Finite Group Databasebeta

← #195486

Subgroups — #195486

Order 512.

1310 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3167C211yesmaximal, characteristic
2#60144C211yesmaximal, characteristic
3#21306C211yesmaximal, characteristic, complemented
4#17574C211yesmaximal, characteristic
5#17486C211yesmaximal, characteristic, complemented
6#9055C218yesmaximal, complemented
7#7897C211yesmaximal, characteristic, complemented
8C2×#1278C211yesmaximal, characteristic, complemented
9C2×#568—21no—
10C2×#533C2211yescharacteristic
11C2×#532C2211yescharacteristic
12#3167—22no—
13#3167—22no—
14#3167C414yescomplemented
15#3167C2214yes—
16C2×#498C411yescharacteristic, complemented
17C2×#492C411yescharacteristic, complemented
18C2×#487C411yescharacteristic, complemented
19C2×#486C411yescharacteristic, complemented
20C2×#465—22no—
21C2×C4×SD16—22no—
22#2935—22no—
23#2935C2214yes—
24D4×M16—22no—
25D4×M16—22no—
26#2932—22no—
27#2931—22no—
28C2×#431C2211yescharacteristic
29C2×#423C2211yescharacteristic
30#2879—21no—
31C2×C4×M16C2211yescharacteristic
32C22×C4×C8C2211yescharacteristic
33#2285C2211yescharacteristic
34#1986—22no—
35#1986C2214yescomplemented
36#1733—21no—
37#1733—21no—
38#1519—22no—
39#1517—22no—
40#1502C2214yes—
41#1493—22no—
42#1487—22no—
43#1481C2214yescomplemented
44#1279C2214yescomplemented
45#1278C2214yescomplemented
46C22×#41—21no—
47C22×SD16—22no—
48#568—44no—
49#568—44no—
50C2×#131—24no—
51C2×#131—24no—
52C22×M16—24no—
53C22×M16—21no—
54C22×M16D411yescharacteristic
55C22×M16D411yescharacteristic
56C23×C8—21no—
57#533—21no—
58#533—21no—
59#533—21no—
60#533—21no—
61#533—21no—
62#533—21no—
63#533C2×C412yes—
64#533C2312yes—
65#532—21no—
66#532—21no—
67#532—21no—
68#532—21no—
69#532C2×C412yes—
70#532D412yes—
71#532D412yes—
72#532C2312yes—
73C2×C4⋊Q8C2×C411yescharacteristic
74C2×C4⋊Q8C2×C411yescharacteristic
75C2×C4⋊Q8C2×C411yescharacteristic
76C2×#127C2×C411yescharacteristic
77C2×#125—21no—
78C2×#124—21no—
79C2×C22⋊Q8—21no—
80C2×C22⋊Q8—21no—
81C2×#121—21no—
82C2×#121D412yes—
83C2×C4×Q8—21no—
84C2×C4×Q8—21no—
85C2×C4×D4—21no—
86C2×C4×D4D412yes—
87C2×#117—21no—
88C2×#117—21no—
89C22×C42C2311yescharacteristic
90#498—22no—
91#498C2×C414yes—
92#492—22no—
93#492C2×C414yes—
94#487—22no—
95#487C2×C414yes—
96#486—22no—
97#486C2×C414yes—
98#465—28no—
99#465—28no—
100C4×SD16—28no—
101C4×SD16—28no—
102#436—24no—
103#436—24no—
104#435—28no—
105#435—28no—
106C8×D4—24no—
107C8×D4—24no—
108#431—21no—
109#431—21no—
110#431—21no—
111#431C2312yes—
112C2×#106—22no—
113#423—21no—
114#423—21no—
115#423D412yes—
116#423D412yes—
117C2×C4⋊C8—21no—
118C2×C4⋊C8—21no—
119C2×C4⋊C8C2312yes—
120C2×#103—22no—
121C2×#102—22no—
122#408—24no—
123#408—24no—
124#407—24no—
125C2×C22⋊C8—21no—
126C4×M16—21no—
127C4×M16—21no—
128C4×M16D412yes—
129C4×M16C2312yes—
130C2×#97—21no—
131C2×C4×C8—21no—
132C2×C4×C8—21no—
133C2×C4×C8—21no—
134C2×C4×C8—21no—
