Finite Group Databasebeta

← #206426

Subgroups — #206426

Order 512.

1068 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60700C211yesmaximal, characteristic
2#21856C211yesmaximal, characteristic, complemented
3#21815C211yesmaximal, characteristic, complemented
4#20366C211yesmaximal, characteristic, complemented
5#17796C211yesmaximal, characteristic, complemented
6#17473C211yesmaximal, characteristic, complemented
7C2×#1492C211yesmaximal, characteristic, complemented
8#9854C212yesmaximal, complemented
9#9852C212yesmaximal, complemented
10#9847C212yesmaximal, complemented
11#9846C212yesmaximal, complemented
12#3444C2211yescharacteristic
13C2×#532C2211yescharacteristic
14#3184C412yescomplemented
15#3182C412yescomplemented
16#3180—22no—
17#3171—22no—
18#3169C2212yes—
19#3167C2212yes—
20#3165C411yescharacteristic, complemented
21#3151C411yescharacteristic, complemented
22#3141C411yescharacteristic, complemented
23#3138C411yescharacteristic, complemented
24#3055—22no—
25C4×AGL(1,ℤ/8)—22no—
26#2920C2211yescharacteristic
27C2×#423C2211yescharacteristic
28C2×C4×M16C2211yescharacteristic
29C22×C4×C8C2211yescharacteristic
30C4×#117C2211yescharacteristic
31#1994C2212yescomplemented
32#1980—22no—
33#1962C2212yes—
34#1962C2212yes—
35#1962C2212yescomplemented
36#1871—22no—
37#1806—22no—
38#1789—22no—
39#1746—22no—
40#1744C2212yescomplemented
41#1744C2212yescomplemented
42#1729—22no—
43#1569—22no—
44#1567C2212yescomplemented
45#1566C2212yescomplemented
46#1565—22no—
47#1494—22no—
48#1493C2211yescharacteristic, complemented
49#1493C2211yescharacteristic, complemented
50#1492—22no—
51#1492C2212yes—
52#1492C2212yes—
53#1492C2211yescharacteristic, complemented
54#1492C2211yescharacteristic, complemented
55#1476—22no—
56C2×D4⋊C8—22no—
57C22×#41—21no—
58C2×AGL(1,ℤ/8)—22no—
59C22×M16—21no—
60C22×M16D412yes—
61C23×C8—21no—
62#533C2×C411yescharacteristic
63#533C2×C411yescharacteristic
64#533C2311yescharacteristic
65#532—21no—
66#532—21no—
67#532—21no—
68#532—21no—
69#532—21no—
70#532—21no—
71#532C2×C411yescharacteristic
72#532C2×C412yes—
73#532C2×C411yescharacteristic
74#532C2312yes—
75#532C2311yescharacteristic
76C2×C4⋊Q8C2×C411yescharacteristic
77C2×#127C2×C411yescharacteristic
78#529—21no—
79#529—21no—
80#529—21no—
81#529—21no—
82#529—21no—
83#529—21no—
84C2×#124—21no—
85#525—21no—
86#525—21no—
87#525—21no—
88C2×#121—21no—
89C2×#121—21no—
90C4×#41—21no—
91C4×#41—21no—
92C4×#41—21no—
93C2×C4×D4—21no—
94C2×C4×D4—21no—
95C2×#117—21no—
96C22×C42C2311yescharacteristic
97#501—22no—
98#500C2×C412yes—
99#498C2×C412yes—
100#496—22no—
101#493C2×C412yes—
102#492C2×C412yes—
103#490—22no—
104#489—22no—
105#487C2×C412yes—
106#486C2×C412yes—
107#486C2×C412yes—
108#486C2×C412yes—
109#469—24no—
110#468—24no—
111#466—24no—
112#465—24no—
113#442—24no—
114#440—24no—
115C4×SD16—24no—
116C4×D8—24no—
117#432—21no—
118#432—21no—
119#431—21no—
120#431—21no—
121#428—22no—
122#423—21no—
123#423—21no—
124#423—21no—
125#423C2311yescharacteristic
126#423C2312yes—
