Finite Group Databasebeta

← #164334

Subgroups — #164334

Order 512.

1140 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#36362C211yesmaximal, characteristic
2#20559C212yesmaximal, complemented
3#7690C212yesmaximal, complemented
4#7684C212yesmaximal, complemented
5#7676C212yesmaximal, complemented
6#7637C211yesmaximal, characteristic, complemented
7#7631C211yesmaximal, characteristic, complemented
8#7626C211yesmaximal, characteristic, complemented
9#7569C211yesmaximal, characteristic, complemented
10#7563C211yesmaximal, characteristic, complemented
11C2×#1270C211yesmaximal, characteristic, complemented
12C2×#553C411yescharacteristic
13C2×#546C411yescharacteristic
14C2×#435—22no—
15#2861C2211yescharacteristic
16#2856C2211yescharacteristic
17#2850C2211yescharacteristic
18#2566C411yescharacteristic
19#2554—22no—
20#2544C411yescharacteristic
21#2502C411yescharacteristic
22#2500C411yescharacteristic
23#2464—22no—
24#2433C411yescharacteristic
25#2431C411yescharacteristic
26#2424—22no—
27#2358—22no—
28C4×#127C2211yescharacteristic
29#2297C2211yescharacteristic
30C2×#390C2211yescharacteristic
31C42×D4C2211yescharacteristic
32#1771—22no—
33C4×C4⋊C8C2212yescomplemented
34#1753—22no—
35C8×C22⋊C4—22no—
36#1749C2212yes—
37C4×C22⋊C8C2212yescomplemented
38#1745C2212yescomplemented
39#1728C2212yescomplemented
40C2×#336C2212yes—
41#1272C2212yescomplemented
42#1272C2212yescomplemented
43#1270C2212yes—
44#1270C2212yes—
45#1270C2211yescharacteristic
46#1270C2211yescharacteristic
47#1270C2211yescharacteristic
48#1270C2211yescharacteristic
49#1270C2212yescomplemented
50#1270C2212yescomplemented
51C22×#41C2×C411yescharacteristic
52C23×D4C2×C411yescharacteristic
53C22×M16—22no—
54C23×C8—22no—
55#553—44no—
56#546—44no—
57C2×#127C2×C411yescharacteristic
58C2×#127C2×C411yescharacteristic
59C2×#126—21no—
60C2×#125C2×C411yescharacteristic
61C2×#124C2×C411yescharacteristic
62C2×#124C2×C411yescharacteristic
63C2×#123—21no—
64C2×#123—21no—
65C2×C22⋊Q8C2×C411yescharacteristic
66C2×#121—21no—
67C2×#121—22no—
68C2×#121—21no—
69C2×#121C2×C411yescharacteristic
70C2×#121C2×C411yescharacteristic
71C2×#121C2×C411yescharacteristic
72C2×#120—21no—
73C2×#120—22no—
74C2×C4×D4—21no—
75C2×C4×D4—22no—
76C2×C4×D4C2×C411yescharacteristic
77C2×C4×D4C2×C411yescharacteristic
78C2×C4×D4C2×C411yescharacteristic
79C2×C4×D4D412yes—
80C2×#117C2×C411yescharacteristic
81C22×C4⋊C4C2×C411yescharacteristic
82C22×C22⋊C4—22no—
83C22×C22⋊C4—21no—
84C22×C42C2311yescharacteristic
85C22×C42C2311yescharacteristic
86#435—216no—
87C2×C4⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×#97—22no—
91C2×#97—22no—
92C2×C4×C8—22no—
93C2×C4×C8—22no—
94#398—22no—
95#396—22no—
96#394—22no—
97#394—22no—
98#392—22no—
99#392—22no—
100#391C2×C411yescharacteristic
101#391C2×C411yescharacteristic
102#391C2311yescharacteristic
103#390—22no—
104#390—22no—
105#390—22no—
106#390C2×C411yescharacteristic
107#390C2×C411yescharacteristic
108#390C2×C411yescharacteristic
109#390C2×C411yescharacteristic
110#390C2311yescharacteristic
111#390C2311yescharacteristic
112#390C2311yescharacteristic
113#388—22no—
114#388—22no—
115#388—22no—
116#388—22no—
117#386—22no—
118#386—22no—
119#386—22no—
120#386—22no—
121#385—22no—
122#385—22no—
