Finite Group Databasebeta

← #215438

Subgroups — #215438

Order 512.

1528 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#35001C211yesmaximal, characteristic
2#21902C211yesmaximal, characteristic, complemented
3#13948C211yesmaximal, characteristic, complemented
4#9715C212yesmaximal, complemented
5#9712C212yesmaximal, complemented
6#9680C212yesmaximal, complemented
7#9680C212yesmaximal, complemented
8#8328C212yesmaximal, complemented
9#8328C212yesmaximal, complemented
10C2×#534—21no—
11C2×#534—21no—
12C22×#121C2212yes—
13#2856C2212yes—
14#2856C2212yes—
15#2656—21no—
16#2656—21no—
17#2539—24no—
18#2501C414yescomplemented
19#2433C414yescomplemented
20C2×#388—21no—
21C2×C4×C22⋊C4C2211yescharacteristic
22#1923—24no—
23#1797C2212yescomplemented
24#1779C2212yes—
25C4×C22⋊C8C2212yescomplemented
26C2×#327C2214yescomplemented
27C2×#327C2214yescomplemented
28C2×#327C2214yescomplemented
29C2×#327C2214yescomplemented
30#1442—21no—
31#1439—21no—
32#1436—21no—
33C4⋊#95C2211yescharacteristic, complemented
34C4⋊#95C2211yescharacteristic, complemented
35#1434C2211yescharacteristic, complemented
36C22⋊#95C2211yescharacteristic
37C22⋊#95C2211yescharacteristic
38C4×#95C2211yescharacteristic, complemented
39C23×D4—22no—
40C23×D4C2×C412yes—
41C24×C4C2311yescharacteristic
42#534—44no—
43#534—44no—
44#534—44no—
45#534—44no—
46C2×#127C2×C412yes—
47C2×#127C2×C412yes—
48C2×#123—24no—
49C2×#123—24no—
50C2×#121—22no—
51C2×#121—22no—
52C2×#121—22no—
53C2×#121—22no—
54C2×#121—22no—
55C2×#121—22no—
56C2×#121—44no—
57C2×#121—44no—
58C2×#121C2×C414yes—
59C2×#121C2314yes—
60C2×#120—24no—
61C2×#120—24no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4C2×C412yes—
65C22×C22⋊C4—22no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4D411yescharacteristic
69C22×C22⋊C4D411yescharacteristic
70C22×C42—21no—
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C2×C4⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8C2314yes—
77C2×C4×C8—22no—
78#399—24no—
79#399—24no—
80#392—24no—
81#392—24no—
82#390—22no—
83#390—22no—
84#390—22no—
85#390—22no—
86#390C2×C414yes—
87#390C2×C414yes—
88#388—44no—
89#384C2×C412yes—
90#384C2×C412yes—
91C4×C22⋊C4C2×C414yes—
92C4×C22⋊C4D414yes—
93C2×#95—21no—
94C2×#95—21no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95D411yescharacteristic
101C2×#95D411yescharacteristic
102C2×#95C2311yescharacteristic
103C2×#95C2311yescharacteristic
104C2×#95C2311yescharacteristic
105C2×#95C2311yescharacteristic
106#336—22no—
107#336—22no—
108#327—44no—
109#327—44no—
110#327—44no—
111#327—44no—
112C25—22no—
113C25C2×D411yescharacteristic
114C22×D4—24no—
115C22×D4—24no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—44no—
122C22×D4—44no—
123C22×D4—44no—
124C22×D4—44no—
125C22×D4—44no—
126C22×D4—44no—
127C22×D4—44no—
128C22×D4—82no—
129C22×D4—82no—
130C22×D4C22×C414yes—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—81no—
139C23×C4—81no—
140C23×C4C22×C412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D412yes—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2411yescharacteristic
148C22×C8—24no—
149#127—82no—
150#127—82no—
151#123—416no—
152#123—416no—
153#121—24no—
154#121—24no—
155#121—24no—
156#121—24no—
157#121—44no—
158#121—44no—
159#121—44no—
160#121—44no—
161#121—48no—
162#121—44no—
163#121—44no—
164#121—44no—
165#121—44no—
166#121—44no—
167#121—44no—
168#121—44no—
169#121—44no—
170#120—416no—
171#120—416no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—81no—
183C2×C4⋊C4—81no—
184C2×C4⋊C4C22×C414yes—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—48no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—81no—
199C2×C22⋊C4—81no—
200C2×C22⋊C4C22⋊C414yes—
201C2×C22⋊C4C22⋊C414yes—
202C2×C22⋊C4D814yes—
203C2×C22⋊C4SD1614yes—
204C2×C42—42no—
205C2×C42—42no—
206C2×C42—81no—
207C2×C42—81no—
208C2×C42—81no—
209C2×C42C22×C412yes—
210C2×C42C22×C412yes—
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C4⋊C8—82no—
214C22⋊C8—24no—
215C22⋊C8—24no—
216C22⋊C8—82no—
217C22⋊C8—82no—
218C4×C8—82no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—42no—
224#95—42no—
225#95—42no—
226#95—42no—
227#95—81no—
228#95—81no—
229#95—81no—
230#95C22⋊C414yes—
231#95C22⋊C414yes—
232C24—21no—
233C24—21no—
234C24—21no—
235C24—21no—
236C24—24no—
237C24—41no—
238C24—41no—
239C24—41no—
240C24—41no—
241C24—44no—
242C24—44no—
243C24—44no—
244C24—82no—
245C24—82no—
246C24—82no—
247C24—82no—
248C24C2×C22⋊C412yes—
249C24C2×D812yes—
250C24C2×SD1612yes—
251C24C22×D411yescharacteristic
