Finite Group Databasebeta

← #6690750

Subgroups — #6690750

Order 512.

1930 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#39634C211yesmaximal, characteristic, complemented
2#36179C211yesmaximal, characteristic
3#33068C211yesmaximal, characteristic, complemented
4#32037C214yesmaximal, complemented
5#30721C214yesmaximal, complemented
6#30220C211yesmaximal, characteristic, complemented
7#29799C211yesmaximal, characteristic, complemented
8#26234C212yesmaximal, complemented
9#25394C212yesmaximal, complemented
10#24446C214yesmaximal, complemented
11#24241C212yesmaximal, complemented
12#24055C214yesmaximal, complemented
13#22431C212yesmaximal, complemented
14#22264C211yesmaximal, characteristic, complemented
15#21117C211yesmaximal, characteristic, complemented
16C2×#547C2212yes—
17C2×#547C2214yes—
18C2×#483C2214yes—
19C2×#470C2214yes—
20C2×#462C2212yes—
21C2×#452C2214yes—
22C2×C4×SD16C2211yescharacteristic
23C2×#436C2212yes—
24C2×#417C2214yes—
25#2868C2211yescharacteristic
26#2859C2211yescharacteristic
27#2856—21no—
28#2780—24no—
29#2647—21no—
30#2542C2214yes—
31#2542C2212yes—
32#2538C2212yes—
33#2501—22no—
34#2501—24no—
35#2487C2214yes—
36#2454—22no—
37#2417C2214yes—
38D4×C4⋊C4C2212yes—
39#2328—21no—
40C4×#127C2211yescharacteristic
41#2295C2211yescharacteristic
42C2×#384—21no—
43C42×D4C2211yescharacteristic
44#2043C2214yes—
45#2043C2214yes—
46#2043C2214yes—
47#2035C2214yes—
48#2035C2214yes—
49#2035C2214yes—
50#2035C2214yes—
51#2016C2214yes—
52#2001C2214yes—
53#2001C2214yes—
54#1981C2212yes—
55#1956C2212yes—
56#1956C2212yes—
57#1954C2212yes—
58#1948C2212yes—
59#1927C2212yes—
60#1920C2214yes—
61#1920C2212yes—
62#1894C2214yes—
63#1894C2214yes—
64#1877C2212yes—
65#1872C2214yes—
66#1872C2214yes—
67#1870C2214yes—
68#1870C2214yes—
69#1864C2214yes—
70#1864C2212yes—
71#1831C2211yescharacteristic
72#1802C2212yes—
73#1788C2214yes—
74#1788C2214yes—
75C4×#106C2211yescharacteristic
76C4×#103C2211yescharacteristic
77C4×#102C2211yescharacteristic
78#1744C2211yescharacteristic
79C22×#41C2314yes—
80C22×Q16C2314yes—
81C22×SD16C2311yescharacteristic
82C22×M16C2314yes—
83#547—28no—
84#547—44no—
85#547—48no—
86#547—48no—
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2314yes—
89C2×#127D411yescharacteristic
90C2×#125—21no—
91C2×#123—24no—
92C2×#123—24no—
93C2×#123—24no—
94C2×C22⋊Q8C2314yes—
95C2×C22⋊Q8C2314yes—
96C2×C22⋊Q8C2314yes—
97C2×C22⋊Q8C2314yes—
98C2×#121—22no—
99C2×#121—24no—
100C2×C4×Q8C2311yescharacteristic
101C2×C4×D4—22no—
102C2×C4×D4C2312yes—
103C2×C4×D4C2311yescharacteristic
104C2×C4×D4C2312yes—
105C2×C4×D4C2314yes—
106C22×C4⋊C4—22no—
107C22×C4⋊C4—24no—
108C22×C4⋊C4C2314yes—
109C22×C42—21no—
110#483—48no—
111#470—48no—
112#462—44no—
113#452—48no—
114C4×SD16—24no—
115#436—44no—
116C2×#106C2312yes—
117C2×#106C2314yes—
118C2×#106C2311yescharacteristic
119C2×C4⋊C8C2312yes—
120#417—48no—
121C2×#103C2312yes—
122C2×#103C2311yescharacteristic
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#102C2314yes—
131C2×#102C2311yescharacteristic
132C2×#102C2312yes—
133C2×C22⋊C8C2314yes—
134C2×C22⋊C8C2314yes—
135C2×C22⋊C8C2314yes—
136C2×C4×C8C2311yescharacteristic
137C2×C4×C8C2311yescharacteristic
138#400—24no—
139#399—24no—
140#399—24no—
141#399—24no—
142#399—24no—
143#398—24no—
144#396—24no—
145#396—24no—
146#391C2312yes—
147#390—22no—
148#390—22no—
149#390—24no—
150#390D412yes—
151#390D414yes—
152#389—22no—
153#389—22no—
154#389—22no—
155#389C2314yes—
