Finite Group Databasebeta

← #206962

Subgroups — #206962

Order 512.

1634 conjugacy classes of subgroups.

The subgroup lattice has 635 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C211yesmaximal, characteristic
2C4×#407C211yesmaximal, characteristic
3C2×C4⋊#95C211yesmaximal, characteristic
4#9887C214yesmaximal, complemented
5#9885C212yesmaximal, complemented
6#9884C212yesmaximal, complemented
7#8020C214yesmaximal
8C2×#533C411yescharacteristic, complemented
9C23×C4⋊C4—21no—
10C23×C42C2211yescharacteristic
11C2×#407C2212yes—
12C2×C4×M16—21no—
13#2857C411yescharacteristic, complemented
14#2857C411yescharacteristic, complemented
15#2857C2211yescharacteristic
16#2857C2211yescharacteristic
17#2855—21no—
18#2855—21no—
19#2851C412yescomplemented
20#2851C2212yes—
21#2848—21no—
22#2848—21no—
23C4×C22⋊Q8—21no—
24C4×C22⋊Q8—21no—
25C4×C22⋊Q8—21no—
26C4×C22⋊Q8—21no—
27#2296C412yescomplemented
28#2285C411yescharacteristic, complemented
29#2280C2212yes—
30C22×#95—21no—
31C22×#95C2212yes—
32#1749—21no—
33#1749C2212yes—
34C4×C22⋊C8—21no—
35C4×C22⋊C8C2212yes—
36#1728—21no—
37C4⋊#95—21no—
38C4⋊#95—21no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
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×C4⋊C4—28no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—44no—
66C22×C4⋊C4—44no—
67C22×C42—44no—
68C22×C42C2×C411yescharacteristic
69C22×C42C2×C411yescharacteristic
70C22×C42C2×C411yescharacteristic
71C22×C42C2×C411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80#407—24no—
81#407Q818yescomplemented
82C2×C22⋊C8—22no—
83C2×C22⋊C8C2314yes—
84C4×M16—44no—
85C2×#97—22no—
86C2×C4×C8—22no—
87#391—21no—
88#391—21no—
89#391—21no—
90#391—21no—
91#389—21no—
92#389—21no—
93#389—21no—
94#389—21no—
95#389C2×C412yescomplemented
96#389C2×C412yescomplemented
97#389C2×C412yescomplemented
98#389C2×C412yescomplemented
99#387—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#385—22no—
104#385—22no—
105#385—22no—
106#385—22no—
107#385—22no—
108#385—22no—
109#384C2×C411yescharacteristic, complemented
110#384C2×C411yescharacteristic, complemented
111#382C2×C412yescomplemented
112#381—22no—
113#380—21no—
114#380—21no—
115#380—21no—
116#380—21no—
117#380—21no—
118#380—21no—
119#380C2×C412yes—
120#380C2×C412yes—
121#380C2312yes—
122#380C2312yes—
123#379—22no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—22no—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—22no—
130C2×#95—28no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95—44no—
135C2×#95C2314yes—
136#336—22no—
137#336—22no—
138C25C2×D411yescharacteristic
139C22×Q8—21no—
140C22×Q8—21no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—22no—
144C23×C4—22no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—28no—
148C23×C4—24no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—44no—
154C23×C4—44no—
155C23×C4—44no—
156C23×C4—44no—
157C23×C4—44no—
158C23×C4—44no—
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22⋊C411yescharacteristic
161C23×C4C22⋊C411yescharacteristic
162C23×C4C22⋊C411yescharacteristic
163C23×C4C22⋊C411yescharacteristic
164C23×C4C22⋊C411yescharacteristic
165C23×C4C22⋊C411yescharacteristic
166C23×C4C22⋊C411yescharacteristic
167C23×C4C22×C411yescharacteristic
168C23×C4C22×C411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×Q814yes—
177C23×C4#4114yes—
178C23×C4C2411yescharacteristic
