Finite Group Databasebeta

← #205517

Subgroups — #205517

Order 512.

1021 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60782C211yesmaximal, characteristic
2#20536C211yesmaximal, characteristic, complemented
3#17806C211yesmaximal, characteristic, complemented
4#17546C211yesmaximal, characteristic, complemented
5C2×#1502C211yesmaximal, characteristic, complemented
6#17463C211yesmaximal, characteristic, complemented
7#17458C211yesmaximal, characteristic, complemented
8#9754C214yesmaximal, complemented
9#9753C214yesmaximal, complemented
10#3444C2211yescharacteristic
11C2×#532C2211yescharacteristic
12#3212—22no—
13#3211—22no—
14#3210C412yescomplemented
15#3209C411yescharacteristic, complemented
16#3208C411yescharacteristic, complemented
17#3207C412yescomplemented
18#3206C411yescharacteristic, complemented
19#3205C411yescharacteristic, complemented
20#3147C2212yes—
21#3131C2212yes—
22#3035—22no—
23#3002—22no—
24#2920C2211yescharacteristic
25C2×#431C2211yescharacteristic
26C2×#431C2211yescharacteristic
27C2×#431C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#1770—22no—
30#1770C2214yescomplemented
31#1749—22no—
32#1574—22no—
33#1573—22no—
34#1564C2214yescomplemented
35#1509—22no—
36#1504—22no—
37#1503C2212yescomplemented
38#1502—22no—
39#1502C2214yes—
40#1502C2212yescomplemented
41C2×#327—22no—
42#1484—22no—
43#1482—22no—
44#1481C2212yescomplemented
45#1480C2214yes—
46#1480C2212yescomplemented
47#1478—22no—
48#1476—22no—
49C22×#41—21no—
50C2×#135—24no—
51#533C2×C411yescharacteristic
52#533C2×C411yescharacteristic
53#533C2311yescharacteristic
54#532—21no—
55#532—21no—
56#532—21no—
57#532—21no—
58#532—21no—
59#532—21no—
60#532C2×C412yes—
61#532C2×C411yescharacteristic
62#532C2×C411yescharacteristic
63#532C2312yes—
64#532C2311yescharacteristic
65C2×C4⋊Q8C2×C411yescharacteristic
66C2×#127C2×C411yescharacteristic
67#529—21no—
68#529—21no—
69#529—21no—
70#529—21no—
71#529—21no—
72#529—21no—
73C2×#124—21no—
74#525—21no—
75#525—21no—
76#525—21no—
77C2×#121—21no—
78C2×#121—21no—
79C4×#41—21no—
80C4×#41—21no—
81C4×#41—21no—
82C2×C4×D4—21no—
83C2×C4×D4—21no—
84C22×C4⋊C4—21no—
85C22×C42—21no—
86C22×C42C2311yescharacteristic
87#495—24no—
88#494C2×C412yescomplemented
89#493C2×C412yescomplemented
90#492C2×C412yescomplemented
91#492C2×C412yescomplemented
92#488—24no—
93#487C2×C412yescomplemented
94#487C2×C412yescomplemented
95#486C2×C412yescomplemented
96#486C2×C412yescomplemented
97#464—24no—
98#463—24no—
99#462—22no—
100#461—22no—
101#460—22no—
102#459—22no—
103#452—24no—
104#451—22no—
105#450—22no—
106#449—24no—
107#448—22no—
108#447—22no—
109#432—21no—
110#432—21no—
111#431—21no—
112#431—21no—
113#431—21no—
114#431—21no—
115#431—21no—
116#431—21no—
117#431C2×C411yescharacteristic
118#431C2×C411yescharacteristic
119#431C2×C412yes—
120#431D412yes—
121#431D412yes—
122#431D412yes—
123#431D412yes—
124#431C2312yes—
125#431C2312yes—
126#423—21no—
127C2×C4⋊C8—21no—
128C2×C4⋊C8D412yes—
129C2×C4⋊C8C2312yes—
130#416—24no—
131#407—21no—
132C2×C22⋊C8—21no—
133C2×C22⋊C8—21no—
134C2×C22⋊C8D412yes—
135C4×M16—21no—
136C2×#97—24no—
137C2×C4×C8—21no—
138C2×C4×C8—21no—
139C2×C4×C8C2312yes—
140C4×C4⋊C4—21no—
141C4×C4⋊C4—21no—
142C4×C4⋊C4D412yes—
143C4×C4⋊C4C2312yes—
144C2×#95D412yes—
145#336—24no—
146#335—24no—
147#334—24no—
148#328—24no—
149#327—28no—
150#327—24no—
151Q8⋊C8—24no—
152D4⋊C8—24no—
153D4⋊C8—28no—
154C2×#41—22no—
155C2×#41—21no—
156C2×#41—21no—
157C2×#41—21no—
158C2×#41—22no—
159C2×#41—41no—
160C2×#41—41no—
161C2×#41—82no—
162C2×#41—82no—
163C22×Q8—21no—
164C22×D4—21no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4—41no—
168C23×C4—22no—
169C23×C4—41no—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4#4111yescharacteristic
173#135—84no—
174#135—84no—
175C2×Q16—24no—
176C2×SD16—24no—
177C2×SD16—24no—
178C2×D8—24no—
