Finite Group Databasebeta

← #6493040

Subgroups — #6493040

Order 512.

1559 conjugacy classes of subgroups.

The subgroup lattice has 595 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35284C211yesmaximal, characteristic, complemented
2#34151C211yesmaximal, characteristic
3#33162C211yesmaximal, characteristic, complemented
4#33114C212yesmaximal, complemented
5#33068C211yesmaximal, characteristic, complemented
6#29999C211yesmaximal, characteristic, complemented
7#29799C211yesmaximal, characteristic, complemented
8#29665C212yesmaximal
9#26956C214yesmaximal
10#26718C214yesmaximal, complemented
11#26639C214yesmaximal, complemented
12#26614C214yesmaximal, complemented
13#22152C212yesmaximal, complemented
14#21926C211yesmaximal, characteristic, complemented
15#21506C212yesmaximal, complemented
16C2×#529C2211yescharacteristic
17C2×#485C2214yes—
18C2×#462C2212yes—
19C2×#461C2212yes—
20C2×#461C2214yes—
21C2×#441C2214yes—
22#2868C2211yescharacteristic
23#2859C2211yescharacteristic
24#2857C2212yes—
25#2647—21no—
26#2629—21no—
27#2507—22no—
28#2497—22no—
29#2446C2214yes—
30#2429C2214yes—
31#2374—22no—
32#2373—22no—
33#2370C2212yes—
34#2367C2212yes—
35#2363C2212yes—
36#2362C2212yes—
37#2330—21no—
38#2314—21no—
39C4×C4⋊Q8C2212yes—
40#2298—21no—
41#2295C2211yescharacteristic
42#2293—21no—
43#2293C2211yescharacteristic
44#2290—21no—
45C42×Q8C2211yescharacteristic
46C42×D4C2211yescharacteristic
47C4×#117—21no—
48C4×#117C2211yescharacteristic
49C4×#117C2211yescharacteristic
50#2087C2214yes—
51#2068C2214yes—
52#2043C2214yes—
53#2026C2214yes—
54#2016C2214yes—
55#2015C2214yes—
56#2001C2214yes—
57#2000C2214yes—
58#1981C2212yes—
59#1981C2214yes—
60#1981C2212yes—
61#1973C2214yes—
62#1948C2212yes—
63#1947C2212yes—
64#1935C2214yes—
65#1928C2214yes—
66#1927C2214yes—
67#1927C2214yes—
68#1924C2214yes—
69#1920C2214yes—
70#1917C2214yes—
71#1880C2214yes—
72#1878C2214yes—
73#1873C2214yes—
74#1864C2214yes—
75#1863C2214yes—
76#1842C2211yescharacteristic
77#1831C2211yescharacteristic
78#1803C2212yes—
79#1802C2212yes—
80#1797C2212yes—
81#1767C2212yes—
82#1763C2214yes—
83C22×Q16C2314yes—
84C22×SD16C2314yes—
85C2×C4⋊Q8C2312yes—
86C2×C4⋊Q8C2314yes—
87#529—24no—
88#529—24no—
89C2×#126—21no—
90C2×#125—21no—
91C2×#125C2311yescharacteristic
92C2×#124C2311yescharacteristic
93C2×#123—21no—
94C2×C22⋊Q8C2311yescharacteristic
95C2×C22⋊Q8C2314yes—
96C2×#121C2311yescharacteristic
97C2×C4×Q8C2312yes—
98C2×C4×Q8C2311yescharacteristic
99C2×C4×Q8C2314yes—
100C2×C4×D4—21no—
101C2×C4×D4C2312yes—
102C2×C4×D4C2311yescharacteristic
103C2×#117—21no—
104C2×#117—21no—
105C2×#117D411yescharacteristic
106C2×#117D411yescharacteristic
107C2×#117C2312yes—
108C22×C42—21no—
109C22×C42C2311yescharacteristic
110#485—48no—
111#462—28no—
112#461—28no—
113#461—48no—
114#441—48no—
115C2×#107C2314yes—
116C2×#106C2314yes—
117C2×C4⋊C8C2312yes—
118C2×C4⋊C8C2312yes—
119C2×C4⋊C8C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#102C2314yes—
128C2×C22⋊C8C2314yes—
129C2×#97C2314yes—
130#400—24no—
131#398—24no—
132#398—24no—
133#396—24no—
134#391C2314yes—
135#391C2312yes—
136#391C2314yes—
137#390—22no—
138#390C2312yes—
139#389—22no—
140#389—22no—
141#389—22no—
142#389—22no—
143#389—22no—
144#389—22no—
145#389—24no—
146#389C2314yes—
147#389C2312yes—
148#389C2312yes—