156#389C2312yes—
157#387—22no—
158#386—24no—
159#386—24no—
160#385—24no—
161#385—24no—
162#384—22no—
163#384—24no—
164#384—21no—
165#384—21no—
166#384—22no—
167#384D411yescharacteristic
168#384C2312yes—
169#382—21no—
170#380—24no—
171#380C2314yes—
172#380C2314yes—
173C4×C4⋊C4—21no—
174C4×C4⋊C4C2311yescharacteristic
175C4×C4⋊C4C2312yes—
176C4×C4⋊C4C2311yescharacteristic
177C4×C22⋊C4—21no—
178C4×C22⋊C4—22no—
179#376—21no—
180C43C2311yescharacteristic
181#340C2314yes—
182#340C2314yes—
183#340C2312yes—
184#340C2312yes—
185#340C2312yes—
186#336C2314yes—
187#336C2312yes—
188#336C2312yes—
189C2×#41—28no—
190C2×#41—28no—
191C2×#41—28no—
192C2×#41—44no—
193C2×#41—44no—
194C2×#41—44no—
195C22×Q8C2411yescharacteristic
196C22×Q8C2414yes—
197C22×D4—24no—
198C22×D4C2×D412yes—
199C22×D4C2×D414yes—
200C22×D4C2411yescharacteristic
201C23×C4—22no—
202C23×C4—21no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4C2414yes—
206C2×Q16—44no—
207C2×Q16—44no—
208C2×Q16—44no—
209C2×SD16—22no—
210C2×SD16—24no—
211C2×M16—44no—
212C2×M16—44no—
213C2×M16—44no—
214C22×C8C2412yes—
215C22×C8C2414yes—
216C22×C8C2411yescharacteristic
217C22×C8C2411yescharacteristic
218C4⋊Q8—24no—
219C4⋊Q8—24no—
220C4⋊Q8—44no—
221C4⋊Q8—44no—
222#127—24no—
223#125—82no—
224#123—416no—
225#123—416no—
226#123—416no—
227C22⋊Q8—28no—
228C22⋊Q8—28no—
229C22⋊Q8—28no—
230C22⋊Q8—44no—
231C22⋊Q8—44no—
232C22⋊Q8—44no—
233C22⋊Q8—44no—
234C22⋊Q8—44no—
235#121—48no—
236#121—416no—
237C4×Q8—22no—
238C4×Q8—22no—
239C4×D4—24no—
240C4×D4—24no—
241C4×D4—24no—
242C4×D4—22no—
243C4×D4—24no—
244C4×D4—44no—
245C4×D4—44no—
246C4×D4—84no—
247C4×D4#4114yes—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—44no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4—48no—
265C2×C4⋊C4—416no—
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2×D414yes—
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2414yes—
270C2×C4⋊C4C2411yescharacteristic
271C2×C4⋊C4C2411yescharacteristic
272C2×C4⋊C4C2412yes—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4C2×D414yes—
282C2×C22⋊C4C2×D414yes—
283C2×C22⋊C4C2×D414yes—
284C2×C42—21no—
285C2×C42—21no—
286C2×C42—22no—
287C2×C42—21no—
288C2×C42—44no—
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D412yes—
291C2×C42C2×D411yescharacteristic
292C2×C42C2412yes—
293C2×C42C2411yescharacteristic
294#106—22no—
295#106—28no—
296#106—24no—
297C4⋊C8—24no—
298#103—22no—
299#103—24no—
300#103—44no—
301#103—44no—
302#103—44no—
303#103—44no—
304#103—44no—
305#103—44no—
306#103—44no—
307#102—22no—
308#102—24no—
309#102—28no—
310C22⋊C8—44no—
311C22⋊C8—44no—
312C22⋊C8—44no—
313C4×C8—22no—
314C4×C8—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95—21no—
319#95—22no—
320#95—21no—
321#95—24no—
322#95—24no—
323#95—24no—
324C24—21no—
325C24C22×D414yes—
326#41—416no—
327#41—88no—
328#41—88no—
329#41—88no—
330C2×Q8—21no—
331C2×Q8—24no—
332C2×Q8—22no—
333C2×Q8—24no—
334C2×Q8—24no—
335C2×Q8—24no—
336C2×Q8—24no—
337C2×Q8—24no—
338C2×Q8—21no—
339C2×Q8—22no—
340C2×D4—24no—
341C2×D4—24no—
342C2×D4—28no—
343C2×D4—24no—
344C2×D4—22no—
345C2×D4—21no—
346C2×D4—44no—
347C2×D4—44no—
348C2×D4—48no—
349C2×D4—48no—
350C2×D4—48no—
351C2×D4C2×#4112yes—
352C2×D4H(5,𝔽2)14yes—
353C22×C4—28no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—24no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—24no—
362C22×C4—24no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—21no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—44no—
