Finite Group Databasebeta

← #6470761

Subgroups — #6470761

Order 512.

2176 conjugacy classes of subgroups.

The subgroup lattice has 591 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3200C212yesmaximal
2#37386C211yesmaximal, characteristic, complemented
3#34265C211yesmaximal, characteristic
4#33117C212yesmaximal, complemented
5#32905C212yesmaximal, complemented
6#29759C212yesmaximal, complemented
7#29723C212yesmaximal, complemented
8#25216C214yesmaximal, complemented
9#25205C214yesmaximal, complemented
10#25099C214yesmaximal, complemented
11#25078C214yesmaximal, complemented
12#21937C211yesmaximal, characteristic, complemented
13#20913C212yesmaximal, complemented
14C2×#536C2212yes—
15C2×#535C2211yescharacteristic
16C2×#520C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18#3200—216no—
19C2×#485C2214yes—
20C2×#482C2214yes—
21C2×#468C2214yes—
22C2×#465C2214yes—
23C2×#461C2214yes—
24C2×#448C2214yes—
25C2×#450C2214yes—
26C2×#419C2212yescomplemented
27C22×#103C2212yescomplemented
28C2×#408C2212yes—
29#2857C2212yes—
30#2734—22no—
31#2724—22no—
32#2717—22no—
33#2717—22no—
34#2621C2212yes—
35#2499C2214yescomplemented
36#2434C2214yescomplemented
37#2367—22no—
38#2366—22no—
39#2363C2214yes—
40#2361C2214yes—
41#2321—22no—
42#2309C2211yescharacteristic
43C4×#120—22no—
44#2297C2211yescharacteristic
45#2296C2211yescharacteristic
46#2274C2211yescharacteristic
47#2084C2212yescomplemented
48#2083C2212yescomplemented
49#2073C2212yes—
50#2073C2212yes—
51#2068C2214yescomplemented
52#2066C2214yescomplemented
53#2055C2214yes—
54#2055C2214yes—
55#2043C2212yescomplemented
56#2043C2212yescomplemented
57#2013C2212yes—
58#2013C2212yes—
59#1981C2214yescomplemented
60#1949C2214yescomplemented
61#1928C2214yescomplemented
62#1927C2214yescomplemented
63#1877C2214yes—
64#1877C2214yes—
65#1873C2214yes—
66#1873C2214yescomplemented
67#1872C2214yes—
68#1872C2214yescomplemented
69#1851C2214yescomplemented
70#1850C2214yescomplemented
71#1802C2214yes—
72#1798C2212yes—
73#1797C2212yes—
74#1736C2212yescomplemented
75#1735C2212yescomplemented
76C22×#41C2311yescharacteristic
77C23×Q8C2312yes—
78C23×D4C2311yescharacteristic
79C24×C4—21no—
80C22×SD16C2314yes—
81C22×SD16C2314yes—
82C22×M16C2312yes—
83C23×C8C2312yes—
84#536—28no—
85#536—28no—
86#535—24no—
87#535D418yes—
88C2×C4⋊Q8—22no—
89C2×C4⋊Q8C2312yes—
90C2×C4⋊Q8C2312yes—
91C2×#127—21no—
92C2×#124—21no—
93C2×#124—22no—
94C2×#123—22no—
95C2×#123—22no—
96C2×C22⋊Q8C2314yes—
97C2×C22⋊Q8C2314yes—
98C2×#121C2314yes—
99C2×#120—22no—
100#520—44no—
101C2×C4×Q8—21no—
102C2×C4×D4—22no—
103C2×C4×D4—21no—
104C2×C4×D4—24no—
105C2×C4×D4—22no—
106C2×C4×D4—44no—
107C2×C4×D4C2314yes—
108C2×#117C2312yes—
109C2×#117C2312yes—
110C2×#117C2311yescharacteristic
111C2×#117C2311yescharacteristic
112C2×#117C2311yescharacteristic
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2312yes—
115C22×C22⋊C4—21no—
116C22×C42C2311yescharacteristic
117#485—216no—
118#482—216no—
119#468—216no—
120#465—216no—
121#461—48no—
122#450—216no—
123#448—216no—
124C2×#107C2314yes—
125C2×#106C2314yes—
126C2×C4⋊C8C2314yes—
127C2×C4⋊C8C2314yes—
128#419—44no—
129C2×#103—44no—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138#408—28no—
139C2×C22⋊C8C2314yes—
140C2×C22⋊C8C2314yes—
141#398—24no—
142#397—24no—
143#397—24no—
144#397—24no—
145#395—24no—
146#395—24no—
147#391D412yes—
148#391C2314yes—
149#391C2314yes—
150#390—22no—
151#390—24no—
152#390D412yes—
153#390C2312yes—
154#390C2312yes—
155#389—22no—
156#389—22no—
157#389C2312yes—
158#389C2312yes—
159#388—24no—
160#388—24no—
161#387—24no—
162#387—24no—
163#386—24no—
164#386—24no—
