Finite Group Databasebeta

← #6470739

Subgroups — #6470739

Order 512.

2190 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3190C212yesmaximal
2#37453C211yesmaximal, characteristic, complemented
3#34200C211yesmaximal, characteristic
4#33117C212yesmaximal, complemented
5#32916C212yesmaximal, complemented
6#29766C212yesmaximal, complemented
7#29726C212yesmaximal, complemented
8#25212C214yesmaximal, complemented
9#25208C214yesmaximal, complemented
10#25082C214yesmaximal, complemented
11#25078C214yesmaximal, complemented
12#21931C211yesmaximal, characteristic, complemented
13C2×#1872C212yesmaximal, complemented
14C2×#525C2211yescharacteristic
15C22×C22⋊Q8C2212yes—
16C2×#520C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18#3190—216no—
19C2×#485C2214yes—
20C2×#483C2212yes—
21C2×#480C2212yes—
22C2×#467C2214yes—
23C2×#465C2214yes—
24C2×#462C2212yescomplemented
25C2×#461C2212yescomplemented
26C2×#452C2214yes—
27C2×#451C2212yes—
28C2×#448C2212yes—
29C22×#103C2214yes—
30C22×C22⋊C8C2212yes—
31#2817—22no—
32#2627—22no—
33#2627—22no—
34#2621C2212yes—
35#2621C2212yes—
36#2503C2214yes—
37#2483—22no—
38#2434C2214yes—
39#2361C2214yes—
40#2360C2214yes—
41#2352—22no—
42D4×C4⋊C4—22no—
43#2322—22no—
44#2317C2211yescharacteristic
45#2311C2211yescharacteristic
46C4×#120—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#1872C2214yes—
66#1872C2214yes—
67#1872C2214yes—
68#1872C2214yes—
69#1850C2214yes—
70#1850C2214yes—
71#1805C2212yescomplemented
72#1804C2212yescomplemented
73#1798C2212yes—
74#1798C2212yes—
75C2×#340C2214yes—
76C22×#41C2311yescharacteristic
77C23×Q8C2312yes—
78C24×C4—21no—
79C24×C4C2311yescharacteristic
80C22×Q16C2314yes—
81C22×SD16C2314yes—
82C23×C8C2314yes—
83C2×C4⋊Q8C2312yes—
84C2×C4⋊Q8C2312yes—
85#525—24no—
86#525D418yes—
87C2×#123—21no—
88C2×C22⋊Q8—22no—
89C2×C22⋊Q8—22no—
90C2×C22⋊Q8—28no—
91C2×C22⋊Q8—28no—
92C2×C22⋊Q8C2314yes—
93C2×C22⋊Q8C2314yes—
94C2×C22⋊Q8C2312yes—
95C2×#121C2312yes—
96C2×#120—21no—
97#520—44no—
98C2×C4×Q8C2312yes—
99C2×C4×D4—22no—
100C2×C4×D4—22no—
101C2×C4×D4—24no—
102C2×C4×D4C2312yes—
103C2×#117—21no—
104C2×#117C2311yescharacteristic
105C2×#117C2311yescharacteristic
106C2×#117C2311yescharacteristic
107C22×C4⋊C4—21no—
108C22×C4⋊C4—22no—
109C22×C4⋊C4—22no—
110C22×C4⋊C4—21no—
111C22×C4⋊C4—44no—
112C22×C4⋊C4C2312yes—
113C22×C4⋊C4C2311yescharacteristic
114C22×C4⋊C4C2314yes—
115C22×C4⋊C4C2311yescharacteristic
116C22×C22⋊C4—22no—
117#485—216no—
118#483—28no—
119#480—28no—
120#467—216no—
121#465—216no—
122#462—44no—
123#461—44no—
124#452—216no—
125#451—28no—
126#448—28no—
127C2×#107C2314yes—
128C2×#106C2314yes—
129C2×C4⋊C8C2314yes—
130C2×C4⋊C8C2312yes—
131C2×C4⋊C8C2312yes—
132C2×#103—48no—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#103C2314yes—
137C2×#103C2314yes—
138C2×#103C2314yes—
139C2×#103C2314yes—
140C2×#102C2314yes—
141C2×C22⋊C8—28no—
142C2×C22⋊C8C2314yes—
143C2×C22⋊C8C2312yes—
144C2×C22⋊C8C2312yes—
145#400—24no—
146#400—24no—
147#398—22no—
148#398—22no—
149#398—24no—
150#397—22no—
151#397—22no—
152#397—24no—
153#395—24no—
154#395—24no—
155#391C2314yes—
156#391C2312yes—
157#391C2314yes—
158#390C2312yes—
159#389—24no—
160#389C2314yes—
161#388—24no—
162#387—24no—
163#386—22no—
164#386—24no—
