Finite Group Databasebeta

← #207684

Subgroups — #207684

Order 512.

1402 conjugacy classes of subgroups.

The subgroup lattice has 573 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Q8C211yesmaximal, characteristic
2#20714C211yesmaximal, characteristic
3C4×#407C211yesmaximal, characteristic
4#9882C212yesmaximal, complemented
5#9654C212yesmaximal, complemented
6#9085C214yesmaximal
7C4×#324C214yesmaximal, complemented
8C22×C22⋊Q8C411yescharacteristic
9C22×C4×Q8C411yescharacteristic
10C23×C42C2211yescharacteristic
11C2×#431—21no—
12C2×#407C2212yes—
13C2×C4×M16—21no—
14C4×C22⋊Q8—21no—
15C4×C22⋊Q8—21no—
16C4×C22⋊Q8—21no—
17C4×C22⋊Q8—21no—
18C4×C22⋊Q8—21no—
19C4×C22⋊Q8—21no—
20C4×C22⋊Q8—21no—
21C4×C22⋊Q8—21no—
22C2×#391C2211yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#387C412yes—
25C2×#385C2212yes—
26C2×#380C411yescharacteristic
27C2×C4×C4⋊C4C411yescharacteristic
28C2×C4×C4⋊C4C2212yes—
29C2×C4×C22⋊C4C412yes—
30#1817—21no—
31#1797—21no—
32#1797C2212yes—
33#1779C2212yes—
34#1749—21no—
35#1749C2212yes—
36C4×C22⋊C8—21no—
37C4×C22⋊C8—21no—
38C4×C22⋊C8C2212yes—
39C4×C22⋊C8C2212yes—
40#1728—21no—
41C2×#324C2218yes—
42#1280C2218yes—
43C23×Q8C2×C411yescharacteristic
44C24×C4C2×C411yescharacteristic
45C24×C4C2×C411yescharacteristic
46C24×C4C2311yescharacteristic
47C22×M16—22no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—28no—
53C2×C4×Q8—21no—
54C2×C4×Q8—21no—
55C2×C4×Q8—21no—
56C2×C4×Q8—21no—
57C2×C4×Q8—44no—
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C412yes—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C412yes—
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C22⋊C4C2×C412yes—
65C22×C22⋊C4C2×C412yes—
66C22×C42—44no—
67C22×C42C2×C412yes—
68C22×C42C2×C411yescharacteristic
69C22×C42C2×C412yes—
70C22×C42C2×C411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42D411yescharacteristic
78C22×C42D411yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81#431—44no—
82C2×C4⋊C8—22no—
83#407—28no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8C2314yes—
88C2×C22⋊C8C2314yes—
89C4×M16—44no—
90C2×#97—22no—
91C2×C4×C8—22no—
92C2×C4×C8—22no—
93#391—21no—
94#391—21no—
95#391—21no—
96#391—21no—
97#389—21no—
98#389—21no—
99#389—21no—
100#389—21no—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#385—22no—
106#385—22no—
107#385—22no—
108#385—22no—
109#380—21no—
110#380—21no—
111#380—21no—
112#380—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C22⋊C4—22no—
122C4×C22⋊C4—22no—
123C4×C22⋊C4—22no—
124C4×C22⋊C4—22no—
125C2×#95C2×C412yes—
126C2×#95C2×C412yes—
127C2×#95C2×C412yes—
128C2×#95C2×C412yes—
129C2×#95C2312yes—
130#336—22no—
131#336—22no—
132#336—22no—
133#336—22no—
134#324—216no—
135C25C22×C411yescharacteristic
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—24no—
139C22×Q8—42no—
140C22×Q8—42no—
141C23×C4—24no—
142C23×C4—24no—
143C23×C4—24no—
144C23×C4—24no—
145C23×C4—42no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—42no—
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×C4C22×C412yes—
160C23×C4C22×C411yescharacteristic
161C23×C4C22×C411yescharacteristic
162C23×C4C22×C411yescharacteristic
163C23×C4C22×C412yes—
164C23×C4C22×C412yes—
165C23×C4C22×C412yes—
166C23×C4C22×C411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2411yescharacteristic
172C2×M16—28no—
173C2×M16—44no—
174C2×M16—44no—
175C22×C8—24no—
176C22×C8—24no—
177C22⋊Q8—28no—
178C22⋊Q8—28no—
179C22⋊Q8—28no—
180C22⋊Q8—28no—
181C4×Q8—44no—
182C4×Q8—44no—
183C4×Q8—44no—
184C4×Q8—44no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4M1618yescomplemented
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—44no—
211C2×C42—24no—
212C2×C42—24no—
213C2×C42—24no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—28no—
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×C42—44no—
