Finite Group Databasebeta

← #6434539

Subgroups — #6434539

Order 512.

1714 conjugacy classes of subgroups.

The subgroup lattice has 571 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35321C211yesmaximal, characteristic, complemented
2#34207C211yesmaximal, characteristic
3#30132C212yesmaximal, complemented
4#30127C212yesmaximal, complemented
5#30090C212yesmaximal, complemented
6#30054C212yesmaximal, complemented
7#29965C212yesmaximal, complemented
8#29908C212yesmaximal, complemented
9#29726C212yesmaximal, complemented
10#29629C212yesmaximal
11#23824C211yesmaximal, characteristic, complemented
12#23820C211yesmaximal, characteristic, complemented
13#23805C211yesmaximal, characteristic, complemented
14#23799C211yesmaximal, characteristic, complemented
15#23694C214yesmaximal, complemented
16#21461C214yesmaximal, complemented
17#20780C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C2×#520C2212yes—
20C2×#452C2214yes—
21C2×#441C2214yes—
22C2×C4×SD16C2214yes—
23#2863C2212yes—
24#2855C2212yes—
25#2647C2211yescharacteristic
26#2642C2211yescharacteristic
27#2629C2211yescharacteristic
28#2621C2211yescharacteristic
29#2585—22no—
30#2573—22no—
31#2524—22no—
32#2515—22no—
33#2361—22no—
34#2361—22no—
35#2354—22no—
36D4×C4⋊C4—22no—
37#2326C2212yes—
38#2326C2212yes—
39#2323C2212yes—
40#2320C2212yes—
41#2311C2212yes—
42C4×C4⋊Q8C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×#127C2211yescharacteristic
46C4×#127C2211yescharacteristic
47C4×#125C2211yescharacteristic
48C2×#384C2211yescharacteristic
49C4×#117C2211yescharacteristic
50#2099C2212yes—
51#2099C2212yes—
52#2096C2212yes—
53#2096C2212yes—
54#2096C2212yes—
55#2094C2212yes—
56#2084C2214yes—
57#2074C2212yes—
58#2073C2212yes—
59#2072C2214yes—
60#2066C2214yes—
61#2065C2214yes—
62#2056C2214yes—
63#2055C2214yes—
64#2043C2212yes—
65#2042C2214yes—
66#2041C2212yes—
67#2026C2212yes—
68#2025C2212yes—
69#2016C2212yes—
70#2016C2212yes—
71#2015C2214yes—
72#2014C2212yes—
73#2013C2212yes—
74#1878C2214yes—
75#1877C2214yes—
76#1872C2214yes—
77#1865C2214yes—
78#1864C2214yes—
79#1853C2214yes—
80#1840C2212yes—
81#1831C2212yes—
82#1817C2212yes—
83#1798C2212yes—
84#1788C2214yes—
85#1768C2212yes—
86#1762C2214yes—
87C4×#103C2214yes—
88#1750C2212yes—
89#1745C2211yescharacteristic
90#1744C2211yescharacteristic
91C22×#41C2312yes—
92C22×Q16C2314yes—
93C22×SD16C2314yes—
94#532—28no—
95C2×C4⋊Q8C2311yescharacteristic
96C2×C4⋊Q8C2311yescharacteristic
97C2×#127C2311yescharacteristic
98C2×#125C2311yescharacteristic
99C2×#124C2312yes—
100C2×C22⋊Q8C2312yes—
101C2×#121—22no—
102#520—48no—
103C2×C4×Q8C2312yes—
104C2×C4×Q8C2312yes—
105C2×C4×Q8C2312yes—
106C2×C4×D4—22no—
107C2×C4×D4—22no—
108C2×C4×D4—22no—
109C2×C4×D4—22no—
110C2×C4×D4—22no—
111C2×C4×D4C2312yes—
112C2×C4×D4C2312yes—
113C2×#117—22no—
114C2×#117C2311yescharacteristic
115C2×#117C2311yescharacteristic
116C22×C4⋊C4—22no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4C2312yes—
119C22×C42C2311yescharacteristic
120#452—216no—
121#441—48no—
122C4×SD16—48no—
123C2×#106C2314yes—
124C2×#106C2314yes—
125C2×C4⋊C8C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×C22⋊C8C2314yes—
135C2×#97C2312yes—
136C2×C4×C8C2312yes—
137#400—22no—
138#400—22no—
139#400—22no—