135C2×C4×C8—21no—
136C2×C4×C8C2×C411yescharacteristic
137C2×C4×C8C2×C411yescharacteristic
138C2×C4×C8C2×C412yes—
139C2×C4×C8C2312yes—
140#384—21no—
141#382—21no—
142#380—21no—
143#380—21no—
144C4×C4⋊C4—21no—
145C4×C4⋊C4C2312yes—
146#339—24no—
147#339—24no—
148#327—28no—
149Q8⋊C8—28no—
150C2×#41—22no—
151C2×#41—22no—
152C2×#41—22no—
153C2×#41—22no—
154C2×#41—42no—
155C2×#41—42no—
156C2×#41—42no—
157C2×#41—42no—
158C22×Q8—21no—
159C22×Q8—21no—
160C22×Q8—21no—
161C22×D4—22no—
162C22×D4—21no—
163C22×D4—21no—
164C22×D4C22⋊C412yes—
165C23×C4—22no—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4#4111yescharacteristic
169C2×SD16—28no—
170C2×SD16—48no—
171#131—416no—
172#131—48no—
173#131—48no—
174C2×M16—21no—
175C2×M16—22no—
176C2×M16—28no—
177C2×M16—22no—
178C2×M16—22no—
179C2×M16—21no—
180C2×M16—22no—
181C2×M16—22no—
182C2×M16—416no—
183C2×M16—42no—
184C2×M16—42no—
185C2×M16SD1614yescomplemented
186C2×M16C2×D412yes—
187C2×M16C2×D412yes—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—28no—
191C22×C8—21no—
192C22×C8—22no—
193C22×C8—21no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—44no—
197C22×C8—41no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C4⋊Q8—21no—
201C4⋊Q8—21no—
202C4⋊Q8—21no—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8C22⋊C412yes—
207C4⋊Q8C22×C412yes—
208C4⋊Q8C22×C412yes—
209C4⋊Q8C22×C412yes—
210#127—21no—
211#127—21no—
212#127C22⋊C412yes—
213#127C22×C412yes—
214#125—22no—
215#125—22no—
216#125—41no—
217#125—41no—
218#124—22no—
219#124—22no—
220#124—22no—
221#124—22no—
222C22⋊Q8—22no—
223C22⋊Q8—22no—
224C22⋊Q8—22no—
225C22⋊Q8—22no—
226C22⋊Q8—41no—
227C22⋊Q8—41no—
228C22⋊Q8—41no—
229C22⋊Q8—41no—
230#121—22no—
231#121—22no—
232#121—22no—
233#121—22no—
234#121—22no—
235#121—22no—
236#121C22⋊C414yes—
237#121C2×D414yes—
238C4×Q8—22no—
239C4×Q8—22no—
240C4×Q8—22no—
241C4×Q8—22no—
242C4×Q8—41no—
243C4×Q8—41no—
244C4×Q8—41no—
245C4×Q8—41no—
246C4×D4—22no—
247C4×D4—22no—
248C4×D4—22no—
249C4×D4—22no—
250C4×D4—22no—
251C4×D4—22no—
252C4×D4C22⋊C414yes—
253C4×D4C2×D414yes—
254#117—22no—
255#117—22no—
256#117—41no—
257#117—41no—
258#117—42no—
259#117—42no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—41no—
267C2×C4⋊C4—41no—
268C2×C4⋊C4—41no—
269C2×C4⋊C4—41no—
270C2×C4⋊C4C22⋊C412yes—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—41no—
274C2×C22⋊C4—41no—
275C2×C42—22no—
276C2×C42—41no—
277C2×C42C22×C411yescharacteristic
278C2×C42C22×C411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2×D411yescharacteristic
284C2×C42C2×D411yescharacteristic
285C2×C42#4111yescharacteristic
286C2×C42#4111yescharacteristic
287C2×C42#4111yescharacteristic
288C2×C42C2411yescharacteristic
289#106—28no—
290C4⋊C8—22no—
291C4⋊C8—22no—
292C4⋊C8—22no—
293C4⋊C8—22no—
294C4⋊C8—41no—
295C4⋊C8C2×D414yes—
296#103—28no—
297#102—28no—
298C22⋊C8—28no—
299C22⋊C8—28no—
300C22⋊C8—22no—
301C22⋊C8—41no—
