Finite Group Databasebeta

← #416104

Subgroups — #416104

Order 512.

1114 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35286C211yesmaximal, characteristic
2#21170C211yesmaximal, characteristic
3C4×#428C211yesmaximal, characteristic
4#13910C211yesmaximal, characteristic
5#11342C212yesmaximal
6#11340C212yesmaximal
7#11217C212yesmaximal
8#11210C212yesmaximal
9#10857C211yesmaximal, characteristic
10#10852C211yesmaximal, characteristic
11#10851C211yesmaximal, characteristic
12C2×#428C2211yescharacteristic
13C2×#428C2211yescharacteristic
14C2×#428C2212yes—
15C2×#423C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17#2859—21no—
18#2859—21no—
19#2647—22no—
20#2571—21no—
21#2571—21no—
22#2552—21no—
23#2552—21no—
24#2442—22no—
25#2442—22no—
26#2317C2212yes—
27#2316C2212yes—
28C2×#389—22no—
29#2285C2211yescharacteristic
30C2×#384C2212yes—
31C2×#384C2211yescharacteristic
32C4×#117C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34C2×C4×C4⋊C4C2211yescharacteristic
35#1949—22no—
36#1948—21no—
37#1947—21no—
38#1776—21no—
39#1776—21no—
40C4×#107—21no—
41C4×#106—21no—
42C2×#343C2212yes—
43C2×#343C2212yes—
44#1736C2212yes—
45#1736C2211yescharacteristic
46#1736C2211yescharacteristic
47C2×#339C2212yes—
48C2×#339C2212yes—
49C2×#339C2212yes—
50C2×#339C2212yes—
51#1444—22no—
52C4⋊#95C2212yes—
53C22×M16C2311yescharacteristic
54C22×M16C2312yes—
55C22×M16C2311yescharacteristic
56C2×#125—22no—
57C2×#125—22no—
58C2×#117C2311yescharacteristic
59C2×#117C2312yes—
60C2×#117C2311yescharacteristic
61C22×C4⋊C4—21no—
62C22×C4⋊C4—24no—
63C22×C4⋊C4D412yes—
64C22×C4⋊C4D412yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C42—21no—
69C22×C42C2312yes—
70C22×C42C2311yescharacteristic
71#428—22no—
72#428—22no—
73#428—22no—
74#428—22no—
75#428—24no—
76#428—42no—
77C2×#107—21no—
78C2×#107—21no—
79C2×#107—22no—
80C2×#106—21no—
81C2×#106—21no—
82C2×#106—22no—
83#423—22no—
84#423Q814yes—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C4×M16—22no—
88C4×M16Q814yes—
89C2×#97—21no—
90C2×C4×C8—21no—
91#400—24no—
92#400—24no—
93#398—24no—
94#398—24no—
95#389—22no—
96#389—22no—
97#389—22no—
98#389—22no—
99#389—22no—
100#389—22no—
101#389—22no—
102#389—22no—
103#389—22no—
104#389—22no—
105#389—48no—
106#385—24no—
107#384—22no—
108#384—24no—
109#384—21no—
110#384—21no—
111#384—21no—
112#384—21no—
113#384—42no—
114#384D411yescharacteristic
115#384D411yescharacteristic
116#382—22no—
117#382—22no—
118#382—22no—
119#382D412yes—
120#380—24no—
121#380—21no—
122#380—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4D411yescharacteristic
131C4×C4⋊C4D411yescharacteristic
132C4×C4⋊C4D411yescharacteristic
133C4×C4⋊C4D411yescharacteristic
134C4×C4⋊C4Q814yes—
135C4×C22⋊C4—21no—
136C4×C22⋊C4—21no—
137#376—21no—
138C2×#95—21no—
139C2×#95—24no—
140C2×#95—21no—
141C43—21no—
142#343—24no—
143#343—42no—
144#340—22no—
145#340—22no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#339—24no—
151#339—42no—
152#339—42no—
153#339—42no—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D414yes—
159C23×C4C2×D411yescharacteristic
160C23×C4C2411yescharacteristic
161C2×M16—21no—
