Finite Group Databasebeta

← #416103

Subgroups — #416103

Order 512.

1022 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35181C211yesmaximal, characteristic
2#21170C211yesmaximal, characteristic
3#21170C211yesmaximal, characteristic
4#13779C211yesmaximal, characteristic
5#11340C212yesmaximal
6#11339C212yesmaximal
7#11217C212yesmaximal
8#11217C212yesmaximal
9#10847C211yesmaximal, characteristic
10#10840C211yesmaximal, characteristic
11#10840C211yesmaximal, characteristic
12C2×#428C2212yes—
13C2×#428C2212yes—
14C2×#423C2211yescharacteristic
15C2×#423C2211yescharacteristic
16#2851—22no—
17#2646—22no—
18#2571—21no—
19#2571—21no—
20#2570—22no—
21#2570—22no—
22#2552—21no—
23#2552—21no—
24#2317C2212yes—
25#2317C2212yes—
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×#376C2211yescharacteristic
33#1949—22no—
34#1949—22no—
35#1948—21no—
36#1948—21no—
37#1947—21no—
38#1947—21no—
39C2×#343C2212yes—
40C2×#343C2212yes—
41#1736C2212yes—
42#1736C2212yes—
43C2×#339C2212yes—
44C2×#339C2212yes—
45C2×#339C2212yes—
46C2×#339C2212yes—
47#1436—22no—
48#1436—22no—
49C4⋊#95C2211yescharacteristic
50C4⋊#95C2211yescharacteristic
51C22×M16C2312yes—
52C22×M16C2312yes—
53C2×#125—22no—
54C2×#117C2312yes—
55C2×#117C2312yes—
56C22×C4⋊C4—22no—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C42—21no—
62C22×C42D412yes—
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66#428—24no—
67#428—24no—
68#428—42no—
69#428—42no—
70C2×#107—22no—
71C2×#107—22no—
72C2×#106—22no—
73C2×#106—22no—
74#423—22no—
75#423—22no—
76#423Q814yes—
77#423Q814yes—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82#400—24no—
83#400—24no—
84#398—24no—
85#398—24no—
86#389—22no—
87#389—22no—
88#389—22no—
89#389—22no—
90#389—22no—
91#389—22no—
92#387—24no—
93#387—24no—
94#385—24no—
95#385—24no—
96#384—22no—
97#384—22no—
98#384—22no—
99#384—22no—
100#384D411yescharacteristic
101#384D411yescharacteristic
102#384D411yescharacteristic
103#384D411yescharacteristic
104#382—22no—
105#382—22no—
106#382—22no—
107#382—22no—
108#382—22no—
109#382D412yes—
110#382D412yes—
111#380—21no—
112#380—21no—
113#380—21no—
114#380—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127#376—22no—
128#376Q814yes—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—22no—
132C2×#95—21no—
133C2×#95—22no—
134C2×#95—22no—
135C2×#95D412yes—
136#343—42no—
137#343—42no—
138#340—22no—
139#340—22no—
140#340—22no—
141#340—22no—
142#339—24no—
143#339—24no—
144#339—42no—
145#339—42no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4C2×D412yes—
151C23×C4C2×D412yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2411yescharacteristic
155C2×M16—22no—
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16C2×Q814yes—
166C2×M16C2×Q814yes—
167C22×C8—22no—
168C22×C8—22no—
169#125—84no—
170#117—22no—
171#117—22no—
172#117—22no—
173#117—22no—
174#117—22no—
175#117—22no—
176#117—22no—
177#117—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—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—21no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—82no—
214C2×C4⋊C4—82no—
215C2×C4⋊C4C2×D412yes—
216C2×C4⋊C4C2×D412yes—
217C2×C4⋊C4C2×D412yes—
218C2×C4⋊C4C2×D412yes—
219C2×C4⋊C4#4112yes—
220C2×C4⋊C4#4112yes—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—21no—
234C2×C42—22no—
235C2×C42—41no—
236C2×C42—41no—
237C2×C42—82no—
238C2×C42—82no—
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×Q812yes—
244C2×C42C2×Q812yes—
245C2×C42C2×Q812yes—
246C2×C42C2×Q812yes—
247C2×C42C2×Q812yes—
248C2×C42#4112yes—
249C2×C42#4112yes—
250C2×C42#4112yes—
251#107—44no—
252#107—44no—
253#106—44no—
254#106—44no—
255C4⋊C8—24no—
256C4⋊C8—24no—
257C4⋊C8—24no—
258C4⋊C8—24no—
259#95—28no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—24no—
271#95—24no—
272#95—41no—
273#95—41no—
274#95—82no—
275#95—82no—
276C24C22×D411yescharacteristic
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4#12011yescharacteristic
320C22×C4#12011yescharacteristic
321C22×C4#12011yescharacteristic
322C22×C4#12011yescharacteristic
