Finite Group Databasebeta

← #278017

Subgroups — #278017

Order 512.

772 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33207C211yesmaximal, characteristic, complemented
2#24836C211yesmaximal, characteristic
3#24751C211yesmaximal, characteristic
4C8×#121C211yesmaximal, characteristic
5C2×#1671C211yesmaximal, characteristic, complemented
6C22×#334C211yesmaximal, characteristic, complemented
7#10598C211yesmaximal, characteristic, complemented
8#10597C211yesmaximal, characteristic, complemented
9#10480C211yesmaximal, characteristic, complemented
10#10477C211yesmaximal, characteristic, complemented
11#9729C211yesmaximal, characteristic, complemented
12#9612C211yesmaximal, characteristic, complemented
13#9131C211yesmaximal, characteristic, complemented
14#9127C211yesmaximal, characteristic, complemented
15#8642C211yesmaximal, characteristic, complemented
16C2×#498C2211yescharacteristic
17C2×#476C2211yescharacteristic
18C2×#465—21no—
19C2×#461C2211yescharacteristic
20C2×C8×D4—21no—
21C2×C8×D4C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C22×C4⋊C8C2211yescharacteristic
24C22×C4×C8C2211yescharacteristic
25#2857C2211yescharacteristic
26#2857C2211yescharacteristic
27C4×#121C2211yescharacteristic
28#2088—21no—
29#2066—21no—
30#2065—21no—
31#2055—21no—
32#1990C2211yescharacteristic
33#1990C2211yescharacteristic
34#1987C2211yescharacteristic
35#1981C2211yescharacteristic
36#1954C2211yescharacteristic
37#1948C2211yescharacteristic
38#1948C2211yescharacteristic
39#1941—21no—
40#1941—21no—
41#1939C2211yescharacteristic
42#1923—21no—
43#1893—21no—
44#1892—21no—
45#1869—21no—
46#1844C2211yescharacteristic
47#1807—21no—
48#1807—21no—
49#1807—21no—
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52#1797C2211yescharacteristic
53C8×C4⋊C4C2211yescharacteristic
54C8×C22⋊C4—21no—
55#1671—24no—
56C2×#334—24no—
57C2×#334C2211yescharacteristic
58C2×#334C2211yescharacteristic
59C2×#334C2211yescharacteristic
60C2×#334C2211yescharacteristic
61C2×#333C2211yescharacteristic
62C2×#333C2211yescharacteristic
63C2×#332C2211yescharacteristic
64C2×#332C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69C22×SD16—21no—
70C23×C8—21no—
71C23×C8—21no—
72C23×C8—21no—
73C23×C8D411yescharacteristic
74C23×C8D411yescharacteristic
75C2×C4⋊Q8C2311yescharacteristic
76C2×C4⋊Q8C2311yescharacteristic
77C2×C22⋊Q8—21no—
78C2×C22⋊Q8—21no—
79C2×#121D411yescharacteristic
80C2×C4×D4—21no—
81C2×C4×D4C2311yescharacteristic
82C22×C42C2311yescharacteristic
83#498—22no—
84#498—22no—
85#476—22no—
86#476—41no—
87#465—44no—
88#461—22no—
89#461—41no—
90C8×D4—22no—
91C8×D4—22no—
92C8×D4—44no—
93C2×#106—21no—
94C2×#106—21no—
95C2×#106—21no—
96C2×#106—21no—
97C2×#106—21no—
98C2×#106—21no—
99C2×#106—21no—
100C2×C4⋊C8—22no—
101C2×C4⋊C8—22no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8—21no—
105C2×C4⋊C8—41no—
106C2×C4⋊C8—41no—
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×C4⋊C8C2311yescharacteristic
112C2×C4⋊C8C2311yescharacteristic
113C2×#103—21no—
114C2×#103—21no—
115C2×#103—21no—
116C2×#102—21no—
117C2×#102—21no—
118C2×#102—21no—
119C2×C22⋊C8—21no—
120C2×C22⋊C8—21no—
121C2×C22⋊C8—21no—
122C2×C22⋊C8—21no—
123C2×C4×C8—22no—
124C2×C4×C8—22no—
125C2×C4×C8—21no—
126C2×C4×C8D411yescharacteristic
127C2×C4×C8D411yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130#391—21no—
131#391—21no—