302#97—22no—
303#97—22no—
304C4×C8—22no—
305C4×C8—21no—
306C4×C8—21no—
307C4×C8—22no—
308C4×C8—22no—
309C4×C8—22no—
310C4×C8—22no—
311C4×C8—41no—
312C4×C8C22×C412yes—
313C4×C8C22×C412yes—
314#95—22no—
315#95—41no—
316C24—22no—
317C24#12111yescharacteristic
318#41—42no—
319#41—42no—
320#41—42no—
321#41—42no—
322#41—44no—
323#41—44no—
324#41—44no—
325#41—44no—
326#41—44no—
327#41—44no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—44no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—42no—
336C2×Q8—42no—
337C2×Q8—82no—
338C2×D4—24no—
339C2×D4—24no—
340C2×D4—22no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—22no—
344C2×D4—22no—
345C2×D4—44no—
346C2×D4—44no—
347C2×D4—44no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—82no—
351C2×D4C2×C22⋊C414yes—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—44no—
358C22×C4—44no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—42no—
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C4×D411yescharacteristic
365C22×C4C4×D411yescharacteristic
366C22×C4C4×D411yescharacteristic
367C22×C4C4×D411yescharacteristic
368C22×C4#12011yescharacteristic
369C22×C4#12011yescharacteristic
370C22×C4#12111yescharacteristic
371C22×C4#12111yescharacteristic
372C22×C4#12111yescharacteristic
373C22×C4#12111yescharacteristic
374C22×C4#12111yescharacteristic
375C22×C4#12311yescharacteristic
376C22×C4#12311yescharacteristic
377C22×C4C2×SD1612yes—
378C22×C4#13512yes—
379C22×C4AGL(1,ℤ/8)12yes—
380C22×C4#13712yes—
381C22×C4C22×D411yescharacteristic
382C22×C4C22×D411yescharacteristic
383C22×C4C2×#4111yescharacteristic
384SD16—416no—
385M16—22no—
386M16—24no—
387M16—22no—
388M16—22no—
389M16—416no—
390M16—44no—
391M16—44no—
392M16—42no—
393M16C2×SD1614yescomplemented
394C2×C8—24no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—24no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—24no—
401C2×C8—44no—
402C2×C8—44no—
403C2×C8—416no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—42no—
407C2×C8—42no—
408C2×C8—42no—
409C2×C8—82no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—44no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—42no—
425C4⋊C4—42no—
426C4⋊C4—82no—
427C4⋊C4—82no—
428C4⋊C4C2×C22⋊C414yes—
429C22⋊C4—24no—
430C22⋊C4—24no—
431C22⋊C4—24no—
432C22⋊C4—24no—
433C22⋊C4—42no—
434C22⋊C4—42no—
435C22⋊C4—82no—
436C42—42no—
437C42—42no—
438C42—42no—
439C42—42no—
440C42C2×C22⋊C411yescharacteristic
441C42C2×C22⋊C411yescharacteristic
442C42C4×D411yescharacteristic
443C42C4×D411yescharacteristic
444C42C4×D411yescharacteristic
445C42C4×D411yescharacteristic
446C42#12011yescharacteristic
447C42#12011yescharacteristic
448C42#12111yescharacteristic
449C42#12111yescharacteristic
450C42#12311yescharacteristic
451C42#12311yescharacteristic
452C42C23×C411yescharacteristic
453C42C22×D411yescharacteristic
454C42C22×D411yescharacteristic
455C42C2×#4111yescharacteristic
