Finite Group Databasebeta

← #164800

Subgroups — #164800

Order 512.

1079 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36289C211yesmaximal, characteristic
2#22000C212yesmaximal, complemented
3#7730C212yesmaximal, complemented
4#7719C212yesmaximal, complemented
5#7690C212yesmaximal, complemented
6#7649C211yesmaximal, characteristic, complemented
7#7640C211yesmaximal, characteristic, complemented
8#7626C211yesmaximal, characteristic, complemented
9#7601C211yesmaximal, characteristic, complemented
10#7592C211yesmaximal, characteristic, complemented
11C2×#1270C211yesmaximal, characteristic, complemented
12C2×#554C411yescharacteristic
13C2×D42C411yescharacteristic
14#2858C2211yescharacteristic
15#2850C2211yescharacteristic
16#2850C2211yescharacteristic
17#2563C411yescharacteristic
18#2548C411yescharacteristic
19#2504C411yescharacteristic
20#2504C411yescharacteristic
21#2490—22no—
22#2472—22no—
23#2472—22no—
24#2435C411yescharacteristic
25#2435C411yescharacteristic
26D4×C22⋊C4—22no—
27C4×#124C2211yescharacteristic
28C2×#390C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C42×D4C2211yescharacteristic
31#1924—22no—
32#1917—22no—
33#1869—22no—
34#1850—22no—
35#1839C2212yescomplemented
36#1831C2212yescomplemented
37#1817C2212yescomplemented
38#1797C2212yescomplemented
39#1749C2212yescomplemented
40C2×#336C2212yescomplemented
41#1272C2212yescomplemented
42#1272C2212yescomplemented
43#1270C2211yescharacteristic
44#1270C2211yescharacteristic
45#1270C2211yescharacteristic
46#1270C2211yescharacteristic
47#1270C2212yescomplemented
48#1270C2212yescomplemented
49#1270C2212yescomplemented
50#1270C2212yescomplemented
51C23×D4—21no—
52C23×D4C2×C411yescharacteristic
53C23×D4C2×C411yescharacteristic
54C23×C8—22no—
55#554—44no—
56D42—44no—
57C2×C4⋊Q8C2×C411yescharacteristic
58C2×#127C2×C411yescharacteristic
59C2×#125—21no—
60C2×#124—22no—
61C2×#124C2×C411yescharacteristic
62C2×#124C2×C411yescharacteristic
63C2×#124C2×C411yescharacteristic
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×C22⋊Q8C2×C411yescharacteristic
67C2×C22⋊Q8C2×C411yescharacteristic
68C2×#121—21no—
69C2×#121C2×C411yescharacteristic
70C2×#121C2×C411yescharacteristic
71C2×#120—21no—
72C2×#120—21no—
73C2×C4×Q8C2×C411yescharacteristic
74C2×C4×D4—21no—
75C2×C4×D4C2×C411yescharacteristic
76C2×C4×D4C2×C411yescharacteristic
77C2×C4×D4D412yes—
78C22×C4⋊C4C2×C411yescharacteristic
79C22×C4⋊C4C2×C411yescharacteristic
80C22×C22⋊C4—22no—
81C22×C22⋊C4—22no—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×#97—22no—
90#397—22no—
91#397—22no—
92#394—22no—
93#394—22no—
94#394—22no—
95#393—22no—
96#391C2311yescharacteristic
97#390—22no—
98#390—22no—
99#390—22no—
100#390—22no—
101#390C2×C411yescharacteristic
102#390C2×C411yescharacteristic
103#390C2×C411yescharacteristic
104#390C2×C411yescharacteristic
105#390C2×C411yescharacteristic
106#390C2×C411yescharacteristic
107#390C2311yescharacteristic
108#390C2311yescharacteristic
109#390C2311yescharacteristic
110#388—22no—
111#388—22no—
112#388—22no—
113#386—22no—
114#386—22no—
115#386—22no—
116#386—22no—
117#382C2×C411yescharacteristic
118#382C2×C411yescharacteristic
119#382C2×C411yescharacteristic
120#380—22no—
121#379—22no—
122#379—22no—
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4D412yes—
125C4×C22⋊C4—22no—
126C4×C22⋊C4C2×C411yescharacteristic
