Finite Group Databasebeta

← #164072

Subgroups — #164072

Order 512.

1201 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35256C211yesmaximal, characteristic
2#22530C211yesmaximal, characteristic, complemented
3#22473C211yesmaximal, characteristic, complemented
4#7724C212yesmaximal, complemented
5#7710C212yesmaximal, complemented
6#7684C212yesmaximal, complemented
7#7670C212yesmaximal, complemented
8#7597C211yesmaximal, characteristic, complemented
9#7572C211yesmaximal, characteristic, complemented
10#7572C211yesmaximal, characteristic, complemented
11#7563C211yesmaximal, characteristic, complemented
12C2×#537C411yescharacteristic
13C2×#535C411yescharacteristic
14C2×#535C411yescharacteristic
15C2×C4×#41C2211yescharacteristic
16C2×C8×Q8—21no—
17C2×#436—21no—
18#2856C2212yes—
19#2850C2212yes—
20#2620C412yes—
21#2619C412yes—
22#2428—21no—
23#2428—21no—
24#2426—21no—
25#2426—21no—
26#2424—22no—
27#2424—22no—
28#2317C411yescharacteristic
29#2297C2211yescharacteristic
30C2×#390C2211yescharacteristic
31#1984—21no—
32#1954—22no—
33#1954—21no—
34#1870—22no—
35#1817C2212yescomplemented
36#1797C2211yescharacteristic
37#1797C2211yescharacteristic
38#1797C2212yescomplemented
39C4×C22⋊C8C2211yescharacteristic
40C4×C22⋊C8C2211yescharacteristic
41C4×C22⋊C8C2212yescomplemented
42#1728C2212yescomplemented
43#1272C2211yescharacteristic, complemented
44#1272C2212yescomplemented
45#1272C2211yescharacteristic, complemented
46#1270C2212yes—
47#1270C2212yes—
48#1270C2212yes—
49#1270C2212yes—
50#1270C2211yescharacteristic, complemented
51#1270C2212yescomplemented
52#1270C2211yescharacteristic, complemented
53C22×#41C2×C411yescharacteristic
54C22×#41C2×C411yescharacteristic
55C23×D4C2×C411yescharacteristic
56C22×M16—22no—
57#537—44no—
58#535—44no—
59#535—44no—
60C2×#127—21no—
61C2×#127C2×C412yes—
62C2×#124—21no—
63C2×#124C2×C412yes—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8C2×C412yes—
66C2×#121—22no—
67C2×#121—21no—
68C2×#121—22no—
69C2×#121—22no—
70C2×#121—21no—
71C2×#121—21no—
72C2×#121C2×C412yes—
73C2×#121C2×C412yes—
74C2×#121C2×C412yes—
75C2×#120—22no—
76C2×#120—22no—
77C4×#41—44no—
78C2×C4×Q8D411yescharacteristic
79C2×C4×Q8D411yescharacteristic
80C2×C4×D4D411yescharacteristic
81C2×C4×D4D412yes—
82C2×C4×D4D412yes—
83C2×C4×D4D411yescharacteristic
84C2×#117C2×C411yescharacteristic
85C2×#117C2×C411yescharacteristic
86C2×#117C2×C412yes—
87C22×C4⋊C4C2×C411yescharacteristic
88C22×C22⋊C4—21no—
89C22×C22⋊C4—21no—
90C22×C42C2311yescharacteristic
91C22×C42C2312yes—
92C8×Q8—44no—
93#436—44no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—22no—
97C2×C22⋊C8—22no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—22no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—22no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8—22no—
107#396—22no—
108#396—22no—
109#394—22no—
110#394—22no—
111#394—22no—
112#394—22no—
113#392—22no—
114#392—22no—
115#391C2×C411yescharacteristic
116#391C2311yescharacteristic
117#390—22no—
118#390—22no—
119#390—21no—
120#390—21no—
121#390C2×C411yescharacteristic
122#390C2×C412yes—
123#390C2312yes—
124#390C2311yescharacteristic
125#389—21no—
126#389—21no—
127#388—22no—
128#388—22no—
129#386—22no—
130#386—22no—
131#386—21no—
132#386—21no—
133#385—22no—
134#384C2×C412yes—
135#382C2×C412yes—
136C4×C22⋊C4—21no—
137C4×C22⋊C4—21no—
138#336—22no—
139#336—22no—
140#336C2312yes—
141#336C2312yes—
142#336C2311yescharacteristic
143#336C2311yescharacteristic
144#336C2311yescharacteristic
145#336C2311yescharacteristic
146C25—21no—
147C2×#41—24no—
148C2×#41—24no—
149C2×#41—24no—
