Finite Group Databasebeta

← #6470750

Subgroups — #6470750

Order 512.

2066 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3196C212yesmaximal
2#37453C211yesmaximal, characteristic, complemented
3#34201C211yesmaximal, characteristic
4#33117C212yesmaximal, complemented
5#32916C212yesmaximal, complemented
6#29766C212yesmaximal, complemented
7#29726C212yesmaximal, complemented
8#25229C214yesmaximal, complemented
9#25225C214yesmaximal, complemented
10#25099C214yesmaximal, complemented
11#25095C214yesmaximal, complemented
12#21935C211yesmaximal, characteristic, complemented
13#20911C212yesmaximal, complemented
14C2×#525C2211yescharacteristic
15C22×C22⋊Q8C2212yes—
16C2×#520C2211yescharacteristic
17C22×#117C2211yescharacteristic
18#3196—216no—
19C2×#484C2212yes—
20C2×#481C2212yes—
21C2×#485C2214yes—
22C2×#470C2214yes—
23C2×#468C2214yes—
24C2×#462C2212yescomplemented
25C2×#461C2212yescomplemented
26C2×#452C2214yes—
27C2×#451C2212yes—
28C2×#448C2212yes—
29C2×#419C2214yes—
30C2×#407C2212yes—
31#2817—22no—
32#2627—22no—
33#2627—22no—
34#2621C2212yes—
35#2621C2212yes—
36#2503C2214yes—
37#2483—22no—
38#2434C2214yes—
39#2363C2214yes—
40#2362C2214yes—
41#2354—22no—
42#2352—22no—
43#2321—22no—
44#2317C2211yescharacteristic
45#2311C2211yescharacteristic
46C4×#123—22no—
47#2280C2211yescharacteristic
48#2274C2211yescharacteristic
49#2085C2214yescomplemented
50#2074C2212yescomplemented
51#2073C2212yescomplemented
52#2068C2214yescomplemented
53#2066C2214yescomplemented
54#2056C2214yescomplemented
55#2055C2214yescomplemented
56#2043C2214yescomplemented
57#2016C2212yescomplemented
58#2013C2212yescomplemented
59#1981C2214yescomplemented
60#1949C2214yescomplemented
61#1928C2214yescomplemented
62#1927C2214yescomplemented
63#1878C2214yescomplemented
64#1877C2214yescomplemented
65#1873C2214yes—
66#1873C2214yes—
67#1872C2214yes—
68#1872C2214yes—
69#1851C2214yes—
70#1851C2214yes—
71#1805C2212yescomplemented
72#1804C2212yescomplemented
73#1798C2212yes—
74#1798C2212yes—
75#1736C2214yes—
76C22×#41C2311yescharacteristic
77C23×Q8C2312yes—
78C24×C4C2311yescharacteristic
79C22×Q16C2314yes—
80C22×SD16C2314yes—
81C22×M16C2314yes—
82C2×C4⋊Q8C2312yes—
83C2×C4⋊Q8C2312yes—
84#525—24no—
85#525D418yes—
86C2×#123—21no—
87C2×C22⋊Q8—22no—
88C2×C22⋊Q8—22no—
89C2×C22⋊Q8—28no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8C2314yes—
92C2×C22⋊Q8C2314yes—
93C2×C22⋊Q8C2312yes—
94C2×#121C2312yes—
95C2×#120—21no—
96#520—44no—
97C2×C4×Q8C2312yes—
98C2×C4×D4—22no—
99C2×C4×D4—22no—
100C2×C4×D4C2312yes—
101C2×#117—22no—
102C2×#117—21no—
103C2×#117—44no—
104C2×#117C2314yes—
105C2×#117C2311yescharacteristic
106C2×#117C2311yescharacteristic
107C2×#117C2311yescharacteristic
108C22×C4⋊C4—22no—
109C22×C4⋊C4—21no—
110C22×C4⋊C4C2312yes—
111C22×C4⋊C4C2311yescharacteristic
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4—21no—
114C22×C22⋊C4—22no—
115C22×C42—21no—
116#485—216no—
117#484—28no—
118#481—28no—
119#470—216no—
120#468—216no—
121#462—44no—
122#461—44no—
123#452—216no—
124#451—28no—
125#448—28no—
126C2×#107C2314yes—
127C2×#106C2314yes—
128C2×C4⋊C8C2314yes—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2312yes—
131#419—48no—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#102C2314yes—
140#407—28no—
141C2×C22⋊C8C2314yes—
142C2×C22⋊C8C2312yes—
143C2×C22⋊C8C2312yes—
144#400—24no—
145#400—24no—
146#398—22no—
147#398—22no—
148#398—24no—
149#397—22no—
150#397—22no—
151#397—24no—
152#395—24no—
153#395—24no—
154#391C2314yes—
155#391C2312yes—
156#391C2314yes—
157#390—24no—
158#390C2312yes—
159#389C2314yes—
160#388—24no—
161#388—24no—
162#388—24no—
163#387—24no—
