Finite Group Databasebeta

← #241313

Subgroups — #241313

Order 512.

802 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34948C211yesmaximal, characteristic
2#21984C211yesmaximal, characteristic
3#21365C211yesmaximal, characteristic
4#21362C211yesmaximal, characteristic
5#10582C211yesmaximal, characteristic, complemented
6#10581C211yesmaximal, characteristic, complemented
7#10580C211yesmaximal, characteristic, complemented
8#10579C211yesmaximal, characteristic, complemented
9#10578C211yesmaximal, characteristic, complemented
10#10578C211yesmaximal, characteristic, complemented
11#10577C211yesmaximal, characteristic, complemented
12#10576C211yesmaximal, characteristic, complemented
13#8046C211yesmaximal, characteristic
14#7902C211yesmaximal, characteristic
15#7892C211yesmaximal, characteristic
16C2×#432C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#431C2211yescharacteristic
19#2571C2211yescharacteristic
20#2571C2211yescharacteristic
21#2570—21no—
22#2570—21no—
23#2569—22no—
24#2552C2211yescharacteristic
25#2552C2211yescharacteristic
26#2442—21no—
27#2442—21no—
28#2402—22no—
29C4×#125—21no—
30C4×#125—21no—
31#2296—21no—
32#2285—21no—
33#2285C2211yescharacteristic
34#2285C2211yescharacteristic
35C4×#117C2211yescharacteristic
36C2×C4×C4⋊C4C2211yescharacteristic
37#1949—22no—
38#1948C2211yescharacteristic
39#1948C2211yescharacteristic
40#1948C2211yescharacteristic
41#1948C2211yescharacteristic
42#1947C2211yescharacteristic
43#1947C2211yescharacteristic
44#1947C2211yescharacteristic
45#1947C2211yescharacteristic
46#1854—22no—
47#1854—21no—
48#1854—21no—
49#1853—21no—
50#1852—21no—
51#1845C2211yescharacteristic
52#1844C2211yescharacteristic
53#1834C2211yescharacteristic
54#1834C2211yescharacteristic
55#1776—21no—
56#1776—21no—
57C4×#107—21no—
58C4×#106—21no—
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1279C2211yescharacteristic
68#1279C2211yescharacteristic
69#1279C2211yescharacteristic
70#1279C2211yescharacteristic
71C2×#125—21no—
72C2×#125—21no—
73C2×#117—21no—
74C2×#117—21no—
75C2×#117D411yescharacteristic
76C2×#117D411yescharacteristic
77C22×C4⋊C4D411yescharacteristic
78C22×C4⋊C4D411yescharacteristic
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81#432—22no—
82#432Q814yes—
83#431—22no—
84#431—22no—
85#431Q814yes—
86#431Q814yes—
87C2×#107—21no—
88C2×#107—21no—
89C2×#107—21no—
90C2×#107—21no—
91C2×#107—22no—
92C2×#106—21no—
93C2×#106—21no—
94C2×#106—21no—
95C2×#106—21no—
96C2×#106—22no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×#97—21no—
106C2×C4×C8—21no—
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109#400—22no—
110#400—22no—
111#400—22no—
112#400—22no—
113#398—22no—
114#398—22no—
115#398—22no—
116#398—22no—
117#389—21no—
118#389—21no—
119#389—21no—
120#389—21no—
121#389—21no—
122#389—21no—
123#389—21no—
124#389—21no—
125#387—22no—
126#387—22no—
127#387—22no—
128#387—22no—
129#385—22no—
130#385—22no—
131#384—21no—
132#384—21no—
133#384C2311yescharacteristic
134#384C2311yescharacteristic
135#384C2311yescharacteristic
136#384C2311yescharacteristic
137#382—21no—
138#382—21no—
139#382—21no—
140#382—21no—
141#382—21no—
142#382—21no—
143#380—22no—
144#380—22no—
145#380—21no—
146#380—21no—
147#380—21no—
148#380D412yes—
149C4×C4⋊C4—22no—
150C4×C4⋊C4—22no—
151C4×C4⋊C4—22no—
152C4×C4⋊C4—22no—
153C4×C4⋊C4—21no—
154C4×C4⋊C4—21no—
155C4×C4⋊C4—21no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4D411yescharacteristic
159C4×C4⋊C4D411yescharacteristic
160C4×C4⋊C4D411yescharacteristic
161C4×C4⋊C4D411yescharacteristic
162C4×C4⋊C4D412yes—
163C4×C4⋊C4C2311yescharacteristic
164C4×C4⋊C4C2311yescharacteristic
165C4×C4⋊C4C2311yescharacteristic
166C4×C4⋊C4C2311yescharacteristic
167C4×C22⋊C4—21no—
168C4×C22⋊C4—21no—
169#376—21no—
170C2×#95—21no—
171C43—21no—
172#340—22no—
173#340—22no—
174#340—21no—
175#340—21no—
176#340—21no—
177#340—21no—
178#340—21no—
179#340—21no—
180#340—21no—
181#340—21no—
182C23×C4—21no—
183C23×C4—21no—
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193#125—42no—
194#125—42no—
195#125—42no—
196#125—42no—
197#117—41no—
198#117—41no—
199#117—41no—
200#117—41no—
201#117—42no—
202#117—42no—
203#117—82no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—41no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4C2×D411yescharacteristic
229C2×C4⋊C4C2×D411yescharacteristic
230C2×C4⋊C4C2×D411yescharacteristic
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C4⋊C4C2×D411yescharacteristic