150C2×#41—42no—
151C2×#41—42no—
152C2×#41—42no—
153C22×Q8C22⋊C411yescharacteristic
154C22×Q8C22×C411yescharacteristic
155C22×D4—21no—
156C22×D4—22no—
157C22×D4—22no—
158C22×D4—22no—
159C22×D4—21no—
160C22×D4—22no—
161C22×D4—22no—
162C22×D4—24no—
163C22×D4—42no—
164C22×D4—42no—
165C22×D4C22⋊C411yescharacteristic
166C22×D4C22⋊C412yes—
167C22×D4C22×C412yes—
168C22×D4C22×C411yescharacteristic
169C23×C4—21no—
170C23×C4—21no—
171C23×C4C22×C412yes—
172C23×C4C22×C411yescharacteristic
173C23×C4C2×D412yes—
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D412yes—
177C2×M16—44no—
178C2×M16—44no—
179C2×M16—82no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—22no—
183C22×C8—21no—
184C22×C8—22no—
185C22×C8—21no—
186#127—44no—
187#127—44no—
188#124—44no—
189#124—44no—
190C22⋊Q8—44no—
191C22⋊Q8—44no—
192#121—44no—
193#121—44no—
194#121—48no—
195#121—44no—
196#121—44no—
197#121—44no—
198#121—44no—
199#121—82no—
200#121—82no—
201#121—82no—
202#121—82no—
203#120—48no—
204#120—48no—
205C4×Q8—24no—
206C4×Q8—42no—
207C4×D4—24no—
208C4×D4—44no—
209C4×D4—42no—
210C4×D4—44no—
211#117—24no—
212#117—44no—
213#117—42no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4C22⋊C411yescharacteristic
220C2×C4⋊C4C22⋊C412yes—
221C2×C4⋊C4C22⋊C412yes—
222C2×C4⋊C4C22⋊C411yescharacteristic
223C2×C4⋊C4C22×C412yes—
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C411yescharacteristic
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—22no—
234C2×C22⋊C4—22no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—22no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—81no—
242C2×C22⋊C4—81no—
243C2×C22⋊C4—81no—
244C2×C22⋊C4—81no—
245C2×C22⋊C4C22⋊C412yes—
246C2×C22⋊C4C22⋊C412yes—
247C2×C22⋊C4C22⋊C412yes—
248C2×C42—21no—
249C2×C42—42no—
250C2×C42—44no—
251C2×C42C22×C412yes—
252C2×C42C22×C411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D412yes—
256C2×C42C2×D412yes—
257C2×C42C2411yescharacteristic
258C4⋊C8—42no—
259C4⋊C8—42no—
260C4⋊C8—44no—
261C22⋊C8—42no—
262C22⋊C8—44no—
263C22⋊C8—42no—
264C22⋊C8—44no—
265C22⋊C8—81no—
266C22⋊C8—82no—
267C22⋊C8—81no—
268C4×C8—42no—
269C4×C8—42no—
270C4×C8—44no—
271#95—21no—
272#95—21no—
273#95—22no—
274#95—22no—
275#95—21no—
276#95—21no—
277C24—22no—
278C24—21no—
279C24—42no—
280C24—42no—
281C24—42no—
282C24—81no—
283C24—81no—
284C24—81no—
285C24—81no—
286C24C2×C22⋊C412yes—
287C24C2×C22⋊C411yescharacteristic
288#41—416no—
289#41—88no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—22no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8#10014yes—
296C2×D4—24no—
297C2×D4—24no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—22no—
301C2×D4—24no—
302C2×D4—22no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—41no—
306C2×D4—42no—
307C2×D4—41no—
308C2×D4—44no—
309C2×D4—42no—
310C2×D4—42no—
311C2×D4—42no—
312C2×D4—44no—
313C2×D4—44no—
314C2×D4—44no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—44no—
318C2×D4—42no—
319C2×D4—88no—
320C2×D4—82no—
321C2×D4—82no—
322C2×D4—82no—
323C2×D4—82no—
324C2×D4—82no—
325C2×D4—82no—
326C2×D4—82no—
327C2×D4—82no—
328C2×D4—82no—
329C2×D4—82no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—22no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—42no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—81no—
360C22×C4—81no—
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C412yes—
364C22×C4C2×C22⋊C412yes—
365C22×C4C2×C22⋊C412yes—
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4#13114yes—
368C22×C4C23×C411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371M16—88no—
372C2×C8—22no—
