Finite Group Databasebeta

← #482611

Subgroups — #482611

Order 512.

928 conjugacy classes of subgroups.

The subgroup lattice has 621 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36033C211yesmaximal, characteristic
2#20689C211yesmaximal, characteristic
3#17367C211yesmaximal, characteristic
4#11845C212yesmaximal, complemented
5#11841C212yesmaximal, complemented
6#11810C212yesmaximal, complemented
7#11801C212yesmaximal, complemented
8#11759C212yesmaximal, complemented
9#11755C212yesmaximal, complemented
10C2×#556C411yescharacteristic, complemented
11C2×#518C2211yescharacteristic
12#2973—21no—
13#2638C411yescharacteristic, complemented
14#2638C411yescharacteristic, complemented
15#2621C2211yescharacteristic
16#2580C411yescharacteristic, complemented
17#2561C2211yescharacteristic
18#2561C2211yescharacteristic
19#2547C411yescharacteristic, complemented
20#2482C2211yescharacteristic
21#2482C2211yescharacteristic
22#2480—21no—
23#2480—21no—
24#2468—21no—
25#2468—21no—
26#2447—21no—
27#2435—21no—
28#2396C2211yescharacteristic
29#2395C411yescharacteristic, complemented
30#2395C411yescharacteristic, complemented
31#2367—21no—
32#2362—21no—
33#2357C411yescharacteristic, complemented
34#2308—21no—
35#1820—21no—
36#1820C2212yes—
37#1819—21no—
38#1819C2212yes—
39C2×#342—21no—
40C2×#342C2212yes—
41#1466—21no—
42#1465—21no—
43#1464C2212yes—
44#1462C2212yes—
45#1459—21no—
46C2×#323C2212yes—
47#1352C2214yes—
48#1351C2214yes—
49#1344C2214yes—
50#1344C2214yes—
51C23×D4D411yescharacteristic
52#556—44no—
53C2×C4⋊Q8C2×C411yescharacteristic, complemented
54C2×#126—21no—
55C2×#126—21no—
56C2×#124—21no—
57C2×#124—21no—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8C2×C411yescharacteristic
60C2×C22⋊Q8C2×C411yescharacteristic, complemented
61#518—44no—
62C2×C4×Q8—21no—
63C2×C4×Q8C2×C411yescharacteristic, complemented
64C2×C4×D4C2×C411yescharacteristic
65C2×C4×D4C2×C411yescharacteristic
66C2×C4×D4C2×C411yescharacteristic
67C2×C4×D4C2×C411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×C4×D4D411yescharacteristic
71C2×C4×D4D411yescharacteristic
72C2×#117C2×C411yescharacteristic
73C2×#117C2×C411yescharacteristic, complemented
74C2×#117D411yescharacteristic
75C2×#117C2311yescharacteristic
76C22×C22⋊C4—22no—
77C22×C22⋊C4D411yescharacteristic
78C22×C22⋊C4D411yescharacteristic
79C22×C22⋊C4C2311yescharacteristic
80C22×C22⋊C4C2311yescharacteristic
81#435—44no—
82#432—22no—
83#407—22no—
84C2×C22⋊C8—22no—
85#400C2×C411yescharacteristic, complemented
86#399C2×C411yescharacteristic
87#399C2×C411yescharacteristic, complemented
88#399C2311yescharacteristic
89#398—21no—
90#398—21no—
91#398C2×C411yescharacteristic, complemented
92#398C2×C411yescharacteristic, complemented
93#397—21no—
94#397—21no—
95#397—21no—
96#397—21no—
97#397—21no—
98#397C2×C411yescharacteristic
99#397C2×C411yescharacteristic
100#397C2×C411yescharacteristic, complemented
101#397C2×C411yescharacteristic, complemented
102#397C2311yescharacteristic
103#397C2311yescharacteristic
104#396—21no—
105#395C2×C411yescharacteristic, complemented
106#394—21no—
107#394—21no—
108#394—21no—
109#394—21no—
110#393C2×C411yescharacteristic
111#393C2×C411yescharacteristic, complemented
112#393C2311yescharacteristic
113#391—21no—
114#391—21no—
115#390—21no—
116#390—21no—
117#389—21no—
118#388—21no—
119#388—21no—
120#388—21no—
121#388—21no—
122#388—21no—
123#388—21no—
124#388—21no—
125#388—21no—
126#388—21no—
127#388—21no—
128#387—21no—
129#387—21no—
130#386—44no—
131#385—21no—
132#385—21no—
133#385—21no—
