Finite Group Databasebeta

← #205612

Subgroups — #205612

Order 512.

1123 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60700C211yesmaximal, characteristic
2#20329C211yesmaximal, characteristic, complemented
3#17797C211yesmaximal, characteristic, complemented
4#17651C211yesmaximal, characteristic, complemented
5#17647C211yesmaximal, characteristic, complemented
6#17541C211yesmaximal, characteristic, complemented
7C2×#1505C211yesmaximal, characteristic, complemented
8#9762C214yesmaximal, complemented
9#9761C214yesmaximal, complemented
10#3444C2211yescharacteristic
11C2×#532C2211yescharacteristic
12#3184C412yescomplemented
13#3182C412yescomplemented
14#3180—22no—
15#3171—22no—
16#3169C2212yes—
17#3167C2212yes—
18#3165C411yescharacteristic, complemented
19#3151C411yescharacteristic, complemented
20#3141C411yescharacteristic, complemented
21#3138C411yescharacteristic, complemented
22#3055—22no—
23C4×AGL(1,ℤ/8)—22no—
24#2920C2211yescharacteristic
25C2×#432C2211yescharacteristic
26C2×C4×M16—22no—
27C2×C4×M16C2211yescharacteristic
28C2×C4×M16C2211yescharacteristic
29C2×#376C2211yescharacteristic
30#1726—22no—
31#1726C2214yescomplemented
32#1569—22no—
33#1568C2214yescomplemented
34#1565—22no—
35#1531C2212yescomplemented
36#1530C2214yes—
37#1530C2212yescomplemented
38#1529—22no—
39#1527—22no—
40#1525—22no—
41#1524—22no—
42#1506C2212yescomplemented
43#1505—22no—
44#1505C2214yes—
45#1505C2212yescomplemented
46#1504—22no—
47#1500—22no—
48C2×#327—22no—
49C22×#41—21no—
50C2×AGL(1,ℤ/8)—22no—
51C22×M16—21no—
52C22×M16D412yes—
53#533C2×C411yescharacteristic
54#533C2×C411yescharacteristic
55#533C2311yescharacteristic
56#532—21no—
57#532—21no—
58#532—21no—
59#532—21no—
60#532—21no—
61#532—21no—
62#532C2×C411yescharacteristic
63#532C2×C412yes—
64#532C2×C411yescharacteristic
65#532C2312yes—
66#532C2311yescharacteristic
67C2×C4⋊Q8C2×C411yescharacteristic
68C2×#127C2×C411yescharacteristic
69#529—21no—
70#529—21no—
71#529—21no—
72#529—21no—
73#529—21no—
74#529—21no—
75C2×#124—21no—
76#525—21no—
77#525—21no—
78#525—21no—
79C2×#121—21no—
80C2×#121—21no—
81C4×#41—21no—
82C4×#41—21no—
83C4×#41—21no—
84C2×C4×D4—21no—
85C2×C4×D4—21no—
86C22×C42—21no—
87C22×C42C2311yescharacteristic
88#501—22no—
89#500C2×C412yescomplemented
90#498C2×C412yescomplemented
91#496—22no—
92#493C2×C412yescomplemented
93#492C2×C412yescomplemented
94#490—22no—
95#489—22no—
96#487C2×C412yescomplemented
97#486C2×C412yescomplemented
98#486C2×C412yescomplemented
99#486C2×C412yescomplemented
100#469—24no—
101#468—24no—
102#466—24no—
103#465—24no—
104#442—24no—
105#440—24no—
106C4×SD16—24no—
107C4×D8—24no—
108#432—21no—
109#432—21no—
110#432—21no—
111#432—21no—
112#432C2312yes—
113#432C2312yes—
114#431—21no—
115#431—21no—
116#428—22no—
117#423—21no—
118C2×C4⋊C8—21no—
119#418—22no—
120#417—22no—
121#407—21no—
122C2×C22⋊C8—21no—
123C4×M16—216no—
124C4×M16—21no—
125C4×M16—21no—
126C4×M16—21no—
127C4×M16C2×C411yescharacteristic
128C4×M16C2×C411yescharacteristic
129C4×M16C2×C412yes—
130C4×M16D412yes—
131C4×M16D412yes—
132C4×M16D412yes—
133C4×M16D412yes—
134C2×#97—24no—
135C2×#97—21no—
136C2×#97D412yes—
137C2×#97C2312yes—
138C2×C4×C8—24no—
139C2×C4×C8—21no—
140C2×C4×C8C2312yes—
141#376—21no—
142#376—21no—
143#376D412yes—
144#376C2312yes—
145C2×#95—21no—
146C2×#95D412yes—
147#335—24no—
148#334—24no—
149#330—24no—
150#329—24no—
151#329—28no—
152#328—24no—
153#327—28no—
154#327—24no—
155C2×#41—21no—
156C2×#41—21no—
157C2×#41—22no—
158C2×#41—21no—
159C2×#41—22no—
160C2×#41—41no—
161C2×#41—41no—
162C2×#41—84no—
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
173AGL(1,ℤ/8)—48no—
174C2×SD16—24no—
175C2×D8—24no—
176C2×M16—28no—
177C2×M16—22no—
178C2×M16—22no—
179C2×M16—28no—
180C2×M16—41no—
181C2×M16C22⋊C412yes—