165#385—22no—
166#385—24no—
167#385—24no—
168#385—24no—
169#385—24no—
170#384—21no—
171#382—21no—
172#382C2312yes—
173#382C2312yes—
174#381—24no—
175#381—24no—
176#381D412yes—
177#380—22no—
178#380D412yes—
179#380D412yes—
180#379D412yes—
181C4×C4⋊C4—22no—
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—22no—
206C22×D4—21no—
207C22×D4C2411yescharacteristic
208C23×C4—21no—
209C23×C4—21no—
210C23×C4—22no—
211C23×C4—24no—
212C23×C4—22no—
213C23×C4—21no—
214C23×C4—22no—
215C23×C4—42no—
216C23×C4—42no—
217C23×C4—44no—
218C23×C4—84no—
219C23×C4C2×D412yes—
220C23×C4C2×D411yescharacteristic
221C23×C4C2×D414yes—
222C23×C4#4114yes—
223C23×C4C2412yes—
224C23×C4C2411yescharacteristic
225C2×Q16—28no—
226C2×Q16—44no—
227C2×Q16—44no—
228C2×SD16—28no—
229C2×SD16—44no—
230C2×SD16—44no—
231C22×C8—28no—
232C22×C8—44no—
233C22×C8—44no—
234C22×C8C2414yes—
235C22×C8C2414yes—
236C4⋊Q8—44no—
237C4⋊Q8—42no—
238C4⋊Q8—42no—
239#123—28no—
240C22⋊Q8—24no—
241C22⋊Q8—28no—
242C22⋊Q8—28no—
243C22⋊Q8—28no—
244C22⋊Q8—28no—
245C22⋊Q8—48no—
246C22⋊Q8—48no—
247C22⋊Q8C2×D418yes—
248#121—24no—
249#121C2×D418yes—
250#120—28no—
251C4×Q8—24no—
252C4×Q8—42no—
253C4×D4—216no—
254C4×D4—24no—
255C4×D4—42no—
256C4×D4—48no—
257C4×D4—88no—
258#117—22no—
259#117—42no—
260#117—42no—
261#117—44no—
262#117C2×D414yes—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—28no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—24no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—44no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—48no—
282C2×C4⋊C4—44no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—48no—
285C2×C4⋊C4C2×D411yescharacteristic
286C2×C4⋊C4C2×D411yescharacteristic
287C2×C4⋊C4C2×D412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2414yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—48no—
300C2×C22⋊C4C2×D414yes—
301C2×C22⋊C4C2×D414yes—
302C2×C22⋊C4C2×D412yes—
303C2×C22⋊C4C2×D412yes—
304C2×C42—21no—
305C2×C42—22no—
306C2×C42—24no—
307C2×C42C2412yes—
308C2×C42C2412yes—
309#107—28no—
310#106—28no—
311C4⋊C8—44no—
312C4⋊C8—42no—
313C4⋊C8—42no—
314#103—28no—
315#103—28no—
316#103—28no—
317#103—28no—
318#103—28no—
319#103—44no—
320#103—44no—
321#102—28no—
322C22⋊C8—24no—
323C22⋊C8—24no—
324C22⋊C8—28no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—22no—
329#95—24no—
330#95—22no—
331#95—24no—
332#95—24no—
333#95—24no—
334#95—24no—
335#95C2×D414yes—
336#95C2×D414yes—
337C24—24no—
338C24—24no—
339C24—21no—
340C24—41no—
341C24—41no—
342C24#12114yes—
343C24C22×D411yescharacteristic
344C24C22×D412yes—
345#41—48no—
346#41—48no—
347C2×Q8—21no—
348C2×Q8—24no—
349C2×Q8—24no—
350C2×Q8—22no—
351C2×Q8—24no—
352C2×Q8—24no—
353C2×Q8—22no—
354C2×Q8—24no—
355C2×Q8—24no—
356C2×Q8—416no—
357C2×Q8—44no—
358C2×Q8—82no—
359C2×Q8—82no—
360C2×Q8—88no—
361C2×Q8C22×D412yes—
362C2×D4—28no—
363C2×D4—24no—
364C2×D4—21no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—44no—
370C2×D4—44no—
371C2×D4C22×D412yes—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—22no—
384C22×C4—48no—
385C22×C4—48no—
386C22×C4—42no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—41no—
390C22×C4—44no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—48no—
394C22×C4—42no—
395C22×C4—44no—
