Finite Group Databasebeta

← #281673

Subgroups — #281673

Order 512.

750 conjugacy classes of subgroups.

The subgroup lattice has 573 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#25813C211yesmaximal, characteristic
2#24846C211yesmaximal, characteristic
3#24302C211yesmaximal, characteristic
4#23050C211yesmaximal, characteristic
5C2×#1676C211yesmaximal, characteristic, complemented
6C2×#1548C211yesmaximal, characteristic
7#10601C211yesmaximal, characteristic, complemented
8#10543C211yesmaximal, characteristic, complemented
9#10483C211yesmaximal, characteristic, complemented
10#10279C211yesmaximal, characteristic, complemented
11#9730C211yesmaximal, characteristic, complemented
12#9567C211yesmaximal, characteristic, complemented
13#9216C211yesmaximal, characteristic, complemented
14#9128C211yesmaximal, characteristic
15#8832C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C2×#500C2211yescharacteristic
18C2×#500C2211yescharacteristic
19C2×#477C2211yescharacteristic
20C2×#477C2211yescharacteristic
21C2×#477C2211yescharacteristic
22C2×C8×Q8C2211yescharacteristic
23C2×#431C2211yescharacteristic
24C2×#423C2211yescharacteristic
25#2857C2211yescharacteristic
26C4×C22⋊Q8C2211yescharacteristic
27#2086—21no—
28#2066—21no—
29#2057—21no—
30#2057—21no—
31#2057—21no—
32#2024—21no—
33#1990C2211yescharacteristic
34#1990C2211yescharacteristic
35#1987C2211yescharacteristic
36#1987C2211yescharacteristic
37#1987C2211yescharacteristic
38#1984C2211yescharacteristic
39#1948C2211yescharacteristic
40#1939C2211yescharacteristic
41#1923—21no—
42#1923—21no—
43#1916—21no—
44#1894—21no—
45#1872—21no—
46#1852—21no—
47#1852—21no—
48#1852—21no—
49#1809—21no—
50#1803C2211yescharacteristic
51#1797C2211yescharacteristic
52C8×C4⋊C4—21no—
53C8×C4⋊C4C2211yescharacteristic
54C4×C22⋊C8C2211yescharacteristic
55#1676—24no—
56C2×#335C2211yescharacteristic
57C2×#335C2211yescharacteristic
58#1548—24no—
59C2×#333C2211yescharacteristic
60C2×#333C2211yescharacteristic
61C2×#333C2211yescharacteristic
62C2×#333C2211yescharacteristic
63C2×#333C2211yescharacteristic
64C2×#333C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1277C2211yescharacteristic
68#1277C2211yescharacteristic
69C23×Q8—21no—
70C22×M16—21no—
71C2×C4⋊Q8—22no—
72C2×C4⋊Q8—22no—
73C2×C4⋊Q8—21no—
74C2×C4⋊Q8—21no—
75C2×C4⋊Q8—21no—
76C2×C4⋊Q8—41no—
77C2×C4⋊Q8—41no—
78C2×C4⋊Q8C2311yescharacteristic
79C2×C4⋊Q8C2311yescharacteristic
80C2×C4⋊Q8C2311yescharacteristic
81C2×C22⋊Q8—21no—
82C2×C22⋊Q8D411yescharacteristic
83C2×C4×Q8C2311yescharacteristic
84C22×C4⋊C4—21no—
85C22×C4⋊C4—21no—
86C22×C42C2311yescharacteristic
87#500—22no—
88#500—22no—
89#500—41no—
90#500—41no—
91#477—22no—
92#477—22no—
93#477—22no—
94#477—41no—
95#477—41no—
96#477—41no—
97C8×Q8—22no—
98C8×Q8—41no—
99#431—22no—
100#431—41no—
101C2×#107—21no—
102C2×#107—21no—
103C2×#107—21no—
104C2×#107—21no—
105C2×#107—21no—
106C2×#107—21no—
107C2×#107—21no—
108C2×#106—21no—
109#423—22no—
110#423—41no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8C2311yescharacteristic
115C2×C4⋊C8C2311yescharacteristic
116C2×C4⋊C8C2311yescharacteristic
117C2×C4⋊C8C2311yescharacteristic
118C2×C4⋊C8C2311yescharacteristic
119C2×C4⋊C8C2311yescharacteristic
120C2×#103—21no—
