Finite Group Databasebeta

← #206449

Subgroups — #206449

Order 512.

1044 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60782C211yesmaximal, characteristic
2#21993C211yesmaximal, characteristic, complemented
3#17504C211yesmaximal, characteristic, complemented
4#17500C211yesmaximal, characteristic, complemented
5#17473C211yesmaximal, characteristic, complemented
6C2×#1492C211yesmaximal, characteristic, complemented
7C2×C8×M16C211yesmaximal, characteristic, complemented
8#9860C212yesmaximal, complemented
9#9858C212yesmaximal, complemented
10#9857C212yesmaximal, complemented
11#9856C212yesmaximal, complemented
12#3444C2211yescharacteristic
13C2×#532C2211yescharacteristic
14#3212—22no—
15#3211—22no—
16#3210C412yes—
17#3209C411yescharacteristic
18#3208C411yescharacteristic
19#3207C412yes—
20#3206C411yescharacteristic
21#3205C411yescharacteristic
22#3147C2212yes—
23#3131C2212yes—
24#3035—22no—
25#3002—22no—
26#2920C2211yescharacteristic
27C2×#431C2211yescharacteristic
28C2×#423C2211yescharacteristic
29C2×C4×M16C2211yescharacteristic
30C22×C4×C8C2211yescharacteristic
31#1950—22no—
32#1855—22no—
33#1848C2212yescomplemented
34#1837C2212yescomplemented
35#1494—22no—
36#1493C2211yescharacteristic, complemented
37#1493C2211yescharacteristic, complemented
38#1493C2212yescomplemented
39#1492—22no—
40#1492C2212yes—
41#1492C2212yes—
42#1492C2212yes—
43#1492C2212yes—
44#1492C2211yescharacteristic, complemented
45#1492C2211yescharacteristic, complemented
46#1492C2212yescomplemented
47#1490—22no—
48#1489—22no—
49#1484—22no—
50#1483—22no—
51#1476—22no—
52C2×D4⋊C8—22no—
53C8×M16—22no—
54C8×M16—22no—
55C2×#322C2212yescomplemented
56C2×C82C2212yescomplemented
57C22×#41—21no—
58C2×#135—24no—
59C22×M16—21no—
60C22×M16—21no—
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×C412yes—
72#532C2×C411yescharacteristic
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—
95C22×C42C2311yescharacteristic
96#495—24no—
97#494C2×C412yes—
98#493C2×C412yes—
99#492C2×C412yes—
100#492C2×C412yes—
101#488—24no—
102#487C2×C412yes—
103#487C2×C412yes—
104#486C2×C412yes—
105#486C2×C412yes—
106#464—24no—
107#463—24no—
108#462—22no—
109#461—22no—
110#460—22no—
111#459—22no—
112#452—24no—
113#451—22no—
114#450—22no—
115#449—24no—
116#448—22no—
117#447—22no—
118#432—21no—
119#432—21no—
120#431—21no—
121#431—21no—
122#431C2×C411yescharacteristic
123#431C2×C411yescharacteristic
124#431C2×C412yes—
125#431D412yes—
126#431D412yes—
127C2×#108—24no—
128#423—21no—
129#423—21no—
130#423—21no—
131#423C2311yescharacteristic
132#423C2312yes—
133#423C2311yescharacteristic
134C2×C4⋊C8—21no—
135C2×C4⋊C8—21no—
136C2×C4⋊C8D412yes—
137#416—24no—
138#407—21no—
139C2×C22⋊C8D412yes—
140C4×M16—21no—
141C4×M16—21no—
142C4×M16—21no—
143C4×M16D412yes—
144C4×M16D412yes—
145C2×#97—21no—
146C2×C4×C8—21no—
147C2×C4×C8—21no—
148C2×C4×C8—21no—
149C2×C4×C8D412yes—
150C2×C4×C8D412yes—
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2312yes—
154C2×C4×C8C2312yes—
155#341—24no—
156Q8⋊C8—22no—
157Q8⋊C8—22no—
158Q8⋊C8—24no—
159D4⋊C8—24no—
160D4⋊C8—24no—
161D4⋊C8—22no—
162D4⋊C8—22no—
163D4⋊C8—24no—
164D4⋊C8—24no—
165D4⋊C8—24no—
166#322—24no—
167C82—24no—
168C2×#41—22no—
169C2×#41—21no—
170C2×#41—21no—
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×C4#4111yescharacteristic
185C23×C4#4111yescharacteristic
186#135—84no—
187#135—84no—
188C2×Q16—24no—
189C2×SD16—24no—
190C2×SD16—24no—
191C2×D8—24no—
192C2×M16—28no—
193C2×M16—22no—
194C2×M16—22no—
195C2×M16—22no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—41no—
199C2×M16—82no—
200C22×C8—21no—
201C22×C8—22no—
202C22×C8—22no—
203C22×C8—22no—
204C22×C8—21no—
205C22×C8—44no—
