteristic
110C22×C4⋊C4—22no—
111C22×C42C2311yescharacteristic
112#478—48no—
113#461—48no—
114#448—216no—
115C2×#107C2314yes—
116C2×#106C2314yes—
117C2×C4⋊C8C2314yes—
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×C22⋊C8C2314yes—
127C2×#97C2312yes—
128C2×C4×C8C2312yes—
129#400—24no—
130#400—24no—
131#400—24no—
132#400—24no—
133#391—24no—
134#391—22no—
135#391—22no—
136#391—22no—
137#391—22no—
138#391—22no—
139#391—22no—
140#391C2314yes—
141#391C2314yes—
142#391C2314yes—
143#391C2314yes—
144#390—22no—
145#390—22no—
146#390C2312yes—
147#390C2312yes—
148#389—22no—
149#389C2312yes—
150#389C2312yes—
151#388—22no—
152#388—22no—
153#388—22no—
154#388—22no—
155#388—22no—
156#387—22no—
157#387—22no—
158#387—22no—
159#386—22no—
160#386—22no—
161#386—22no—
162#386—22no—
163#385—22no—
164#384C2311yescharacteristic
165#382—22no—
166#382—22no—
167#382—22no—
168#382—22no—
169#382C2311yescharacteristic
170#381—24no—
171#381—22no—
172#381—22no—
173#380C2312yes—
174C4×C4⋊C4—22no—
175C4×C4⋊C4D412yes—
176C4×C4⋊C4D412yes—
177C4×C4⋊C4C2312yes—
178C4×C4⋊C4C2312yes—
179C4×C4⋊C4C2312yes—
180C4×C22⋊C4—22no—
181C4×C22⋊C4—22no—
182C4×C22⋊C4—22no—
183C4×C22⋊C4—22no—
184C4×C22⋊C4—22no—
185C4×C22⋊C4D412yes—
186C4×C22⋊C4D412yes—
187#376C2311yescharacteristic
188#376C2311yescharacteristic
189C2×#95—22no—
190C43C2311yescharacteristic
191C43C2311yescharacteristic
192#340C2314yes—
193#340C2314yes—
194#340C2314yes—
195#340C2314yes—
196#336C2312yes—
197#336C2312yes—
198#336C2314yes—
199#336C2312yes—
200#336C2312yes—
201#336C2314yes—
202C22×Q8—22no—
203C22×Q8—22no—
204C22×Q8—28no—
205C22×Q8—42no—
206C22×Q8—42no—
207C22×Q8—44no—
208C22×Q8C2414yes—
209C22×D4—22no—
210C22×D4—22no—
211C22×D4C2412yes—
212C23×C4—22no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4C2×D412yes—
217C23×C4C2411yescharacteristic
218C2×SD16—28no—
219C2×SD16—28no—
220C2×SD16—44no—
221C2×SD16—44no—
222C2×SD16—44no—
223C2×SD16—44no—
224C22×C8C2414yes—
225C22×C8C2414yes—
226C4⋊Q8—28no—
227C4⋊Q8—44no—
228C4⋊Q8—44no—
229C4⋊Q8—48no—
230#127—24no—
231#125—24no—
232#125—28no—
233#123—216no—
234C22⋊Q8—48no—
235#121—216no—
236#121—24no—
237#121#4118yes—
238C4×Q8—48no—
239C4×D4—24no—
240C4×D4—24no—
241C4×D4—28no—
242C4×D4—44no—
243C4×D4—84no—
244C4×D4—84no—
245#117—24no—
246#117—42no—
247#117—42no—
248#117—42no—
249#117—84no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—48no—
260C2×C4⋊C4C2×D412yes—
261C2×C4⋊C4C2×D411yescharacteristic
262C2×C4⋊C4C2×D411yescharacteristic
263C2×C4⋊C4C2414yes—
264C2×C4⋊C4C2412yes—
265C2×C4⋊C4C2412yes—
266C2×C4⋊C4C2412yes—
267C2×C22⋊C4—22no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4C2×D412yes—
278C2×C22⋊C4C2×D411yescharacteristic
279C2×C22⋊C4C2×D411yescharacteristic
280C2×C42—24no—
281C2×C42—22no—
282C2×C42—22no—
283C2×C42—22no—
284C2×C42—22no—
285C2×C42C2×D412yes—
286C2×C42C2×D412yes—
287C2×C42C2×D412yes—
288C2×C42C2×D412yes—
289C2×C42C2×D412yes—
290C2×C42C2411yescharacteristic
291C2×C42C2411yescharacteristic
292C2×C42C2411yescharacteristic
293C2×C42C2411yescharacteristic
294C2×C42C2411yescharacteristic
295C2×C42C2411yescharacteristic
296#107—44no—
297#106—44no—
298C4⋊C8—28no—
299#103—28no—
300#103—28no—
301#103—44no—
302#103—44no—
303#103—44no—
304#103—44no—
305#102—44no—
306#102—44no—
307C22⋊C8—28no—
308#97—42no—
309C4×C8—42no—
310#95—22no—
311#95—22no—
312#95—22no—
313#95—22no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—24no—
318#95—24no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95C2×D412yes—
326#95C2×D412yes—
327#95C2×D412yes—