121C2×#103—21no—
122C2×#103—21no—
123C2×#103—21no—
124C2×C22⋊C8—21no—
125C2×C22⋊C8—21no—
126C2×C22⋊C8D411yescharacteristic
127C2×C22⋊C8D411yescharacteristic
128C2×C4×C8—21no—
129C2×C4×C8—21no—
130C2×C4×C8—21no—
131C2×C4×C8C2311yescharacteristic
132C2×C4×C8C2311yescharacteristic
133#391—21no—
134#391D411yescharacteristic
135#389—21no—
136#389—21no—
137#389D411yescharacteristic
138#387—21no—
139#385—21no—
140#384C2311yescharacteristic
141#380—21no—
142#380—21no—
143#380D411yescharacteristic
144C4×C4⋊C4—21no—
145C4×C4⋊C4C2311yescharacteristic
146C4×C22⋊C4—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152#340—21no—
153#340—21no—
154#340—21no—
155#340—21no—
156#340—21no—
157#340—21no—
158#340—21no—
159#336—21no—
160#336—21no—
161#336—21no—
162#336—21no—
163#336—21no—
164#336D411yescharacteristic
165#336D411yescharacteristic
166#335—22no—
167#335—41no—
168#333—22no—
169#333—22no—
170#333—22no—
171#333—22no—
172#333—41no—
173#333D414yes—
174C22×Q8—24no—
175C22×Q8—21no—
176C22×Q8—21no—
177C22×Q8—21no—
178C22×Q8—42no—
179C22×Q8—41no—
180C22×Q8—82no—
181C22×Q8—82no—
182C22×Q8C2×D411yescharacteristic
183C23×C4—41no—
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4#4111yescharacteristic
187C2×M16—42no—
188C2×M16—42no—
189C2×M16—81no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8C2×D411yescharacteristic
197C22×C8C2×D411yescharacteristic
198C22×C8C2×D411yescharacteristic
199C22×C8C2×D411yescharacteristic
200C4⋊Q8—24no—
201C4⋊Q8—21no—
202C4⋊Q8—21no—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8—42no—
212C4⋊Q8—42no—
213C4⋊Q8—42no—
214C4⋊Q8—42no—
215C4⋊Q8—42no—
216C4⋊Q8C2×D412yes—
217C4⋊Q8#4112yes—
218C22⋊Q8—22no—
219C22⋊Q8—82no—
220C22⋊Q8Q1614yes—
221C4×Q8—21no—
222C4×Q8—21no—
223C4×Q8—21no—
224C4×Q8—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C4⋊C4C2×D411yescharacteristic
246C2×C4⋊C4C2×D411yescharacteristic
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—41no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42C2×D411yescharacteristic
257C2×C42C2×D411yescharacteristic
258C2×C42#4111yescharacteristic
259C2×C42C2411yescharacteristic
260#107—42no—
261#107—42no—
262#107—42no—
263#107—42no—
264#107—42no—
265#107—42no—
266#107—42no—
267#106—81no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—42no—
279C4⋊C8—42no—
280C4⋊C8—42no—
281C4⋊C8C2×D412yes—
282C4⋊C8C2×D412yes—
283#103—42no—
284#103—42no—
285#103—42no—
286#103—81no—
287C22⋊C8—22no—
288C22⋊C8—22no—
289C22⋊C8—42no—
290C22⋊C8—42no—
291C4×C8—21no—
292C4×C8—21no—
293C4×C8—21no—
294C4×C8—21no—
295C4×C8—42no—
296C4×C8—42no—
297C4×C8—42no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—41no—
304#95—41no—
305C24#12111yescharacteristic
306C2×Q8—21no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8—44no—
314C2×Q8—41no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×Q8—41no—
318C2×Q8—82no—
319C2×Q8—81no—
320C2×Q8—81no—
321C2×Q8—81no—
322C2×Q8—81no—
323C2×Q8—81no—
324C2×Q8—81no—
325C2×Q8—88no—
326C2×Q8—88no—
327C2×Q8—82no—
328C2×Q8C2×Q1612yes—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—41no—
347C22×C4—44no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—82no—
353C22×C4—82no—
354C22×C4#12111yescharacteristic
355C22×C4#12111yescharacteristic