149#388—22no—
150#388—24no—
151#388—22no—
152#388—22no—
153#388—22no—
154#388—22no—
155#388—22no—
156#387—24no—
157#387—22no—
158#387—22no—
159#387—24no—
160#387—22no—
161#387—22no—
162#386—22no—
163#385—22no—
164#384—21no—
165#384—21no—
166#384—22no—
167#384C2312yes—
168#382—21no—
169#380—22no—
170#380—22no—
171C4×C4⋊C4—22no—
172C4×C4⋊C4—22no—
173C4×C4⋊C4—22no—
174C4×C4⋊C4—21no—
175C4×C4⋊C4—21no—
176C4×C4⋊C4—22no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4D412yes—
179C4×C4⋊C4C2312yes—
180C4×C4⋊C4C2312yes—
181C4×C4⋊C4C2312yes—
182C4×C4⋊C4C2311yescharacteristic
183C4×C4⋊C4C2311yescharacteristic
184C4×C4⋊C4C2312yes—
185C4×C22⋊C4—21no—
186C4×C22⋊C4—21no—
187C4×C22⋊C4—22no—
188C4×C22⋊C4—21no—
189C4×C22⋊C4—21no—
190C4×C22⋊C4—22no—
191C4×C22⋊C4—22no—
192C4×C22⋊C4D412yes—
193C4×C22⋊C4C2312yes—
194#376—21no—
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—
209C22×Q8C2414yes—
210C22×Q8C2411yescharacteristic
211C22×D4—21no—
212C22×D4C2411yescharacteristic
213C23×C4—21no—
214C23×C4—22no—
215C23×C4C2×D411yescharacteristic
216C23×C4C2×D411yescharacteristic
217C23×C4C2411yescharacteristic
218C2×Q16—28no—
219C2×Q16—44no—
220C2×Q16—44no—
221C2×SD16—28no—
222C2×SD16—44no—
223C2×SD16—44no—
224C22×C8C2414yes—
225C22×C8C2414yes—
226C4⋊Q8—24no—
227C4⋊Q8—24no—
228C4⋊Q8—44no—
229C4⋊Q8—44no—
230#126—28no—
231#125—22no—
232#125—22no—
233#125—44no—
234#124—22no—
235#124—22no—
236#123—28no—
237C22⋊Q8—22no—
238C22⋊Q8—22no—
239C22⋊Q8—44no—
240C22⋊Q8—44no—
241#121—22no—
242#121—22no—
243C4×Q8—22no—
244C4×Q8—24no—
245C4×Q8—42no—
246C4×Q8—44no—
247C4×Q8—44no—
248C4×Q8#4114yes—
249C4×D4—28no—
250C4×D4—24no—
251C4×D4—22no—
252C4×D4—42no—
253C4×D4#4114yes—
254#117—28no—
255#117—24no—
256#117—24no—
257#117—42no—
258#117—44no—
259#117—42no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2414yes—
280C2×C4⋊C4C2411yescharacteristic
281C2×C4⋊C4C2411yescharacteristic
282C2×C4⋊C4C2412yes—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4C2×D412yes—
294C2×C22⋊C4C2×D412yes—
295C2×C42—21no—
296C2×C42—22no—
297C2×C42—22no—
298C2×C42—21no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—21no—
302C2×C42—22no—
303C2×C42—24no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—44no—
307C2×C42C2×D411yescharacteristic
308C2×C42C2×D411yescharacteristic
309C2×C42C2×D411yescharacteristic
310C2×C42C2×D411yescharacteristic
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D412yes—
313C2×C42C2×D412yes—
314C2×C42C2×D411yescharacteristic
315C2×C42C2412yes—
316C2×C42C2412yes—
317C2×C42C2411yescharacteristic
318C2×C42C2411yescharacteristic
319#107—44no—
320#106—28no—
321C4⋊C8—24no—
322C4⋊C8—24no—
323C4⋊C8—44no—
324#103—28no—
325#103—28no—
326#103—28no—
327#103—28no—
328#103—44no—
329#103—44no—
330#103—44no—
331#102—44no—
332C22⋊C8—44no—
333#97—44no—
334#95—21no—
335#95—22no—
336#95—22no—
337#95—22no—
338#95—21no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
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
354C2×Q8—21no—
355C2×Q8—24no—
356C2×Q8—24no—
357C2×Q8—22no—
358C2×Q8—24no—
359C2×Q8—24no—
360C2×Q8—22no—
361C2×Q8—24no—
362C2×Q8—24no—
363C2×Q8C2×#4112yes—
364C2×D4—24no—
365C2×D4—21no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—82no—
369C2×D4C2×#4112yes—