182C2×M16C22⋊C414yes—
183C2×M16C22⋊C412yes—
184C22×C8—22no—
185C22×C8—41no—
186C22×C8C2×D414yes—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8C22×C412yes—
191C4⋊Q8C22×C411yescharacteristic
192C4⋊Q8C22×C411yescharacteristic
193C4⋊Q8C22×C411yescharacteristic
194#127—21no—
195#127—21no—
196#127—21no—
197#127C22×C412yes—
198#127C22×C411yescharacteristic
199#126—41no—
200#126—41no—
201#125—21no—
202#125—21no—
203#125—21no—
204#124—21no—
205#124—22no—
206#124—21no—
207#124—22no—
208#124—21no—
209#124—41no—
210#124—41no—
211#123—41no—
212#123—41no—
213#123—41no—
214C22⋊Q8—21no—
215C22⋊Q8—21no—
216C22⋊Q8—21no—
217C22⋊Q8—21no—
218C22⋊Q8—21no—
219C22⋊Q8—21no—
220#121—22no—
221#121—21no—
222#121—22no—
223#121—21no—
224#121—21no—
225#121—21no—
226#121—22no—
227#121—22no—
228#121—21no—
229#121—21no—
230#121—41no—
231#121—41no—
232#121—41no—
233#121—41no—
234#120—41no—
235C4×Q8—21no—
236C4×Q8—21no—
237C4×Q8—21no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×Q8—21no—
241C4×D4—22no—
242C4×D4—21no—
243C4×D4—22no—
244C4×D4—21no—
245C4×D4—21no—
246C4×D4—21no—
247C4×D4—22no—
248C4×D4—21no—
249C4×D4—22no—
250C4×D4—21no—
251C4×D4—41no—
252C4×D4—41no—
253C4×D4—41no—
254C4×D4—41no—
255C4×D4—41no—
256C4×D4—41no—
257C4×D4—41no—
258C4×D4—41no—
259C4×D4—41no—
260C4×D4—41no—
261#117—21no—
262#117—21no—
263#117—21no—
264#117—41no—
265#117—41no—
266#117—41no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C22⋊C4—41no—
270C2×C22⋊C4—41no—
271C2×C42—28no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—41no—
275C2×C42C22×C411yescharacteristic
276C2×C42C22×C411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42#4111yescharacteristic
284C2×C42#4111yescharacteristic
285C2×C42#4111yescharacteristic
286C2×C42C2411yescharacteristic
287#107—24no—
288#106—24no—
289C4⋊C8—22no—
290C4⋊C8—22no—
291C4⋊C8—41no—
292C4⋊C8—41no—
293#103—24no—
294#102—24no—
295#102—24no—
296#102—24no—
297C22⋊C8—22no—
298C22⋊C8—22no—
299C22⋊C8—41no—
300C22⋊C8—41no—
301#97—216no—
302#97—22no—
303#97—22no—
304#97—22no—
305#97—41no—
306#97C22⋊C414yes—
307#97C22⋊C412yes—
308#97C22⋊C412yes—
309#97C2×D414yes—
310C4×C8—216no—
311C4×C8—22no—
312C4×C8—22no—
313C4×C8—41no—
314C4×C8C22×C412yes—
315C4×C8C22×C412yes—
316C4×C8C2×D414yes—
317#95—22no—
318#95—22no—
319#95—22no—
320#95C2×D414yes—
321C24—41no—
322C24#12111yescharacteristic
323#41—44no—
324#41—84no—
325#41—82no—
326#41—84no—
327#41—84no—
328#41—82no—
329#41—84no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×Q8—22no—
334C2×Q8—41no—
335C2×Q8—82no—
336C2×D4—21no—
337C2×D4—22no—
338C2×D4—21no—
339C2×D4—21no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—41no—
343C2×D4—42no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—42no—
347C2×D4—41no—
348C2×D4—41no—
349C2×D4—41no—
350C2×D4—44no—
351C2×D4—82no—
352C2×D4—82no—
353C2×D4—84no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—28no—
357C22×C4—42no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—42no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—81no—
365C22×C4—82no—
366C22×C4—82no—
367C22×C4—84no—
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C4×D411yescharacteristic
371C22×C4C4×D411yescharacteristic
372C22×C4C4×D411yescharacteristic
373C22×C4C4×D411yescharacteristic
374C22×C4#12111yescharacteristic
375C22×C4#12111yescharacteristic
376C22×C4#12111yescharacteristic
377C22×C4#12111yescharacteristic
378C22×C4#12111yescharacteristic
379C22×C4#12411yescharacteristic
380C22×C4#12411yescharacteristic
381C22×C4#12711yescharacteristic
382C22×C4#12711yescharacteristic
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C2×#4111yescharacteristic
386SD16—48no—
387D8—48no—
388M16—28no—
389M16—28no—
390M16—24no—
391M16—28no—
392M16—82no—
393M16#10418yes—
394C2×C8—28no—
