Finite Group Databasebeta

← #164807

Subgroups — #164807

Order 512.

1026 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36362C211yesmaximal, characteristic
2#21998C212yesmaximal, complemented
3#7730C212yesmaximal, complemented
4#7716C212yesmaximal, complemented
5#7684C212yesmaximal, complemented
6#7651C211yesmaximal, characteristic, complemented
7#7640C211yesmaximal, characteristic, complemented
8#7631C211yesmaximal, characteristic, complemented
9#7603C211yesmaximal, characteristic, complemented
10#7592C211yesmaximal, characteristic, complemented
11#7563C211yesmaximal, characteristic, complemented
12C2×#553C411yescharacteristic
13C2×#546C411yescharacteristic
14#2861C2211yescharacteristic
15#2856C2211yescharacteristic
16#2850C2211yescharacteristic
17#2566C411yescharacteristic
18#2554—22no—
19#2544C411yescharacteristic
20#2502C411yescharacteristic
21#2500C411yescharacteristic
22#2464—22no—
23#2433C411yescharacteristic
24#2431C411yescharacteristic
25#2424—22no—
26#2358—22no—
27C4×#127C2211yescharacteristic
28#2297C2211yescharacteristic
29C2×#390C2211yescharacteristic
30C42×D4C2211yescharacteristic
31#1923—22no—
32#1916—22no—
33#1870—22no—
34#1851—22no—
35#1839C2212yescomplemented
36#1831C2212yescomplemented
37#1817C2212yes—
38#1797C2212yes—
39C4×C22⋊C8C2212yescomplemented
40#1728C2212yescomplemented
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—
54#553—44no—
55#546—44no—
56C2×#127C2×C411yescharacteristic
57C2×#127C2×C411yescharacteristic
58C2×#126—21no—
59C2×#125C2×C411yescharacteristic
60C2×#124C2×C411yescharacteristic
61C2×#124C2×C411yescharacteristic
62C2×#123—21no—
63C2×#123—21no—
64C2×C22⋊Q8C2×C411yescharacteristic
65C2×#121—21no—
66C2×#121—22no—
67C2×#121—21no—
68C2×#121C2×C411yescharacteristic
69C2×#121C2×C411yescharacteristic
70C2×#121C2×C411yescharacteristic
71C2×#120—21no—
72C2×#120—22no—
73C2×C4×D4—21no—
74C2×C4×D4—22no—
75C2×C4×D4C2×C411yescharacteristic
76C2×C4×D4C2×C411yescharacteristic
77C2×C4×D4C2×C411yescharacteristic
78C2×C4×D4D412yes—
79C2×#117C2×C411yescharacteristic
80C22×C4⋊C4C2×C411yescharacteristic
81C22×C22⋊C4—22no—
82C22×C22⋊C4—21no—
83C22×C42C2311yescharacteristic
84C22×C42C2311yescharacteristic
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C4×C8—22no—
91#398—22no—
92#396—22no—
93#394—22no—
94#394—22no—
95#392—22no—
96#392—22no—
97#391C2×C411yescharacteristic
98#391C2×C411yescharacteristic
99#391C2311yescharacteristic
100#390—22no—
101#390—22no—
102#390—22no—
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#388—22no—
114#386—22no—
115#386—22no—
116#386—22no—
117#386—22no—
118#385—22no—
119#385—22no—
120#384C2×C411yescharacteristic
121#384C2×C411yescharacteristic
122#382C2×C411yescharacteristic
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—
141C2×#41—22no—
142C2×#41—41no—
143C2×#41—42no—
144C2×#41—81no—
145C22×Q8C22×C411yescharacteristic
146C22×D4—22no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—22no—
152C22×D4—21no—
153C22×D4—22no—
154C22×D4—41no—
155C22×D4—42no—
156C22×D4—81no—
157C22×D4C22⋊C411yescharacteristic
158C22×D4C22×C411yescharacteristic
159C22×D4C22×C411yescharacteristic
160C22×D4C22×C411yescharacteristic
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4C22×C411yescharacteristic
166C23×C4C22×C411yescharacteristic
167C23×C4C2×D412yes—
168C23×C4C2×D412yes—
169C2×M16—44no—
170C2×M16—82no—
171C2×M16—82no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176#127—22no—
177#127—41no—
178#127—81no—
179#126—44no—
180#125—42no—
181#124—42no—
182#124—81no—
183#123—44no—
184#123—44no—
185C22⋊Q8—42no—
186#121—44no—
187#121—42no—
188#121—44no—
189#121—42no—
190#121—42no—
191#121—82no—
192#121—82no—
193#120—48no—
194#120—44no—
195C4×D4—24no—
196C4×D4—42no—
197C4×D4—42no—
198C4×D4—44no—
199C4×D4—44no—
200C4×D4—82no—
201C4×D4—82no—
202#117—42no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4C22⋊C411yescharacteristic
211C2×C4⋊C4C22⋊C412yes—
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—22no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—22no—