328#95C2×D412yes—
329C24—22no—
330C24C22×D411yescharacteristic
331C2×Q8—24no—
332C2×Q8—24no—
333C2×Q8—24no—
334C2×Q8—24no—
335C2×Q8—24no—
336C2×Q8—24no—
337C2×Q8—44no—
338C2×Q8—44no—
339C2×Q8—44no—
340C2×Q8—44no—
341C2×Q8—44no—
342C2×Q8—44no—
343C2×Q8—88no—
344C2×Q8—88no—
345C2×D4—28no—
346C2×D4—28no—
347C2×D4—22no—
348C2×D4—24no—
349C2×D4—24no—
350C2×D4—82no—
351C2×D4—82no—
352C2×D4—82no—
353C2×D4—82no—
354C2×D4C2×#4114yes—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—28no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—42no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—82no—
372C22×C4—82no—
373C22×C4—82no—
374C22×C4—82no—
375C22×C4—82no—
376C22×C4—82no—
377C22×C4C22×D411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D412yes—
380C22×C4C22×D412yes—
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D412yes—
385C22×C4C22×D412yes—
386C22×C4C2×#4114yes—
387C22×C4C2511yescharacteristic
388SD16—88no—
389SD16—88no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C4⋊C4—28no—
397C4⋊C4—24no—
398C4⋊C4—22no—
399C4⋊C4—24no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—48no—
409C4⋊C4—48no—
410C4⋊C4—44no—
411C4⋊C4—82no—
412C4⋊C4—82no—
413C4⋊C4C2×#4114yes—
414C22⋊C4—28no—
415C22⋊C4—24no—
416C22⋊C4—28no—
417C22⋊C4—41no—
418C22⋊C4—41no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C22⋊C4—44no—
422C22⋊C4—44no—
423C22⋊C4—44no—
424C22⋊C4—82no—
425C22⋊C4—82no—
426C22⋊C4—82no—
427C42—24no—
428C42—24no—
429C42—24no—
430C42—22no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—42no—
436C42—42no—
437C42—44no—
438C42—82no—
439C42—82no—
440C42—82no—
441C42H(5,𝔽2)14yes—
442C23—22no—
443C23—21no—
444C23—21no—
445C23—22no—
446C23—24no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23#52512yes—
454C23C23×D411yescharacteristic
455Q8—44no—
456Q8—44no—
457Q8—44no—
458Q8—44no—
459Q8—88no—
460Q8—88no—
461D4—28no—
462D4—44no—
463D4—88no—
464D4—88no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—28no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—48no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—88no—
503C2×C4—88no—
504C2×C4—164no—
505C2×C4—164no—
506C2×C4#52512yes—
507C2×C4#53514yes—
508C2×C4#57714yes—
509C2×C4C22×#4112yes—
510C2×C4C2×H(5,𝔽2)12yes—
511C2×C4#58512yes—
512C8—44no—
513C8—44no—
514C22—22no—
515C22—24no—
516C22—41no—
517C22—41no—
518C22—42no—
519C22—44no—
520C22—44no—
521C22—44no—
522C22—82no—
523C22—82no—
524C22—82no—
525C22—82no—
526C22C2×#52511yescharacteristic
527C22C2×#53511yescharacteristic
528C22#344911yescharacteristic
529C22C2×#57712yes—
530C22#358612yes—
531C4—22no—
532C4—24no—
533C4—24no—
534C4—41no—
535C4—41no—
536C4—41no—
537C4—41no—
538C4—42no—
539C4—42no—
540C4—44no—
541C4—84no—
542C4—82no—
543C4—82no—
544C4—84no—
545C4#351414yes—
546C2—42no—
547C2—84no—
548C2#3512911yescharacteristic
549C2#6059112yes—
550C2#6081312yes—
551C2#6101211yescharacteristic
552C2#6102011yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#540C22
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. #35320C2semidirect
3. #33040C2semidirect
4. #30067C2semidirect
5. #30039C2semidirect
6. #29560C2semidirect
7. #29554C2semidirect
8. #29361C2semidirect
9. #28388C2semidirect
10. #27592C2semidirect
11. #24304C2semidirect
12. #24209C2semidirect
14. #22058C2semidirect
15. #20785C2semidirect

Maximal subgroups

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

subgroupindex
#353202
#342802
#330402
#300672
#300392
#295602
#295542
#293612
#283882
#275922
#243042
#242092
#239402
#220582
#207852