Finite Group Databasebeta

← #205030

Subgroups — #205030

Order 512.

1601 conjugacy classes of subgroups.

The subgroup lattice has 593 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35078C211yesmaximal, characteristic
2#21932C211yesmaximal, characteristic, complemented
3C2×#1504C211yesmaximal, characteristic, complemented
4C2×#1507C211yesmaximal, characteristic, complemented
5C2×#1501C211yesmaximal, characteristic, complemented
6C22×#327C211yesmaximal, characteristic, complemented
7#13749C211yesmaximal, characteristic, complemented
8#9713C214yesmaximal, complemented
9#9712C214yesmaximal, complemented
10C2×#525C2211yescharacteristic
11C22×#121C2211yescharacteristic
12C2×#407C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14#2856C2212yes—
15#2850C2212yes—
16#2661—22no—
17#2543—22no—
18#2504C411yescharacteristic, complemented
19#2502C411yescharacteristic, complemented
20#2500C412yescomplemented
21#2435C411yescharacteristic, complemented
22#2433C411yescharacteristic, complemented
23#2431C412yescomplemented
24#2410—22no—
25#2321—22no—
26C2×C4×C22⋊C4—22no—
27C2×C4×C22⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29C22×#95C2211yescharacteristic
30#1851—22no—
31#1850—22no—
32#1797C2214yescomplemented
33#1507—24no—
34#1504—24no—
35#1501—24no—
36C2×#328C2212yescomplemented
37C2×#328C2212yescomplemented
38C2×#327—24no—
39C2×#327C2214yes—
40C2×#327C2214yes—
41C2×#327C2212yescomplemented
42C2×#327C2212yescomplemented
43C4×#95—22no—
44C4×#95C2214yescomplemented
45C22×#41C2×C411yescharacteristic
46C23×D4—21no—
47C23×D4C2×C411yescharacteristic
48C24×C4—21no—
49C24×C4C2311yescharacteristic
50C22×M16—21no—
51C23×C8—21no—
52C2×#127C2×C412yescomplemented
53C2×#124C2×C412yescomplemented
54#525—22no—
55#525—22no—
56#525—22no—
57#525—22no—
58C2×#123—21no—
59C2×C22⋊Q8C2×C411yescharacteristic
60C2×C22⋊Q8C2311yescharacteristic
61C2×#121—21no—
62C2×#121—21no—
63C2×#121—24no—
64C2×#121—22no—
65C2×#121—22no—
66C2×#121—22no—
67C2×#121—22no—
68C2×#121C2×C412yes—
69C2×#121C2×C411yescharacteristic
70C2×#121C2312yes—
71C2×#121C2311yescharacteristic
72C2×#120—21no—
73C2×#120—22no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×#117C2×C411yescharacteristic
77C22×C4⋊C4C2×C411yescharacteristic
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4D412yes—
81C22×C42—21no—
82C22×C42—24no—
83C22×C42C2312yes—
84C2×C4⋊C8—24no—
85#407—22no—
86#407—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8C2312yes—
92C2×C22⋊C8C2312yes—
93#398—22no—
94#397—22no—
95#394—24no—
96#394—22no—
97#392—24no—
98#392—22no—
99#391C2×C412yescomplemented
100#390—24no—
101#390C2×C412yescomplemented
102#390C2×C412yescomplemented
103#390C2×C412yescomplemented
104#388—24no—
105#388—24no—
106#386—24no—
107#386—22no—
108#386—24no—
109#385—22no—
110#384C2×C412yescomplemented
111#382C2×C412yescomplemented
112#379—22no—
113C4×C22⋊C4—216no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4C2×C412yes—
116C4×C22⋊C4C2×C411yescharacteristic
117C4×C22⋊C4C2×C411yescharacteristic
118C4×C22⋊C4D412yes—
119C4×C22⋊C4D412yes—
120C4×C22⋊C4D414yescomplemented
121C2×#95—21no—
122C2×#95—24no—
123C2×#95—22no—
124C2×#95D412yes—
125C2×#95D414yescomplemented
126C2×#95C2312yes—
127C2×#95C2312yes—
128#336—24no—
129#328—24no—
130#328—24no—
131#327—28no—
132#327—28no—
133#327—24no—
134#327—24no—
135C25—21no—
136C25C2×D411yescharacteristic
137C2×#41—22no—
138C2×#41—22no—
139C2×#41—82no—
140C22×Q8C22×C411yescharacteristic
141C22×D4—21no—
142C22×D4—21no—
143C22×D4—22no—
144C22×D4—21no—
145C22×D4—22no—
146C22×D4—24no—
147C22×D4—21no—
148C22×D4—22no—
149C22×D4—22no—
150C22×D4—42no—
151C22×D4—42no—
152C22×D4—82no—
153C22×D4—82no—
154C22×D4—82no—
155C22×D4C22×C412yes—