132#390D411yescharacteristic
133#389—21no—
134#389—21no—
135#389—21no—
136#389—21no—
137#389D411yescharacteristic
138#388—21no—
139#386—21no—
140#384C2311yescharacteristic
141#384C2311yescharacteristic
142#380—21no—
143#380—21no—
144#380—21no—
145#380—21no—
146#380D411yescharacteristic
147C4×C4⋊C4C2311yescharacteristic
148C4×C22⋊C4—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#334—22no—
163#334—22no—
164#334—22no—
165#334D414yescomplemented
166#333—22no—
167#333—41no—
168#332—22no—
169#332—41no—
170C22×Q8—21no—
171C22×Q8—21no—
172C22×D4—21no—
173C22×D4C2×D411yescharacteristic
174C23×C4—21no—
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4#4111yescharacteristic
178C2×SD16—42no—
179C2×SD16—42no—
180C2×SD16—81no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8—22no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—22no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—41no—
192C22×C8—42no—
193C22×C8—42no—
194C22×C8—42no—
195C22×C8—42no—
196C22×C8—42no—
197C22×C8—42no—
198C22×C8—42no—
199C22×C8—81no—
200C22×C8—81no—
201C22×C8SD1614yescomplemented
202C22×C8C2×D411yescharacteristic
203C22×C8C2×D411yescharacteristic
204C22×C8C2×D411yescharacteristic
205C22×C8C2×D411yescharacteristic
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C22⋊Q8—42no—
215C22⋊Q8—42no—
216C22⋊Q8—82no—
217#121—22no—
218#121SD1614yes—
219C4×D4—21no—
220C4×D4—21no—
221C4×D4—21no—
222C4×D4—21no—
223C4×D4—42no—
224C4×D4—42no—
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—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4C2×D411yescharacteristic
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C4⋊C4C2×D411yescharacteristic
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—41no—
247C2×C22⋊C4—41no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42#4111yescharacteristic
256C2×C42C2411yescharacteristic
257#106—42no—
258#106—42no—
259#106—42no—
260#106—42no—
261#106—42no—
262#106—42no—
263#106—81no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—42no—
275C4⋊C8—42no—
276C4⋊C8—42no—
277C4⋊C8C2×D412yes—
278C4⋊C8C2×D412yes—
279#103—42no—
280#103—42no—
281#103—81no—
282#102—42no—
283#102—42no—
284#102—81no—
285C22⋊C8—42no—
286C22⋊C8—42no—
287C22⋊C8—42no—
288C22⋊C8—42no—
289C4×C8—22no—
290C4×C8—22no—
291C4×C8—21no—
292C4×C8—21no—
293C4×C8—42no—
294C4×C8C2×D412yes—
295C4×C8#4112yes—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—41no—
300#95—41no—
301#95—41no—
302#95—41no—
303C24—21no—
304C24#12111yescharacteristic
305C2×Q8—41no—
306C2×Q8—41no—
307C2×Q8—41no—
308C2×Q8—41no—
309C2×Q8—41no—
310C2×Q8—41no—
311C2×Q8—41no—
312C2×Q8—41no—
313C2×Q8—81no—
314C2×Q8—81no—
315C2×D4—24no—
316C2×D4—21no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4C2×SD1612yes—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4#12111yescharacteristic
350C22×C4#12111yescharacteristic
351C22×C4#12111yescharacteristic
352C22×C4C2×SD1612yes—
353C22×C4C2×SD1612yes—
354C22×C4#13512yes—
355C22×C4#13712yes—
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358C22×C4C2×#4111yescharacteristic
359SD16—84no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—28no—
365C2×C8—21no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—44no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
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—84no—
388C2×C8—84no—
389C2×C8—84no—
390C2×C8#12112yes—