164#386—22no—
165#386—24no—
166#385—22no—
167#385—24no—
168#385—24no—
169#384—21no—
170#382—21no—
171#382C2312yes—
172#382C2312yes—
173#381—24no—
174#381—24no—
175#381D412yes—
176#380—22no—
177#380D412yes—
178#380D412yes—
179#379D412yes—
180C4×C4⋊C4—22no—
181C4×C4⋊C4—24no—
182C4×C4⋊C4C2314yes—
183C4×C22⋊C4—22no—
184C4×C22⋊C4—24no—
185C4×C22⋊C4C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#336C2312yes—
191#336C2312yes—
192#336C2314yes—
193C25C2×D411yescharacteristic
194C2×#41—22no—
195C2×#41—42no—
196C2×#41—42no—
197C2×#41C2×D414yes—
198C22×Q8—22no—
199C22×Q8—24no—
200C22×Q8—24no—
201C22×Q8—44no—
202C22×Q8—44no—
203C22×Q8C2414yes—
204C22×Q8C2411yescharacteristic
205C22×D4—21no—
206C22×D4—22no—
207C22×D4C2411yescharacteristic
208C23×C4—21no—
209C23×C4—21no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4—21no—
213C23×C4—22no—
214C23×C4—42no—
215C23×C4—42no—
216C23×C4C2×D412yes—
217C23×C4C2×D411yescharacteristic
218C23×C4C2×D414yes—
219C23×C4#4114yes—
220C23×C4C2411yescharacteristic
221C23×C4C2412yes—
222C2×Q16—28no—
223C2×Q16—44no—
224C2×Q16—44no—
225C2×SD16—28no—
226C2×SD16—44no—
227C2×SD16—44no—
228C2×M16—28no—
229C2×M16—44no—
230C2×M16—44no—
231C22×C8C2414yes—
232C22×C8C2414yes—
233C4⋊Q8—44no—
234C4⋊Q8—42no—
235C4⋊Q8—42no—
236#123—28no—
237C22⋊Q8—24no—
238C22⋊Q8—28no—
239C22⋊Q8—28no—
240C22⋊Q8—28no—
241C22⋊Q8—28no—
242C22⋊Q8—48no—
243C22⋊Q8—48no—
244C22⋊Q8C2×D418yes—
245#121—24no—
246#121C2×D418yes—
247#120—28no—
248C4×Q8—24no—
249C4×Q8—42no—
250C4×D4—216no—
251C4×D4—24no—
252C4×D4—42no—
253C4×D4—48no—
254#117—22no—
255#117—28no—
256#117—42no—
257#117—42no—
258#117—44no—
259#117—48no—
260#117—44no—
261#117C2×D414yes—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—42no—
277C2×C4⋊C4—44no—
278C2×C4⋊C4—48no—
279C2×C4⋊C4C2×D411yescharacteristic
280C2×C4⋊C4C2×D411yescharacteristic
281C2×C4⋊C4C2×D412yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2414yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—44no—
295C2×C22⋊C4—48no—
296C2×C22⋊C4C2×D412yes—
297C2×C22⋊C4C2×D412yes—
298C2×C22⋊C4C2×D414yes—
299C2×C22⋊C4C2×D414yes—
300C2×C42—21no—
301C2×C42—24no—
302C2×C42—22no—
303C2×C42—24no—
304C2×C42—44no—
305C2×C42C2412yes—
306C2×C42C2412yes—
307#107—28no—
308#106—28no—
309C4⋊C8—44no—
310C4⋊C8—42no—
311C4⋊C8—42no—
312#103—28no—
313#103—28no—
314#103—28no—
315#103—28no—
316#103—28no—
317#103—44no—
318#103—44no—
319#102—28no—
320C22⋊C8—24no—
321C22⋊C8—24no—
322C22⋊C8—28no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—22no—
328#95—22no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95C2×D414yes—
334#95C2×D414yes—
335C24—21no—
336C24—41no—
337C24—41no—
338C24—42no—
339C24—42no—
340C24#12114yes—
341C24C22×D412yes—
342C24C22×D411yescharacteristic
343#41—48no—
344#41—48no—
345C2×Q8—21no—
346C2×Q8—24no—
347C2×Q8—24no—
348C2×Q8—22no—
349C2×Q8—24no—
350C2×Q8—24no—
351C2×Q8—22no—
352C2×Q8—24no—
353C2×Q8—24no—
354C2×Q8—416no—
355C2×Q8—44no—
356C2×Q8—82no—
357C2×Q8—82no—
358C2×Q8—88no—
359C2×Q8C22×D412yes—
360C2×D4—24no—
361C2×D4—28no—
362C2×D4—21no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—42no—
366C2×D4—42no—
367C2×D4—82no—
368C2×D4—82no—
369C2×D4C22×D412yes—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—21no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—48no—
383C22×C4—48no—
384C22×C4—41no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—41no—
388C22×C4—44no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—44no—
393C22×C4—44no—
394C22×C4—44no—
395C22×C4—44no—
396C22×C4—84no—
397C22×C4—82no—
398C22×C4—82no—
399C22×C4—82no—
400C22×C4—82no—
401C22×C4#12118yes—