140#398—22no—
141#398—22no—
142#398—22no—
143#397—22no—
144#396—22no—
145#391C2312yes—
146#391C2312yes—
147#391C2312yes—
148#391C2312yes—
149#391C2312yes—
150#391C2312yes—
151#390—22no—
152#390—22no—
153#390C2312yes—
154#390C2312yes—
155#389—22no—
156#389—22no—
157#389—22no—
158#389—22no—
159#389C2312yes—
160#389C2312yes—
161#389C2312yes—
162#389C2312yes—
163#388—22no—
164#388—22no—
165#388—24no—
166#388—24no—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#386—24no—
177#385—22no—
178#385—24no—
179#385—24no—
180#384—22no—
181#384C2311yescharacteristic
182#384C2311yescharacteristic
183#384C2311yescharacteristic
184#384C2312yes—
185#383—22no—
186#383—22no—
187#382—22no—
188#382C2311yescharacteristic
189#381—22no—
190#380—24no—
191#380C2312yes—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4D412yes—
197C4×C4⋊C4D412yes—
198C4×C4⋊C4C2312yes—
199C4×C22⋊C4D412yes—
200C4×C22⋊C4D412yes—
201#376C2311yescharacteristic
202#376C2311yescharacteristic
203C43C2311yescharacteristic
204C43C2311yescharacteristic
205#340C2314yes—
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#336C2312yes—
210#336C2312yes—
211#336C2314yes—
212#336C2312yes—
213#336C2312yes—
214#336C2314yes—
215C2×#41—24no—
216C2×#41—44no—
217C2×#41—44no—
218C2×#41#4118yes—
219C22×Q8C2412yes—
220C22×Q8C2412yes—
221C22×D4—22no—
222C22×D4—22no—
223C22×D4C2412yes—
224C23×C4—22no—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4C2×D412yes—
229C23×C4C2411yescharacteristic
230C2×Q16—28no—
231C2×Q16—44no—
232C2×Q16—44no—
233C2×SD16—28no—
234C2×SD16—44no—
235C2×SD16—44no—
236C22×C8C2414yes—
237C22×C8C2414yes—
238C4⋊Q8—24no—
239C4⋊Q8—42no—
240#127—24no—
241#125—42no—
242#124—24no—
243#124—24no—
244C22⋊Q8—28no—
245#121—216no—
246C4×Q8—28no—
247C4×Q8—28no—
248C4×Q8—44no—
249C4×D4—216no—
250C4×D4—28no—
251C4×D4—44no—
252C4×D4—48no—
253C4×D4—48no—
254C4×D4—84no—
255C4×D4—84no—
256#117—24no—
257#117—42no—
258#117—48no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—22no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—44no—
276C2×C4⋊C4—48no—
277C2×C4⋊C4—84no—
278C2×C4⋊C4C2×D411yescharacteristic
279C2×C4⋊C4C2×D411yescharacteristic
280C2×C4⋊C4C2×D412yes—
281C2×C4⋊C4C2412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4C2×D411yescharacteristic
294C2×C22⋊C4C2×D411yescharacteristic
295C2×C22⋊C4C2×D412yes—
296C2×C42—24no—
297C2×C42—22no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42C2×D412yes—
302C2×C42C2×D412yes—
303C2×C42C2×D412yes—
304C2×C42C2×D412yes—
305C2×C42C2×D412yes—
306C2×C42C2411yescharacteristic
307C2×C42C2411yescharacteristic
308C2×C42C2411yescharacteristic
309C2×C42C2411yescharacteristic
310C2×C42C2411yescharacteristic
311C2×C42C2411yescharacteristic
312#106—44no—
313#106—44no—
314C4⋊C8—44no—
315#103—28no—
316#103—44no—
317#103—44no—
318#103—44no—
319#103—44no—
320#102—28no—
321#102—44no—
322#102—44no—
323C22⋊C8—28no—
324#97—24no—
325C4×C8—24no—
326#95—24no—
327#95—24no—
328#95—22no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95C2×D412yes—
344#95C2×D412yes—
345#95C2×D412yes—
346#95C2×D412yes—
347C24—22no—
348C24C22×D411yescharacteristic
349#41—416no—
350#41—416no—
351C2×Q8—22no—
352C2×Q8—24no—
353C2×Q8—24no—
354C2×Q8—24no—
355C2×Q8—24no—
