Finite Group Databasebeta

← #207918

Subgroups — #207918

Order 512.

1294 conjugacy classes of subgroups.

The subgroup lattice has 583 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C211yesmaximal, characteristic
2C4×#407C211yesmaximal, characteristic
3#14682C211yesmaximal, characteristic
4#9885C212yesmaximal, complemented
5#9679C212yesmaximal, complemented
6#9115C214yesmaximal
7#8217C214yesmaximal, complemented
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#2855C412yescomplemented
18#2851—21no—
19#2851—21no—
20#2848—21no—
21#2848C2212yes—
22C4×C22⋊Q8—21no—
23C4×C22⋊Q8—21no—
24C4×C22⋊Q8C412yescomplemented
25C4×C22⋊Q8C2212yes—
26#2296—21no—
27#2285C411yescharacteristic, complemented
28C2×#382—21no—
29#2280—21no—
30C2×#380—21no—
31#1749—21no—
32#1749C2212yes—
33C4×C22⋊C8—21no—
34C4×C22⋊C8C2212yes—
35#1728—21no—
36C2×#328C2218yes—
37C4⋊#95C2212yes—
38#1434—21no—
39#1434C2212yes—
40C22⋊#95—21no—
41C22⋊#95C2212yes—
42#1294C2218yes—
43C24×C4D411yescharacteristic
44C24×C4D411yescharacteristic
45C24×C4C2311yescharacteristic
46C22×M16—22no—
47#533—44no—
48C2×C4⋊Q8C2×C411yescharacteristic
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×#125—21no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8C2×C412yes—
55C2×C22⋊Q8C2×C412yes—
56C2×C22⋊Q8C2312yes—
57C2×C4×Q8—21no—
58C2×C4×Q8C2×C412yes—
59C2×#117—21no—
60C2×#117—21no—
61C22×C4⋊C4—22no—
62C22×C22⋊C4—22no—
63C22×C42—44no—
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C411yescharacteristic
67C22×C42C2×C411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42C2311yescharacteristic
75C22×C42C2311yescharacteristic
76#407—28no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8C2314yes—
79C4×M16—44no—
80C2×#97—22no—
81C2×C4×C8—22no—
82#391—21no—
83#391—21no—
84#391C2×C412yes—
85#391C2×C412yes—
86#389—21no—
87#389—21no—
88#389—21no—
89#389—21no—
90#389—21no—
91#389—21no—
92#389C2×C412yes—
93#389C2×C412yes—
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
105#384C2×C411yescharacteristic
106#382—21no—
107#382—44no—
108#381—22no—
109#380—21no—
110#380—21no—
111#380—21no—
112#380—21no—
113#380—21no—
114#380—21no—
115#380—21no—
116#380—44no—
117#380C2×C412yes—
118#380C2×C412yes—
119#380C2312yes—
120#379—22no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4C2×C412yes—
125C4×C22⋊C4—22no—
126C4×C22⋊C4—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95—22no—
131C2×#95C2314yes—
132#336—22no—
133#336—22no—
134#328—216no—
135C25C2×D411yescharacteristic
136C22×Q8—21no—
137C22×Q8C22×C412yes—
138C23×C4—22no—
139C23×C4—24no—
140C23×C4—22no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4C22⋊C411yescharacteristic
150C23×C4C22⋊C411yescharacteristic
151C23×C4C22⋊C411yescharacteristic
152C23×C4C22⋊C411yescharacteristic
153C23×C4C22⋊C411yescharacteristic
154C23×C4C22⋊C411yescharacteristic
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22×C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2411yescharacteristic
167C2×M16—28no—
168C2×M16—44no—
169C2×M16—44no—
170C22×C8—24no—
171C4⋊Q8—42no—
172C4⋊Q8—42no—
173#125—44no—
174C22⋊Q8—28no—
175C22⋊Q8—44no—
176C22⋊Q8—44no—
177C22⋊Q8—42no—
178C22⋊Q8—42no—
179C22⋊Q8—44no—
180C22⋊Q8—44no—
181C4×Q8—44no—
182C4×Q8—44no—
183#117—44no—
184#117—44no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4C22×C412yes—
202C2×C4⋊C4C22×C412yes—
203C2×C4⋊C4C22×C412yes—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—44no—
211C2×C22⋊C4—44no—
212C2×C22⋊C4—44no—
213C2×C42—24no—
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×C42C22⋊C411yescharacteristic
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×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C22⋊C8—28no—
244C22⋊C8—24no—
245C22⋊C8#4118yes—
246#97—44no—