323C22×C4C22⋊Q812yes—
324C22×C4C22⋊Q812yes—
325C22×C4C4⋊Q812yes—
326C22×C4C4⋊Q812yes—
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C22×Q811yescharacteristic
330C22×C4C22×Q811yescharacteristic
331C22×C4C22×Q811yescharacteristic
332C22×C4C2×#4111yescharacteristic
333C22×C4C2×#4111yescharacteristic
334C22×C4C2×#4111yescharacteristic
335C22×C4C2×#4111yescharacteristic
336C22×C4C2×#4111yescharacteristic
337C22×C4H(5,𝔽2)11yescharacteristic
338C22×C4H(5,𝔽2)11yescharacteristic
339C22×C42−1+411yescharacteristic
340C22×C42−1+411yescharacteristic
341M16—24no—
342M16—24no—
343M16—24no—
344M16—24no—
345M16—42no—
346M16—42no—
347M16—42no—
348M16—42no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—81no—
384C4⋊C4—81no—
385C4⋊C4—81no—
386C4⋊C4—81no—
387C4⋊C4—84no—
388C4⋊C4—84no—
389C4⋊C4—88no—
390C4⋊C4—84no—
391C4⋊C4—84no—
392C22⋊C4—42no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C22⋊C4—42no—
396C42—22no—
397C42—22no—
398C42—22no—
399C42—22no—
400C42—22no—
401C42—44no—
402C42—41no—
403C42—41no—
404C42—42no—
405C42—42no—
406C42—42no—
407C42—42no—
408C42—82no—
409C42—82no—
410C42—82no—
411C42C22⋊Q812yes—
412C42C22⋊Q812yes—
413C42C22⋊Q812yes—
414C42C22⋊Q812yes—
415C42#12514yescomplemented
416C42C4⋊Q812yes—
417C42C4⋊Q812yes—
418C42C4⋊Q812yes—
419C42C4⋊Q812yes—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23C2×#12011yescharacteristic
425C23C2×C22⋊Q811yescharacteristic
426C23#52511yescharacteristic
427C23#52511yescharacteristic
428C23C2×C4⋊Q811yescharacteristic
429C23#53711yescharacteristic
430C23#53711yescharacteristic
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—24no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4#39512yes—
490C2×C4#39512yes—
491C2×C4#39512yes—
492C2×C4#39512yes—
493C2×C4C2×C22⋊Q811yescharacteristic
494C2×C4C2×C22⋊Q811yescharacteristic
495C2×C4C2×#12511yescharacteristic
496C2×C4#52911yescharacteristic
497C2×C4#52911yescharacteristic
498C2×C4C2×C4⋊Q811yescharacteristic
499C2×C4C2×C4⋊Q811yescharacteristic
500C2×C4#53311yescharacteristic
501C2×C4#54011yescharacteristic
502C2×C4#54111yescharacteristic
503C2×C4#54411yescharacteristic
504C2×C4#54411yescharacteristic
505C2×C4#54611yescharacteristic
506C2×C4#54611yescharacteristic
507C2×C4#54711yescharacteristic
508C2×C4#54711yescharacteristic
509C2×C4#54811yescharacteristic
510C2×C4#54811yescharacteristic
511C2×C4#55011yescharacteristic
512C2×C4#55011yescharacteristic
513C2×C4#55411yescharacteristic
514C2×C4#55411yescharacteristic
515C2×C4#55411yescharacteristic
516C2×C4#55411yescharacteristic
517C2×C4#55711yescharacteristic
518C2×C4#55711yescharacteristic
519C2×C4#55711yescharacteristic
520C2×C4#55711yescharacteristic
521C8—44no—
522C8—44no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22C2×#39511yescharacteristic
536C22#240711yescharacteristic
537C22#240711yescharacteristic
538C22#258411yescharacteristic
539C22#258411yescharacteristic
540C22#262911yescharacteristic
541C22#285111yescharacteristic
542C22#301911yescharacteristic
543C22#302011yescharacteristic
544C22#302111yescharacteristic
545C22#302211yescharacteristic
546C4—22no—
547C4—22no—
548C4—22no—
549C4—22no—
550C4—44no—
551C4—82no—
552C4—82no—
553C4—82no—
554C4—82no—
555C4—82no—
556C4—82no—
557C4—82no—
558C4—82no—
559C4—82no—
560C4#245311yescharacteristic
561C4#245311yescharacteristic
562C4#245411yescharacteristic
563C4#245411yescharacteristic
564C4#255011yescharacteristic
565C4#255011yescharacteristic
566C4#255211yescharacteristic
567C4#255211yescharacteristic
568C2—21no—
569C2—21no—
570C2—21no—
571C2—21no—
572C2#1521811yescharacteristic
573C2#2356111yescharacteristic
574C2#2356211yescharacteristic
575C2#2418011yescharacteristic
576C2#2418111yescharacteristic
577C2#2418211yescharacteristic
578C2#2418311yescharacteristic
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
415. C42#125semidirect

Maximal subgroups

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

subgroupindex
#351812
#211702
#137792
#113402
#113392
#112172
#108472
#108402