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8C2×#4114yes—
359C2×D4—28no—
360C2×D4—28no—
361C2×D4—22no—
362C2×D4—24no—
363C2×D4—24no—
364C2×D4—44no—
365C2×D4—44no—
366C2×D4—44no—
367C2×D4—44no—
368C2×D4C2×#4114yes—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—28no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—42no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—84no—
388C22×C4—84no—
389C22×C4—84no—
390C22×C4C22×D412yes—
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D412yes—
393C22×C4C22×D412yes—
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D412yes—
399C22×C4C2×#4114yes—
400C22×C4C2511yescharacteristic
401Q16—416no—
402SD16—416no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C4⋊C4—28no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—42no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—82no—
430C4⋊C4—82no—
431C4⋊C4—82no—
432C4⋊C4—82no—
433C4⋊C4—84no—
434C22⋊C4—216no—
435C22⋊C4—22no—
436C22⋊C4—22no—
437C22⋊C4—42no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—84no—
442C22⋊C4—84no—
443C22⋊C4—82no—
444C42—24no—
445C42—24no—
446C42—24no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—42no—
453C42—42no—
454C42—44no—
455C42—44no—
456C42—44no—
457C42—82no—
458C42H(5,𝔽2)14yes—
459C23—22no—
460C23—21no—
461C23—21no—
462C23—22no—
463C23—24no—
464C23—44no—
465C23—42no—
466C23—42no—
467C23—44no—
468C23#52512yes—
469C23C23×D411yescharacteristic
470Q8—28no—
471Q8—44no—
472Q8—44no—
473Q8—44no—
474D4—28no—
475D4—416no—
476D4—416no—
477D4—44no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—28no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—48no—
500C2×C4—416no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—164no—
520C2×C4—164no—
521C2×C4—164no—
522C2×C4#52512yes—
523C2×C4#53512yes—
524C2×C4#53512yes—
525C2×C4#57714yes—
526C2×C4C22×#4112yes—
527C2×C4C2×H(5,𝔽2)12yes—
528C2×C4#58512yes—
529C8—44no—
530C8—44no—
531C22—22no—
532C22—24no—
533C22—42no—
534C22—41no—
535C22—41no—
536C22—44no—
537C22—48no—
538C22—84no—
539C22—82no—
540C22—82no—
541C22C2×#52511yescharacteristic
542C22C2×#53511yescharacteristic
543C22#344911yescharacteristic
544C22C2×#57712yes—
545C22#358612yes—
546C4—22no—
547C4—22no—
548C4—22no—
549C4—24no—
550C4—44no—
551C4—42no—
552C4—42no—
553C4—41no—
554C4—41no—
555C4—42no—
556C4—82no—
557C4—82no—
558C4—82no—
559C4—82no—
560C4—82no—
561C4—82no—
562C4#345914yes—
563C2—42no—
564C2—84no—
565C2#3459311yescharacteristic
566C2#6058512yes—
567C2#6081811yescharacteristic
568C2#6083111yescharacteristic
569C2#6099212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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. #35321C2semidirect
3. #30132C2semidirect
4. #30127C2semidirect
5. #30090C2semidirect
6. #30054C2semidirect
7. #29965C2semidirect
8. #29908C2semidirect
9. #29726C2semidirect
11. #23824C2semidirect
12. #23820C2semidirect
13. #23805C2semidirect
14. #23799C2semidirect
15. #23694C2semidirect
16. #21461C2semidirect
17. #20780C2semidirect

Maximal subgroups

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

subgroupindex
#353212
#342072
#301322
#301272
#300902
#300542
#299652
#299082
#297262
#296292
#238242
#238202
#238052
#237992
#236942
#214612
#207802