123#384C2×C411yescharacteristic
124#384C2×C411yescharacteristic
125#382C2×C411yescharacteristic
126C4×C4⋊C4C2×C411yescharacteristic
127C4×C4⋊C4D412yes—
128C4×C22⋊C4—22no—
129C4×C22⋊C4C2×C411yescharacteristic
130C4×C22⋊C4C2×C411yescharacteristic
131C4×C22⋊C4D412yes—
132C4×C22⋊C4D412yes—
133C43C2311yescharacteristic
134#336—22no—
135#336—22no—
136#336—22no—
137#336C2312yes—
138#336C2312yes—
139#336C2312yes—
140#336C2312yes—
141C25—21no—
142C2×#41—22no—
143C2×#41—41no—
144C2×#41—42no—
145C2×#41—81no—
146C22×Q8C22×C411yescharacteristic
147C22×D4—22no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—22no—
153C22×D4—21no—
154C22×D4—22no—
155C22×D4—41no—
156C22×D4—42no—
157C22×D4—81no—
158C22×D4C22⋊C411yescharacteristic
159C22×D4C22×C411yescharacteristic
160C22×D4C22×C411yescharacteristic
161C22×D4C22×C411yescharacteristic
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4—21no—
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C411yescharacteristic
168C23×C4C2×D412yes—
169C23×C4C2×D412yes—
170C2×M16—28no—
171C2×M16—82no—
172C2×M16—82no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—28no—
178C22×C8—82no—
179C22×C8—82no—
180#127—22no—
181#127—41no—
182#127—81no—
183#126—44no—
184#125—42no—
185#124—42no—
186#124—81no—
187#123—44no—
188#123—44no—
189C22⋊Q8—42no—
190#121—44no—
191#121—42no—
192#121—44no—
193#121—42no—
194#121—42no—
195#121—82no—
196#121—82no—
197#120—44no—
198#120—48no—
199C4×D4—24no—
200C4×D4—28no—
201C4×D4—42no—
202C4×D4—42no—
203C4×D4—44no—
204C4×D4—82no—
205C4×D4—82no—
206#117—42no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4C22⋊C411yescharacteristic
215C2×C4⋊C4C22⋊C412yes—
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—44no—
238C2×C22⋊C4—82no—
239C2×C22⋊C4—82no—
240C2×C22⋊C4C22⋊C411yescharacteristic
241C2×C22⋊C4C22⋊C412yes—
242C2×C22⋊C4C22⋊C411yescharacteristic
243C2×C22⋊C4C22⋊C412yes—
244C2×C42—21no—
245C2×C42—81no—
246C2×C42—81no—
247C2×C42C22×C411yescharacteristic
248C2×C42C22×C411yescharacteristic
249C2×C42C22×C411yescharacteristic
250C2×C42C22×C411yescharacteristic
251C2×C42C2×D412yes—
252C2×C42C2×D412yes—
253C2×C42C2×D412yes—
254C2×C42C2×D412yes—
255C2×C42C2411yescharacteristic
256C4⋊C8—28no—
257C22⋊C8—28no—
258C22⋊C8—28no—
259#97—28no—
260#97—82no—
261C4×C8—82no—
262C4×C8—82no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95C22⋊C412yes—
268#95C22⋊C412yes—
269#95C22⋊C412yes—
270C24—21no—
271C24—21no—
272C24—21no—
273C24—42no—
274C24—42no—
275C24—44no—
276C24—82no—
277C24—81no—
278C24C2×C22⋊C411yescharacteristic
279C24C2×C22⋊C411yescharacteristic
280#41—84no—
281#41—84no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—41no—
285C2×Q8—41no—
286C2×D4—22no—
287C2×D4—24no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—44no—
296C2×D4—42no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—44no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—44no—
309C2×D4—44no—
310C2×D4—82no—
311C2×D4—81no—
312C2×D4—82no—
313C2×D4—82no—
314C2×D4—84no—
315C2×D4—84no—
316C2×D4—81no—
317C2×D4—82no—
318C2×D4—82no—
319C2×D4#13112yes—
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—22no—
331C22×C4—24no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—81no—
347C22×C4—82no—
348C22×C4—82no—
349C22×C4—82no—
