Finite Group Databasebeta

← #482689

Subgroups — #482689

Order 512.

950 conjugacy classes of subgroups.

The subgroup lattice has 639 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36033C211yesmaximal, characteristic
2#20633C211yesmaximal, characteristic
3#17367C211yesmaximal, characteristic
4#11897C212yesmaximal, complemented
5#11878C212yesmaximal, complemented
6#11826C212yesmaximal, complemented
7#11822C212yesmaximal, complemented
8#11759C212yesmaximal, complemented
9#11755C212yesmaximal, complemented
10C2×#556C411yescharacteristic, complemented
11C2×#518C2211yescharacteristic
12#2973—21no—
13C22×#99—21no—
14#2638C411yescharacteristic, complemented
15#2638C2211yescharacteristic
16#2621C411yescharacteristic, complemented
17#2580C2211yescharacteristic
18#2561C411yescharacteristic, complemented
19#2561C2211yescharacteristic
20#2547C2211yescharacteristic
21#2482C411yescharacteristic, complemented
22#2482C2211yescharacteristic
23#2480—21no—
24#2480—21no—
25#2468—21no—
26#2468—21no—
27#2447—21no—
28#2435—21no—
29#2396C411yescharacteristic, complemented
30#2395C411yescharacteristic, complemented
31#2395C2211yescharacteristic
32#2367—21no—
33#2362—21no—
34#2357C411yescharacteristic, complemented
35#2315—21no—
36#1799—21no—
37#1799—21no—
38#1799C2212yes—
39#1799C2212yes—
40C2×#342C2212yes—
41#1466C2212yes—
42#1465—21no—
43#1464—21no—
44#1462C2212yes—
45#1459C2212yes—
46C2×#323—21no—
47#1352C2214yescomplemented
48#1351C2214yescomplemented
49#1344C2214yescomplemented
50#1344C2214yescomplemented
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×D4D411yescharacteristic
67C2×C4×D4D411yescharacteristic
68C2×C4×D4D411yescharacteristic
69C2×C4×D4D411yescharacteristic
70C2×C4×D4C2311yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C2×#117C2×C411yescharacteristic
73C2×#117C2×C411yescharacteristic
74C2×#117C2×C411yescharacteristic, complemented
75C2×#117D411yescharacteristic
76C22×C22⋊C4—22no—
77C22×C22⋊C4C2×C411yescharacteristic
78C22×C22⋊C4D411yescharacteristic
79C22×C22⋊C4D411yescharacteristic
80C22×C22⋊C4C2311yescharacteristic
81#435—44no—
82#432—22no—
83C2×#99—22no—
84C2×#99—22no—
85C2×#99—44no—
86#407—22no—
87C2×C22⋊C8—22no—
88#400C2×C411yescharacteristic, complemented
89#399C2×C411yescharacteristic
90#399C2×C411yescharacteristic, complemented
91#399C2311yescharacteristic
92#398—21no—
93#398—21no—
94#398C2×C411yescharacteristic, complemented
95#398C2×C411yescharacteristic, complemented
96#397—21no—
97#397—21no—
98#397—21no—
99#397—21no—
100#397—21no—
101#397C2×C411yescharacteristic
102#397C2×C411yescharacteristic
103#397C2×C411yescharacteristic, complemented
104#397C2×C411yescharacteristic, complemented
105#397C2311yescharacteristic
106#397C2311yescharacteristic
107#396—21no—
108#395C2×C411yescharacteristic, complemented
109#394—21no—
110#394—21no—
111#394—21no—
112#394—21no—
113#393C2×C411yescharacteristic
114#393C2×C411yescharacteristic, complemented
115#393C2311yescharacteristic
116#391—21no—
117#391—21no—
118#390—21no—
119#390—21no—
120#389—21no—
121#388—21no—
122#388—21no—
123#388—21no—
124#388—21no—
125#388—21no—
126#388—21no—
127#388—21no—
128#388—21no—
129#388—21no—
130#388—21no—
131#387—21no—
132#387—21no—
133#385—21no—
134#385—21no—
135#385—21no—
136#385—21no—
137#385—21no—
138#385—21no—
139#385—44no—
140#382C2×C411yescharacteristic, complemented
141#381—21no—
142#381—21no—
143#381C2×C411yescharacteristic
144#381C2×C411yescharacteristic, complemented
145#380—21no—
146#380—21no—
147#380—22no—
148#380—22no—
149#379—21no—
150#379—21no—
151#379—21no—
152#379—21no—
153C4×C4⋊C4—21no—
154C4×C4⋊C4—21no—
155C4×C4⋊C4—21no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4C2×C411yescharacteristic, complemented
