Finite Group Databasebeta

← #206963

Subgroups — #206963

Order 512.

1482 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C211yesmaximal, characteristic
2C4×#407C211yesmaximal, characteristic
3C2×#1434C211yesmaximal, characteristic
4#9889C212yesmaximal, complemented
5#9888C212yesmaximal, complemented
6#9886C214yesmaximal, complemented
7#8020C214yesmaximal
8C2×#533C411yescharacteristic, complemented
9C23×C42C2211yescharacteristic
10C2×#407C2212yes—
11C2×C4×M16—21no—
12#2857C411yescharacteristic, complemented
13#2857C411yescharacteristic, complemented
14#2857C2211yescharacteristic
15#2857C2211yescharacteristic
16#2855—21no—
17#2855—21no—
18#2851C412yescomplemented
19#2851C2212yes—
20#2848—21no—
21#2848—21no—
22C4×C22⋊Q8—21no—
23C4×C22⋊Q8—21no—
24C4×C22⋊Q8—21no—
25C4×C22⋊Q8—21no—
26#2296C412yescomplemented
27#2285C411yescharacteristic, complemented
28#2280C2212yes—
29C22×#95—21no—
30C22×#95—21no—
31C22×#95C2212yes—
32#1749—21no—
33#1749C2212yes—
34C4×C22⋊C8—21no—
35C4×C22⋊C8C2212yes—
36#1728—21no—
37#1434—21no—
38#1434—21no—
39#1434C2212yes—
40#1434C2212yes—
41#1294C2218yes—
42#1294C2218yes—
43C24×C4—22no—
44C24×C4D411yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4C2311yescharacteristic
47C22×M16—22no—
48#533—44no—
49C2×C4⋊Q8C2×C411yescharacteristic, complemented
50C2×C4⋊Q8C2×C411yescharacteristic, complemented
51C2×#125C2×C412yescomplemented
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×#117C2×C412yes—
61C2×#117C2×C412yes—
62C22×C42—44no—
63C22×C42C2×C411yescharacteristic
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75#407—24no—
76#407Q818yescomplemented
77C2×C22⋊C8—22no—
78C2×C22⋊C8C2314yes—
79C4×M16—44no—
80C2×#97—22no—
81C2×C4×C8—22no—
82#391—21no—
83#391—21no—
84#391—21no—
85#391—21no—
86#389—21no—
87#389—21no—
88#389—21no—
89#389—21no—
90#389C2×C412yescomplemented
91#389C2×C412yescomplemented
92#389C2×C412yescomplemented
93#389C2×C412yescomplemented
94#387—22no—
95#387—22no—
96#387—22no—
97#387—22no—
98#385—22no—
99#385—22no—
100#385—22no—
101#385—22no—
102#385—22no—
103#385—22no—
104#384C2×C411yescharacteristic, complemented
105#384C2×C411yescharacteristic, complemented
106#382C2×C412yescomplemented
107#381—22no—
108#380—21no—
109#380—21no—
110#380—21no—
111#380—21no—
112#380—21no—
113#380—21no—
114#380C2×C412yes—
115#380C2×C412yes—
116#380C2312yes—
117#380C2312yes—
118#379—22no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125C2×#95—28no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—28no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95—22no—
132C2×#95—44no—
133C2×#95C2314yes—
134#336—22no—
135#336—22no—
136C25C2×D411yescharacteristic
137C22×Q8—21no—
138C22×Q8—21no—
139C23×C4—24no—
140C23×C4—24no—
141C23×C4—28no—
142C23×C4—24no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—44no—
150C23×C4—44no—
151C23×C4—44no—
152C23×C4—44no—
153C23×C4—44no—
154C23×C4—44no—
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22⋊C411yescharacteristic
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×Q814yes—
173C23×C4#4114yes—
174C23×C4C2411yescharacteristic
175C2×M16—28no—
176C2×M16—44no—
177C2×M16—44no—
178C22×C8—24no—
179C4⋊Q8—42no—
180C4⋊Q8—42no—
181#125—44no—
182C22⋊Q8—44no—
183C22⋊Q8—44no—
184C22⋊Q8—44no—
185C22⋊Q8—44no—
186C22⋊Q8—44no—
187C22⋊Q8—44no—
188C4×Q8—44no—
189C4×Q8—44no—
190#117—44no—
191#117—44no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4C22×C412yes—
205C2×C4⋊C4C22×C412yes—
206C2×C4⋊C4C22×C412yes—
207C2×C4⋊C4C22×C412yes—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C42—24no—
215C2×C42—24no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—42no—
219C2×C42—42no—
220C2×C42—42no—
221C2×C42—42no—
222C2×C42—42no—
223C2×C42—42no—
224C2×C42—42no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42C22⋊C411yescharacteristic
229C2×C42C22⋊C411yescharacteristic
230C2×C42C22⋊C411yescharacteristic
231C2×C42C22⋊C411yescharacteristic
232C2×C42C22⋊C411yescharacteristic
233C2×C42C22⋊C411yescharacteristic
234C2×C42C22⋊C411yescharacteristic
235C2×C42C22⋊C411yescharacteristic
236C2×C42C22×C411yescharacteristic
237C2×C42C22×C411yescharacteristic
238C2×C42C22×C411yescharacteristic
239C2×C42C22×C411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C22⋊C8—28no—
245C22⋊C8—24no—
246C22⋊C8C2×Q818yes—