391C2×C8#12112yes—
392C2×C8C2×SD1612yes—
393C2×C8C2×SD1612yes—
394C2×C8#13512yes—
395C4⋊C4—21no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—42no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—81no—
418C4⋊C4—81no—
419C4⋊C4—82no—
420C4⋊C4—82no—
421C4⋊C4C2×SD1612yes—
422C22⋊C4—24no—
423C22⋊C4—42no—
424C22⋊C4—42no—
425C22⋊C4—41no—
426C22⋊C4—41no—
427C22⋊C4—42no—
428C22⋊C4—82no—
429C22⋊C4—82no—
430C42—21no—
431C42—21no—
432C42—21no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—41no—
437C42—41no—
438C42C22×D411yescharacteristic
439C42C2×#4111yescharacteristic
440C42H(5,𝔽2)11yescharacteristic
441C422−1+411yescharacteristic
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—22no—
447C23—41no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23#46112yes—
454C23#46512yes—
455C23#47112yes—
456C23C2×#12111yescharacteristic
457Q8—81no—
458Q8—81no—
459Q8—81no—
460Q8—81no—
461Q8—161no—
462Q8—161no—
463D4—22no—
464D4—22no—
465D4—48no—
466D4—41no—
467D4—41no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4#46112yes—
513C2×C4#46512yes—
514C2×C4C2×#12111yescharacteristic
515C2×C4#53711yescharacteristic
516C2×C4D4211yescharacteristic
517C2×C4#54711yescharacteristic
518C2×C4#54811yescharacteristic
519C2×C4#55311yescharacteristic
520C2×C4C22×SD1611yescharacteristic
521C2×C4C22×SD1611yescharacteristic
522C2×C4C2×#13511yescharacteristic
523C2×C4C2×#13511yescharacteristic
524C2×C4C2×#13711yescharacteristic
525C2×C4#57511yescharacteristic
526C2×C4#57711yescharacteristic
527C2×C4#57711yescharacteristic
528C8—22no—
529C8—22no—
530C8—42no—
531C8—42no—
532C8—41no—
533C8—81no—
534C8—81no—
535C8—81no—
536C8—81no—
537C8#46514yescomplemented
538C22—22no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—81no—
548C22—81no—
549C22—81no—
550C22—81no—
551C22#167114yescomplemented
552C22#241611yescharacteristic
553C22C2×#46111yescharacteristic
554C22#303511yescharacteristic
555C22C2×#46511yescharacteristic
556C22#306011yescharacteristic
557C22C2×#47111yescharacteristic
558C22#306711yescharacteristic
559C4—22no—
560C4—22no—
561C4—21no—
562C4—21no—
563C4—42no—
564C4—41no—
565C4—81no—
566C4—81no—
567C4—161no—
568C4—161no—
569C4—161no—
570C4—161no—
571C4C2×#46511yescharacteristic
572C4#305611yescharacteristic
573C4#324611yescharacteristic
574C4#324811yescharacteristic
575C4D4×SD1611yescharacteristic
576C4#328211yescharacteristic
577C4#333511yescharacteristic
578C4#333711yescharacteristic
579C2—24no—
580C2—82no—
581C2C2×#167111yescharacteristic
582C2#1912411yescharacteristic
583C2#1919411yescharacteristic
584C2#1922111yescharacteristic
585C2#2475111yescharacteristic
586C2#2477611yescharacteristic
587C2#2480211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#121D4
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
1. #33207C2semidirect
5. C2×#1671C2semidirect
6. C22×#334C2semidirect
7. #10598C2semidirect
8. #10597C2semidirect
9. #10480C2semidirect
10. #10477C2semidirect
11. #9729C2semidirect
12. #9612C2semidirect
13. #9131C2semidirect
14. #9127C2semidirect
15. #8642C2semidirect
165. #334D4semidirect
201. C22×C8SD16semidirect
537. C8#465semidirect
551. C22#1671semidirect

Maximal subgroups

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

subgroupindex
#332072
#248362
#247512
C8×#1212
C2×#16712
C22×#3342
#105982
#105972
#104802
#104772
#97292
#96122
#91312
#91272
#86422