158C4×C22⋊C4—21no—
159C4×C22⋊C4—21no—
160C4×C22⋊C4—21no—
161C4×C22⋊C4C2×C411yescharacteristic
162C4×C22⋊C4C2×C411yescharacteristic, complemented
163#342—22no—
164#342C2314yes—
165#324—22no—
166#323—22no—
167#323—22no—
168#323C2314yes—
169C25C22⋊C411yescharacteristic
170C25C2×D411yescharacteristic
171C22×Q8—21no—
172C22×Q8C22×C411yescharacteristic
173C22×D4—22no—
174C22×D4—22no—
175C22×D4—42no—
176C22×D4—42no—
177C22×D4C22⋊C411yescharacteristic
178C22×D4C22⋊C411yescharacteristic
179C22×D4C2×D411yescharacteristic
180C22×D4C2×D411yescharacteristic
181C23×C4—42no—
182C23×C4C22⋊C411yescharacteristic
183C23×C4C22⋊C411yescharacteristic
184C23×C4C22⋊C411yescharacteristic
185C23×C4C22×C411yescharacteristic
186C23×C4C22×C411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2×D411yescharacteristic
191C2×M16—42no—
192C22×C8—42no—
193C4⋊Q8—42no—
194#126—44no—
195#126—44no—
196#124—44no—
197#124—42no—
198#124—42no—
199C22⋊Q8—44no—
200C22⋊Q8—42no—
201C22⋊Q8—42no—
202C4×Q8—42no—
203C4×Q8—44no—
204C4×D4—24no—
205C4×D4—24no—
206C4×D4—42no—
207C4×D4—42no—
208C4×D4—42no—
209C4×D4—42no—
210C4×D4—42no—
211C4×D4—42no—
212#117—42no—
213#117—42no—
214#117—42no—
215#117—42no—
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⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4C22⋊C411yescharacteristic
231C2×C4⋊C4C22⋊C411yescharacteristic
232C2×C4⋊C4C22×C411yescharacteristic
233C2×C4⋊C4C22×C411yescharacteristic
234C2×C4⋊C4C22×C411yescharacteristic
235C2×C4⋊C4C22×C411yescharacteristic
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C22⋊C4—24no—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—42no—
261C2×C22⋊C4—42no—
262C2×C22⋊C4—44no—
263C2×C22⋊C4—44no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4C22⋊C411yescharacteristic
266C2×C22⋊C4C22⋊C411yescharacteristic
267C2×C22⋊C4C22⋊C411yescharacteristic
268C2×C22⋊C4C22⋊C411yescharacteristic
269C2×C22⋊C4C22⋊C411yescharacteristic
270C2×C22⋊C4C22⋊C411yescharacteristic
271C2×C22⋊C4C22×C411yescharacteristic
272C2×C22⋊C4C22×C411yescharacteristic
273C2×C22⋊C4C2×D411yescharacteristic
274C2×C22⋊C4C2×D411yescharacteristic
275C2×C22⋊C4C2×D411yescharacteristic
276C2×C22⋊C4C2411yescharacteristic
277C2×C42—21no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—21no—
281C2×C42—21no—
282C2×C42—21no—
283C2×C42C22⋊C411yescharacteristic
284C2×C42C22⋊C411yescharacteristic
285C2×C42C22×C411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C4⋊C8—42no—
288#99—44no—
289#99—44no—
290C22⋊C8—24no—
291C22⋊C8—42no—
292C22⋊C8—42no—
293#97—42no—
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—21no—
307#95—21no—
308#95—21no—
309#95—21no—
310#95—21no—
311#95—42no—
312#95—42no—
313#95C22×C411yescharacteristic
314#95C22×C411yescharacteristic
315#95C22×C411yescharacteristic
316#95C22×C411yescharacteristic
317C24—22no—
318C24—21no—
319C24—21no—
320C24—22no—
321C24—22no—
322C24—22no—
323C24—41no—
324C24—81no—
325C24—81no—
326C24—81no—
327C24—81no—
328C24C2×C22⋊C411yescharacteristic
329C24C2×C22⋊C411yescharacteristic
330C24C2×C22⋊C411yescharacteristic
331C24C2×C22⋊C411yescharacteristic
332C24C22×D411yescharacteristic
333C2×Q8—42no—
334C2×Q8—42no—
335C2×Q8—41no—
336C2×Q8—42no—
337C2×Q8—81no—
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—48no—
351C2×D4—48no—
352C2×D4AGL(1,ℤ/8)12yes—
353C2×D4#13712yes—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—44no—
374C22×C4—81no—
375C22×C4—81no—
376C22×C4—81no—
377C22×C4—81no—
378C22×C4—81no—
379C22×C4—81no—
380C22×C4—81no—
381C22×C4—81no—
382C22×C4—84no—
383C22×C4—162no—
384C22×C4—162no—
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C23×C411yescharacteristic