247C4×C8—44no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—28no—
259#95—44no—
260#95—44no—
261#95—44no—
262#95—44no—
263C24—21no—
264C24—21no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—41no—
270C24—41no—
271C24C2×C22⋊C411yescharacteristic
272C24C2×C22⋊C411yescharacteristic
273C24C22×D411yescharacteristic
274C2×Q8—24no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—21no—
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—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—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—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—416no—
337C22×C4—416no—
338C22×C4—48no—
339C22×C4—48no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4AGL(1,ℤ/8)12yes—
358C22×C4#13712yes—
359C22×C4#13714yes—
360C22×C4C23×C411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C2×#4114yes—
363M16—416no—
364C2×C8—28no—
365C2×C8—44no—
366C2×C8—44no—
367C4⋊C4—24no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—44no—
375C4⋊C4—44no—
376C4⋊C4—44no—
377C4⋊C4—44no—
378C4⋊C4—42no—
379C4⋊C4—44no—
380C4⋊C4—42no—
381C4⋊C4—44no—
382C4⋊C4—42no—
383C4⋊C4—88no—
384C22⋊C4—44no—
385C22⋊C4—44no—
386C22⋊C4—44no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C22⋊C4—44no—
390C22⋊C4—44no—
391C22⋊C4—88no—
392C42—22no—
393C42—22no—
394C42—22no—
395C42—22no—
396C42—22no—
397C42—22no—
398C42—22no—
399C42—22no—
400C42—22no—
401C42—41no—
402C42—41no—
403C42—41no—
404C42—41no—
405C42—41no—
406C42—41no—
407C42—44no—
408C42—44no—
409C42#9914yes—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—24no—
415C23—24no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—44no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23#41012yes—
437C23#42112yes—
438C23C22×C22⋊C411yescharacteristic
439C23#53212yes—
440C23C2×AGL(1,ℤ/8)11yescharacteristic
441C23C2×#13712yes—
442C23C2×#13711yescharacteristic
443Q8—84no—
444Q8—84no—
445C2×C4—28no—
446C2×C4—28no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
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—24no—
462C2×C4—24no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—84no—
504C2×C4—84no—
505C2×C4—84no—
506C2×C4—84no—
507C2×C4—84no—
508C2×C4—84no—
509C2×C4—84no—
510C2×C4C2×#9912yes—
511C2×C4C2×#9912yes—
512C2×C4C2×#9912yes—
513C2×C4#41712yes—
514C2×C4#41712yes—
515C2×C4#41912yes—
516C2×C4#41912yes—
517C2×C4#42112yes—
518C2×C4C4×#4112yes—
519C2×C4#52512yes—
520C8—48no—
521C22—22no—
522C22—21no—
523C22—21no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—21no—
528C22—21no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—48no—
536C22—48no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22#229711yescharacteristic
546C22C22×#9911yescharacteristic
547C22C2×#41011yescharacteristic
548C22C2×#41911yescharacteristic
549C22C2×#41711yescharacteristic
550C22C2×#42111yescharacteristic
551C22C2×#42111yescharacteristic
552C22#318212yes—
553C22#318512yes—
554C4—22no—
555C4—22no—
556C4—22no—
557C4—22no—
558C4—22no—
559C4—22no—
560C4—22no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—82no—
566C4—82no—
567C4—82no—
568C4—82no—
569C4—164no—
570C4#151314yes—
571C2—22no—
572C2—22no—
573C2—42no—
574C2—42no—
575C2C2×#151311yescharacteristic
576C2#1756211yescharacteristic
577C2#1757811yescharacteristic
578C2#1758811yescharacteristic
579C2#2138311yescharacteristic
580C2#2140911yescharacteristic
581C2#2142011yescharacteristic
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. #9885C2semidirect
5. #9679C2semidirect
7. #8217C2semidirect
8. C2×#533C4semidirect
12. #2857C4semidirect
13. #2857C4semidirect
17. #2855C4semidirect
24. C4×C22⋊Q8C4semidirect
27. #2285C4semidirect

Maximal subgroups

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

subgroupindex
#366252
C4×#4072
#146822
#98852
#96792
#91152
#82172