156C22×D4C22×C411yescharacteristic
157C23×C4—22no—
158C23×C4—22no—
159C23×C4—28no—
160C23×C4—24no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—82no—
164C23×C4—82no—
165C23×C4—82no—
166C23×C4C22×C411yescharacteristic
167C23×C4C22×C411yescharacteristic
168C23×C4C2×D414yes—
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D412yes—
173C23×C4C2×D412yes—
174C23×C4#4112yes—
175C23×C4#4112yes—
176C23×C4C2411yescharacteristic
177C2×M16—42no—
178C2×M16—82no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—42no—
182C22×C8—82no—
183#127—82no—
184#124—82no—
185#123—42no—
186#123—42no—
187C22⋊Q8—22no—
188C22⋊Q8—22no—
189C22⋊Q8—21no—
190C22⋊Q8—21no—
191C22⋊Q8—21no—
192C22⋊Q8—21no—
193#121—24no—
194#121—24no—
195#121—22no—
196#121—22no—
197#121—22no—
198#121—22no—
199#121—21no—
200#121—21no—
201#121—22no—
202#121—22no—
203#121—21no—
204#121—21no—
205#121—42no—
206#121—42no—
207#121—42no—
208#121—42no—
209#121—84no—
210#121—84no—
211#120—42no—
212#120—42no—
213#120—48no—
214C4×D4—42no—
215C4×D4—42no—
216C4×D4—42no—
217C4×D4—42no—
218#117—22no—
219#117—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4C22×C412yes—
227C2×C4⋊C4C22×C411yescharacteristic
228C2×C4⋊C4C22×C411yescharacteristic
229C2×C22⋊C4—28no—
230C2×C22⋊C4—24no—
231C2×C22⋊C4—28no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—22no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—42no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4C22⋊C412yes—
246C2×C22⋊C4C22⋊C414yes—
247C2×C22⋊C4C22⋊C412yes—
248C2×C42—21no—
249C2×C42—216no—
250C2×C42—24no—
251C2×C42—82no—
252C2×C42—84no—
253C2×C42—84no—
254C2×C42—82no—
255C2×C42C22×C412yes—
256C2×C42C22×C412yes—
257C2×C42C2×D414yes—
258C4⋊C8—84no—
259C22⋊C8—22no—
260C22⋊C8—22no—
261C22⋊C8—22no—
262C22⋊C8—22no—
263C22⋊C8—42no—
264C22⋊C8—42no—
265#95—216no—
266#95—24no—
267#95—24no—
268#95—24no—
269#95—22no—
270#95—22no—
271#95C22⋊C414yes—
272#95C22⋊C412yes—
273#95C22⋊C412yes—
274#95C2×D414yes—
275#95C2×D414yes—
276C24—22no—
277C24—22no—
278C24—24no—
279C24—22no—
280C24—21no—
281C24—42no—
282C24—42no—
283C24—44no—
284C24—82no—
285C24—82no—
286C24C2×C22⋊C411yescharacteristic
287C24C2×C22⋊C411yescharacteristic
288C24#12112yes—
289C24#12112yes—
290C24C22×D411yescharacteristic
291#41—84no—
292#41—84no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—42no—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—42no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—42no—
306C2×D4—42no—
307C2×D4—42no—
308C2×D4—44no—
309C2×D4—42no—
310C2×D4—44no—
311C2×D4—42no—
312C2×D4—42no—
313C2×D4—44no—
314C2×D4—44no—
315C2×D4—42no—
316C2×D4—48no—
317C2×D4—82no—
318C2×D4—84no—
319C2×D4—84no—
320C2×D4—84no—
321C2×D4—82no—
322C2×D4—84no—
323C2×D4—84no—
324C2×D4—84no—
325C2×D4—84no—
326C2×D4—88no—
327C22×C4—24no—
328C22×C4—28no—
329C22×C4—24no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—216no—
335C22×C4—28no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—44no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—42no—
350C22×C4—44no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—82no—
355C22×C4—82no—
356C22×C4—82no—
357C22×C4—82no—
358C22×C4—82no—
359C22×C4—84no—
360C22×C4—82no—
361C22×C4—816no—
362C22×C4—84no—
363C22×C4C2×C22⋊C414yes—
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C414yes—
367C22×C4C4×D412yes—
368C22×C4C4×D412yes—
369C22×C4C4×D412yes—
370C22×C4C4×D412yes—
371C22×C4#12112yes—
372C22×C4#12112yes—
373C22×C4#12412yes—
374C22×C4#12412yes—
375C22×C4#12712yes—
376C22×C4#12712yes—
377C22×C4C23×C411yescharacteristic
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D411yescharacteristic
380C22×C4C22×D411yescharacteristic
381C22×C4C2×#4111yescharacteristic
