Finite Group Databasebeta

← #416105

Subgroups — #416105

Order 512.

1042 conjugacy classes of subgroups.

The subgroup lattice has 586 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#13911C211yesmaximal, characteristic
5#11342C212yesmaximal
6#11339C212yesmaximal
7#11217C212yesmaximal
8#11211C212yesmaximal
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—
28#2285C2211yescharacteristic
29C2×#384C2211yescharacteristic
30C2×#382C2212yes—
31C4×#117C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1949—22no—
35#1948—21no—
36#1947—21no—
37#1776—21no—
38#1776—21no—
39C4×#107—21no—
40C4×#106—21no—
41C2×#343C2212yes—
42C2×#343C2212yes—
43#1736C2212yes—
44#1736C2211yescharacteristic
45#1736C2211yescharacteristic
46C2×#339C2212yes—
47C2×#339C2212yes—
48C2×#339C2212yes—
49C2×#339C2212yes—
50#1442—22no—
51C4⋊#95—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⋊C4D412yes—
63C22×C4⋊C4D412yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C42—21no—
68C22×C42C2312yes—
69C22×C42C2311yescharacteristic
70#428—22no—
71#428—22no—
72#428—22no—
73#428—22no—
74#428—24no—
75#428—42no—
76C2×#107—21no—
77C2×#107—21no—
78C2×#107—22no—
79C2×#106—21no—
80C2×#106—21no—
81C2×#106—22no—
82#423—22no—
83#423Q814yes—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C4×M16—22no—
87C4×M16Q814yes—
88C2×#97—21no—
89C2×C4×C8—21no—
90#400—24no—
91#400—24no—
92#398—24no—
93#398—24no—
94#389—22no—
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#385—24no—
105#384—22no—
106#384—21no—
107#384—21no—
108#384—21no—
109#384—21no—
110#384D411yescharacteristic
111#384D411yescharacteristic
112#382—22no—
113#382—22no—
114#382—22no—
115#382—24no—
116#382—42no—
117#382D412yes—
118#380—24no—
119#380—21no—
120#380—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4D411yescharacteristic
129C4×C4⋊C4D411yescharacteristic
130C4×C4⋊C4D411yescharacteristic
131C4×C4⋊C4D411yescharacteristic
132C4×C4⋊C4Q814yes—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—21no—
135#376—21no—
136C2×#95—21no—
137C2×#95—24no—
138C2×#95—21no—
139C2×#95—24no—
140C43—21no—
141#343—24no—
142#343—42no—
143#340—22no—
144#340—22no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#339—24no—
150#339—42no—
151#339—42no—
152#339—42no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D414yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2411yescharacteristic
160C2×M16—21no—
161C2×M16—21no—
162C2×M16—21no—
163C2×M16—21no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—22no—
167C2×M16—21no—
168C2×M16—22no—
169C2×M16—22no—
170C2×M16—21no—
171C2×M16—21no—
172C2×M16—21no—
173C2×M16C2×Q812yes—
174C2×M16C2×Q814yes—
175C2×M16C2×Q812yes—
176C2×M16#4112yes—
177C2×M16#4112yes—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—22no—
181#125—84no—
182#125—84no—
183#117—22no—
184#117—22no—
185#117—21no—
186#117—21no—
187#117—21no—
188#117—21no—
189#117—21no—
190#117—21no—
191#117—22no—
192#117—22no—
193#117—21no—
194#117—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—82no—
228C2×C4⋊C4C2×D411yescharacteristic
229C2×C4⋊C4C2×D411yescharacteristic
230C2×C4⋊C4C2×D411yescharacteristic
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C4⋊C4C2×D412yes—
233C2×C4⋊C4C2×D412yes—
234C2×C4⋊C4#4112yes—
235C2×C4⋊C4#4112yes—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—22no—
239C2×C42—22no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—22no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—44no—
252C2×C42—41no—
253C2×C42—41no—
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42C2×Q814yes—
259C2×C42C2×Q812yes—
260C2×C42C2×Q812yes—
261C2×C42C2×Q812yes—
262C2×C42#4112yes—
263#107—24no—
264#107—24no—
265#107—44no—
266#106—24no—
267#106—24no—
268#106—44no—
269C4⋊C8—24no—
270C4⋊C8—24no—
271#97—24no—
272C4×C8—24no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—24no—
277#95—21no—
278#95—21no—
279#95—21no—
280#95—21no—
281#95—22no—
282#95—22no—
283#95—24no—
284#95—48no—
285#95—41no—
286#95—41no—
287#95—84no—
288C24C22×D411yescharacteristic
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—21no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—44no—
320C22×C4—44no—
321C22×C4#12011yescharacteristic
322C22×C4#12011yescharacteristic
323C22×C4#12011yescharacteristic
324C22×C4#12011yescharacteristic
325C22×C4C22⋊Q812yes—
326C22×C4C22⋊Q812yes—