165#386D412yes—
166#386D412yes—
167#385—21no—
168#385—21no—
169#385—24no—
170#385—24no—
171#385—24no—
172#384C2312yes—
173#382C2312yes—
174#380—22no—
175#380—22no—
176#379—22no—
177C4×C4⋊C4—22no—
178C4×C4⋊C4—22no—
179C4×C4⋊C4C2314yes—
180C4×C22⋊C4—22no—
181C4×C22⋊C4—24no—
182C4×C22⋊C4C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#336C2312yes—
188#336C2312yes—
189#336C2312yes—
190#336C2312yes—
191C25—21no—
192C2×#41—42no—
193C2×#41—42no—
194C2×#41C2×D414yes—
195C2×#41#4114yes—
196C22×Q8—22no—
197C22×Q8—24no—
198C22×Q8—24no—
199C22×Q8—42no—
200C22×Q8—42no—
201C22×Q8—42no—
202C22×Q8—42no—
203C22×Q8C2×D411yescharacteristic
204C22×Q8C2414yes—
205C22×D4—24no—
206C22×D4—21no—
207C22×D4—22no—
208C22×D4—42no—
209C22×D4—42no—
210C22×D4C2×D414yes—
211C22×D4C2×D412yes—
212C22×D4C2×D411yescharacteristic
213C22×D4C2412yes—
214C23×C4—21no—
215C23×C4—22no—
216C23×C4—22no—
217C23×C4—24no—
218C23×C4—44no—
219C23×C4—84no—
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D411yescharacteristic
222C23×C4C2411yescharacteristic
223C23×C4C2412yes—
224C2×SD16—28no—
225C2×SD16—28no—
226C2×SD16—44no—
227C2×SD16—44no—
228C2×SD16—44no—
229C2×SD16—44no—
230C2×M16—24no—
231C2×M16—42no—
232C2×M16—42no—
233C22×C8—24no—
234C22×C8—42no—
235C22×C8—42no—
236C22×C8C2414yes—
237C22×C8C2414yes—
238C4⋊Q8—44no—
239C4⋊Q8—44no—
240C4⋊Q8—48no—
241#127—28no—
242#124—28no—
243#124—48no—
244#123—48no—
245#123—48no—
246C22⋊Q8—28no—
247C22⋊Q8—28no—
248C22⋊Q8—28no—
249C22⋊Q8—28no—
250#121—28no—
251#121C2×D4116yes—
252#120—216no—
253C4×Q8—44no—
254C4×D4—28no—
255C4×D4—44no—
256C4×D4—48no—
257C4×D4—44no—
258C4×D4—48no—
259C4×D4—88no—
260#117—22no—
261#117—24no—
262#117—24no—
263#117—24no—
264#117—42no—
265#117—42no—
266#117—42no—
267#117C2×D414yes—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—24no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2411yescharacteristic
284C2×C4⋊C4C2411yescharacteristic
285C2×C4⋊C4C2414yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—44no—
299C2×C22⋊C4C2×D414yes—
300C2×C22⋊C4C2×D412yes—
301C2×C22⋊C4C2×D414yes—
302C2×C42—21no—
303C2×C42—22no—
304C2×C42—21no—
305C2×C42—22no—
306C2×C42—24no—
307C2×C42—42no—
308C2×C42C2×D412yes—
309C2×C42C2412yes—
310C2×C42C2412yes—
311#107—28no—
312#106—28no—
313C4⋊C8—44no—
314C4⋊C8—44no—
315#103—28no—
316#103—28no—
317#103—28no—
318#103—28no—
319#103—44no—
320#103—44no—
321#102—28no—
322#102—28no—
323C22⋊C8—28no—
324C22⋊C8—28no—
325#95—22no—
326#95—24no—
327#95—22no—
328#95—24no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95C2×D414yes—
334#95C2×D414yes—
335C24—22no—
336C24—24no—
337C24—42no—
338C24—42no—
339C24—82no—
340C24—82no—
341C24C22×D411yescharacteristic
342C24C22×D412yes—
343#41—48no—
344#41—48no—
345C2×Q8—24no—
346C2×Q8—24no—
347C2×Q8—24no—
348C2×Q8—24no—
349C2×Q8—24no—
350C2×Q8—24no—
351C2×Q8—41no—
352C2×Q8—41no—
353C2×Q8—48no—
354C2×Q8—48no—
355C2×Q8—44no—
356C2×Q8—82no—
357C2×Q8—82no—
358C2×Q8—84no—
359C2×Q8—84no—
360C2×Q8#12114yes—
361C2×D4—216no—
362C2×D4—24no—
363C2×D4—22no—
364C2×D4—24no—
365C2×D4—24no—
366C2×D4—48no—
367C2×D4—41no—
368C2×D4—42no—
369C2×D4—41no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—48no—
374C2×D4—84no—
375C2×D4—84no—
376C2×D4#12118yes—
377C2×D4#12114yes—
378C2×D4C22×D414yes—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—24no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—42no—
396C22×C4—48no—
397C22×C4—44no—
398C22×C4—44no—
399C22×C4—48no—
400C22×C4—84no—
401C22×C4—82no—
402C22×C4—82no—
403C22×C4—82no—
404C22×C4—84no—
405C22×C4—816no—
406C22×C4C22×D414yes—
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C22×D411yescharacteristic