206C22×C8—41no—
207C22×C8C2×D414yes—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8C22×C412yes—
212C4⋊Q8C22×C411yescharacteristic
213C4⋊Q8C22×C411yescharacteristic
214C4⋊Q8C22×C411yescharacteristic
215#127—21no—
216#127—21no—
217#127—21no—
218#127C22×C412yes—
219#127C22×C411yescharacteristic
220#126—41no—
221#126—41no—
222#125—21no—
223#125—21no—
224#125—21no—
225#124—22no—
226#124—21no—
227#124—21no—
228#124—22no—
229#124—21no—
230#124—41no—
231#124—41no—
232#123—41no—
233#123—41no—
234#123—41no—
235C22⋊Q8—21no—
236C22⋊Q8—21no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239C22⋊Q8—21no—
240C22⋊Q8—21no—
241#121—21no—
242#121—21no—
243#121—22no—
244#121—22no—
245#121—21no—
246#121—21no—
247#121—22no—
248#121—22no—
249#121—21no—
250#121—21no—
251#121—41no—
252#121—41no—
253#121—41no—
254#121—41no—
255#120—41no—
256C4×Q8—21no—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×Q8—21no—
260C4×Q8—21no—
261C4×Q8—21no—
262C4×D4—21no—
263C4×D4—21no—
264C4×D4—22no—
265C4×D4—22no—
266C4×D4—21no—
267C4×D4—21no—
268C4×D4—22no—
269C4×D4—22no—
270C4×D4—21no—
271C4×D4—21no—
272C4×D4—41no—
273C4×D4—41no—
274C4×D4—41no—
275C4×D4—41no—
276C4×D4—41no—
277C4×D4—41no—
278C4×D4—41no—
279C4×D4—41no—
280C4×D4—41no—
281C4×D4—41no—
282#117—21no—
283#117—21no—
284#117—21no—
285#117—41no—
286#117—41no—
287#117—41no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C22⋊C4—41no—
291C2×C22⋊C4—41no—
292C2×C42—41no—
293C2×C42C22×C411yescharacteristic
294C2×C42C22×C411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42#4111yescharacteristic
303C2×C42#4111yescharacteristic
304C2×C42C2411yescharacteristic
305#108—84no—
306C4⋊C8—22no—
307C4⋊C8—21no—
308C4⋊C8—21no—
309C4⋊C8—22no—
310C4⋊C8—21no—
311C4⋊C8—21no—
312C4⋊C8—41no—
313C4⋊C8—41no—
314C4⋊C8C22⋊C412yes—
315C4⋊C8C22⋊C412yes—
316C4⋊C8C22⋊C414yes—
317#103—24no—
318#103—24no—
319#102—24no—
320#102—24no—
321C22⋊C8—41no—
322C22⋊C8—41no—
323C22⋊C8C22⋊C412yes—
324C22⋊C8C22⋊C412yes—
325C22⋊C8C22⋊C414yes—
326#97—22no—
327#97—22no—
328#97—41no—
329C4×C8—22no—
330C4×C8—22no—
331C4×C8—21no—
332C4×C8—22no—
333C4×C8—21no—
334C4×C8—41no—
335C4×C8C22×C412yes—
336C4×C8C22×C412yes—
337C4×C8C2×D412yes—
338C4×C8C2×D412yes—
339C4×C8C2×D414yes—
340C4×C8C2×D414yes—
341C24—41no—
342C24#12411yescharacteristic
343#41—42no—
344#41—44no—
345#41—42no—
346#41—44no—
347#41—44no—
348#41—82no—
349#41—84no—
350#41—84no—
351#41—82no—
352#41—82no—
353C2×Q8—22no—
354C2×Q8—21no—
355C2×Q8—21no—
356C2×Q8—21no—
357C2×Q8—41no—
358C2×Q8—82no—
359C2×D4—22no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—41no—
366C2×D4—42no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—42no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—44no—
374C2×D4—82no—
375C2×D4—82no—
376C2×D4—84no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—42no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—42no—
384C22×C4—41no—
385C22×C4—41no—
386C22×C4—42no—
387C22×C4—82no—
388C22×C4—82no—
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C4×D411yescharacteristic
392C22×C4C4×D411yescharacteristic
393C22×C4C4×D411yescharacteristic
394C22×C4C4×D411yescharacteristic
395C22×C4#12111yescharacteristic
396C22×C4#12111yescharacteristic
397C22×C4#12111yescharacteristic
398C22×C4#12111yescharacteristic
399C22×C4#12411yescharacteristic
400C22×C4#12411yescharacteristic
401C22×C4#12411yescharacteristic
402C22×C4#12711yescharacteristic
403C22×C4#12711yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C2×#4111yescharacteristic
406C22×C4C2×#4111yescharacteristic
407Q16—84no—
408SD16—84no—
409SD16—84no—
410D8—84no—
411M16—28no—
412M16—28no—
413M16—42no—
414M16—44no—
415M16—42no—
416M16—82no—
