Finite Group Databasebeta

← #205519

Subgroups — #205519

Order 512.

1013 conjugacy classes of subgroups.

The subgroup lattice has 585 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60782C211yesmaximal, characteristic
2#20547C211yesmaximal, characteristic, complemented
3#17806C211yesmaximal, characteristic, complemented
4#17544C211yesmaximal, characteristic, complemented
5#17533C211yesmaximal, characteristic, complemented
6#17466C211yesmaximal, characteristic, complemented
7C2×#1480C211yesmaximal, 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
27C22×C4⋊C8C2211yescharacteristic
28C4×#117C2211yescharacteristic
29#1770C2214yescomplemented
30#1767—22no—
31#1752—22no—
32#1574—22no—
33#1573—22no—
34#1564C2214yescomplemented
35#1507—22no—
36#1505—22no—
37#1503C2212yescomplemented
38#1502C2214yes—
39#1502C2212yescomplemented
40#1501—22no—
41#1500—22no—
42#1483—22no—
43#1481C2212yescomplemented
44#1480—22no—
45#1480C2214yes—
46#1480C2212yescomplemented
47#1477—22no—
48C2×D4⋊C8—22no—
49C22×#41—21no—
50C2×#135—24no—
51C23×C8—21no—
52#533C2×C411yescharacteristic
53#533C2×C411yescharacteristic
54#533C2311yescharacteristic
55#532—21no—
56#532—21no—
57#532—21no—
58#532—21no—
59#532—21no—
60#532—21no—
61#532C2×C412yes—
62#532C2×C411yescharacteristic
63#532C2×C411yescharacteristic
64#532C2312yes—
65#532C2311yescharacteristic
66C2×C4⋊Q8C2×C411yescharacteristic
67C2×#127C2×C411yescharacteristic
68#529—21no—
69#529—21no—
70#529—21no—
71#529—21no—
72#529—21no—
73#529—21no—
74C2×#124—21no—
75#525—21no—
76#525—21no—
77#525—21no—
78C2×#121—21no—
79C2×#121—21no—
80C4×#41—21no—
81C4×#41—21no—
82C4×#41—21no—
83C2×C4×D4—21no—
84C2×C4×D4—21no—
85C2×#117—21no—
86C22×C42C2311yescharacteristic
87#495—24no—
88#494C2×C412yes—
89#493C2×C412yes—
90#492C2×C412yes—
91#492C2×C412yes—
92#488—24no—
93#487C2×C412yes—
94#487C2×C412yes—
95#486C2×C412yes—
96#486C2×C412yes—
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#431C2×C411yescharacteristic
116#431C2×C411yescharacteristic
117#431C2×C412yes—
118#431D412yes—
119#431D412yes—
120#431C2312yes—
121#431C2312yes—
122#423—21no—
123C2×C4⋊C8—21no—
124C2×C4⋊C8—21no—
125C2×C4⋊C8—21no—
126C2×C4⋊C8—21no—
127C2×C4⋊C8D412yes—
128C2×C4⋊C8D412yes—
129C2×C4⋊C8D412yes—
130C2×C4⋊C8C2312yes—
131#416—24no—
132#407—21no—
133C2×C22⋊C8—21no—
134C2×C22⋊C8D412yes—
135C4×M16—21no—
136C2×#97—24no—
137C2×C4×C8—21no—
138C2×C4×C8C2312yes—
139C4×C4⋊C4—21no—
140C4×C4⋊C4C2312yes—
141C4×C22⋊C4—21no—
142C4×C22⋊C4D412yes—
143#376D412yes—
144C43—21no—
145#336—24no—
146#335—24no—
147#334—24no—
148#328—24no—
149#327—24no—
150#327—28no—
151Q8⋊C8—24no—
152D4⋊C8—28no—
153D4⋊C8—24no—
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—41no—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4#4111yescharacteristic
172#135—84no—
173#135—84no—
174C2×Q16—24no—
175C2×SD16—24no—
176C2×SD16—24no—
177C2×D8—24no—
178C2×M16—41no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—41no—
183C22×C8—82no—