162C2×M16—21no—
163C2×M16—21no—
164C2×M16—21no—
165C2×M16—22no—
166C2×M16—22no—
167C2×M16—22no—
168C2×M16—21no—
169C2×M16—22no—
170C2×M16—22no—
171C2×M16—21no—
172C2×M16—21no—
173C2×M16—21no—
174C2×M16C2×Q812yes—
175C2×M16C2×Q814yes—
176C2×M16C2×Q812yes—
177C2×M16#4112yes—
178C2×M16#4112yes—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—22no—
182#125—84no—
183#125—84no—
184#117—22no—
185#117—22no—
186#117—21no—
187#117—21no—
188#117—21no—
189#117—21no—
190#117—21no—
191#117—21no—
192#117—22no—
193#117—22no—
194#117—21no—
195#117—21no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—48no—
229C2×C4⋊C4—48no—
230C2×C4⋊C4—82no—
231C2×C4⋊C4—84no—
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×C4⋊C4#4112yes—
239C2×C4⋊C4#4112yes—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—44no—
256C2×C42—41no—
257C2×C42—41no—
258C2×C42C2×D411yescharacteristic
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×Q814yes—
263C2×C42C2×Q812yes—
264C2×C42C2×Q812yes—
265C2×C42C2×Q812yes—
266C2×C42#4112yes—
267#107—24no—
268#107—24no—
269#107—44no—
270#106—24no—
271#106—24no—
272#106—44no—
273C4⋊C8—24no—
274C4⋊C8—24no—
275#97—24no—
276C4×C8—24no—
277#95—24no—
278#95—24no—
279#95—24no—
280#95—24no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—22no—
286#95—22no—
287#95—24no—
288#95—48no—
289#95—41no—
290#95—41no—
291C24C22×D411yescharacteristic
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—21no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—44no—
323C22×C4—44no—
324C22×C4#12011yescharacteristic
325C22×C4#12011yescharacteristic
326C22×C4#12011yescharacteristic
327C22×C4#12011yescharacteristic
328C22×C4C22⋊Q812yes—
329C22×C4C22⋊Q812yes—
330C22×C4C4⋊Q812yes—
331C22×C4C4⋊Q812yes—
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C22×Q811yescharacteristic
335C22×C4C22×Q811yescharacteristic
336C22×C4C22×Q811yescharacteristic
337C22×C4C2×#4111yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4C2×#4111yescharacteristic
340C22×C4C2×#4111yescharacteristic
341C22×C4C2×#4111yescharacteristic
342C22×C4H(5,𝔽2)11yescharacteristic
343C22×C4H(5,𝔽2)11yescharacteristic
344C22×C42−1+411yescharacteristic
345C22×C42−1+411yescharacteristic
346M16—22no—
347M16—22no—
348M16—22no—
349M16—22no—
350M16—22no—
351M16—24no—
352M16—24no—
353M16—42no—
354M16—41no—
355M16—42no—
356M16—41no—
357M16#12514yescomplemented
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—24no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—42no—
366C2×C8—42no—
367C4⋊C4—216no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—42no—
385C4⋊C4—44no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—42no—
391C4⋊C4—41no—
392C4⋊C4—42no—
393C4⋊C4—41no—
394C4⋊C4—42no—
395C4⋊C4—44no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—82no—
399C4⋊C4—82no—
400C4⋊C4—84no—
401C4⋊C4—84no—
402C4⋊C4—84no—
403C4⋊C4—84no—
404C4⋊C4—816no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C22⋊C4—41no—
408C22⋊C4—42no—
409C22⋊C4—41no—
410C22⋊C4—42no—
411C42—24no—
412C42—24no—
413C42—42no—
414C42—42no—
415C42—42no—
416C42—42no—
417C42—42no—
418C42—42no—
419C42—41no—