179C2×M16—28no—
180C2×M16—44no—
181C2×M16—44no—
182C22×C8—24no—
183C4⋊Q8—42no—
184C4⋊Q8—42no—
185#125—44no—
186C22⋊Q8—44no—
187C22⋊Q8—44no—
188C22⋊Q8—44no—
189C22⋊Q8—44no—
190C22⋊Q8—42no—
191C22⋊Q8—42no—
192C22⋊Q8—42no—
193C22⋊Q8—42no—
194C4×Q8—44no—
195C4×Q8—44no—
196#117—44no—
197#117—44no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—432no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4C22×C412yes—
218C2×C4⋊C4C22×C412yes—
219C2×C4⋊C4C22×C412yes—
220C2×C4⋊C4C22×C412yes—
221C2×C22⋊C4—22no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—42no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42—42no—
235C2×C42—42no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42—42no—
239C2×C42—42no—
240C2×C42—42no—
241C2×C42C22⋊C411yescharacteristic
242C2×C42C22⋊C411yescharacteristic
243C2×C42C22⋊C411yescharacteristic
244C2×C42C22⋊C411yescharacteristic
245C2×C42C22⋊C411yescharacteristic
246C2×C42C22⋊C411yescharacteristic
247C2×C42C22⋊C411yescharacteristic
248C2×C42C22⋊C411yescharacteristic
249C2×C42C22×C411yescharacteristic
250C2×C42C22×C411yescharacteristic
251C2×C42C22×C411yescharacteristic
252C2×C42C22×C411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C2×C42C2×D411yescharacteristic
257C22⋊C8—28no—
258C22⋊C8—24no—
259C22⋊C8C2×Q818yes—
260#97—44no—
261C4×C8—44no—
262#95—22no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—28no—
273#95—44no—
274#95—44no—
275#95—44no—
276C24—21no—
277C24—21no—
278C24—21no—
279C24—21no—
280C24—22no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24C2×C22⋊C411yescharacteristic
285C24C2×C22⋊C411yescharacteristic
286C24#12412yes—
287C24C4⋊Q812yes—
288C24C22×D411yescharacteristic
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C22×C4—216no—
296C22×C4—216no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—28no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—48no—
348C22×C4—48no—
349C22×C4—41no—
350C22×C4—48no—
351C22×C4—48no—
352C22×C4—416no—
353C22×C4—44no—
354C22×C4—416no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4—416no—
358C22×C4—44no—
359C22×C4—44no—
360C22×C4—44no—
361C22×C4—88no—
362C22×C4—88no—
363C22×C4—88no—
364C22×C4—88no—
365C22×C4#9914yes—
366C22×C4#10414yes—
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C2×C22⋊C411yescharacteristic
372C22×C4C2×C22⋊C411yescharacteristic
373C22×C4C2×C22⋊C411yescharacteristic
374C22×C4C2×C22⋊C411yescharacteristic
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C4×Q814yes—
378C22×C4C4×Q814yes—
379C22×C4C22⋊Q814yes—
380C22×C4C22⋊Q814yes—
381C22×C4#12412yes—
382C22×C4C4⋊Q812yes—
383C22×C4AGL(1,ℤ/8)12yes—
384C22×C4AGL(1,ℤ/8)12yes—
385C22×C4#13712yes—
386C22×C4#13712yes—
387C22×C4C23×C411yescharacteristic
388C22×C4C22×D411yescharacteristic
389C22×C4C22×Q812yes—
390C22×C4C2×#4112yes—
391M16—416no—
392C2×C8—28no—
393C2×C8—44no—
394C2×C8—44no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—416no—
400C4⋊C4—416no—
401C4⋊C4—44no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—44no—
405C4⋊C4—44no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—42no—
412C4⋊C4—44no—
413C22⋊C4—42no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—42no—
417C22⋊C4—44no—
418C22⋊C4—44no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C42—22no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—22no—
427C42—22no—
428C42—22no—
429C42—41no—
430C42—41no—