134#385—21no—
135#385—21no—
136#385—21no—
137#382C2×C411yescharacteristic, complemented
138#381—21no—
139#381—21no—
140#381—22no—
141#381C2×C411yescharacteristic
142#381C2×C411yescharacteristic, complemented
143#380—21no—
144#380—21no—
145#379—21no—
146#379—21no—
147#379—21no—
148#379—21no—
149#379—22no—
150C4×C4⋊C4—21no—
151C4×C4⋊C4—21no—
152C4×C4⋊C4—21no—
153C4×C4⋊C4—21no—
154C4×C4⋊C4C2×C411yescharacteristic, complemented
155C4×C22⋊C4—21no—
156C4×C22⋊C4—21no—
157C4×C22⋊C4—21no—
158C4×C22⋊C4C2×C411yescharacteristic
159C4×C22⋊C4C2×C411yescharacteristic, complemented
160#342—22no—
161#342—22no—
162#342—22no—
163#342C2314yes—
164#324—22no—
165#323—22no—
166#323—22no—
167#323C2314yes—
168C25C2×D411yescharacteristic
169C25C2×D411yescharacteristic
170C22×Q8—21no—
171C22×Q8C22×C411yescharacteristic
172C22×D4—24no—
173C22×D4—42no—
174C22×D4—42no—
175C22×D4C22⋊C411yescharacteristic
176C22×D4C22⋊C411yescharacteristic
177C22×D4C22⋊C411yescharacteristic
178C22×D4C22⋊C411yescharacteristic
179C23×C4—42no—
180C23×C4C22⋊C411yescharacteristic
181C23×C4C22⋊C411yescharacteristic
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22⋊C411yescharacteristic
184C23×C4C22×C411yescharacteristic
185C23×C4C22×C411yescharacteristic
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C2×M16—42no—
190C22×C8—42no—
191C4⋊Q8—42no—
192#126—44no—
193#126—44no—
194#124—44no—
195#124—44no—
196C22⋊Q8—44no—
197C22⋊Q8—42no—
198C22⋊Q8—42no—
199C4×Q8—42no—
200C4×Q8—44no—
201C4×D4—24no—
202C4×D4—24no—
203C4×D4—42no—
204C4×D4—42no—
205C4×D4—42no—
206C4×D4—42no—
207C4×D4—42no—
208C4×D4—42no—
209#117—42no—
210#117—42no—
211#117—42no—
212#117—42no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4C22⋊C411yescharacteristic
226C2×C4⋊C4C22⋊C411yescharacteristic
227C2×C4⋊C4C22×C411yescharacteristic
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C4⋊C4C22×C411yescharacteristic
230C2×C4⋊C4C22×C411yescharacteristic
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4—42no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—44no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4C22⋊C411yescharacteristic
263C2×C22⋊C4C22⋊C411yescharacteristic
264C2×C22⋊C4C22⋊C411yescharacteristic
265C2×C22⋊C4C22⋊C411yescharacteristic
266C2×C22⋊C4C22×C411yescharacteristic
267C2×C22⋊C4C22×C411yescharacteristic
268C2×C22⋊C4C2×D411yescharacteristic
269C2×C22⋊C4C2×D411yescharacteristic
270C2×C22⋊C4C2×D411yescharacteristic
271C2×C22⋊C4C2×D411yescharacteristic
272C2×C22⋊C4C2×D411yescharacteristic
273C2×C22⋊C4C2411yescharacteristic
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42C22⋊C411yescharacteristic
281C2×C42C22⋊C411yescharacteristic
282C2×C42C22×C411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C4⋊C8—42no—
285C22⋊C8—24no—
286C22⋊C8—42no—
287C22⋊C8—42no—
288#97—42no—
289#95—21no—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—21no—
298#95—21no—
299#95—21no—
300#95—21no—
301#95—21no—
302#95—21no—
303#95—21no—
304#95—21no—
305#95—21no—
306#95—44no—
307#95C22×C411yescharacteristic
308#95C22×C411yescharacteristic
309#95C22×C411yescharacteristic
310#95C22×C411yescharacteristic
311C24—22no—
312C24—22no—
313C24—22no—
314C24—22no—
315C24—22no—
316C24—41no—
317C24—81no—
318C24—81no—
319C24—81no—
320C24—81no—
321C24C2×C22⋊C411yescharacteristic
322C24C2×C22⋊C411yescharacteristic
323C24C2×C22⋊C411yescharacteristic
324C24C2×C22⋊C411yescharacteristic
325C24C22×D411yescharacteristic
326C2×Q8—42no—
327C2×Q8—42no—
328C2×Q8—41no—
329C2×Q8—42no—
330C2×Q8—81no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—48no—
