Finite Group Databasebeta

← #482415

Subgroups — #482415

Order 512.

1140 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2#20722C211yesmaximal, characteristic
3#17362C211yesmaximal, characteristic
4#11900C214yesmaximal, complemented
5#11808C214yesmaximal, complemented
6#11787C214yesmaximal, complemented
7C22×C4⋊Q8C411yescharacteristic, complemented
8C22×C4×D4C2211yescharacteristic
9C2×#407—21no—
10#2563—21no—
11#2563—21no—
12#2563—21no—
13#2563—21no—
14#2563C412yescomplemented
15#2563C412yescomplemented
16#2563C2212yes—
17#2563C2212yes—
18C2×#393C412yescomplemented
19C2×#393C2212yes—
20C2×#390—21no—
21C2×#382—21no—
22#2280—21no—
23C2×#380—21no—
24C2×#380—21no—
25C2×C4×C4⋊C4C411yescharacteristic, complemented
26#1819C2212yes—
27#1799—21no—
28#1799C2212yes—
29#1781—21no—
30C4×#99—21no—
31C4×#99C2212yes—
32#1465—21no—
33#1465C2212yes—
34#1463—21no—
35#1462C2212yes—
36#1459—21no—
37#1459C2212yes—
38#1351C22116yescomplemented
39C23×Q8C2×C412yes—
40C23×D4D411yescharacteristic
41C24×C4D411yescharacteristic
42C24×C4D411yescharacteristic
43C22×M16—41no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—41no—
47C2×C4⋊Q8—41no—
48C2×C4⋊Q8—41no—
49C2×C4⋊Q8—41no—
50C2×C4⋊Q8C2×C412yes—
51C2×C4⋊Q8C2×C412yes—
52C2×C4×D4—41no—
53C2×C4×D4—41no—
54C2×C4×D4—41no—
55C2×C4×D4—41no—
56C2×C4×D4C2×C411yescharacteristic
57C2×C4×D4C2×C411yescharacteristic
58C2×C4×D4D411yescharacteristic
59C2×C4×D4D411yescharacteristic
60C2×C4×D4D411yescharacteristic
61C2×C4×D4D411yescharacteristic
62C2×C4×D4C2311yescharacteristic
63C2×C4×D4C2311yescharacteristic
64C2×#117—41no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4D411yescharacteristic
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4C2×C411yescharacteristic
72C22×C22⋊C4C2311yescharacteristic
73C22×C42—21no—
74C22×C42C2×C411yescharacteristic
75C2×#99—22no—
76C2×#99—22no—
77#407—22no—
78#407—22no—
79#407—41no—
80#407—41no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—41no—
83#393—22no—
84#393—22no—
85#393—22no—
86#393—22no—
87#393C2×C414yes—
88#393C2×C414yes—
89#393C2×C414yes—
90#393C2314yes—
91#390—22no—
92#390—22no—
93#390—22no—
94#390—22no—
95#382—22no—
96#382—22no—
97#382—22no—
98#382—22no—
99#381—41no—
100#380—22no—
101#380—22no—
102#380—22no—
103#380—22no—
104#380—22no—
105#380—22no—
106#380—22no—
107#380—22no—
108#380—22no—
109#380—41no—
110#379—22no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4C2×C412yes—
114C4×C4⋊C4C2×C412yes—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—41no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95C2×C412yes—
120#342—22no—
121#342C2314yes—
122#324—22no—
123#324—22no—
124#323—22no—
125#323C2314yes—
126C25C22⋊C411yescharacteristic
127C25C2×D411yescharacteristic
128C22×Q8—22no—
129C22×Q8—42no—
130C22×Q8—42no—
131C22×Q8—84no—
132C22×Q8C22×C414yes—
133C22×D4—24no—
134C22×D4—42no—
135C22×D4—41no—
136C22×D4—41no—
137C22×D4C22⋊C411yescharacteristic
138C22×D4C22⋊C411yescharacteristic
139C22×D4C2×D411yescharacteristic
140C22×D4C2×D411yescharacteristic
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—44no—
152C23×C4—44no—
153C23×C4C22⋊C411yescharacteristic
154C23×C4C22⋊C411yescharacteristic
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22×C411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C2×M16—42no—
167C2×M16—42no—
168C2×M16—84no—
169C22×C8—42no—
170C4⋊Q8—42no—
171C4⋊Q8—42no—
172C4⋊Q8—42no—
173C4⋊Q8—42no—
174C4⋊Q8—42no—
175C4⋊Q8—42no—
176C4⋊Q8—42no—
177C4⋊Q8—42no—
178C4×D4—41no—
179C4×D4—41no—
180C4×D4—41no—
181C4×D4—41no—
182C4×D4—41no—
183C4×D4—41no—
184C4×D4—41no—
185C4×D4—41no—
186C4×D4—41no—
187C4×D4—41no—
188C4×D4—41no—
189C4×D4—41no—
190C4×D4—41no—
191C4×D4—41no—
192C4×D4—41no—
193C4×D4—41no—
194#117—84no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4C22⋊C411yescharacteristic
212C2×C4⋊C4C22⋊C411yescharacteristic
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—42no—
226C2×C22⋊C4—84no—
227C2×C22⋊C4—84no—
228C2×C22⋊C4C22⋊C411yescharacteristic
229C2×C22⋊C4C22⋊C411yescharacteristic
230C2×C22⋊C4C22×C411yescharacteristic
231C2×C22⋊C4C22×C411yescharacteristic
232C2×C22⋊C4C22×C411yescharacteristic
233C2×C22⋊C4C2×D411yescharacteristic
234C2×C22⋊C4C2×D411yescharacteristic
235C2×C22⋊C4C2411yescharacteristic
