Finite Group Databasebeta

← #6493467

Subgroups — #6493467

Order 512.

1599 conjugacy classes of subgroups.

The subgroup lattice has 599 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35284C211yesmaximal, characteristic, complemented
2#34125C211yesmaximal, characteristic
3#33163C211yesmaximal, characteristic, complemented
4#33067C211yesmaximal, characteristic, complemented
5#32907C212yesmaximal
6#29994C211yesmaximal, characteristic, complemented
7#29926C212yesmaximal, complemented
8#29798C211yesmaximal, characteristic, complemented
9#27401C214yesmaximal, complemented
10#27201C214yesmaximal, complemented
11#26167C214yesmaximal, complemented
12#25151C214yesmaximal
13#22146C212yesmaximal, complemented
14#21922C211yesmaximal, characteristic, complemented
15C4×#462C212yesmaximal, complemented
16C2×C4×#41C2211yescharacteristic
17C2×#462C2212yes—
18C2×#462C2214yes—
19C2×#461C2212yes—
20C2×#452C2214yes—
21C2×C4×Q16C2214yes—
22#2868C2211yescharacteristic
23#2859C2211yescharacteristic
24#2857C2212yes—
25#2647—21no—
26#2629—21no—
27#2507—22no—
28#2497—22no—
29#2446C2214yes—
30#2429C2214yes—
31#2357—22no—
32#2357C2212yes—
33#2351C2212yes—
34#2347—22no—
35#2347C2212yes—
36#2341C2212yes—
37#2330—21no—
38#2314—21no—
39C4×C4⋊Q8C2212yes—
40#2293—21no—
41#2293—21no—
42#2293C2211yescharacteristic
43#2293C2211yescharacteristic
44#2283—21no—
45#2283C2211yescharacteristic
46C42×Q8C2211yescharacteristic
47C42×Q8C2211yescharacteristic
48C4×#117—21no—
49C4×#117C2211yescharacteristic
50#2097C2214yes—
51#2086C2214yes—
52#2074C2214yes—
53#2067C2214yes—
54#2043C2214yes—
55#2036C2214yes—
56#2035C2214yes—
57#2013C2214yes—
58#1981C2214yes—
59#1981C2212yes—
60#1981C2212yes—
61#1968C2214yes—
62#1948C2212yes—
63#1947C2212yes—
64#1933C2214yes—
65#1928C2214yes—
66#1923C2214yes—
67#1919C2214yes—
68#1916C2214yes—
69#1880C2214yes—
70#1878C2214yes—
71#1878C2214yes—
72#1877C2214yes—
73#1873C2214yes—
74#1865C2214yes—
75#1865C2214yes—
76#1842C2211yescharacteristic
77#1831C2211yescharacteristic
78#1803C2212yes—
79#1802C2212yes—
80#1797C2212yes—
81C4×C4⋊C8C2212yes—
82C4×#103C2214yes—
83C22×#41C2311yescharacteristic
84C22×Q16C2314yes—
85C22×SD16C2314yes—
86C2×C4⋊Q8C2312yes—
87C2×C4⋊Q8C2314yes—
88C2×#125—21no—
89C2×C22⋊Q8C2314yes—
90C4×#41—24no—
91C4×#41—24no—
92C2×C4×Q8C2311yescharacteristic
93C2×C4×Q8C2312yes—
94C2×C4×Q8C2312yes—
95C2×C4×Q8C2311yescharacteristic
96C2×C4×Q8C2314yes—
97C2×C4×D4—21no—
98C2×C4×D4—21no—
99C2×C4×D4C2311yescharacteristic
100C2×C4×D4C2311yescharacteristic
101C2×#117—21no—
102C2×#117—21no—
103C2×#117—21no—
104C2×#117D411yescharacteristic
105C2×#117D411yescharacteristic
106C2×#117C2311yescharacteristic
107C22×C42—21no—
108C22×C42C2311yescharacteristic
109#462—28no—
110#462—48no—
111#461—28no—
112#452—48no—
113C4×Q16—48no—
114C2×#107C2314yes—
115C2×#106C2314yes—
116C2×C4⋊C8C2312yes—
117C2×C4⋊C8C2312yes—
118C2×C4⋊C8C2314yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#102C2314yes—
127C2×C22⋊C8C2314yes—
128C2×C4×C8C2314yes—
129#400—24no—
130#398—24no—
131#398—24no—
132#396—24no—
133#391C2314yes—
134#391C2314yes—
135#390—22no—
136#390—22no—
137#390C2312yes—
138#390C2312yes—
139#389—22no—
140#389—22no—
141#389—22no—
142#389—22no—
143#389—22no—
144#389—22no—
145#389—22no—
146#389—24no—
147#389C2314yes—
148#389C2312yes—
149#389C2312yes—
150#388—22no—
151#388—22no—
152#388—24no—
153#388—22no—
154#388—22no—
155#388—22no—