395C2×C8—24no—
396C2×C8—28no—
397C2×C8—42no—
398C2×C8—42no—
399C2×C8—82no—
400C2×C8C2×C22⋊C414yes—
401C2×C8C2×C22⋊C414yes—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—21no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C22⋊C4—41no—
419C22⋊C4—41no—
420C22⋊C4—41no—
421C22⋊C4—42no—
422C22⋊C4—41no—
423C22⋊C4—42no—
424C22⋊C4—41no—
425C22⋊C4—42no—
426C22⋊C4—42no—
427C22⋊C4—41no—
428C22⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—41no—
431C42—24no—
432C42—24no—
433C42—28no—
434C42—41no—
435C42—41no—
436C42—41no—
437C42C2×C22⋊C411yescharacteristic
438C42C2×C22⋊C411yescharacteristic
439C42C4×D411yescharacteristic
440C42C4×D411yescharacteristic
441C42C4×D411yescharacteristic
442C42C4×D411yescharacteristic
443C42#12111yescharacteristic
444C42#12111yescharacteristic
445C42#12411yescharacteristic
446C42#12411yescharacteristic
447C42#12711yescharacteristic
448C42#12711yescharacteristic
449C42C23×C411yescharacteristic
450C42C22×D411yescharacteristic
451C42C22×D411yescharacteristic
452C42C2×#4111yescharacteristic
453C23—22no—
454C23—22no—
455C23—41no—
456C23—41no—
457C23—42no—
458C23—82no—
459C23—82no—
460C23—82no—
461C23#39011yescharacteristic
462C23#39011yescharacteristic
463C23C2×#12111yescharacteristic
464Q8—42no—
465Q8—82no—
466Q8—81no—
467Q8—81no—
468Q8—82no—
469D4—42no—
470D4—44no—
471D4—82no—
472D4—82no—
473D4—81no—
474D4—81no—
475D4—82no—
476D4—82no—
477D4—84no—
478D4—82no—
479D4—82no—
480D4—84no—
481D4—82no—
482D4—82no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—28no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—84no—
508C2×C4—84no—
509C2×C4—82no—
510C2×C4—82no—
511C2×C4—88no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4#39011yescharacteristic
515C2×C4#39011yescharacteristic
516C2×C4#39011yescharacteristic
517C2×C4#39011yescharacteristic
518C2×C4#39011yescharacteristic
519C2×C4#39011yescharacteristic
520C2×C4C2×#10414yes—
521C2×C4C22×C22⋊C411yescharacteristic
522C2×C4C2×C4×D411yescharacteristic
523C2×C4C2×C4×D411yescharacteristic
524C2×C4C2×#12111yescharacteristic
525C2×C4C2×#12411yescharacteristic
526C2×C4C2×#12711yescharacteristic
527C8—28no—
528C8—24no—
529C8—82no—
530C8—82no—
531C8C2×#10418yes—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—44no—
540C22—82no—
541C22—81no—
542C22—81no—
543C22—82no—
544C22—81no—
545C22—82no—
546C22—164no—
547C22#196212yes—
548C22C2×#39011yescharacteristic
549C22#306812yes—
550C4—21no—
551C4—21no—
552C4—21no—
553C4—21no—
554C4—21no—
555C4—21no—
556C4—21no—
557C4—21no—
558C4—21no—
559C4—21no—
560C4—48no—
561C4—82no—
562C4—81no—
563C4—81no—
564C4—81no—
565C4#196212yes—
566C4C22×#10412yes—
567C2—22no—
568C2—22no—
569C2—41no—
570C2—161no—
571C2—162no—
572C2C2×#196211yescharacteristic
573C2#2124411yescharacteristic
574C2#2124911yescharacteristic
subgroupG/H
center, socleC22C2×#390
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radical#60700C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20329C2semidirect
3. #17797C2semidirect
4. #17651C2semidirect
5. #17647C2semidirect
6. #17541C2semidirect
7. C2×#1505C2semidirect
8. #9762C2semidirect
9. #9761C2semidirect
12. #3184C4semidirect
13. #3182C4semidirect
18. #3165C4semidirect
19. #3151C4semidirect
20. #3141C4semidirect
21. #3138C4semidirect
31. #1726C22semidirect
33. #1568C22semidirect
35. #1531C22semidirect
37. #1530C22semidirect
42. #1506C22semidirect
45. #1505C22semidirect
89. #500C2×C4semidirect
90. #498C2×C4semidirect
92. #493C2×C4semidirect
93. #492C2×C4semidirect
96. #487C2×C4semidirect
97. #486C2×C4semidirect
98. #486C2×C4semidirect
99. #486C2×C4semidirect

Maximal subgroups

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

subgroupindex
#607002
#203292
#177972
#176512
#176472
#175412
C2×#15052
#97622
#97612