127#423C2311yescharacteristic
128C2×C4⋊C8—21no—
129C2×C4⋊C8—21no—
130C2×#104—24no—
131C2×#104—24no—
132C2×#104—24no—
133#418—22no—
134#417—22no—
135#407—21no—
136C4×M16—21no—
137C4×M16C2×C411yescharacteristic
138C4×M16C2×C411yescharacteristic
139C4×M16C2×C412yes—
140C4×M16D412yes—
141C4×M16D412yes—
142C2×#97D412yes—
143C2×C4×C8—21no—
144C2×C4×C8—21no—
145C2×C4×C8—21no—
146C2×C4×C8D412yes—
147C2×C4×C8D412yes—
148C2×C4×C8C2311yescharacteristic
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2312yes—
151C4×C4⋊C4—21no—
152C4×C4⋊C4D412yes—
153C4×C22⋊C4—21no—
154C4×C22⋊C4D412yes—
155#376—21no—
156C43C2312yes—
157#339—24no—
158#336—24no—
159#336—24no—
160#335—24no—
161#334—24no—
162Q8⋊C8—22no—
163Q8⋊C8—22no—
164D4⋊C8—24no—
165D4⋊C8—24no—
166D4⋊C8—22no—
167D4⋊C8—22no—
168C2×#41—21no—
169C2×#41—21no—
170C2×#41—22no—
171C2×#41—21no—
172C2×#41—22no—
173C2×#41—41no—
174C2×#41—41no—
175C2×#41—82no—
176C2×#41—82no—
177C22×Q8—21no—
178C22×D4—21no—
179C22×D4—21no—
180C22×D4—21no—
181C22×D4—41no—
182C23×C4—41no—
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4#4111yescharacteristic
186AGL(1,ℤ/8)—48no—
187C2×SD16—24no—
188C2×D8—24no—
189C2×M16—22no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—44no—
193C2×M16—41no—
194C2×M16—82no—
195C2×M16C22⋊C412yes—
196C2×M16C22⋊C412yes—
197C2×M16C22⋊C414yes—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—44no—
203C22×C8—41no—
204C22×C8C2×D414yes—
205C4⋊Q8—21no—
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8C22×C412yes—
209C4⋊Q8C22×C411yescharacteristic
210C4⋊Q8C22×C411yescharacteristic
211C4⋊Q8C22×C411yescharacteristic
212#127—21no—
213#127—21no—
214#127—21no—
215#127C22×C412yes—
216#127C22×C411yescharacteristic
217#126—41no—
218#126—41no—
219#125—21no—
220#125—21no—
221#125—21no—
222#124—21no—
223#124—22no—
224#124—21no—
225#124—22no—
226#124—21no—
227#124—41no—
228#124—41no—
229#123—41no—
230#123—41no—
231#123—41no—
232C22⋊Q8—21no—
233C22⋊Q8—21no—
234C22⋊Q8—21no—
235C22⋊Q8—21no—
236C22⋊Q8—21no—
237C22⋊Q8—21no—
238#121—22no—
239#121—21no—
240#121—22no—
241#121—21no—
242#121—21no—
243#121—21no—
244#121—22no—
245#121—21no—
246#121—22no—
247#121—21no—
248#121—41no—
249#121—41no—
250#121—41no—
251#121—41no—
252#120—41no—
253C4×Q8—21no—
254C4×Q8—21no—
255C4×Q8—21no—
256C4×Q8—21no—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×D4—22no—
260C4×D4—21no—
261C4×D4—22no—
262C4×D4—21no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—22no—
266C4×D4—22no—
267C4×D4—21no—
268C4×D4—21no—
269C4×D4—41no—
270C4×D4—41no—
271C4×D4—41no—
272C4×D4—41no—
273C4×D4—41no—
274C4×D4—41no—
275C4×D4—41no—
276C4×D4—41no—
277C4×D4—41no—
278C4×D4—41no—
279#117—21no—
280#117—21no—
281#117—21no—
282#117—44no—
283#117—41no—
284#117—41no—
285#117—41no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—41no—
291C2×C22⋊C4—41no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42—41no—
295C2×C42C22×C411yescharacteristic
296C2×C42C22×C411yescharacteristic
297C2×C42C2×D414yes—
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42C2×D411yescharacteristic