127C4×C22⋊C4C2×C411yescharacteristic
128C4×C22⋊C4D412yes—
129C4×C22⋊C4D412yes—
130C43C2311yescharacteristic
131#336—22no—
132#336—22no—
133#336—22no—
134#336—22no—
135#336—22no—
136#336C2312yes—
137#336C2312yes—
138#336C2312yes—
139#336C2312yes—
140C25—21no—
141C25—21no—
142C22×Q8—21no—
143C22×Q8C22×C411yescharacteristic
144C22×D4—22no—
145C22×D4—22no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—41no—
153C22×D4—41no—
154C22×D4—42no—
155C22×D4—42no—
156C22×D4—48no—
157C22×D4—44no—
158C22×D4—81no—
159C22×D4—81no—
160C22×D4C22⋊C411yescharacteristic
161C22×D4C22×C411yescharacteristic
162C22×D4C22×C411yescharacteristic
163C22×D4C22×C411yescharacteristic
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4C22×C411yescharacteristic
168C23×C4C22×C411yescharacteristic
169C23×C4C2×D412yes—
170C23×C4C2×D412yes—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—44no—
176C22×C8—82no—
177C22×C8—82no—
178C4⋊Q8—42no—
179#127—42no—
180#125—44no—
181#124—22no—
182#124—41no—
183#124—82no—
184#124—82no—
185#124—81no—
186#124—81no—
187C22⋊Q8—44no—
188C22⋊Q8—44no—
189C22⋊Q8—42no—
190C22⋊Q8—42no—
191#121—44no—
192#121—42no—
193#121—42no—
194#120—44no—
195#120—44no—
196C4×Q8—42no—
197C4×D4—24no—
198C4×D4—42no—
199C4×D4—44no—
200C4×D4—44no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4C22⋊C411yescharacteristic
210C2×C4⋊C4C22⋊C412yes—
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—22no—
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×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—82no—
233C2×C22⋊C4—82no—
234C2×C22⋊C4—82no—
235C2×C22⋊C4—82no—
236C2×C22⋊C4C22⋊C411yescharacteristic
237C2×C22⋊C4C22⋊C411yescharacteristic
238C2×C22⋊C4C22⋊C412yes—
239C2×C22⋊C4C22⋊C412yes—
240C2×C42—21no—
241C2×C42—81no—
242C2×C42—81no—
243C2×C42C22×C411yescharacteristic
244C2×C42C22×C411yescharacteristic
245C2×C42C22×C411yescharacteristic
246C2×C42C22×C411yescharacteristic
247C2×C42C2×D412yes—
248C2×C42C2×D412yes—
249C2×C42C2×D412yes—
250C2×C42C2×D412yes—
251C2×C42C2411yescharacteristic
252C4⋊C8—82no—
253C4⋊C8—82no—
254C22⋊C8—44no—
255C22⋊C8—82no—
256C22⋊C8—82no—
257#97—82no—
258#95—22no—
259#95—22no—
260#95—22no—
261#95—22no—
262#95C22⋊C412yes—
263#95C22⋊C412yes—
264#95C22⋊C412yes—
265C24—21no—
266C24—21no—
267C24—21no—
268C24—42no—
269C24—42no—
270C24—44no—
271C24—42no—
272C24—42no—
273C24—44no—
274C24—82no—
275C24—82no—
276C24—81no—
277C24—81no—
278C24C2×C22⋊C411yescharacteristic
279C24C2×C22⋊C411yescharacteristic
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—42no—
283C2×Q8—41no—
284C2×Q8—41no—
285C2×Q8—82no—
286C2×D4—24no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—44no—
303C2×D4—42no—
304C2×D4—44no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—44no—
308C2×D4—42no—
309C2×D4—81no—
310C2×D4—81no—
311C2×D4—82no—
312C2×D4—84no—
313C2×D4—84no—
314C2×D4—84no—
315C2×D4—84no—
316C2×D4—816no—
317C2×D4#10014yes—
318C2×D4#13112yes—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—24no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—44no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—82no—
345C22×C4—82no—
346C22×C4C2×C22⋊C412yes—
347C22×C4C2×C22⋊C412yes—
348C22×C4C2×C22⋊C412yes—
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4C23×C411yescharacteristic