184C22×C8C2×D414yes—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8C22×C412yes—
189C4⋊Q8C22×C411yescharacteristic
190C4⋊Q8C22×C411yescharacteristic
191C4⋊Q8C22×C411yescharacteristic
192#127—21no—
193#127—21no—
194#127—21no—
195#127C22×C412yes—
196#127C22×C411yescharacteristic
197#126—41no—
198#126—41no—
199#125—21no—
200#125—21no—
201#125—21no—
202#124—22no—
203#124—21no—
204#124—21no—
205#124—22no—
206#124—21no—
207#124—41no—
208#124—41no—
209#123—41no—
210#123—41no—
211#123—41no—
212C22⋊Q8—21no—
213C22⋊Q8—21no—
214C22⋊Q8—21no—
215C22⋊Q8—21no—
216C22⋊Q8—21no—
217C22⋊Q8—21no—
218#121—21no—
219#121—21no—
220#121—22no—
221#121—22no—
222#121—21no—
223#121—21no—
224#121—22no—
225#121—21no—
226#121—22no—
227#121—21no—
228#121—41no—
229#121—41no—
230#121—41no—
231#121—41no—
232#120—41no—
233C4×Q8—21no—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×Q8—21no—
237C4×Q8—21no—
238C4×Q8—21no—
239C4×D4—21no—
240C4×D4—21no—
241C4×D4—22no—
242C4×D4—22no—
243C4×D4—21no—
244C4×D4—21no—
245C4×D4—22no—
246C4×D4—21no—
247C4×D4—22no—
248C4×D4—21no—
249C4×D4—41no—
250C4×D4—41no—
251C4×D4—41no—
252C4×D4—41no—
253C4×D4—41no—
254C4×D4—41no—
255C4×D4—41no—
256C4×D4—41no—
257C4×D4—41no—
258C4×D4—41no—
259#117—21no—
260#117—21no—
261#117—21no—
262#117—44no—
263#117—41no—
264#117—41no—
265#117—41no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—22no—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—41no—
271C2×C22⋊C4—41no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—41no—
276C2×C42C22×C411yescharacteristic
277C2×C42C22×C411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42#4111yescharacteristic
285C2×C42#4111yescharacteristic
286C2×C42#4111yescharacteristic
287C2×C42C2411yescharacteristic
288C4⋊C8—22no—
289C4⋊C8—22no—
290C4⋊C8—22no—
291C4⋊C8—22no—
292C4⋊C8—22no—
293C4⋊C8—41no—
294C4⋊C8—41no—
295C4⋊C8C22⋊C412yes—
296C4⋊C8C22⋊C412yes—
297C4⋊C8C22⋊C414yes—
298C4⋊C8C2×D414yes—
299#103—24no—
300#103—24no—
301#102—24no—
302#102—24no—
303C22⋊C8—22no—
304C22⋊C8—22no—
305C22⋊C8—41no—
306C22⋊C8—41no—
307C22⋊C8C22⋊C412yes—
308C22⋊C8C22⋊C412yes—
309C22⋊C8C22⋊C414yes—
310#97—41no—
311#97—84no—
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—84no—
329#41—82no—
330#41—82no—
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—84no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—82no—
367C22×C4—82no—
368C22×C4—81no—
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C4×D411yescharacteristic
372C22×C4C4×D411yescharacteristic
373C22×C4C4×D411yescharacteristic
374C22×C4C4×D411yescharacteristic
375C22×C4#12111yescharacteristic
376C22×C4#12111yescharacteristic
377C22×C4#12111yescharacteristic
378C22×C4#12111yescharacteristic
379C22×C4#12111yescharacteristic
380C22×C4#12411yescharacteristic
381C22×C4#12411yescharacteristic
382C22×C4#12711yescharacteristic
383C22×C4#12711yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C2×#4111yescharacteristic
387Q16—84no—
388SD16—84no—
389SD16—84no—
390D8—84no—
391M16—82no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—44no—