370C22×C4—22no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—42no—
387C22×C4—44no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—44no—
391C22×C4—82no—
392C22×C4C22×D411yescharacteristic
393C22×C4C22×D411yescharacteristic
394C22×C4C22×D412yes—
395C22×C4C22×D412yes—
396C22×C4C22×D412yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D412yes—
400C22×C4C22×D411yescharacteristic
401C22×C4C2511yescharacteristic
402Q16—416no—
403SD16—88no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—22no—
422C4⋊C4—42no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—48no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—82no—
431C4⋊C4—84no—
432C4⋊C4—84no—
433C4⋊C4—84no—
434C4⋊C4—84no—
435C4⋊C4C2×#4112yes—
436C4⋊C4C2×#4112yes—
437C4⋊C4H(5,𝔽2)14yes—
438C22⋊C4—24no—
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—24no—
444C22⋊C4—42no—
445C22⋊C4—44no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—82no—
449C22⋊C4—84no—
450C42—22no—
451C42—22no—
452C42—24no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—21no—
457C42—21no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—42no—
463C42—42no—
464C42—44no—
465C42—44no—
466C42—44no—
467C42—42no—
468C42—42no—
469C42—82no—
470C42—82no—
471C42—84no—
472C42#13714yes—
473C42C2×#4112yes—
474C42C2×#4112yes—
475C23—22no—
476C23—22no—
477C23—21no—
478C23—22no—
479C23—21no—
480C23—21no—
481C23—42no—
482C23—42no—
483C23—41no—
484C23—41no—
485C23C23×D411yescharacteristic
486Q8—24no—
487Q8—41no—
488Q8—41no—
489Q8—44no—
490Q8—44no—
491Q8—44no—
492Q8—44no—
493D4—22no—
494D4—22no—
495D4—42no—
496D4—84no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—21no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—44no—
527C2×C4—48no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—82no—
541C2×C4—88no—
542C2×C4#52512yes—
543C2×C4#52512yes—
544C2×C4C2×#13714yes—
545C2×C4C2×#13712yes—
546C2×C4#57512yes—
547C2×C4C22×#4111yescharacteristic
548C2×C4C22×#4111yescharacteristic
549C2×C4C2×H(5,𝔽2)12yes—
550C2×C4#58512yes—
551C8—44no—
552C8—44no—
553C22—42no—
554C22—42no—
555C22—41no—
556C22—42no—
557C22—41no—
558C22—42no—
559C22—41no—
560C22—42no—
561C22—82no—
562C22—82no—
563C22C2×#52511yescharacteristic
564C22#348812yes—
565C22C22×#13711yescharacteristic
566C22C2×#57511yescharacteristic
567C22#358612yes—
568C4—24no—
569C4—22no—
570C4—42no—
571C4—44no—
572C4—42no—
573C4—42no—
574C4—41no—
575C4—42no—
576C4—41no—
577C4—42no—
578C4—41no—
579C4—42no—
580C4—82no—
581C4—84no—
582C4—84no—
583C4#311614yes—
584C4#346012yes—
585C4#350811yescharacteristic
586C4#351211yescharacteristic
587C2—81no—
588C2—82no—
589C2#3525911yescharacteristic
590C2C2×#311611yescharacteristic
591C2#6056911yescharacteristic
592C2#6196012yes—
593C2#6196912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#121C23
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. #33162C2semidirect
4. #33114C2semidirect
5. #33068C2semidirect
6. #29999C2semidirect
7. #29799C2semidirect
10. #26718C2semidirect
11. #26639C2semidirect
12. #26614C2semidirect
13. #22152C2semidirect
14. #21926C2semidirect
15. #21506C2semidirect

Maximal subgroups

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

subgroupindex
#352842
#341512
#331622
#331142
#330682
#299992
#297992
#296652
#269562
#267182
#266392
#266142
#221522
#219262
#215062