369C22×C4—48no—
370C22×C4—48no—
371C22×C4—48no—
372C22×C4—44no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—84no—
376C22×C4—84no—
377C22×C4—84no—
378C22×C4C22×D412yes—
379C22×C4C22×D414yes—
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D412yes—
382C22×C4C22×D411yescharacteristic
383C22×C4C2511yescharacteristic
384Q16—88no—
385SD16—44no—
386M16—88no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—22no—
396C2×C8—22no—
397C4⋊C4—24no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—22no—
402C4⋊C4—24no—
403C4⋊C4—21no—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—44no—
407C4⋊C4—48no—
408C4⋊C4—48no—
409C4⋊C4—48no—
410C4⋊C4—48no—
411C4⋊C4—48no—
412C4⋊C4—44no—
413C4⋊C4—42no—
414C4⋊C4—44no—
415C4⋊C4—44no—
416C4⋊C4—82no—
417C4⋊C4—82no—
418C4⋊C4—84no—
419C4⋊C4—84no—
420C4⋊C4—84no—
421C4⋊C4C2×#4112yes—
422C4⋊C4H(5,𝔽2)14yes—
423C22⋊C4—28no—
424C22⋊C4—24no—
425C22⋊C4—28no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—48no—
429C22⋊C4—48no—
430C22⋊C4—48no—
431C22⋊C4—84no—
432C22⋊C4—84no—
433C22⋊C4—84no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—42no—
438C42—42no—
439C42—44no—
440C42—42no—
441C42—42no—
442C42#13714yes—
443C42C2×#4112yes—
444C42H(5,𝔽2)12yes—
445C23—22no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—42no—
455C23—41no—
456C23—41no—
457C23C23×D411yescharacteristic
458Q8—41no—
459Q8—44no—
460Q8—41no—
461Q8—44no—
462Q8—42no—
463Q8—44no—
464Q8—44no—
465D4—22no—
466D4—24no—
467D4—48no—
468D4—88no—
469D4—816no—
470D4#54014yes—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—22no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—416no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—48no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—44no—
505C2×C4—816no—
506C2×C4—168no—
507C2×C4#54612yes—
508C2×C4#54612yes—
509C2×C4C2×#13714yes—
510C2×C4C2×#13712yes—
511C2×C4#57712yes—
512C2×C4C22×#4111yescharacteristic
513C2×C4C2×H(5,𝔽2)12yes—
514C2×C4C2×H(5,𝔽2)12yes—
515C2×C4C2×H(5,𝔽2)11yescharacteristic
516C8—41no—
517C8—41no—
518C8—42no—
519C8—44no—
520C22—24no—
521C22—42no—
522C22—44no—
523C22—42no—
524C22—44no—
525C22—44no—
526C22—44no—
527C22—44no—
528C22—44no—
529C22—44no—
530C22—44no—
531C22—44no—
532C22C2×#54611yescharacteristic
533C22#348412yes—
534C22C22×#13711yescharacteristic
535C22C2×#57711yescharacteristic
536C22#358612yes—
537C4—22no—
538C4—24no—
539C4—44no—
540C4—44no—
541C4—42no—
542C4—42no—
543C4—81no—
544C4—84no—
545C4—81no—
546C4—84no—
547C4—82no—
548C4#333114yes—
549C4C2×#54011yescharacteristic
550C4#350811yescharacteristic
551C4#354312yes—
552C2—44no—
553C2—84no—
554C2#3769411yescharacteristic
555C2C2×#333111yescharacteristic
556C2#6122411yescharacteristic
557C2#6158612yes—
558C2#6169212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36179C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#127 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #39634C2semidirect
3. #33068C2semidirect
4. #32037C2semidirect
5. #30721C2semidirect
6. #30220C2semidirect
7. #29799C2semidirect
8. #26234C2semidirect
9. #25394C2semidirect
10. #24446C2semidirect
11. #24241C2semidirect
12. #24055C2semidirect
13. #22431C2semidirect
14. #22264C2semidirect
15. #21117C2semidirect

Maximal subgroups

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

subgroupindex
#396342
#361792
#330682
#320372
#307212
#302202
#297992
#262342
#253942
#244462
#242412
#240552
#224312
#222642
#211172