233C2×C4⋊C4C2×D411yescharacteristic
234C2×C4⋊C4C2×D411yescharacteristic
235C2×C4⋊C4C2×D411yescharacteristic
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4C2×D412yes—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—41no—
242C2×C42—22no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—42no—
254C2×C42—42no—
255C2×C42—41no—
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2×Q812yes—
260C2×C42C2×Q812yes—
261C2×C42C2×Q812yes—
262C2×C42#4112yes—
263C2×C42C2411yescharacteristic
264#107—42no—
265#107—42no—
266#107—42no—
267#107—42no—
268#107—82no—
269#106—42no—
270#106—42no—
271#106—42no—
272#106—42no—
273#106—82no—
274C4⋊C8—24no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8C2×Q812yes—
280C4⋊C8C2×Q812yes—
281C4⋊C8C2×Q812yes—
282C4⋊C8C2×Q812yes—
283C22⋊C8—24no—
284C22⋊C8—24no—
285C22⋊C8—24no—
286#97—24no—
287C4×C8—24no—
288C4×C8—21no—
289C4×C8—21no—
290C4×C8C2×Q812yes—
291C4×C8C2×Q812yes—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—22no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—22no—
305#95—22no—
306#95—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—41no—
311#95—41no—
312C24#12011yescharacteristic
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—81no—
340C22×C4—81no—
341C22×C4#12011yescharacteristic
342C22×C4#12011yescharacteristic
343C22×C4#12011yescharacteristic
344C22×C4C22⋊Q812yes—
345C22×C4C22⋊Q812yes—
346C22×C4C22⋊Q812yes—
347C22×C4C4⋊Q812yes—
348C22×C4C4⋊Q812yes—
349C22×C4C4⋊Q812yes—
350C22×C4C22×D411yescharacteristic
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—42no—
394C4⋊C4—42no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—82no—
412C4⋊C4—82no—
413C4⋊C4—82no—
414C4⋊C4—82no—
415C4⋊C4#13512yes—
416C4⋊C4#13512yes—
417C22⋊C4—42no—
418C22⋊C4—42no—
419C22⋊C4—44no—
420C22⋊C4—82no—
421C42—42no—
422C42—42no—
423C42—42no—
424C42—42no—
425C42—44no—
426C42—41no—
427C42—41no—
428C42—41no—
429C42—41no—
430C42—41no—
431C42—41no—
432C42—41no—
433C42—41no—
434C42—81no—
435C42—81no—
436C42—82no—
437C42C22⋊Q812yes—
438C42C22⋊Q812yes—
439C42C22⋊Q812yes—
440C42C4⋊Q812yes—
441C42C4⋊Q812yes—
442C42C4⋊Q812yes—
443C42C22×Q811yescharacteristic
444C42C22×Q811yescharacteristic
445C42C22×Q811yescharacteristic
446C42C2×#4111yescharacteristic
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23#39512yes—
454C23C2×#12011yescharacteristic
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—42no—
502C2×C4—41no—
503C2×C4—84no—
504C2×C4—84no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4#39512yes—
512C2×C4#39512yes—
513C2×C4#39512yes—
514C2×C4C2×C22⋊Q811yescharacteristic
515C2×C4C2×C22⋊Q811yescharacteristic
516C2×C4C2×C22⋊Q811yescharacteristic
517C2×C4C2×C4⋊Q811yescharacteristic
518C2×C4C2×C4⋊Q811yescharacteristic
519C2×C4C2×C4⋊Q811yescharacteristic
520C2×C4C2×#13511yescharacteristic
521C2×C4C2×#13511yescharacteristic
522C2×C4#57511yescharacteristic
523C2×C4#57511yescharacteristic
524C2×C4#57511yescharacteristic
525C2×C4#57511yescharacteristic
526C2×C4#57511yescharacteristic
527C2×C4#57511yescharacteristic
528C8—42no—
529C8—42no—
530C8—42no—
531C8—42no—
532C8—42no—
533C8—42no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22C2×#39511yescharacteristic
542C22#300211yescharacteristic
543C22#300211yescharacteristic
544C22#300511yescharacteristic
545C22#300511yescharacteristic
546C22#302111yescharacteristic
547C22#302211yescharacteristic
548C4—22no—
549C4—21no—
550C4—21no—
551C4—22no—
552C4—22no—
553C4—42no—
554C4—42no—
555C4—81no—
556C4—81no—
557C4—81no—
558C4—82no—
559C4—81no—
560C4—81no—
561C4—81no—
562C4—82no—
563C4—161no—
564C4—161no—
565C4#307611yescharacteristic
566C4#307611yescharacteristic
567C4#308011yescharacteristic
568C4#308011yescharacteristic
569C4#316011yescharacteristic
570C4#316211yescharacteristic
571C4#316611yescharacteristic
572C4#316611yescharacteristic
573C2—42no—
574C2#1879511yescharacteristic
575C2#1879611yescharacteristic
576C2#1880711yescharacteristic
577C2#1880811yescharacteristic
578C2#2355111yescharacteristic
579C2#2355311yescharacteristic
580C2#2356211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10582C2semidirect
6. #10581C2semidirect
7. #10580C2semidirect
8. #10579C2semidirect
9. #10578C2semidirect
10. #10578C2semidirect
11. #10577C2semidirect
12. #10576C2semidirect

Maximal subgroups

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

subgroupindex
#349482
#219842
#213652
#213622
#105822
#105812
#105802
#105792
#105782
#105772
#105762
#80462
#79022
#78922