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×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2×D411yescharacteristic
242C2×C42C2×D411yescharacteristic
243C2×C42C2×D411yescharacteristic
244C2×C42C2×D411yescharacteristic
245C4⋊C8—44no—
246C22⋊C8—28no—
247C22⋊C8—24no—
248C22⋊C8—28no—
249C22⋊C8—44no—
250C22⋊C8—44no—
251C22⋊C8#4118yes—
252#97—44no—
253C4×C8—44no—
254C4×C8—44no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—22no—
264#95—22no—
265C24—22no—
266C24—22no—
267C24—22no—
268C24—22no—
269C24—22no—
270C24—41no—
271C24—41no—
272C24C2×C22⋊C411yescharacteristic
273C24C2×C22⋊C411yescharacteristic
274C24C23×C411yescharacteristic
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8—42no—
278C2×Q8—416no—
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—24no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
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—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—24no—
325C22×C4—24no—
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—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—416no—
343C22×C4—416no—
344C22×C4—416no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4#10114yes—
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×C4C2×M1614yes—
363C22×C4C2×M1614yes—
364C22×C4C2×M1614yes—
365C22×C4#13114yes—
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C2×#4114yes—
369M16—416no—
370C2×C8—28no—
371C2×C8—44no—
372C2×C8—44no—
373C2×C8—44no—
374C2×C8—44no—
375C2×C8—44no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—28no—
379C4⋊C4—28no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C22⋊C4—28no—
389C22⋊C4—28no—
390C22⋊C4—44no—
391C22⋊C4—44no—
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—44no—
410C42—44no—
411C42#9914yes—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—28no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—44no—
435C23#41012yes—
436C23C2×#10112yes—
437C23C22×C22⋊C411yescharacteristic
438C23C4×#4112yes—
439C23C22×M1612yes—
440C23C2×#13112yes—
441Q8—48no—
442Q8—48no—
443C2×C4—28no—
444C2×C4—28no—
445C2×C4—21no—
446C2×C4—21no—
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—21no—
462C2×C4—21no—
463C2×C4—28no—
464C2×C4—28no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—28no—
473C2×C4—21no—
474C2×C4—44no—
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—44no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—84no—
502C2×C4—84no—
503C2×C4#40712yes—
504C2×C4#40712yes—
505C2×C4#40812yes—
506C2×C4#40812yes—
507C2×C4C2×#9912yes—
508C2×C4C2×#9912yes—
509C2×C4C2×#9912yes—
510C2×C4C2×#10112yes—
511C2×C4C2×#10112yes—
512C2×C4#41312yes—
513C2×C4#52512yes—
514C2×C4#53212yes—
515C8—48no—
516C8—84no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—416no—
530C22—41no—
531C22—41no—
532C22—41no—
533C22—41no—
534C22—41no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22#229711yescharacteristic
539C22C2×#40711yescharacteristic
540C22C2×#40811yescharacteristic
541C22C22×#9911yescharacteristic
542C22C2×#41011yescharacteristic
543C22C22×#10111yescharacteristic
544C22C2×#41311yescharacteristic
545C22#296514yes—
546C4—22no—
547C4—22no—
548C4—22no—
549C4—22no—
550C4—22no—
551C4—22no—
552C4—22no—
553C4—22no—
554C4—22no—
555C4—44no—
556C4—41no—
557C4—44no—
558C4—41no—
559C4—82no—
560C4—82no—
561C4#147214yes—
562C2—24no—
563C2—42no—
564C2—42no—
565C2#1736211yescharacteristic
566C2#1736511yescharacteristic
567C2#1737611yescharacteristic
568C2C2×#147211yescharacteristic
569C2#2137411yescharacteristic
570C2#2142011yescharacteristic
571C2#2142511yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC25C22×C4
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9882C2semidirect
5. #9654C2semidirect
7. C4×#324C2semidirect
203. C2×C4⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Q82
#207142
C4×#4072
#98822
#96542
#90852
C4×#3242