394C22×C4C22×D411yescharacteristic
395M16—162no—
396C2×C8—44no—
397C2×C8—162no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—42no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—42no—
412C4⋊C4—81no—
413C4⋊C4—81no—
414C4⋊C4—81no—
415C4⋊C4—81no—
416C4⋊C4—81no—
417C4⋊C4—162no—
418C4⋊C4—162no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—41no—
425C22⋊C4—41no—
426C22⋊C4—41no—
427C22⋊C4—41no—
428C22⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—42no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—42no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—81no—
445C22⋊C4—81no—
446C22⋊C4—81no—
447C22⋊C4—81no—
448C22⋊C4—81no—
449C22⋊C4—81no—
450C22⋊C4—81no—
451C22⋊C4—81no—
452C22⋊C4—81no—
453C22⋊C4—81no—
454C22⋊C4—84no—
455C22⋊C4#9914yes—
456C42—22no—
457C42—22no—
458C42—41no—
459C42—41no—
460C42—42no—
461C42—42no—
462C42—42no—
463C42—42no—
464C42—81no—
465C42—81no—
466C23—24no—
467C23—24no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—21no—
477C23—21no—
478C23—21no—
479C23—41no—
480C23—41no—
481C23—44no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23—82no—
491C23—82no—
492C23—161no—
493C23—161no—
494C23—162no—
495C23—161no—
496C23—161no—
497C23—162no—
498C23—162no—
499C23C2×#9912yes—
500C23#41712yes—
501C23#42112yes—
502C23C22×C22⋊C411yescharacteristic
503C23C2×AGL(1,ℤ/8)11yescharacteristic
504C23C2×#13711yescharacteristic
505Q8—84no—
506Q8—82no—
507D4—42no—
508D4—42no—
509D4—44no—
510D4—44no—
511D4—42no—
512D4—42no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—44no—
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—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4—82no—
560C2×C4—82no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—84no—
564C2×C4—84no—
565C2×C4—84no—
566C2×C4—161no—
567C2×C4—161no—
568C2×C4—168no—
569C2×C4C2×#9912yes—
570C2×C4C2×#9912yes—
571C2×C4#41712yes—
572C8—84no—
573C22—21no—
574C22—21no—
575C22—22no—
576C22—22no—
577C22—22no—
578C22—21no—
579C22—21no—
580C22—22no—
581C22—22no—
582C22—42no—
583C22—41no—
584C22—41no—
585C22—42no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—84no—
603C22—84no—
604C22—164no—
605C22C22×#9911yescharacteristic
606C22C2×#41711yescharacteristic
607C22C2×#42111yescharacteristic
608C4—22no—
609C4—22no—
610C4—22no—
611C4—41no—
612C4—41no—
613C4—41no—
614C4—41no—
615C4—41no—
616C4—41no—
617C4—41no—
618C4—81no—
619C4—81no—
620C4—81no—
621C4—81no—
622C4—81no—
623C4—81no—
624C4—81no—
625C4—81no—
626C4—164no—
627C2—21no—
628C2—21no—
629C2—41no—
630C2—42no—
631C2—41no—
632C2—42no—
633C2—42no—
634C2—81no—
635C2#1759211yescharacteristic
636C2#3335011yescharacteristic
637C2#3336111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ 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. #11897C2semidirect
5. #11878C2semidirect
6. #11826C2semidirect
7. #11822C2semidirect
8. #11759C2semidirect
9. #11755C2semidirect
10. C2×#556C4semidirect
14. #2638C4semidirect
16. #2621C4semidirect
18. #2561C4semidirect
21. #2482C4semidirect
29. #2396C4semidirect
30. #2395C4semidirect
34. #2357C4semidirect
47. #1352C22semidirect
48. #1351C22semidirect
49. #1344C22semidirect
50. #1344C22semidirect
53. C2×C4⋊Q8C2×C4semidirect
60. C2×C22⋊Q8C2×C4semidirect
63. C2×C4×Q8C2×C4semidirect
74. C2×#117C2×C4semidirect
88. #400C2×C4semidirect
90. #399C2×C4semidirect
94. #398C2×C4semidirect
95. #398C2×C4semidirect
103. #397C2×C4semidirect
104. #397C2×C4semidirect
108. #395C2×C4semidirect
114. #393C2×C4semidirect
140. #382C2×C4semidirect
144. #381C2×C4semidirect
157. C4×C4⋊C4C2×C4semidirect
162. 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
#206332
#173672
#118972
#118782
#118262
#118222
#117592
#117552