303C2×C42C2×D411yescharacteristic
304C2×C42#4111yescharacteristic
305C2×C42#4111yescharacteristic
306C2×C42#4111yescharacteristic
307C2×C42C2411yescharacteristic
308#107—24no—
309#106—24no—
310C4⋊C8—22no—
311C4⋊C8—21no—
312C4⋊C8—21no—
313C4⋊C8—21no—
314C4⋊C8—22no—
315C4⋊C8—21no—
316C4⋊C8—41no—
317C4⋊C8—41no—
318#104—84no—
319#104—84no—
320#104—84no—
321#103—24no—
322#102—24no—
323#102—24no—
324#102—24no—
325C22⋊C8—41no—
326C22⋊C8—41no—
327#97—41no—
328#97C22⋊C414yes—
329#97C22⋊C412yes—
330#97C22⋊C412yes—
331C4×C8—22no—
332C4×C8—22no—
333C4×C8—21no—
334C4×C8—21no—
335C4×C8—41no—
336C4×C8C22×C412yes—
337C4×C8C22×C412yes—
338C4×C8C2×D412yes—
339C4×C8C2×D412yes—
340C4×C8C2×D414yes—
341#95—22no—
342#95—22no—
343C24—41no—
344C24#12111yescharacteristic
345#41—44no—
346#41—42no—
347#41—44no—
348#41—44no—
349#41—42no—
350#41—44no—
351#41—84no—
352#41—84no—
353#41—82no—
354#41—82no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—21no—
358C2×Q8—22no—
359C2×Q8—41no—
360C2×Q8—82no—
361C2×D4—21no—
362C2×D4—22no—
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—44no—
368C2×D4—41no—
369C2×D4—42no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—42no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4—44no—
377C2×D4—82no—
378C2×D4—84no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—42no—
390C22×C4—82no—
391C22×C4—82no—
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C4×D411yescharacteristic
395C22×C4C4×D411yescharacteristic
396C22×C4C4×D411yescharacteristic
397C22×C4C4×D411yescharacteristic
398C22×C4#12111yescharacteristic
399C22×C4#12111yescharacteristic
400C22×C4#12111yescharacteristic
401C22×C4#12111yescharacteristic
402C22×C4#12111yescharacteristic
403C22×C4#12411yescharacteristic
404C22×C4#12411yescharacteristic
405C22×C4#12711yescharacteristic
406C22×C4#12711yescharacteristic
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D411yescharacteristic
409C22×C4C2×#4111yescharacteristic
410SD16—48no—
411D8—48no—
412M16—24no—
413M16—24no—
414M16—44no—
415M16—42no—
416M16—44no—
417M16—42no—
418M16—82no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—44no—
422C2×C8—44no—
423C2×C8—42no—
424C2×C8—42no—
425C2×C8—44no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8—42no—
429C2×C8—82no—
430C2×C8C2×C22⋊C414yes—
431C2×C8C2×C22⋊C414yes—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—44no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C22⋊C4—44no—
450C22⋊C4—41no—
451C22⋊C4—41no—
452C22⋊C4—41no—
453C22⋊C4—42no—
454C22⋊C4—41no—
455C22⋊C4—42no—
456C22⋊C4—41no—
457C22⋊C4—42no—
458C22⋊C4—42no—
459C22⋊C4—41no—
460C22⋊C4—41no—
461C22⋊C4—41no—
462C22⋊C4—41no—
463C42—24no—
464C42—24no—
465C42—44no—
466C42—44no—
467C42—41no—
468C42—41no—
469C42—41no—
470C42C2×C22⋊C411yescharacteristic
471C42C2×C22⋊C411yescharacteristic
472C42C4×D411yescharacteristic
473C42C4×D411yescharacteristic
474C42C4×D411yescharacteristic
475C42C4×D411yescharacteristic
476C42#12111yescharacteristic
477C42#12111yescharacteristic