396C2×C8—42no—
397C2×C8—42no—
398C2×C8—82no—
399C2×C8—82no—
400C2×C8—82no—
401C2×C8C2×C22⋊C414yes—
402C2×C8C2×C22⋊C414yes—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—44no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C22⋊C4—44no—
421C22⋊C4—41no—
422C22⋊C4—41no—
423C22⋊C4—41no—
424C22⋊C4—42no—
425C22⋊C4—41no—
426C22⋊C4—42no—
427C22⋊C4—41no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—41no—
431C22⋊C4—41no—
432C22⋊C4—41no—
433C22⋊C4—41no—
434C42—44no—
435C42—41no—
436C42—42no—
437C42—42no—
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—22no—
459C23—41no—
460C23—41no—
461C23—42no—
462C23—82no—
463C23—82no—
464C23—82no—
465C23#39011yescharacteristic
466C23#39011yescharacteristic
467C23C2×#12111yescharacteristic
468Q8—42no—
469Q8—44no—
470Q8—42no—
471Q8—81no—
472Q8—82no—
473D4—42no—
474D4—44no—
475D4—44no—
476D4—42no—
477D4—82no—
478D4—81no—
479D4—82no—
480D4—82no—
481D4—84no—
482D4—82no—
483D4—82no—
484D4—84no—
485D4—82no—
486D4—82no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—42no—
512C2×C4—84no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—81no—
518C2×C4—82no—
519C2×C4—81no—
520C2×C4#39011yescharacteristic
521C2×C4#39011yescharacteristic
522C2×C4#39011yescharacteristic
523C2×C4#39011yescharacteristic
524C2×C4#39011yescharacteristic
525C2×C4#39011yescharacteristic
526C2×C4#44414yes—
527C2×C4C22×C22⋊C411yescharacteristic
528C2×C4C2×C4×D411yescharacteristic
529C2×C4C2×C4×D411yescharacteristic
530C2×C4C2×#12111yescharacteristic
531C2×C4C2×#12411yescharacteristic
532C2×C4C2×#12711yescharacteristic
533C2×C4#57612yes—
534C2×C4#57712yes—
535C8—24no—
536C8—24no—
537C8—44no—
538C8—82no—
539C8—82no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—44no—
548C22—82no—
549C22—81no—
550C22—81no—
551C22—82no—
552C22—81no—
553C22—82no—
554C22—164no—
555C22#197512yes—
556C22C2×#39011yescharacteristic
557C22#304512yes—
558C4—21no—
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—21no—
564C4—21no—
565C4—21no—
566C4—84no—
567C4—82no—
568C4—81no—
569C4—81no—
570C4—81no—
571C4#197512yes—
572C4#289712yes—
573C4C2×#44412yes—
574C4#304111yescharacteristic
575C4#304311yescharacteristic
576C2—22no—
577C2—22no—
578C2—41no—
579C2—161no—
580C2—162no—
581C2C2×#197511yescharacteristic
582C2#2124411yescharacteristic
583C2#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. #20547C2semidirect
3. #17806C2semidirect
4. #17544C2semidirect
5. #17533C2semidirect
6. #17466C2semidirect
7. C2×#1480C2semidirect
8. #9754C2semidirect
9. #9753C2semidirect
14. #3210C4semidirect
15. #3209C4semidirect
16. #3208C4semidirect
17. #3207C4semidirect
18. #3206C4semidirect
19. #3205C4semidirect
29. #1770C22semidirect
34. #1564C22semidirect
37. #1503C22semidirect
39. #1502C22semidirect
43. #1481C22semidirect
46. #1480C22semidirect

Maximal subgroups

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

subgroupindex
#607822
#205472
#178062
#175442
#175332
#174662
C2×#14802
#97542
#97532