420C42—41no—
421C42—41no—
422C42—41no—
423C42—42no—
424C42—42no—
425C42—82no—
426C42—82no—
427C42C22⋊Q812yes—
428C42C22⋊Q812yes—
429C42C22⋊Q812yes—
430C42C22⋊Q812yes—
431C42C4⋊Q812yes—
432C42C4⋊Q812yes—
433C42C4⋊Q812yes—
434C42C4⋊Q812yes—
435C23—21no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23C2×#12011yescharacteristic
440C23C2×C22⋊Q811yescharacteristic
441C23C2×C4⋊Q811yescharacteristic
442C23#54611yescharacteristic
443C23#54711yescharacteristic
444C23#54811yescharacteristic
445C23#55011yescharacteristic
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—28no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—44no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—84no—
495C2×C4—84no—
496C2×C4#39512yes—
497C2×C4#39512yes—
498C2×C4#39512yes—
499C2×C4#39512yes—
500C2×C4#45312yes—
501C2×C4#45612yes—
502C2×C4C2×C22⋊Q811yescharacteristic
503C2×C4C2×C22⋊Q811yescharacteristic
504C2×C4#52511yescharacteristic
505C2×C4#52511yescharacteristic
506C2×C4C2×#12511yescharacteristic
507C2×C4#52911yescharacteristic
508C2×C4#52911yescharacteristic
509C2×C4C2×C4⋊Q811yescharacteristic
510C2×C4C2×C4⋊Q811yescharacteristic
511C2×C4#53311yescharacteristic
512C2×C4#53711yescharacteristic
513C2×C4#53711yescharacteristic
514C2×C4#54011yescharacteristic
515C2×C4#54111yescharacteristic
516C2×C4#54411yescharacteristic
517C2×C4#54411yescharacteristic
518C2×C4#54611yescharacteristic
519C2×C4#54711yescharacteristic
520C2×C4#54811yescharacteristic
521C2×C4#55011yescharacteristic
522C2×C4#55411yescharacteristic
523C2×C4#55411yescharacteristic
524C2×C4#55411yescharacteristic
525C2×C4#55411yescharacteristic
526C2×C4#55711yescharacteristic
527C2×C4#55711yescharacteristic
528C2×C4#55711yescharacteristic
529C2×C4#55711yescharacteristic
530C8—24no—
531C8—42no—
532C8—44no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22C2×#39511yescharacteristic
546C22#240711yescharacteristic
547C22#240711yescharacteristic
548C22#245311yescharacteristic
549C22#245411yescharacteristic
550C22#255011yescharacteristic
551C22#255211yescharacteristic
552C22C2×#45311yescharacteristic
553C22C2×#45611yescharacteristic
554C22#301911yescharacteristic
555C22#302011yescharacteristic
556C4—22no—
557C4—22no—
558C4—22no—
559C4—48no—
560C4—81no—
561C4—84no—
562C4—82no—
563C4—81no—
564C4—81no—
565C4—82no—
566C4—82no—
567C4—81no—
568C4—82no—
569C4#206014yes—
570C4#245311yescharacteristic
571C4#245411yescharacteristic
572C4#255011yescharacteristic
573C4#255211yescharacteristic
574C4#258411yescharacteristic
575C4#258411yescharacteristic
576C4#262911yescharacteristic
577C4#285111yescharacteristic
578C2—21no—
579C2—21no—
580C2—21no—
581C2—21no—
582C2#1521811yescharacteristic
583C2C2×#206011yescharacteristic
584C2#2356111yescharacteristic
585C2#2414411yescharacteristic
586C2#2414711yescharacteristic
587C2#2417011yescharacteristic
588C2#2417311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
357. M16#125semidirect

Maximal subgroups

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

subgroupindex
#352862
#211702
C4×#4282
#139102
#113422
#113402
#112172
#112102
#108572
#108522
#108512