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—22no—
241C2×C42—22no—
242C2×C42—44no—
243C2×C42—42no—
244C2×C42C22⋊C411yescharacteristic
245C2×C42C22⋊C411yescharacteristic
246C2×C42C22×C411yescharacteristic
247C2×C42C22×C411yescharacteristic
248#99—44no—
249#99—44no—
250C22⋊C8—24no—
251C22⋊C8—42no—
252C22⋊C8—42no—
253C22⋊C8—42no—
254#95—24no—
255#95—24no—
256#95—24no—
257#95—22no—
258#95—24no—
259#95—42no—
260#95—42no—
261#95C22×C414yes—
262C24—22no—
263C24—22no—
264C24—21no—
265C24—21no—
266C24—21no—
267C24—21no—
268C24—41no—
269C24—41no—
270C24—42no—
271C24—42no—
272C24—42no—
273C24—42no—
274C24C2×C22⋊C411yescharacteristic
275C24C2×C22⋊C411yescharacteristic
276C24C2×C22⋊C411yescharacteristic
277C24C2×C22⋊C411yescharacteristic
278C24C22×D411yescharacteristic
279C2×Q8—44no—
280C2×Q8—44no—
281C2×Q8—84no—
282C2×Q8—84no—
283C2×Q8—88no—
284C2×Q8—88no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—48no—
297C2×D4—48no—
298C2×D4#13714yes—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—24no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—44no—
337C22×C4—44no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—44no—
344C22×C4—44no—
345C22×C4—44no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—44no—
349C22×C4—82no—
350C22×C4—82no—
351C22×C4—82no—
352C22×C4—82no—
353C22×C4—82no—
354C22×C4—82no—
355C22×C4#9911yescharacteristic
356C22×C4#9911yescharacteristic
357C22×C4#9911yescharacteristic
358C22×C4#9911yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C23×C411yescharacteristic
368C22×C4C22×D411yescharacteristic
369M16—84no—
370M16—84no—
371C2×C8—44no—
372C2×C8—84no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—44no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4—44no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—84no—
386C22⋊C4—41no—
387C22⋊C4—41no—
388C22⋊C4—41no—
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—44no—
395C22⋊C4—44no—
396C22⋊C4—84no—
397C22⋊C4—84no—
398C22⋊C4—84no—
399C22⋊C4—84no—
400C22⋊C4—84no—
401C22⋊C4—84no—
402C42—21no—
403C42—21no—
404C42—21no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42—21no—
409C42—21no—
410C42—44no—
411C42—44no—
412C42—84no—
413C23—22no—
414C23—22no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—48no—
440C23—48no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—81no—
446C23—81no—
447C23—81no—
448C23—81no—
449C23—84no—
450C23—84no—
451C23C2×#9911yescharacteristic
452C23C2×#9911yescharacteristic
453C23#41912yes—
454C23C22×C22⋊C411yescharacteristic
455C23C2×#13712yes—
456Q8—88no—
457Q8—88no—
458D4—44no—
459D4—44no—
460D4—44no—
461D4—44no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—44no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—88no—
514C2×C4—84no—
515C2×C4—84no—
516C2×C4—84no—
517C2×C4—84no—
518C2×C4—84no—
519C2×C4#35414yes—
520C2×C4C2×#9911yescharacteristic
521C2×C4C2×#9911yescharacteristic
522C2×C4C2×#9911yescharacteristic
523C2×C4C2×#9911yescharacteristic
524C2×C4#41912yes—
525C2×C4#42112yes—
526C8—84no—
527C22—24no—
528C22—24no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—22no—
536C22—22no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—41no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22—41no—
550C22—41no—
551C22—81no—
552C22—81no—
553C22—84no—
554C22—84no—
555C22—84no—
556C22—84no—
557C22—81no—
558C22—81no—
559C22#212212yes—
560C22C2×#35412yes—
561C22C22×#9911yescharacteristic
562C22C2×#41911yescharacteristic
563C22C2×#42111yescharacteristic
564C4—21no—
565C4—21no—
566C4—22no—
567C4—22no—
568C4—41no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—41no—
573C4—41no—
574C4—84no—
575C4—84no—
576C4—164no—
577C4C2×#35412yes—
578C4C2×#35412yes—
579C2—21no—
580C2—21no—
581C2—22no—
582C2—22no—
583C2—42no—
584C2—42no—
585C2—81no—
586C2—81no—
587C2#1757811yescharacteristic
588C2C2×#212211yescharacteristic
589C2C22×#35411yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11900C2semidirect
5. #11808C2semidirect
6. #11787C2semidirect
7. C22×C4⋊Q8C4semidirect
14. #2563C4semidirect
15. #2563C4semidirect
18. C2×#393C4semidirect
25. C2×C4×C4⋊C4C4semidirect
38. #1351C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
#207222
#173622
#119002
#118082
#117872