396C22×C4—44no—
397C22×C4—48no—
398C22×C4—44no—
399C22×C4—44no—
400C22×C4—84no—
401C22×C4—82no—
402C22×C4—84no—
403C22×C4—82no—
404C22×C4—82no—
405C22×C4—82no—
406C22×C4—816no—
407C22×C4#12118yes—
408C22×C4#12114yes—
409C22×C4C22×D412yes—
410C22×C4C22×D414yes—
411C22×C4C22×D412yes—
412C22×C4C22×D412yes—
413C22×C4C22×D411yescharacteristic
414C22×C4C22×D412yes—
415C22×C4C22×D414yes—
416C22×C4C2×#4114yes—
417C22×C4C2×#4112yes—
418C22×C4C2511yescharacteristic
419Q16—88no—
420SD16—88no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—88no—
428C4⋊C4—24no—
429C4⋊C4—28no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—41no—
436C4⋊C4—42no—
437C4⋊C4—41no—
438C4⋊C4—48no—
439C4⋊C4—44no—
440C4⋊C4—42no—
441C4⋊C4—44no—
442C4⋊C4—48no—
443C4⋊C4—48no—
444C4⋊C4—84no—
445C4⋊C4—82no—
446C4⋊C4—82no—
447C4⋊C4—84no—
448C4⋊C4C22×D414yes—
449C4⋊C4H(5,𝔽2)14yes—
450C22⋊C4—24no—
451C22⋊C4—28no—
452C22⋊C4—28no—
453C22⋊C4—28no—
454C22⋊C4—28no—
455C22⋊C4—42no—
456C22⋊C4—42no—
457C22⋊C4—44no—
458C22⋊C4—48no—
459C22⋊C4—48no—
460C22⋊C4—84no—
461C42—24no—
462C42—22no—
463C42—22no—
464C42—24no—
465C42—44no—
466C42—84no—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—21no—
472C23—21no—
473C23—216no—
474C23—22no—
475C23—22no—
476C23—22no—
477C23—44no—
478C23—44no—
479C23—48no—
480C23—44no—
481C23—41no—
482C23—41no—
483C23—42no—
484C23—42no—
485C23C2×#12112yes—
486C23C2×#12114yes—
487C23#57714yes—
488C23#57814yes—
489C23C23×D411yescharacteristic
490Q8—24no—
491Q8—42no—
492Q8—44no—
493Q8—44no—
494Q8—44no—
495Q8—44no—
496Q8—88no—
497D4—24no—
498D4—48no—
499D4—416no—
500D4—42no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—28no—
506C2×C4—24no—
507C2×C4—28no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—48no—
520C2×C4—44no—
521C2×C4—48no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—42no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—84no—
536C2×C4—816no—
537C2×C4—88no—
538C2×C4—84no—
539C2×C4—88no—
540C2×C4—164no—
541C2×C4C2×#12112yes—
542C2×C4C2×#12114yes—
543C2×C4C23×D411yescharacteristic
544C2×C4C22×#4111yescharacteristic
545C2×C4C2×H(5,𝔽2)12yes—
546C2×C4#58512yes—
547C8—44no—
548C8—44no—
549C22—24no—
550C22—24no—
551C22—24no—
552C22—24no—
553C22—24no—
554C22—24no—
555C22—24no—
556C22—22no—
557C22—416no—
558C22—42no—
559C22—41no—
560C22—41no—
561C22—44no—
562C22—42no—
563C22—82no—
564C22—81no—
565C22—81no—
566C22#304618yes—
567C22C22×#12111yescharacteristic
568C22#348411yescharacteristic
569C22#348511yescharacteristic
570C22C2×#57712yes—
571C22C2×#57812yes—
572C4—22no—
573C4—24no—
574C4—44no—
575C4—44no—
576C4—42no—
577C4—42no—
578C4—44no—
579C4—42no—
580C4—84no—
581C4—84no—
582C4—82no—
583C4#345912yes—
584C4#349612yes—
585C2—28no—
586C2—42no—
587C2—82no—
588C2#3498411yescharacteristic
589C2C2×#304612yes—
590C2#6163111yescharacteristic
591C2#6163411yescharacteristic
592C2#6176712yes—
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. #25212C2semidirect
9. #25208C2semidirect
10. #25082C2semidirect
11. #25078C2semidirect
12. #21931C2semidirect
13. C2×#1872C2semidirect
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×#31902
#374532
#342002
#331172
#329162
#297662
#297262
#252122
#252082
#250822
#250782
#219312
C2×#18722