356C22×C4#12111yescharacteristic
357C22×C4C2×Q1612yes—
358C22×C4#13512yes—
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C2×#4111yescharacteristic
362M16—84no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8#13712yes—
392C2×C8#13712yes—
393C4⋊C4—21no—
394C4⋊C4—24no—
395C4⋊C4—22no—
396C4⋊C4—21no—
397C4⋊C4—22no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—42no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—81no—
421C4⋊C4—82no—
422C4⋊C4C2×Q1612yes—
423C22⋊C4—24no—
424C22⋊C4—81no—
425C22⋊C4—82no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—41no—
433C42—41no—
434C42—41no—
435C42—41no—
436C42C22×D411yescharacteristic
437C42C2×#4111yescharacteristic
438C42H(5,𝔽2)11yescharacteristic
439C42H(5,𝔽2)11yescharacteristic
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23#46712yes—
447C23C2×#12111yescharacteristic
448Q8—22no—
449Q8—22no—
450Q8—41no—
451Q8—41no—
452Q8—82no—
453Q8—82no—
454Q8—88no—
455Q8—82no—
456Q8—82no—
457Q8—161no—
458Q8—161no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—42no—
475C2×C4—42no—
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—88no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4#46712yes—
504C2×C4#47212yes—
505C2×C4#52511yescharacteristic
506C2×C4#53811yescharacteristic
507C2×C4D4211yescharacteristic
508C2×C4D4211yescharacteristic
509C2×C4#54611yescharacteristic
510C2×C4#54611yescharacteristic
511C2×C4C22×Q1611yescharacteristic
512C2×C4C2×#13511yescharacteristic
513C2×C4C2×#13711yescharacteristic
514C2×C4C2×#13711yescharacteristic
515C2×C4#57711yescharacteristic
516C2×C4#57811yescharacteristic
517C2×C4#57811yescharacteristic
518C2×C4#57811yescharacteristic
519C8—22no—
520C8—22no—
521C8—41no—
522C8—41no—
523C8—42no—
524C8—42no—
525C8—81no—
526C8—81no—
527C8—81no—
528C8—81no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—41no—
533C22—41no—
534C22—41no—
535C22—41no—
536C22#242411yescharacteristic
537C22#303911yescharacteristic
538C22#304411yescharacteristic
539C22C2×#46711yescharacteristic
540C22#306311yescharacteristic
541C22C2×#47211yescharacteristic
542C22#306811yescharacteristic
543C4—22no—
544C4—21no—
545C4—21no—
546C4—22no—
547C4—41no—
548C4—42no—
549C4—42no—
550C4—81no—
551C4—82no—
552C4—82no—
553C4—81no—
554C4—161no—
555C4—161no—
556C4#327011yescharacteristic
557C4#327611yescharacteristic
558C4#328211yescharacteristic
559C4D4×Q1611yescharacteristic
560C4#329911yescharacteristic
561C4#330011yescharacteristic
562C4#331211yescharacteristic
563C4#332811yescharacteristic
564C2—42no—
565C2#1927211yescharacteristic
566C2#1937911yescharacteristic
567C2#1940511yescharacteristic
568C2#1957311yescharacteristic
569C2#2509511yescharacteristic
570C2#2511011yescharacteristic
571C2#2514011yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
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. C2×#1676C2semidirect
7. #10601C2semidirect
8. #10543C2semidirect
9. #10483C2semidirect
10. #10279C2semidirect
11. #9730C2semidirect
12. #9567C2semidirect
13. #9216C2semidirect

Maximal subgroups

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

subgroupindex
#258132
#248462
#243022
#230502
C2×#16762
C2×#15482
#106012
#105432
#104832
#102792
#97302
#95672
#92162
#91282
#88322