252C2×D4—24no—
253C2×D4—24no—
254C2×D4—44no—
255C2×D4—44no—
256C2×D4—44no—
257C2×D4—44no—
258C2×D4—48no—
259C2×D4—48no—
260C2×D4—44no—
261C2×D4—44no—
262C2×D4—44no—
263C2×D4—44no—
264C2×D4—44no—
265C2×D4—44no—
266C2×D4—48no—
267C2×D4—44no—
268C2×D4—44no—
269C2×D4—44no—
270C2×D4—82no—
271C2×D4—82no—
272C2×D4—82no—
273C2×D4—82no—
274C2×D4—82no—
275C2×D4—82no—
276C2×D4—82no—
277C2×D4—82no—
278C2×D4—82no—
279C2×D4—82no—
280C2×D4—816no—
281C2×D4—88no—
282C2×D4—88no—
283C2×D4—88no—
284C2×D4—88no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—44no—
316C22×C4—41no—
317C22×C4—44no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—44no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—84no—
340C22×C4—84no—
341C22×C4C2×C22⋊C412yes—
342C22×C4C2×C22⋊C412yes—
343C22×C4C2×C22⋊C412yes—
344C22×C4C2×C22⋊C412yes—
345C22×C4C2×C22⋊C412yes—
346C22×C4C2×D812yes—
347C22×C4C2×D812yes—
348C22×C4C2×SD1612yes—
349C22×C4C2×SD1612yes—
350C22×C4AGL(1,ℤ/8)12yes—
351C22×C4AGL(1,ℤ/8)12yes—
352C22×C4C23×C411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4H(5,𝔽2)12yes—
355C22×C4H(5,𝔽2)12yes—
356C2×C8—44no—
357C2×C8—44no—
358C2×C8—44no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—44no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—44no—
365C4⋊C4—44no—
366C4⋊C4—84no—
367C4⋊C4—84no—
368C22⋊C4—28no—
369C22⋊C4—48no—
370C22⋊C4—48no—
371C22⋊C4—44no—
372C22⋊C4—44no—
373C22⋊C4—48no—
374C22⋊C4—48no—
375C22⋊C4—48no—
376C22⋊C4—84no—
377C22⋊C4—84no—
378C22⋊C4#102116yes—
379C42—42no—
380C42—42no—
381C42—42no—
382C42—42no—
383C42—84no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—22no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—44no—
403C23—44no—
404C23—44no—
405C23—44no—
406C23—44no—
407C23—44no—
408C23—44no—
409C23—81no—
410C23—81no—
411C23—81no—
412C23—81no—
413C23—81no—
414C23—81no—
415C23—81no—
416C23—81no—
417C23—88no—
418C23—88no—
419C23—88no—
420C23—164no—
421C23—164no—
422C23—164no—
423C23—164no—
424C23C2×#10214yes—
425C23C22×C22⋊C411yescharacteristic
426C23#53511yescharacteristic
427C23#53511yescharacteristic
428C23C22×D811yescharacteristic
429C23C22×SD1611yescharacteristic
430C23C2×AGL(1,ℤ/8)11yescharacteristic
431C23C2×AGL(1,ℤ/8)11yescharacteristic
432D4—84no—
433D4—84no—
434D4—88no—
435D4—816no—
436D4—84no—
437D4—84no—
438D4—84no—
439D4—84no—
440D4—164no—
441D4—164no—
442D4—164no—
443D4—164no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—48no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—84no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—88no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4C2×#10214yes—
507C2×C4C2×#10214yes—
508C2×C4#41814yes—
509C2×C4#51812yes—
510C2×C4#53512yes—
511C8—84no—
512C22—21no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—24no—
525C22—24no—
526C22—24no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22—82no—
532C22—82no—
533C22—82no—
534C22—82no—
535C22—84no—
536C22—84no—
537C22—84no—
538C22—84no—
539C22—84no—
540C22—84no—
541C22—84no—
542C22—1616no—
543C22#230911yescharacteristic
544C22#288311yescharacteristic
545C22#288311yescharacteristic
546C22C22×#10211yescharacteristic
547C22C2×#41811yescharacteristic
548C22#290111yescharacteristic
549C22#290111yescharacteristic
550C22#320511yescharacteristic
551C22#320511yescharacteristic
552C22#320811yescharacteristic
553C22#320811yescharacteristic
554C4—22no—
555C4—22no—
556C4—24no—
557C4—24no—
558C4—24no—
559C4—42no—
560C4—42no—
561C4—82no—
562C4—82no—
563C4—84no—
564C4—161no—
565C4—161no—
566C2—21no—
567C2—21no—
568C2—21no—
569C2—21no—
570C2—44no—
571C2—164no—
572C2#1761311yescharacteristic
573C2#1761311yescharacteristic
574C2#1762011yescharacteristic
575C2#1762611yescharacteristic
576C2#2179511yescharacteristic
577C2#2180311yescharacteristic
578C2#2181711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#35001C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21902C2semidirect
3. #13948C2semidirect
4. #9715C2semidirect
5. #9712C2semidirect
6. #9680C2semidirect
7. #9680C2semidirect
8. #8328C2semidirect
9. #8328C2semidirect
18. #2501C4semidirect
19. #2433C4semidirect
23. #1797C22semidirect
25. C4×C22⋊C8C22semidirect
26. C2×#327C22semidirect
27. C2×#327C22semidirect
28. C2×#327C22semidirect
29. C2×#327C22semidirect
33. C4⋊#95C22semidirect
34. C4⋊#95C22semidirect
35. #1434C22semidirect
38. C4×#95C22semidirect

Maximal subgroups

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

subgroupindex
#350012
#219022
#139482
#97152
#97122
#96802
#83282