156#388—22no—
157#387—24no—
158#387—24no—
159#384—21no—
160#384—21no—
161#384—22no—
162#384C2312yes—
163#383—21no—
164#383—22no—
165#382—21no—
166#382—22no—
167#382C2312yes—
168#381—21no—
169#381—22no—
170#380—22no—
171#380—22no—
172#380—22no—
173#380C2312yes—
174C4×C4⋊C4—22no—
175C4×C4⋊C4—22no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—21no—
179C4×C4⋊C4—21no—
180C4×C4⋊C4—22no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4D412yes—
183C4×C4⋊C4C2312yes—
184C4×C4⋊C4C2312yes—
185C4×C4⋊C4C2311yescharacteristic
186C4×C4⋊C4C2311yescharacteristic
187C4×C4⋊C4C2312yes—
188C4×C22⋊C4—21no—
189C4×C22⋊C4—21no—
190C4×C22⋊C4—21no—
191C4×C22⋊C4—22no—
192C4×C22⋊C4—22no—
193C4×C22⋊C4D412yes—
194C4×C22⋊C4C2312yes—
195#376—21no—
196#376D411yescharacteristic
197#376D411yescharacteristic
198C43—21no—
199C43—21no—
200C43C2311yescharacteristic
201C43C2311yescharacteristic
202#340C2314yes—
203#340C2314yes—
204#340C2314yes—
205#340C2314yes—
206#336C2314yes—
207#336C2314yes—
208#336C2314yes—
209C2×#41—22no—
210C2×#41—22no—
211C2×#41—24no—
212C2×#41—42no—
213C22×Q8C2414yes—
214C22×Q8C2411yescharacteristic
215C22×D4—21no—
216C22×D4C2411yescharacteristic
217C23×C4—21no—
218C23×C4—21no—
219C23×C4—21no—
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2411yescharacteristic
223C2×Q16—28no—
224C2×Q16—28no—
225C2×Q16—28no—
226C2×SD16—28no—
227C2×SD16—44no—
228C2×SD16—44no—
229C22×C8C2414yes—
230C22×C8C2414yes—
231C4⋊Q8—24no—
232C4⋊Q8—24no—
233C4⋊Q8—44no—
234C4⋊Q8—44no—
235#125—44no—
236C22⋊Q8—44no—
237C22⋊Q8—44no—
238C4×Q8—22no—
239C4×Q8—22no—
240C4×Q8—22no—
241C4×Q8—24no—
242C4×Q8—24no—
243C4×Q8—42no—
244C4×Q8—42no—
245C4×Q8—44no—
246C4×Q8—44no—
247C4×Q8#4114yes—
248C4×D4—28no—
249C4×D4—28no—
250C4×D4—22no—
251C4×D4—22no—
252C4×D4—22no—
253C4×D4#4114yes—
254#117—28no—
255#117—22no—
256#117—22no—
257#117—24no—
258#117—24no—
259#117—44no—
260#117—44no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2414yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C4⋊C4C2411yescharacteristic
282C2×C4⋊C4C2411yescharacteristic
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4C2×D412yes—
293C2×C22⋊C4C2×D412yes—
294C2×C42—28no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—22no—
299C2×C42—22no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—22no—
303C2×C42—22no—
304C2×C42—24no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D412yes—
312C2×C42C2×D412yes—
313C2×C42C2×D411yescharacteristic
314C2×C42C2×D411yescharacteristic
315C2×C42C2411yescharacteristic
316C2×C42C2412yes—
317C2×C42C2412yes—
318C2×C42C2411yescharacteristic
319#107—44no—
320#106—44no—
321C4⋊C8—24no—
322C4⋊C8—24no—
323C4⋊C8—44no—
324#103—28no—
325#103—28no—
326#103—28no—
327#103—44no—
328#103—44no—
329#103—44no—
330#103—44no—
331#102—44no—
332C22⋊C8—44no—
333C4×C8—44no—
334#95—22no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—21no—
342#95—21no—
343#95—24no—
344#95—24no—
345#95—24no—
346#95—24no—
347#95—24no—
348#95C2×D412yes—
349#95C2×D412yes—
350#95C2×D412yes—
351#95C2×D412yes—
352C24—21no—
353C24C22×D411yescharacteristic
354#41—48no—
355#41—44no—
356#41—44no—
357C2×Q8—21no—
358C2×Q8—22no—
359C2×Q8—24no—
360C2×Q8—24no—
361C2×Q8—24no—
362C2×Q8—24no—
363C2×Q8—24no—
364C2×Q8—24no—
365C2×Q8C2×#4112yes—
366C2×Q8H(5,𝔽2)14yes—
367C2×D4—24no—
368C2×D4—21no—
369C2×D4—22no—
370C2×D4—22no—
371C2×D4—44no—
372C2×D4C2×#4112yes—
373C22×C4—21no—