179C2×M16—41no—
180C22×C8—22no—
181C22×C8—41no—
182C22×C8C2×D414yes—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8C22×C412yes—
187C4⋊Q8C22×C411yescharacteristic
188C4⋊Q8C22×C411yescharacteristic
189C4⋊Q8C22×C411yescharacteristic
190#127—21no—
191#127—21no—
192#127—21no—
193#127C22×C412yes—
194#127C22×C411yescharacteristic
195#126—41no—
196#126—41no—
197#125—21no—
198#125—21no—
199#125—21no—
200#124—22no—
201#124—21no—
202#124—21no—
203#124—22no—
204#124—21no—
205#124—41no—
206#124—41no—
207#123—41no—
208#123—41no—
209#123—41no—
210C22⋊Q8—21no—
211C22⋊Q8—21no—
212C22⋊Q8—21no—
213C22⋊Q8—21no—
214C22⋊Q8—21no—
215C22⋊Q8—21no—
216#121—21no—
217#121—21no—
218#121—22no—
219#121—22no—
220#121—21no—
221#121—21no—
222#121—22no—
223#121—21no—
224#121—22no—
225#121—21no—
226#121—41no—
227#121—41no—
228#121—41no—
229#121—41no—
230#120—41no—
231C4×Q8—21no—
232C4×Q8—21no—
233C4×Q8—21no—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×Q8—21no—
237C4×D4—21no—
238C4×D4—21no—
239C4×D4—22no—
240C4×D4—22no—
241C4×D4—21no—
242C4×D4—21no—
243C4×D4—22no—
244C4×D4—21no—
245C4×D4—22no—
246C4×D4—21no—
247C4×D4—41no—
248C4×D4—41no—
249C4×D4—41no—
250C4×D4—41no—
251C4×D4—41no—
252C4×D4—41no—
253C4×D4—41no—
254C4×D4—41no—
255C4×D4—41no—
256C4×D4—41no—
257#117—21no—
258#117—21no—
259#117—21no—
260#117—41no—
261#117—41no—
262#117—41no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—44no—
268C2×C22⋊C4—41no—
269C2×C22⋊C4—41no—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—41no—
273C2×C42—82no—
274C2×C42C22×C411yescharacteristic
275C2×C42C22×C411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42#4111yescharacteristic
283C2×C42#4111yescharacteristic
284C2×C42#4111yescharacteristic
285C2×C42C2411yescharacteristic
286C4⋊C8—22no—
287C4⋊C8—22no—
288C4⋊C8—22no—
289C4⋊C8—41no—
290C4⋊C8—41no—
291C4⋊C8C22⋊C412yes—
292C4⋊C8C22⋊C412yes—
293C4⋊C8C22⋊C414yes—
294C4⋊C8C2×D414yes—
295#103—24no—
296#103—24no—
297#102—24no—
298#102—24no—
299C22⋊C8—22no—
300C22⋊C8—22no—
301C22⋊C8—22no—
302C22⋊C8—22no—
303C22⋊C8—41no—
304C22⋊C8—41no—
305C22⋊C8C22⋊C412yes—
306C22⋊C8C22⋊C412yes—
307C22⋊C8C22⋊C414yes—
308#97—41no—
309#97—84no—
310C4×C8—22no—
311C4×C8—22no—
312C4×C8—22no—
313C4×C8—22no—
314C4×C8—41no—
315C4×C8C22×C412yes—
316C4×C8C22×C412yes—
317C4×C8C2×D414yes—
318#95—22no—
319#95C2×D414yes—
320C24—41no—
321C24#12111yescharacteristic
322#41—42no—
323#41—44no—
324#41—42no—
325#41—44no—
326#41—44no—
327#41—84no—
328#41—82no—
329#41—82no—
330#41—84no—
331#41—82no—
332C2×Q8—22no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—21no—
336C2×Q8—41no—
337C2×Q8—82no—
338C2×D4—22no—
339C2×D4—21no—
340C2×D4—21no—
341C2×D4—22no—
342C2×D4—21no—
343C2×D4—22no—
344C2×D4—41no—
345C2×D4—42no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—42no—
349C2×D4—41no—
350C2×D4—41no—
351C2×D4—41no—
352C2×D4—44no—
353C2×D4—82no—
354C2×D4—82no—
355C2×D4—82no—
356C2×D4—82no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—44no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—42no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—42no—
368C22×C4—84no—
369C22×C4—81no—
370C22×C4—82no—
371C22×C4—82no—
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C4×D411yescharacteristic
375C22×C4C4×D411yescharacteristic
376C22×C4C4×D411yescharacteristic
377C22×C4C4×D411yescharacteristic
378C22×C4#12111yescharacteristic
379C22×C4#12111yescharacteristic
380C22×C4#12111yescharacteristic
381C22×C4#12111yescharacteristic
382C22×C4#12111yescharacteristic
383C22×C4#12411yescharacteristic
384C22×C4#12411yescharacteristic
385C22×C4#12711yescharacteristic
386C22×C4#12711yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C2×#4111yescharacteristic
390Q16—84no—
391SD16—84no—
392SD16—84no—
393D8—84no—
394M16—82no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—42no—