410C22×C4C22×D412yes—
411C22×C4C22×D414yes—
412C22×C4C22×D412yes—
413C22×C4C2×#4114yes—
414C22×C4C2×#4112yes—
415C22×C4C2511yescharacteristic
416SD16—88no—
417SD16—88no—
418M16—84no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—84no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—42no—
434C4⋊C4—44no—
435C4⋊C4—44no—
436C4⋊C4—48no—
437C4⋊C4—48no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—82no—
441C4⋊C4—82no—
442C4⋊C4—82no—
443C4⋊C4—84no—
444C4⋊C4C22×D414yes—
445C4⋊C4H(5,𝔽2)14yes—
446C22⋊C4—24no—
447C22⋊C4—216no—
448C22⋊C4—28no—
449C22⋊C4—44no—
450C22⋊C4—42no—
451C22⋊C4—48no—
452C22⋊C4—44no—
453C22⋊C4—42no—
454C22⋊C4—44no—
455C22⋊C4—48no—
456C22⋊C4—48no—
457C22⋊C4—84no—
458C22⋊C4—84no—
459C42—24no—
460C42—24no—
461C42—24no—
462C42—42no—
463C42—44no—
464C42—42no—
465C42—44no—
466C42—84no—
467C23—24no—
468C23—22no—
469C23—21no—
470C23—22no—
471C23—21no—
472C23—22no—
473C23—24no—
474C23—22no—
475C23—24no—
476C23—48no—
477C23—44no—
478C23—42no—
479C23—42no—
480C23—44no—
481C23—88no—
482C23—84no—
483C23—88no—
484C23—84no—
485C23C2×#12112yes—
486C23C2×#12114yes—
487C23C23×D411yescharacteristic
488Q8—44no—
489Q8—44no—
490Q8—44no—
491Q8—44no—
492Q8—44no—
493Q8—88no—
494D4—28no—
495D4—44no—
496D4—48no—
497D4—48no—
498D4—44no—
499D4—816no—
500C2×C4—22no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—24no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—24no—
516C2×C4—22no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—44no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—44no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—44no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—816no—
535C2×C4—84no—
536C2×C4—82no—
537C2×C4—88no—
538C2×C4—164no—
539C2×C4C2×#12112yes—
540C2×C4C2×#12114yes—
541C2×C4#57714yes—
542C2×C4C23×D411yescharacteristic
543C2×C4C22×#4111yescharacteristic
544C2×C4C2×H(5,𝔽2)12yes—
545C2×C4#58512yes—
546C8—44no—
547C8—44no—
548C22—28no—
549C22—24no—
550C22—24no—
551C22—22no—
552C22—44no—
553C22—42no—
554C22—41no—
555C22—42no—
556C22—41no—
557C22—42no—
558C22—48no—
559C22—44no—
560C22—816no—
561C22—84no—
562C22—82no—
563C22—82no—
564C22C22×#12111yescharacteristic
565C22#348411yescharacteristic
566C22#348511yescharacteristic
567C22C2×#57712yes—
568C22#358612yes—
569C4—22no—
570C4—24no—
571C4—24no—
572C4—44no—
573C4—44no—
574C4—44no—
575C4—42no—
576C4—84no—
577C4—84no—
578C4—81no—
579C4—81no—
580C4#345912yes—
581C4#349612yes—
582C2—44no—
583C2—42no—
584C2—84no—
585C2#3498411yescharacteristic
586C2#6042912yes—
587C2#6163111yescharacteristic
588C2#6165211yescharacteristic
589C2#6180312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2297 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #37386C2semidirect
4. #33117C2semidirect
5. #32905C2semidirect
6. #29759C2semidirect
7. #29723C2semidirect
8. #25216C2semidirect
9. #25205C2semidirect
10. #25099C2semidirect
11. #25078C2semidirect
12. #21937C2semidirect
13. #20913C2semidirect
26. C2×#419C22semidirect
27. C22×#103C22semidirect
35. #2499C22semidirect
36. #2434C22semidirect
47. #2084C22semidirect
48. #2083C22semidirect
51. #2068C22semidirect
52. #2066C22semidirect
55. #2043C22semidirect
56. #2043C22semidirect
59. #1981C22semidirect
60. #1949C22semidirect
61. #1928C22semidirect
62. #1927C22semidirect
66. #1873C22semidirect
68. #1872C22semidirect
69. #1851C22semidirect
70. #1850C22semidirect
74. #1736C22semidirect
75. #1735C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32002
#373862
#342652
#331172
#329052
#297592
#297232
#252162
#252052
#250992
#250782
#219372
#209132