456C23—24no—
457C23—41no—
458C23—41no—
459C23—44no—
460C23—82no—
461C23—82no—
462C23#38611yescharacteristic
463C23#38611yescharacteristic
464C23#46112yes—
465C23#46312yes—
466C23C2×#12111yescharacteristic
467Q8—44no—
468Q8—44no—
469Q8—42no—
470Q8—42no—
471Q8—44no—
472Q8—82no—
473Q8—82no—
474D4—44no—
475D4—48no—
476D4—44no—
477D4—44no—
478D4—44no—
479D4—44no—
480D4—42no—
481D4—42no—
482D4—44no—
483D4—82no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—24no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—44no—
498C2×C4—48no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—82no—
507C2×C4—82no—
508C2×C4#38611yescharacteristic
509C2×C4#38611yescharacteristic
510C2×C4#38611yescharacteristic
511C2×C4#38611yescharacteristic
512C2×C4#38611yescharacteristic
513C2×C4#38611yescharacteristic
514C2×C4C4×SD1612yes—
515C2×C4C4×SD1612yes—
516C2×C4#44012yes—
517C2×C4#44012yes—
518C2×C4#45012yes—
519C2×C4#45012yes—
520C2×C4#45212yes—
521C2×C4#45212yes—
522C2×C4#46112yes—
523C2×C4#46312yes—
524C2×C4C22×C22⋊C411yescharacteristic
525C2×C4C2×C4×D411yescharacteristic
526C2×C4C2×C4×D411yescharacteristic
527C2×C4C2×#12011yescharacteristic
528C2×C4C2×#12111yescharacteristic
529C2×C4C2×#12311yescharacteristic
530C2×C4C22×SD1611yescharacteristic
531C2×C4C2×#13511yescharacteristic
532C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
533C2×C4C2×#13711yescharacteristic
534C8—24no—
535C8—24no—
536C8—42no—
537C8—44no—
538C8—42no—
539C8—82no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—42no—
547C22—42no—
548C22—44no—
549C22—44no—
550C22—44no—
551C22—164no—
552C22#187712yes—
553C22#187712yes—
554C22C2×#38611yescharacteristic
555C22C2×#46111yescharacteristic
556C22C2×#46311yescharacteristic
557C4—21no—
558C4—21no—
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—44no—
564C4—82no—
565C4—82no—
566C4—162no—
567C4#187712yes—
568C4#187712yes—
569C4#188412yes—
570C4C2×C4×SD1611yescharacteristic
571C4C2×#44011yescharacteristic
572C4#295712yes—
573C4C2×#45011yescharacteristic
574C4C2×#45211yescharacteristic
575C2—21no—
576C2—22no—
577C2—22no—
578C2—21no—
579C2—84no—
580C2#2091911yescharacteristic
581C2C2×#187711yescharacteristic
582C2C2×#188411yescharacteristic
subgroupG/H
centerC2×C4C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radicalC2×#532C22
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
3. #21306C2semidirect
5. #17486C2semidirect
6. #9055C2semidirect
7. #7897C2semidirect
8. C2×#1278C2semidirect
14. #3167C4semidirect
16. C2×#498C4semidirect
17. C2×#492C4semidirect
18. C2×#487C4semidirect
19. C2×#486C4semidirect
35. #1986C22semidirect
43. #1481C22semidirect
44. #1279C22semidirect
45. #1278C22semidirect
185. C2×M16SD16semidirect
393. M16C2×SD16semidirect

Maximal subgroups

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

subgroupindex
C2×#31672
#601442
#213062
#175742
#174862
#90552
#78972
C2×#12782