354C22×C4C22×D412yes—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C2×C8—42no—
360C2×C8—42no—
361C2×C8—42no—
362C2×C8—42no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—88no—
368C4⋊C4—22no—
369C4⋊C4—42no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C22⋊C4—22no—
381C22⋊C4—22no—
382C22⋊C4—24no—
383C22⋊C4—42no—
384C22⋊C4—44no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C22⋊C4—42no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—42no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C22⋊C4—42no—
396C22⋊C4—82no—
397C22⋊C4—82no—
398C22⋊C4—82no—
399C22⋊C4—82no—
400C22⋊C4—82no—
401C22⋊C4—82no—
402C22⋊C4—82no—
403C22⋊C4—82no—
404C42—21no—
405C42—21no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42—42no—
410C42—42no—
411C42—41no—
412C42—41no—
413C42—41no—
414C42—42no—
415C42—42no—
416C23—21no—
417C23—21no—
418C23—22no—
419C23—21no—
420C23—21no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—41no—
425C23—41no—
426C23—42no—
427C23—42no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—84no—
439C23—84no—
440C23—88no—
441C23—84no—
442C23—88no—
443C23—84no—
444C23—162no—
445C23—164no—
446C23—162no—
447C23—164no—
448C23C2×#10012yes—
449C23C2×#10012yes—
450C23#42114yes—
451C23C22×C22⋊C411yescharacteristic
452Q8—81no—
453Q8—81no—
454Q8—162no—
455D4—28no—
456D4—42no—
457D4—42no—
458D4—81no—
459D4—81no—
460D4—81no—
461D4—81no—
462D4—84no—
463D4—84no—
464D4—84no—
465D4—84no—
466D4—84no—
467D4—162no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—82no—
508C2×C4—82no—
509C2×C4—84no—
510C2×C4—162no—
511C2×C4—162no—
512C2×C4C2×#10012yes—
513C2×C4C2×#13111yescharacteristic
514C2×C4#56811yescharacteristic
515C2×C4#57612yes—
516C2×C4#57712yes—
517C8—82no—
518C8—82no—
519C8—82no—
520C8—82no—
521C22—22no—
522C22—22no—
523C22—24no—
524C22—24no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—44no—
532C22—81no—
533C22—82no—
534C22—81no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—82no—
540C22—82no—
541C22—82no—
542C22—82no—
543C22—81no—
544C22—81no—
545C22—168no—
546C22—168no—
547C22#153314yes—
548C22#288011yescharacteristic
549C22#288311yescharacteristic
550C22C22×#10011yescharacteristic
551C22#289712yes—
552C22C2×#42112yes—
553C4—21no—
554C4—21no—
555C4—22no—
556C4—22no—
557C4—22no—
558C4—41no—
559C4—41no—
560C4—41no—
561C4—42no—
562C4—42no—
563C4—42no—
564C4—81no—
565C4—81no—
566C4—81no—
567C4—81no—
568C4—81no—
569C2—24no—
570C2—42no—
571C2—82no—
572C2—161no—
573C2—161no—
574C2#1742111yescharacteristic
575C2#1752112yes—
576C2#1759712yes—
577C2C2×#153311yescharacteristic
578C2#1777011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×D42C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22000C2semidirect
3. #7730C2semidirect
4. #7719C2semidirect
5. #7690C2semidirect
6. #7649C2semidirect
7. #7640C2semidirect
8. #7626C2semidirect
9. #7601C2semidirect
10. #7592C2semidirect
11. C2×#1270C2semidirect
35. #1839C22semidirect
36. #1831C22semidirect
37. #1817C22semidirect
38. #1797C22semidirect
39. #1749C22semidirect
40. C2×#336C22semidirect
41. #1272C22semidirect
42. #1272C22semidirect
47. #1270C22semidirect
48. #1270C22semidirect
49. #1270C22semidirect
50. #1270C22semidirect

Maximal subgroups

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

subgroupindex
#362892
#220002
#77302
#77192
#76902
#76492
#76402
#76262
#76012
#75922
C2×#12702