382C22×C4C2×#4111yescharacteristic
383M16—84no—
384C2×C8—44no—
385C2×C8—44no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—84no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—84no—
400C22⋊C4—216no—
401C22⋊C4—24no—
402C22⋊C4—28no—
403C22⋊C4—41no—
404C22⋊C4—41no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—41no—
409C22⋊C4—42no—
410C22⋊C4—41no—
411C22⋊C4—44no—
412C22⋊C4—42no—
413C22⋊C4—48no—
414C22⋊C4—84no—
415C22⋊C4—84no—
416C22⋊C4—84no—
417C22⋊C4#10418yes—
418C42—44no—
419C42—42no—
420C42—42no—
421C42—42no—
422C42—84no—
423C42—816no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—22no—
430C23—22no—
431C23—24no—
432C23—24no—
433C23—22no—
434C23—22no—
435C23—24no—
436C23—42no—
437C23—42no—
438C23—48no—
439C23—48no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—44no—
445C23—44no—
446C23—42no—
447C23—41no—
448C23—42no—
449C23—84no—
450C23—84no—
451C23—84no—
452C23—168no—
453C23#39012yes—
454C23#39012yes—
455C23C2×#10414yes—
456C23C22×C22⋊C411yescharacteristic
457C23C2×#12111yescharacteristic
458C23C2×#12111yescharacteristic
459C23#57612yes—
460C23#57712yes—
461Q8—81no—
462Q8—81no—
463D4—82no—
464D4—82no—
465D4—81no—
466D4—81no—
467D4—84no—
468D4—88no—
469D4—84no—
470D4—84no—
471D4—81no—
472D4—82no—
473D4—81no—
474D4—82no—
475D4—82no—
476D4—84no—
477D4—168no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—28no—
485C2×C4—28no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—48no—
504C2×C4—48no—
505C2×C4—48no—
506C2×C4—44no—
507C2×C4—48no—
508C2×C4—48no—
509C2×C4—48no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—84no—
522C2×C4—816no—
523C2×C4—84no—
524C2×C4#39012yes—
525C2×C4#39012yes—
526C2×C4C2×#10418yes—
527C2×C4C2×C4×D411yescharacteristic
528C2×C4C2×C4×D411yescharacteristic
529C2×C4C2×#12411yescharacteristic
530C2×C4C2×#12711yescharacteristic
531C8—82no—
532C8—82no—
533C22—22no—
534C22—24no—
535C22—24no—
536C22—28no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—44no—
546C22—48no—
547C22—44no—
548C22—44no—
549C22—82no—
550C22—81no—
551C22—84no—
552C22—84no—
553C22—84no—
554C22—81no—
555C22—82no—
556C22—81no—
557C22—82no—
558C22—168no—
559C22#196214yes—
560C22C2×#39011yescharacteristic
561C22#288312yes—
562C22#289712yes—
563C22C22×#10412yes—
564C22#304111yescharacteristic
565C22#304311yescharacteristic
566C22#304512yes—
567C4—28no—
568C4—21no—
569C4—22no—
570C4—21no—
571C4—21no—
572C4—21no—
573C4—22no—
574C4—48no—
575C4—48no—
576C4—44no—
577C4—82no—
578C4—81no—
579C4—81no—
580C4—81no—
581C2—21no—
582C2—22no—
583C2—24no—
584C2—21no—
585C2—42no—
586C2—161no—
587C2—162no—
588C2#1754714yes—
589C2C2×#196211yescharacteristic
590C2#2121311yescharacteristic
591C2#2126311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#35078C2
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. #21932C2semidirect
3. C2×#1504C2semidirect
4. C2×#1507C2semidirect
5. C2×#1501C2semidirect
6. C22×#327C2semidirect
7. #13749C2semidirect
8. #9713C2semidirect
9. #9712C2semidirect
18. #2504C4semidirect
19. #2502C4semidirect
20. #2500C4semidirect
21. #2435C4semidirect
22. #2433C4semidirect
23. #2431C4semidirect
32. #1797C22semidirect
36. C2×#328C22semidirect
37. C2×#328C22semidirect
41. C2×#327C22semidirect
42. C2×#327C22semidirect
44. C4×#95C22semidirect
52. C2×#127C2×C4semidirect
53. C2×#124C2×C4semidirect
99. #391C2×C4semidirect
101. #390C2×C4semidirect
102. #390C2×C4semidirect
103. #390C2×C4semidirect
110. #384C2×C4semidirect
111. #382C2×C4semidirect
120. C4×C22⋊C4D4semidirect
125. C2×#95D4semidirect

Maximal subgroups

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

subgroupindex
#350782
#219322
C2×#15042
C2×#15072
C2×#15012
C22×#3272
#137492
#97132
#97122