350C22×C4C2×C22⋊C412yes—
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C412yes—
356C22×C4C2×C22⋊C412yes—
357C22×C4C23×C411yescharacteristic
358C22×C4C22×D412yes—
359M16—88no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—88no—
373C4⋊C4—22no—
374C4⋊C4—24no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—82no—
384C4⋊C4—82no—
385C22⋊C4—22no—
386C22⋊C4—24no—
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—42no—
397C22⋊C4—42no—
398C22⋊C4—42no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—44no—
403C22⋊C4—44no—
404C22⋊C4—82no—
405C22⋊C4—82no—
406C22⋊C4—82no—
407C22⋊C4#10014yes—
408C22⋊C4#10418yes—
409C42—21no—
410C42—21no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—24no—
415C42—42no—
416C42—42no—
417C42—41no—
418C42—41no—
419C42—41no—
420C42—42no—
421C23—21no—
422C23—21no—
423C23—22no—
424C23—21no—
425C23—22no—
426C23—21no—
427C23—21no—
428C23—22no—
429C23—22no—
430C23—41no—
431C23—41no—
432C23—42no—
433C23—42no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—84no—
445C23—88no—
446C23—84no—
447C23—162no—
448C23—164no—
449C23C2×#10012yes—
450C23C2×#10414yes—
451C23C22×C22⋊C411yescharacteristic
452Q8—81no—
453Q8—81no—
454D4—42no—
455D4—42no—
456D4—44no—
457D4—88no—
458D4—81no—
459D4—81no—
460D4—81no—
461D4—81no—
462D4—84no—
463D4—84no—
464D4—84no—
465D4—164no—
466D4—162no—
467D4—162no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—84no—
507C2×C4—162no—
508C2×C4—162no—
509C2×C4—162no—
510C2×C4C2×#10012yes—
511C2×C4C2×#10012yes—
512C2×C4C2×#10414yes—
513C2×C4C2×#10414yes—
514C2×C4C2×#13111yescharacteristic
515C2×C4#56811yescharacteristic
516C2×C4#57612yes—
517C2×C4#57712yes—
518C8—44no—
519C8—44no—
520C8—44no—
521C8—44no—
522C22—22no—
523C22—22no—
524C22—24no—
525C22—24no—
526C22—41no—
527C22—42no—
528C22—41no—
529C22—44no—
530C22—42no—
531C22—42no—
532C22—41no—
533C22—81no—
534C22—82no—
535C22—81no—
536C22—81no—
537C22—81no—
538C22—81no—
539C22—81no—
540C22—82no—
541C22—82no—
542C22—82no—
543C22—82no—
544C22—81no—
545C22—81no—
546C22—168no—
547C22#152514yes—
548C22#288011yescharacteristic
549C22#288311yescharacteristic
550C22C22×#10011yescharacteristic
551C22#289712yes—
552C22C22×#10412yes—
553C4—21no—
554C4—21no—
555C4—22no—
556C4—22no—
557C4—24no—
558C4—22no—
559C4—22no—
560C4—24no—
561C4—41no—
562C4—42no—
563C4—41no—
564C4—81no—
565C4—81no—
566C4—81no—
567C4—81no—
568C4—81no—
569C2—24no—
570C2—81no—
571C2—82no—
572C2—161no—
573C2—161no—
574C2#1742111yescharacteristic
575C2#1751712yes—
576C2#1754712yes—
577C2C2×#152511yescharacteristic
578C2#1777011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36362C2
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. #20559C2semidirect
3. #7690C2semidirect
4. #7684C2semidirect
5. #7676C2semidirect
6. #7637C2semidirect
7. #7631C2semidirect
8. #7626C2semidirect
9. #7569C2semidirect
10. #7563C2semidirect
11. C2×#1270C2semidirect
33. C4×C4⋊C8C22semidirect
37. C4×C22⋊C8C22semidirect
38. #1745C22semidirect
39. #1728C22semidirect
41. #1272C22semidirect
42. #1272C22semidirect
49. #1270C22semidirect
50. #1270C22semidirect

Maximal subgroups

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

subgroupindex
#363622
#205592
#76902
#76842
#76762
#76372
#76312
#76262
#75692
#75632
C2×#12702