399C2×C8—42no—
400C2×C8—42no—
401C2×C8—42no—
402C2×C8—82no—
403C2×C8C2×C22⋊C414yes—
404C2×C8C2×C22⋊C414yes—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—44no—
414C4⋊C4—44no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C22⋊C4—41no—
424C22⋊C4—41no—
425C22⋊C4—41no—
426C22⋊C4—42no—
427C22⋊C4—41no—
428C22⋊C4—42no—
429C22⋊C4—41no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C42—41no—
437C42—44no—
438C42—41no—
439C42—41no—
440C42—82no—
441C42C2×C22⋊C411yescharacteristic
442C42C2×C22⋊C411yescharacteristic
443C42C4×D411yescharacteristic
444C42C4×D411yescharacteristic
445C42C4×D411yescharacteristic
446C42C4×D411yescharacteristic
447C42#12111yescharacteristic
448C42#12111yescharacteristic
449C42#12411yescharacteristic
450C42#12411yescharacteristic
451C42#12711yescharacteristic
452C42#12711yescharacteristic
453C42C23×C411yescharacteristic
454C42C22×D411yescharacteristic
455C42C22×D411yescharacteristic
456C42C2×#4111yescharacteristic
457C23—22no—
458C23—21no—
459C23—21no—
460C23—41no—
461C23—41no—
462C23—42no—
463C23—82no—
464C23—82no—
465C23—82no—
466C23#39011yescharacteristic
467C23#39011yescharacteristic
468C23C2×#12111yescharacteristic
469Q8—42no—
470Q8—44no—
471Q8—42no—
472Q8—81no—
473Q8—82no—
474D4—42no—
475D4—44no—
476D4—44no—
477D4—42no—
478D4—82no—
479D4—81no—
480D4—82no—
481D4—82no—
482D4—84no—
483D4—82no—
484D4—82no—
485D4—82no—
486D4—81no—
487D4—82no—
488D4—81no—
489D4—82no—
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—44no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—84no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—88no—
518C2×C4—82no—
519C2×C4—84no—
520C2×C4—82no—
521C2×C4#39011yescharacteristic
522C2×C4#39011yescharacteristic
523C2×C4#39011yescharacteristic
524C2×C4#39011yescharacteristic
525C2×C4#39011yescharacteristic
526C2×C4#39011yescharacteristic
527C2×C4#44414yes—
528C2×C4C22×C22⋊C411yescharacteristic
529C2×C4C2×C4×D411yescharacteristic
530C2×C4C2×C4×D411yescharacteristic
531C2×C4C2×#12111yescharacteristic
532C2×C4C2×#12411yescharacteristic
533C2×C4C2×#12711yescharacteristic
534C2×C4#57612yes—
535C2×C4#57712yes—
536C8—24no—
537C8—24no—
538C8—44no—
539C8—82no—
540C8—82no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—42no—
549C22—42no—
550C22—82no—
551C22—81no—
552C22—81no—
553C22—82no—
554C22—81no—
555C22—82no—
556C22—164no—
557C22#197512yes—
558C22C2×#39011yescharacteristic
559C22#304111yescharacteristic
560C22#304311yescharacteristic
561C4—21no—
562C4—21no—
563C4—21no—
564C4—21no—
565C4—21no—
566C4—21no—
567C4—21no—
568C4—21no—
569C4—84no—
570C4—82no—
571C4—81no—
572C4—81no—
573C4—81no—
574C4#197512yes—
575C4#289712yes—
576C4C2×#44412yes—
577C4#304512yes—
578C2—21no—
579C2—22no—
580C2—21no—
581C2—41no—
582C2—161no—
583C2—162no—
584C2C2×#197511yescharacteristic
585C2#2124411yescharacteristic
586C2#2126311yescharacteristic
subgroupG/H
centerC2×C4C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#390
prime radical#60782C2
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. #20536C2semidirect
3. #17806C2semidirect
4. #17546C2semidirect
5. C2×#1502C2semidirect
6. #17463C2semidirect
7. #17458C2semidirect
8. #9754C2semidirect
9. #9753C2semidirect
14. #3210C4semidirect
15. #3209C4semidirect
16. #3208C4semidirect
17. #3207C4semidirect
18. #3206C4semidirect
19. #3205C4semidirect
30. #1770C22semidirect
34. #1564C22semidirect
37. #1503C22semidirect
40. #1502C22semidirect
44. #1481C22semidirect
46. #1480C22semidirect
88. #494C2×C4semidirect
89. #493C2×C4semidirect
90. #492C2×C4semidirect
91. #492C2×C4semidirect
93. #487C2×C4semidirect
94. #487C2×C4semidirect
95. #486C2×C4semidirect
96. #486C2×C4semidirect

Maximal subgroups

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

subgroupindex
#607822
#205362
#178062
#175462
C2×#15022
#174632
#174582
#97542
#97532