233C2×C22⋊C4—44no—
234C2×C22⋊C4—82no—
235C2×C22⋊C4—82no—
236C2×C22⋊C4C22⋊C411yescharacteristic
237C2×C22⋊C4C22⋊C412yes—
238C2×C22⋊C4C22⋊C412yes—
239C2×C22⋊C4C22⋊C411yescharacteristic
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—
257C4×C8—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—44no—
270C24—42no—
271C24—82no—
272C24—81no—
273C24C2×C22⋊C411yescharacteristic
274C24C2×C22⋊C411yescharacteristic
275#41—84no—
276#41—84no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×D4—22no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—42no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—44no—
291C2×D4—42no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—44no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—44no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—44no—
305C2×D4—82no—
306C2×D4—81no—
307C2×D4—82no—
308C2×D4—82no—
309C2×D4—84no—
310C2×D4—84no—
311C2×D4—81no—
312C2×D4—82no—
313C2×D4—82no—
314C2×D4#13112yes—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—81no—
342C22×C4—82no—
343C22×C4—82no—
344C22×C4—82no—
345C22×C4C2×C22⋊C412yes—
346C22×C4C2×C22⋊C412yes—
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C22⋊C412yes—
352C22×C4C23×C411yescharacteristic
353C22×C4C22×D412yes—
354M16—88no—
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—
367C4⋊C4—22no—
368C4⋊C4—24no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—82no—
378C4⋊C4—82no—
379C22⋊C4—22no—
380C22⋊C4—24no—
381C22⋊C4—42no—
382C22⋊C4—42no—
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—42no—
397C22⋊C4—42no—
398C22⋊C4—44no—
399C22⋊C4—82no—
400C22⋊C4—82no—
401C22⋊C4—82no—
402C22⋊C4#10014yes—
403C42—21no—
404C42—21no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—42no—
409C42—42no—
410C42—41no—
411C42—41no—
412C42—41no—
413C42—42no—
414C42—42no—
415C23—21no—
416C23—21no—
417C23—22no—
418C23—22no—
419C23—21no—
420C23—22no—
421C23—21no—
422C23—21no—
423C23—22no—
424C23—22no—
425C23—41no—
426C23—41no—
427C23—42no—
428C23—42no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—42no—
439C23—84no—
440C23—84no—
441C23—88no—
442C23—162no—
443C23—164no—
444C23C2×#10012yes—
445C23C22×C22⋊C411yescharacteristic
446Q8—81no—
447Q8—81no—
448D4—42no—
449D4—42no—
450D4—44no—
451D4—81no—
452D4—81no—
453D4—81no—
454D4—81no—
455D4—88no—
456D4—84no—
457D4—84no—
458D4—84no—
459D4—164no—
460D4—162no—
461D4—162no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—84no—
502C2×C4—162no—
503C2×C4—162no—
504C2×C4—162no—
505C2×C4C2×#10012yes—
506C2×C4C2×#10012yes—
507C2×C4#42114yes—
508C2×C4C2×#13111yescharacteristic
509C2×C4#56811yescharacteristic
510C2×C4#57612yes—
511C2×C4#57712yes—
512C8—82no—
513C8—82no—
514C8—82no—
515C8—82no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—42no—
520C22—42no—
521C22—41no—
522C22—42no—
523C22—41no—
524C22—44no—
525C22—42no—
526C22—42no—
527C22—41no—
528C22—81no—
529C22—82no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22—81no—
534C22—81no—
535C22—82no—
536C22—82no—
537C22—82no—
538C22—82no—
539C22—81no—
540C22—81no—
541C22—168no—
542C22#288011yescharacteristic
543C22#288311yescharacteristic
544C22C22×#10011yescharacteristic
545C22#289712yes—
546C22C2×#42112yes—
547C4—21no—
548C4—21no—
549C4—22no—
550C4—22no—
551C4—22no—
552C4—22no—
553C4—24no—
554C4—41no—
555C4—42no—
556C4—42no—
557C4—41no—
558C4—81no—
559C4—81no—
560C4—81no—
561C4—81no—
562C4—81no—
563C2—42no—
564C2—81no—
565C2—82no—
566C2—161no—
567C2—161no—
568C2#1742111yescharacteristic
569C2#1752212yes—
570C2#1759712yes—
571C2#1766311yescharacteristic
572C2#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. #21998C2semidirect
3. #7730C2semidirect
4. #7716C2semidirect
5. #7684C2semidirect
6. #7651C2semidirect
7. #7640C2semidirect
8. #7631C2semidirect
9. #7603C2semidirect
10. #7592C2semidirect
11. #7563C2semidirect
35. #1839C22semidirect
36. #1831C22semidirect
39. C4×C22⋊C8C22semidirect
40. #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
#219982
#77302
#77162
#76842
#76512
#76402
#76312
#76032
#75922
#75632