327C22×C4C4⋊Q812yes—
328C22×C4C4⋊Q812yes—
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C22×Q811yescharacteristic
332C22×C4C22×Q811yescharacteristic
333C22×C4C22×Q811yescharacteristic
334C22×C4C2×#4111yescharacteristic
335C22×C4C2×#4111yescharacteristic
336C22×C4C2×#4111yescharacteristic
337C22×C4C2×#4111yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4H(5,𝔽2)11yescharacteristic
340C22×C4H(5,𝔽2)11yescharacteristic
341C22×C42−1+411yescharacteristic
342C22×C42−1+411yescharacteristic
343M16—22no—
344M16—22no—
345M16—22no—
346M16—22no—
347M16—22no—
348M16—24no—
349M16—24no—
350M16—42no—
351M16—41no—
352M16—42no—
353M16—41no—
354M16#12514yescomplemented
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—24no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—42no—
363C2×C8—42no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—42no—
381C4⋊C4—48no—
382C4⋊C4—44no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—42no—
388C4⋊C4—41no—
389C4⋊C4—42no—
390C4⋊C4—41no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—44no—
394C4⋊C4—42no—
395C4⋊C4—82no—
396C4⋊C4—82no—
397C4⋊C4—84no—
398C4⋊C4—84no—
399C4⋊C4—84no—
400C4⋊C4—84no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C22⋊C4—41no—
404C22⋊C4—42no—
405C22⋊C4—41no—
406C22⋊C4—42no—
407C42—24no—
408C42—24no—
409C42—42no—
410C42—42no—
411C42—42no—
412C42—42no—
413C42—42no—
414C42—42no—
415C42—41no—
416C42—41no—
417C42—41no—
418C42—41no—
419C42—42no—
420C42—42no—
421C42—82no—
422C42—82no—
423C42C22⋊Q812yes—
424C42C22⋊Q812yes—
425C42C22⋊Q812yes—
426C42C22⋊Q812yes—
427C42C4⋊Q812yes—
428C42C4⋊Q812yes—
429C42C4⋊Q812yes—
430C42C4⋊Q812yes—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23C2×#12011yescharacteristic
436C23C2×C22⋊Q811yescharacteristic
437C23C2×C4⋊Q811yescharacteristic
438C23#54611yescharacteristic
439C23#54711yescharacteristic
440C23#54811yescharacteristic
441C23#55011yescharacteristic
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—84no—
491C2×C4—84no—
492C2×C4#39512yes—
493C2×C4#39512yes—
494C2×C4#39512yes—
495C2×C4#39512yes—
496C2×C4#45412yes—
497C2×C4#45512yes—
498C2×C4C2×C22⋊Q811yescharacteristic
499C2×C4C2×C22⋊Q811yescharacteristic
500C2×C4#52511yescharacteristic
501C2×C4#52511yescharacteristic
502C2×C4C2×#12511yescharacteristic
503C2×C4#52911yescharacteristic
504C2×C4#52911yescharacteristic
505C2×C4C2×C4⋊Q811yescharacteristic
506C2×C4C2×C4⋊Q811yescharacteristic
507C2×C4#53311yescharacteristic
508C2×C4#53711yescharacteristic
509C2×C4#53711yescharacteristic
510C2×C4#54011yescharacteristic
511C2×C4#54111yescharacteristic
512C2×C4#54411yescharacteristic
513C2×C4#54411yescharacteristic
514C2×C4#54611yescharacteristic
515C2×C4#54711yescharacteristic
516C2×C4#54811yescharacteristic
517C2×C4#55011yescharacteristic
518C2×C4#55411yescharacteristic
519C2×C4#55411yescharacteristic
520C2×C4#55411yescharacteristic
521C2×C4#55411yescharacteristic
522C2×C4#55711yescharacteristic
523C2×C4#55711yescharacteristic
524C2×C4#55711yescharacteristic
525C2×C4#55711yescharacteristic
526C8—24no—
527C8—42no—
528C8—44no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22C2×#39511yescharacteristic
542C22#240711yescharacteristic
543C22#240711yescharacteristic
544C22#245311yescharacteristic
545C22#245411yescharacteristic
546C22#255011yescharacteristic
547C22#255211yescharacteristic
548C22C2×#45511yescharacteristic
549C22C2×#45411yescharacteristic
550C22#301911yescharacteristic
551C22#302011yescharacteristic
552C4—22no—
553C4—22no—
554C4—22no—
555C4—84no—
556C4—81no—
557C4—82no—
558C4—81no—
559C4—84no—
560C4—81no—
561C4—82no—
562C4—82no—
563C4—81no—
564C4—82no—
565C4#206114yes—
566C4#245311yescharacteristic
567C4#245411yescharacteristic
568C4#255011yescharacteristic
569C4#255211yescharacteristic
570C4#258411yescharacteristic
571C4#258411yescharacteristic
572C4#262911yescharacteristic
573C4#285111yescharacteristic
574C2—21no—
575C2—21no—
576C2—21no—
577C2—21no—
578C2#1521811yescharacteristic
579C2C2×#206111yescharacteristic
580C2#2356111yescharacteristic
581C2#2414511yescharacteristic
582C2#2414611yescharacteristic
583C2#2417111yescharacteristic
584C2#2417211yescharacteristic
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
354. 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
#139112
#113422
#113392
#112172
#112112
#108572
#108522
#108512