373C2×C8—22no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—41no—
388C2×C8—42no—
389C2×C8—41no—
390C4⋊C4—22no—
391C4⋊C4—24no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—42no—
397C4⋊C4—44no—
398C4⋊C4—82no—
399C4⋊C4—82no—
400C4⋊C4#9914yes—
401C22⋊C4—24no—
402C22⋊C4—24no—
403C22⋊C4—42no—
404C22⋊C4—44no—
405C22⋊C4—42no—
406C22⋊C4—44no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C22⋊C4—41no—
410C22⋊C4—41no—
411C22⋊C4—42no—
412C22⋊C4—42no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—44no—
419C22⋊C4—81no—
420C22⋊C4—81no—
421C22⋊C4—82no—
422C22⋊C4—82no—
423C42—22no—
424C42—24no—
425C42—22no—
426C42—24no—
427C42—41no—
428C42—41no—
429C42—42no—
430C42—42no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—21no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—41no—
445C23—42no—
446C23—42no—
447C23—41no—
448C23—41no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—41no—
453C23—41no—
454C23—41no—
455C23—41no—
456C23—42no—
457C23—42no—
458C23—84no—
459C23—84no—
460C23—84no—
461C23—162no—
462C23—162no—
463C23—162no—
464C23—162no—
465C23#40812yes—
466C23C22×C22⋊C411yescharacteristic
467Q8—28no—
468Q8—82no—
469D4—48no—
470D4—44no—
471D4—84no—
472D4—82no—
473D4—84no—
474D4—84no—
475D4—164no—
476D4—164no—
477D4—164no—
478D4—164no—
479D4—164no—
480C2×C4—22no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—28no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—48no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—82no—
519C2×C4—81no—
520C2×C4—82no—
521C2×C4—84no—
522C2×C4—162no—
523C2×C4#40812yes—
524C2×C4C2×#9914yes—
525C2×C4C2×#9912yes—
526C2×C4C2×#10014yes—
527C2×C4C2×#10012yes—
528C2×C4C2×#13112yes—
529C2×C4#57611yescharacteristic
530C2×C4#57611yescharacteristic
531C2×C4#57711yescharacteristic
532C2×C4#57711yescharacteristic
533C8—42no—
534C8—42no—
535C8—81no—
536C8—81no—
537C8—82no—
538C8—82no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—41no—
543C22—41no—
544C22—41no—
545C22—41no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—82no—
554C22—81no—
555C22—81no—
556C22—82no—
557C22—81no—
558C22—82no—
559C22—81no—
560C22—82no—
561C22—81no—
562C22—82no—
563C22—81no—
564C22—82no—
565C22—81no—
566C22—82no—
567C22—168no—
568C22C2×#40811yescharacteristic
569C22C22×#9911yescharacteristic
570C22C22×#10011yescharacteristic
571C22#289711yescharacteristic
572C22#289711yescharacteristic
573C22#290111yescharacteristic
574C22#290111yescharacteristic
575C4—21no—
576C4—21no—
577C4—24no—
578C4—21no—
579C4—21no—
580C4—22no—
581C4—24no—
582C4—41no—
583C4—41no—
584C4—42no—
585C4—81no—
586C4—82no—
587C4—81no—
588C4—81no—
589C4#147114yes—
590C2—42no—
591C2—82no—
592C2—161no—
593C2—162no—
594C2C2×#147111yescharacteristic
595C2#1746811yescharacteristic
596C2#1748311yescharacteristic
597C2#1760311yescharacteristic
598C2#1761111yescharacteristic
599C2#1768811yescharacteristic
600C2#1778211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#35256C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22530C2semidirect
3. #22473C2semidirect
4. #7724C2semidirect
5. #7710C2semidirect
6. #7684C2semidirect
7. #7670C2semidirect
8. #7597C2semidirect
9. #7572C2semidirect
10. #7572C2semidirect
11. #7563C2semidirect
35. #1817C22semidirect
38. #1797C22semidirect
41. C4×C22⋊C8C22semidirect
42. #1728C22semidirect
43. #1272C22semidirect
44. #1272C22semidirect
45. #1272C22semidirect
50. #1270C22semidirect
51. #1270C22semidirect
52. #1270C22semidirect

Maximal subgroups

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

subgroupindex
#352562
#225302
#224732
#77242
#77102
#76842
#76702
#75972
#75722
#75632