431C42—41no—
432C42—41no—
433C42—41no—
434C42—41no—
435C42—41no—
436C42—41no—
437C42—44no—
438C42—44no—
439C23—24no—
440C23—24no—
441C23—24no—
442C23—24no—
443C23—24no—
444C23—24no—
445C23—24no—
446C23—24no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—22no—
452C23—22no—
453C23—21no—
454C23—21no—
455C23—21no—
456C23—21no—
457C23—21no—
458C23—21no—
459C23—21no—
460C23—21no—
461C23—22no—
462C23—22no—
463C23—24no—
464C23—24no—
465C23—24no—
466C23—24no—
467C23#39112yes—
468C23#39112yes—
469C23C2×#9912yes—
470C23#41012yes—
471C23C2×#10412yes—
472C23#42112yes—
473C23C22×C22⋊C411yescharacteristic
474C23C2×#12411yescharacteristic
475C23C2×C4⋊Q811yescharacteristic
476C23C2×AGL(1,ℤ/8)11yescharacteristic
477C23C2×AGL(1,ℤ/8)11yescharacteristic
478C23C2×#13711yescharacteristic
479C23C2×#13711yescharacteristic
480Q8—84no—
481Q8—84no—
482C2×C4—24no—
483C2×C4—28no—
484C2×C4—24no—
485C2×C4—28no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—28no—
505C2×C4—28no—
506C2×C4—24no—
507C2×C4—28no—
508C2×C4—28no—
509C2×C4—24no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—48no—
531C2×C4—48no—
532C2×C4—48no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—832no—
546C2×C4—84no—
547C2×C4—84no—
548C2×C4—84no—
549C2×C4—84no—
550C2×C4#39112yes—
551C2×C4#39112yes—
552C2×C4C2×#9912yes—
553C2×C4C2×#9912yes—
554C2×C4#41712yes—
555C2×C4#41712yes—
556C2×C4#41712yes—
557C2×C4#41712yes—
558C2×C4C2×#10412yes—
559C2×C4C2×#10412yes—
560C2×C4C2×C4×Q812yes—
561C2×C4C2×C22⋊Q812yes—
562C8—48no—
563C22—28no—
564C22—28no—
565C22—28no—
566C22—28no—
567C22—21no—
568C22—21no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—21no—
573C22—21no—
574C22—22no—
575C22—21no—
576C22—21no—
577C22—22no—
578C22—22no—
579C22—21no—
580C22—21no—
581C22—22no—
582C22—22no—
583C22—21no—
584C22—21no—
585C22—22no—
586C22—22no—
587C22—21no—
588C22—21no—
589C22—22no—
590C22—22no—
591C22#198614yes—
592C22#199614yes—
593C22C2×#39111yescharacteristic
594C22C22×#9911yescharacteristic
595C22C2×#41011yescharacteristic
596C22C2×#41711yescharacteristic
597C22C2×#41711yescharacteristic
598C22C22×#10411yescharacteristic
599C22C2×#42111yescharacteristic
600C22#313811yescharacteristic
601C22#313911yescharacteristic
602C22#316511yescharacteristic
603C22#316511yescharacteristic
604C4—22no—
605C4—22no—
606C4—22no—
607C4—22no—
608C4—22no—
609C4—22no—
610C4—22no—
611C4—22no—
612C4—22no—
613C4—41no—
614C4—41no—
615C4—88no—
616C4—82no—
617C4—82no—
618C4—82no—
619C4—82no—
620C4#150414yes—
621C2—21no—
622C2—21no—
623C2—21no—
624C2—21no—
625C2—24no—
626C2—24no—
627C2C2×#150411yescharacteristic
628C2#1754611yescharacteristic
629C2#1758811yescharacteristic
630C2#1759211yescharacteristic
631C2C2×#198611yescharacteristic
632C2#2132211yescharacteristic
633C2C2×#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. #9887C2semidirect
5. #9885C2semidirect
6. #9884C2semidirect
8. C2×#533C4semidirect
13. #2857C4semidirect
14. #2857C4semidirect
19. #2851C4semidirect
27. #2296C4semidirect
28. #2285C4semidirect
49. C2×C4⋊Q8C2×C4semidirect
50. C2×C4⋊Q8C2×C4semidirect
51. C2×#125C2×C4semidirect
81. #407Q8semidirect
95. #389C2×C4semidirect
96. #389C2×C4semidirect
97. #389C2×C4semidirect
98. #389C2×C4semidirect
109. #384C2×C4semidirect
110. #384C2×C4semidirect
111. #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×C4⋊#952
#98872
#98852
#98842
#80202