478C42#12411yescharacteristic
479C42#12411yescharacteristic
480C42#12711yescharacteristic
481C42#12711yescharacteristic
482C42C23×C411yescharacteristic
483C42C22×D411yescharacteristic
484C42C22×D411yescharacteristic
485C42C2×#4111yescharacteristic
486C23—22no—
487C23—22no—
488C23—41no—
489C23—41no—
490C23—42no—
491C23—82no—
492C23—82no—
493C23—82no—
494C23#39011yescharacteristic
495C23#39011yescharacteristic
496C23C2×#12111yescharacteristic
497Q8—42no—
498Q8—44no—
499Q8—42no—
500Q8—42no—
501Q8—82no—
502D4—42no—
503D4—44no—
504D4—44no—
505D4—42no—
506D4—44no—
507D4—42no—
508D4—82no—
509D4—82no—
510D4—84no—
511D4—82no—
512D4—82no—
513D4—84no—
514D4—82no—
515D4—82no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—84no—
542C2×C4—82no—
543C2×C4—82no—
544C2×C4—82no—
545C2×C4—81no—
546C2×C4—82no—
547C2×C4—81no—
548C2×C4—82no—
549C2×C4#39011yescharacteristic
550C2×C4#39011yescharacteristic
551C2×C4#39011yescharacteristic
552C2×C4#39011yescharacteristic
553C2×C4#39011yescharacteristic
554C2×C4#39011yescharacteristic
555C2×C4#42114yes—
556C2×C4C22×C22⋊C411yescharacteristic
557C2×C4C2×C4×D411yescharacteristic
558C2×C4C2×C4×D411yescharacteristic
559C2×C4C2×#12111yescharacteristic
560C2×C4C2×#12411yescharacteristic
561C2×C4C2×#12711yescharacteristic
562C2×C4#57612yes—
563C2×C4#57712yes—
564C8—24no—
565C8—24no—
566C8—42no—
567C8—42no—
568C8—81no—
569C8—82no—
570C8—81no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—21no—
578C22—44no—
579C22—81no—
580C22—82no—
581C22—82no—
582C22—81no—
583C22—81no—
584C22—82no—
585C22—164no—
586C22#196312yes—
587C22C2×#39011yescharacteristic
588C22#306212yes—
589C4—21no—
590C4—21no—
591C4—21no—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—21no—
596C4—84no—
597C4—81no—
598C4—82no—
599C4—81no—
600C4—81no—
601C4#196312yes—
602C4C2×#42112yes—
603C4#294612yes—
604C4#295512yes—
605C4#315411yescharacteristic
606C4#315511yescharacteristic
607C2—22no—
608C2—22no—
609C2—41no—
610C2—161no—
611C2—162no—
612C2C2×#196311yescharacteristic
613C2#2123511yescharacteristic
614C2#2124211yescharacteristic
subgroupG/H
centerC2×C4C2×#127
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#390
prime radical#60700C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21856C2semidirect
3. #21815C2semidirect
4. #20366C2semidirect
5. #17796C2semidirect
6. #17473C2semidirect
7. C2×#1492C2semidirect
8. #9854C2semidirect
9. #9852C2semidirect
10. #9847C2semidirect
11. #9846C2semidirect
14. #3184C4semidirect
15. #3182C4semidirect
20. #3165C4semidirect
21. #3151C4semidirect
22. #3141C4semidirect
23. #3138C4semidirect
31. #1994C22semidirect
35. #1962C22semidirect
40. #1744C22semidirect
41. #1744C22semidirect
44. #1567C22semidirect
45. #1566C22semidirect
48. #1493C22semidirect
49. #1493C22semidirect
53. #1492C22semidirect
54. #1492C22semidirect

Maximal subgroups

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

subgroupindex
#607002
#218562
#218152
#203662
#177962
#174732
C2×#14922
#98542
#98522
#98472
#98462