247#97—44no—
248C4×C8—44no—
249#95—28no—
250#95—28no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—28no—
262#95—44no—
263#95—44no—
264#95—44no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—22no—
271C24C2×C22⋊C411yescharacteristic
272C24C2×C22⋊C411yescharacteristic
273C24#12412yes—
274C24C4⋊Q812yes—
275C24C22×D411yescharacteristic
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—28no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—48no—
333C22×C4—48no—
334C22×C4—41no—
335C22×C4—416no—
336C22×C4—416no—
337C22×C4—44no—
338C22×C4—416no—
339C22×C4—416no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—416no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—88no—
347C22×C4—88no—
348C22×C4—88no—
349C22×C4—88no—
350C22×C4#9914yes—
351C22×C4#10414yes—
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C4×Q814yes—
363C22×C4C4×Q814yes—
364C22×C4C22⋊Q814yes—
365C22×C4C22⋊Q814yes—
366C22×C4#12412yes—
367C22×C4C4⋊Q812yes—
368C22×C4C23×C411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×Q812yes—
371C22×C4C2×#4112yes—
372M16—416no—
373C2×C8—28no—
374C2×C8—44no—
375C2×C8—44no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—44no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—44no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—44no—
392C22⋊C4—44no—
393C22⋊C4—44no—
394C22⋊C4—44no—
395C22⋊C4—44no—
396C22⋊C4—44no—
397C22⋊C4—44no—
398C42—22no—
399C42—22no—
400C42—22no—
401C42—22no—
402C42—22no—
403C42—22no—
404C42—22no—
405C42—22no—
406C42—41no—
407C42—41no—
408C42—41no—
409C42—41no—
410C42—41no—
411C42—41no—
412C42—41no—
413C42—41no—
414C42—44no—
415C42—44no—
416C23—28no—
417C23—28no—
418C23—28no—
419C23—28no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—24no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—24no—
434C23—24no—
435C23—24no—
436C23—24no—
437C23—24no—
438C23#39112yes—
439C23#39112yes—
440C23C2×#9912yes—
441C23#41012yes—
442C23C2×#10412yes—
443C23#42112yes—
444C23C22×C22⋊C411yescharacteristic
445C23C2×#12411yescharacteristic
446C23C2×C4⋊Q811yescharacteristic
447C23#57512yes—
448C23#57512yes—
449Q8—84no—
450Q8—84no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—28no—
468C2×C4—28no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—44no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—48no—
500C2×C4—48no—
501C2×C4—48no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—832no—
515C2×C4—84no—
516C2×C4—84no—
517C2×C4—84no—
518C2×C4—84no—
519C2×C4#39112yes—
520C2×C4#39112yes—
521C2×C4C2×#9912yes—
522C2×C4C2×#9912yes—
523C2×C4C2×#10412yes—
524C2×C4C2×#10412yes—
525C2×C4C2×C4×Q812yes—
526C2×C4C2×C22⋊Q812yes—
527C8—48no—
528C22—216no—
529C22—216no—
530C22—21no—
531C22—21no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—21no—
536C22—21no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—22no—
547C22—22no—
548C22—22no—
549C22—22no—
550C22#198614yes—
551C22#199614yes—
552C22C2×#39111yescharacteristic
553C22C22×#9911yescharacteristic
554C22C2×#41011yescharacteristic
555C22#289611yescharacteristic
556C22#289611yescharacteristic
557C22C22×#10411yescharacteristic
558C22C2×#42111yescharacteristic
559C22#314012yes—
560C22#316612yes—
561C4—22no—
562C4—22no—
563C4—22no—
564C4—22no—
565C4—22no—
566C4—22no—
567C4—22no—
568C4—22no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—41no—
573C4—88no—
574C4—82no—
575C4—82no—
576C4—82no—
577C4—82no—
578C2—22no—
579C2—22no—
580C2—24no—
581C2—24no—
582C2#1754111yescharacteristic
583C2#1754211yescharacteristic
584C2#1758711yescharacteristic
585C2#1759311yescharacteristic
586C2C2×#198611yescharacteristic
587C2#2132311yescharacteristic
588C2C2×#199611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9889C2semidirect
5. #9888C2semidirect
6. #9886C2semidirect
8. C2×#533C4semidirect
12. #2857C4semidirect
13. #2857C4semidirect
18. #2851C4semidirect
26. #2296C4semidirect
27. #2285C4semidirect
49. C2×C4⋊Q8C2×C4semidirect
50. C2×C4⋊Q8C2×C4semidirect
51. C2×#125C2×C4semidirect
76. #407Q8semidirect
90. #389C2×C4semidirect
91. #389C2×C4semidirect
92. #389C2×C4semidirect
93. #389C2×C4semidirect
104. #384C2×C4semidirect
105. #384C2×C4semidirect
106. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
#366252
C4×#4072
C2×#14342
#98892
#98882
#98862
#80202