343C2×D4—48no—
344C2×D4#9914yes—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—21no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—44no—
365C22×C4—81no—
366C22×C4—81no—
367C22×C4—81no—
368C22×C4—81no—
369C22×C4—81no—
370C22×C4—81no—
371C22×C4—81no—
372C22×C4—81no—
373C22×C4—84no—
374C22×C4—162no—
375C22×C4—162no—
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C23×C411yescharacteristic
385C22×C4C22×D411yescharacteristic
386M16—162no—
387C2×C8—44no—
388C2×C8—162no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—81no—
404C4⋊C4—81no—
405C4⋊C4—81no—
406C4⋊C4—81no—
407C4⋊C4—81no—
408C22⋊C4—22no—
409C22⋊C4—22no—
410C22⋊C4—22no—
411C22⋊C4—22no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—41no—
417C22⋊C4—41no—
418C22⋊C4—41no—
419C22⋊C4—41no—
420C22⋊C4—42no—
421C22⋊C4—42no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—42no—
425C22⋊C4—42no—
426C22⋊C4—42no—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—81no—
433C22⋊C4—81no—
434C22⋊C4—81no—
435C22⋊C4—81no—
436C22⋊C4—81no—
437C22⋊C4—81no—
438C22⋊C4—81no—
439C22⋊C4—81no—
440C22⋊C4—81no—
441C22⋊C4—81no—
442C22⋊C4—84no—
443C22⋊C4—162no—
444C22⋊C4—162no—
445C42—22no—
446C42—22no—
447C42—41no—
448C42—41no—
449C42—42no—
450C42—42no—
451C42—42no—
452C42—42no—
453C42—81no—
454C42—81no—
455C23—24no—
456C23—24no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—41no—
472C23—44no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—82no—
480C23—82no—
481C23—82no—
482C23—82no—
483C23—162no—
484C23—162no—
485C23—162no—
486C23—162no—
487C23C2×#9912yes—
488C23C2×#9912yes—
489C23#42112yes—
490C23C22×C22⋊C411yescharacteristic
491C23#57512yes—
492Q8—84no—
493Q8—82no—
494D4—44no—
495D4—44no—
496D4—44no—
497D4—44no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—44no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—82no—
547C2×C4—82no—
548C2×C4—84no—
549C2×C4—84no—
550C2×C4—84no—
551C2×C4—161no—
552C2×C4—161no—
553C2×C4—161no—
554C2×C4—161no—
555C2×C4—168no—
556C2×C4C2×#9912yes—
557C8—84no—
558C22—21no—
559C22—21no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22—81no—
585C22—84no—
586C22—84no—
587C22—164no—
588C22C22×#9911yescharacteristic
589C22#289611yescharacteristic
590C22C2×#42111yescharacteristic
591C4—22no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—81no—
602C4—81no—
603C4—81no—
604C4—81no—
605C4—81no—
606C4—81no—
607C4—81no—
608C4—81no—
609C4—164no—
610C2—21no—
611C2—21no—
612C2—42no—
613C2—42no—
614C2—42no—
615C2—42no—
616C2—81no—
617C2#1758711yescharacteristic
618C2#3335011yescharacteristic
619C2#3336111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#518 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11845C2semidirect
5. #11841C2semidirect
6. #11810C2semidirect
7. #11801C2semidirect
8. #11759C2semidirect
9. #11755C2semidirect
10. C2×#556C4semidirect
13. #2638C4semidirect
14. #2638C4semidirect
16. #2580C4semidirect
19. #2547C4semidirect
29. #2395C4semidirect
30. #2395C4semidirect
33. #2357C4semidirect
53. C2×C4⋊Q8C2×C4semidirect
60. C2×C22⋊Q8C2×C4semidirect
63. C2×C4×Q8C2×C4semidirect
73. C2×#117C2×C4semidirect
85. #400C2×C4semidirect
87. #399C2×C4semidirect
91. #398C2×C4semidirect
92. #398C2×C4semidirect
100. #397C2×C4semidirect
101. #397C2×C4semidirect
105. #395C2×C4semidirect
111. #393C2×C4semidirect
137. #382C2×C4semidirect
142. #381C2×C4semidirect
154. C4×C4⋊C4C2×C4semidirect
159. C4×C22⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#360332
#206892
#173672
#118452
#118412
#118102
#118012
#117592
#117552