402C22×C4#12114yes—
403C22×C4C22×D414yes—
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4C22×D412yes—
407C22×C4C22×D411yescharacteristic
408C22×C4C22×D412yes—
409C22×C4C22×D414yes—
410C22×C4C2×#4114yes—
411C22×C4C2×#4112yes—
412C22×C4C2511yescharacteristic
413Q16—88no—
414SD16—88no—
415M16—88no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C4⋊C4—24no—
423C4⋊C4—28no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—41no—
430C4⋊C4—42no—
431C4⋊C4—41no—
432C4⋊C4—44no—
433C4⋊C4—42no—
434C4⋊C4—44no—
435C4⋊C4—48no—
436C4⋊C4—48no—
437C4⋊C4—84no—
438C4⋊C4—82no—
439C4⋊C4—82no—
440C4⋊C4—84no—
441C4⋊C4C22×D414yes—
442C4⋊C4H(5,𝔽2)14yes—
443C22⋊C4—24no—
444C22⋊C4—28no—
445C22⋊C4—28no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—42no—
449C22⋊C4—42no—
450C22⋊C4—44no—
451C22⋊C4—48no—
452C22⋊C4—48no—
453C22⋊C4—48no—
454C22⋊C4—84no—
455C42—24no—
456C42—22no—
457C42—22no—
458C42—24no—
459C42—48no—
460C42—44no—
461C42—84no—
462C23—22no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—21no—
470C23—21no—
471C23—22no—
472C23—22no—
473C23—44no—
474C23—48no—
475C23—42no—
476C23—42no—
477C23—41no—
478C23—41no—
479C23—84no—
480C23—82no—
481C23—82no—
482C23C2×#12112yes—
483C23C2×#12114yes—
484C23C23×D411yescharacteristic
485Q8—24no—
486Q8—42no—
487Q8—44no—
488Q8—44no—
489Q8—44no—
490Q8—44no—
491Q8—88no—
492D4—24no—
493D4—48no—
494D4—42no—
495D4—88no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—22no—
500C2×C4—28no—
501C2×C4—24no—
502C2×C4—28no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—24no—
514C2×C4—48no—
515C2×C4—44no—
516C2×C4—48no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—44no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—84no—
531C2×C4—88no—
532C2×C4—88no—
533C2×C4—84no—
534C2×C4—164no—
535C2×C4C2×#12112yes—
536C2×C4C2×#12114yes—
537C2×C4C23×D411yescharacteristic
538C2×C4C22×#4111yescharacteristic
539C2×C4C2×H(5,𝔽2)12yes—
540C2×C4#58512yes—
541C8—44no—
542C8—44no—
543C22—24no—
544C22—24no—
545C22—24no—
546C22—24no—
547C22—22no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—41no—
552C22—42no—
553C22—41no—
554C22—42no—
555C22—44no—
556C22—42no—
557C22—88no—
558C22—82no—
559C22—81no—
560C22—81no—
561C22C22×#12111yescharacteristic
562C22#348411yescharacteristic
563C22#348511yescharacteristic
564C22#358612yes—
565C22#358612yes—
566C4—22no—
567C4—24no—
568C4—44no—
569C4—44no—
570C4—42no—
571C4—42no—
572C4—44no—
573C4—42no—
574C4—84no—
575C4—84no—
576C4—82no—
577C4#345912yes—
578C4#349612yes—
579C2—44no—
580C2—42no—
581C2—82no—
582C2#3498411yescharacteristic
583C2#6042712yes—
584C2#6164911yescharacteristic
585C2#6165211yescharacteristic
586C2#6177912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2280 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #37453C2semidirect
4. #33117C2semidirect
5. #32916C2semidirect
6. #29766C2semidirect
7. #29726C2semidirect
8. #25229C2semidirect
9. #25225C2semidirect
10. #25099C2semidirect
11. #25095C2semidirect
12. #21935C2semidirect
13. #20911C2semidirect
24. C2×#462C22semidirect
25. C2×#461C22semidirect
49. #2085C22semidirect
50. #2074C22semidirect
51. #2073C22semidirect
52. #2068C22semidirect
53. #2066C22semidirect
54. #2056C22semidirect
55. #2055C22semidirect
56. #2043C22semidirect
57. #2016C22semidirect
58. #2013C22semidirect
59. #1981C22semidirect
60. #1949C22semidirect
61. #1928C22semidirect
62. #1927C22semidirect
63. #1878C22semidirect
64. #1877C22semidirect
71. #1805C22semidirect
72. #1804C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31962
#374532
#342012
#331172
#329162
#297662
#297262
#252292
#252252
#250992
#250952
#219352
#209112