374C22×C4—24no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—21no—
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—44no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—82no—
396C22×C4C22×D411yescharacteristic
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D412yes—
399C22×C4C22×D412yes—
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D412yes—
403C22×C4C22×D412yes—
404C22×C4C22×D411yescharacteristic
405C22×C4C2511yescharacteristic
406Q16—416no—
407SD16—88no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—24no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—24no—
426C4⋊C4—44no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—44no—
432C4⋊C4—48no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—84no—
436C4⋊C4—84no—
437C4⋊C4—84no—
438C4⋊C4—84no—
439C4⋊C4C2×#4112yes—
440C4⋊C4C2×#4112yes—
441C22⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—24no—
445C22⋊C4—24no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—82no—
449C22⋊C4—82no—
450C22⋊C4—82no—
451C22⋊C4—84no—
452C42—24no—
453C42—24no—
454C42—22no—
455C42—24no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—21no—
460C42—21no—
461C42—22no—
462C42—22no—
463C42—22no—
464C42—22no—
465C42—42no—
466C42—42no—
467C42—44no—
468C42—44no—
469C42—42no—
470C42—42no—
471C42—82no—
472C42—82no—
473C42—84no—
474C42#13514yes—
475C42#13714yes—
476C42C2×#4112yes—
477C42C2×#4112yes—
478C23—22no—
479C23—21no—
480C23—22no—
481C23—22no—
482C23—21no—
483C23—21no—
484C23—42no—
485C23—42no—
486C23—41no—
487C23—41no—
488C23C23×D411yescharacteristic
489Q8—22no—
490Q8—24no—
491Q8—44no—
492Q8—44no—
493Q8—44no—
494Q8—44no—
495Q8#54014yes—
496D4—24no—
497D4—48no—
498D4—41no—
499D4—41no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—48no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—44no—
527C2×C4—48no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—82no—
544C2×C4—84no—
545C2×C4#52512yes—
546C2×C4#52512yes—
547C2×C4C2×#13514yes—
548C2×C4C2×#13512yes—
549C2×C4C2×#13714yes—
550C2×C4C2×#13712yes—
551C2×C4C22×#4111yescharacteristic
552C2×C4C22×#4111yescharacteristic
553C2×C4C2×H(5,𝔽2)12yes—
554C2×C4#58512yes—
555C8—44no—
556C8—44no—
557C22—42no—
558C22—42no—
559C22—41no—
560C22—42no—
561C22—41no—
562C22—42no—
563C22—42no—
564C22—41no—
565C22—82no—
566C22—82no—
567C22C2×#52511yescharacteristic
568C22#348812yes—
569C22C22×#13511yescharacteristic
570C22C22×#13711yescharacteristic
571C22#358612yes—
572C4—24no—
573C4—24no—
574C4—22no—
575C4—24no—
576C4—42no—
577C4—42no—
578C4—41no—
579C4—42no—
580C4—41no—
581C4—42no—
582C4—41no—
583C4—82no—
584C4—84no—
585C4—82no—
586C4—84no—
587C4#310714yes—
588C4C2×#54011yescharacteristic
589C4#345511yescharacteristic
590C4#349412yes—
591C2—81no—
592C2—82no—
593C2#3526111yescharacteristic
594C2C2×#310711yescharacteristic
595C2#6056911yescharacteristic
596C2#6195412yes—
597C2#6196412yes—
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. #33163C2semidirect
4. #33067C2semidirect
6. #29994C2semidirect
7. #29926C2semidirect
8. #29798C2semidirect
9. #27401C2semidirect
10. #27201C2semidirect
11. #26167C2semidirect
13. #22146C2semidirect
14. #21922C2semidirect
15. C4×#462C2semidirect

Maximal subgroups

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

subgroupindex
#352842
#341252
#331632
#330672
#329072
#299942
#299262
#297982
#274012
#272012
#261672
#251512
#221462
#219222
C4×#4622