417C2×C8—24no—
418C2×C8—28no—
419C2×C8—22no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—22no—
424C2×C8—44no—
425C2×C8—44no—
426C2×C8—42no—
427C2×C8—41no—
428C2×C8—41no—
429C2×C8—42no—
430C2×C8—82no—
431C2×C8C2×C22⋊C414yes—
432C2×C8C2×C22⋊C414yes—
433C4⋊C4—21no—
434C4⋊C4—21no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
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—41no—
450C22⋊C4—41no—
451C22⋊C4—41no—
452C22⋊C4—42no—
453C22⋊C4—42no—
454C22⋊C4—41no—
455C22⋊C4—41no—
456C22⋊C4—42no—
457C22⋊C4—42no—
458C22⋊C4—41no—
459C22⋊C4—41no—
460C22⋊C4—41no—
461C22⋊C4—41no—
462C42—41no—
463C42—41no—
464C42—41no—
465C42C2×C22⋊C411yescharacteristic
466C42C2×C22⋊C411yescharacteristic
467C42C4×D411yescharacteristic
468C42C4×D411yescharacteristic
469C42C4×D411yescharacteristic
470C42C4×D411yescharacteristic
471C42#12111yescharacteristic
472C42#12111yescharacteristic
473C42#12111yescharacteristic
474C42#12111yescharacteristic
475C42#12711yescharacteristic
476C42#12711yescharacteristic
477C42C23×C411yescharacteristic
478C42C22×D411yescharacteristic
479C42C22×D411yescharacteristic
480C42C22×D411yescharacteristic
481C23—22no—
482C23—22no—
483C23—41no—
484C23—41no—
485C23—42no—
486C23—82no—
487C23—82no—
488C23—82no—
489C23#39011yescharacteristic
490C23#39011yescharacteristic
491C23C2×#12411yescharacteristic
492Q8—42no—
493Q8—44no—
494Q8—42no—
495Q8—42no—
496Q8—82no—
497D4—42no—
498D4—44no—
499D4—44no—
500D4—42no—
501D4—44no—
502D4—42no—
503D4—82no—
504D4—82no—
505D4—84no—
506D4—82no—
507D4—84no—
508D4—82no—
509D4—82no—
510D4—82no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—42no—
533C2×C4—84no—
534C2×C4—82no—
535C2×C4—82no—
536C2×C4—82no—
537C2×C4—81no—
538C2×C4—82no—
539C2×C4—81no—
540C2×C4—82no—
541C2×C4#39011yescharacteristic
542C2×C4#39011yescharacteristic
543C2×C4#39011yescharacteristic
544C2×C4#39011yescharacteristic
545C2×C4#39011yescharacteristic
546C2×C4#39011yescharacteristic
547C2×C4#44314yes—
548C2×C4C22×C22⋊C411yescharacteristic
549C2×C4C2×C4×D411yescharacteristic
550C2×C4C2×C4×D411yescharacteristic
551C2×C4C2×#12111yescharacteristic
552C2×C4C2×#12111yescharacteristic
553C2×C4C2×#12711yescharacteristic
554C2×C4#57612yes—
555C2×C4#57712yes—
556C8—24no—
557C8—28no—
558C8—24no—
559C8—24no—
560C8—42no—
561C8—81no—
562C8—82no—
563C8—81no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—44no—
572C22—82no—
573C22—81no—
574C22—81no—
575C22—82no—
576C22—81no—
577C22—82no—
578C22—164no—
579C22#197412yes—
580C22C2×#39011yescharacteristic
581C22#313612yes—
582C4—21no—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—82no—
590C4—81no—
591C4—81no—
592C4—81no—
593C4#197412yes—
594C4#289712yes—
595C4C2×#44312yes—
596C4#295612yes—
597C4#315411yescharacteristic
598C4#315511yescharacteristic
599C2—22no—
600C2—22no—
601C2—41no—
602C2—161no—
603C2—162no—
604C2#2120611yescharacteristic
605C2C2×#197411yescharacteristic
606C2#2124111yescharacteristic
subgroupG/H
centerC2×C4C2×#127
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. #21993C2semidirect
3. #17504C2semidirect
4. #17500C2semidirect
5. #17473C2semidirect
6. C2×#1492C2semidirect
7. C2×C8×M16C2semidirect
8. #9860C2semidirect
9. #9858C2semidirect
10. #9857C2semidirect
11. #9856C2semidirect
33. #1848C22semidirect
34. #1837C22semidirect
36. #1493C22semidirect
37. #1493C22semidirect
38. #1493C22semidirect
44. #1492C22semidirect
45. #1492C22semidirect
46. #1492C22semidirect
55. C2×#322C22semidirect
56. C2×C82C22semidirect

Maximal subgroups

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

subgroupindex
#607822
#219932
#175042
#175002
#174732
C2×#14922
C2×C8×M162
#98602
#98582
#98572
#98562