Finite Group Databasebeta

← #6432490

Subgroups — #6432490

Order 512.

1890 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34065C211yesmaximal, characteristic
2C2×#2314C211yesmaximal, characteristic, complemented
3#23772C214yesmaximal, complemented
4#23770C214yesmaximal, complemented
5#23768C214yesmaximal, complemented
6#23766C214yesmaximal, complemented
7#23745C211yesmaximal, characteristic, complemented
8#23734C211yesmaximal, characteristic, complemented
9#23659C211yesmaximal, characteristic, complemented
10#23642C212yesmaximal, complemented
11#23631C211yesmaximal, characteristic, complemented
12#23482C212yesmaximal
13C4×#428C212yesmaximal, complemented
14C4×#417C212yesmaximal, complemented
15#20377C211yesmaximal, characteristic, complemented
16C2×#525C2211yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C22×#117C2211yescharacteristic
19C2×#428C2212yes—
20C2×#428C2212yes—
21C2×#417C2212yes—
22C2×#417C2212yes—
23C2×#407C2212yes—
24C2×C4×M16C2212yes—
25#2324C2212yes—
26#2324C2212yes—
27#2323C2212yes—
28#2323C2212yes—
29#2322—22no—
30#2321—22no—
31#2314—22no—
32#2314—22no—
33#2314C2214yes—
34#2314C2214yes—
35#2297C2211yescharacteristic
36#2296C2211yescharacteristic
37#2293—22no—
38#2290—22no—
39C2×#388—22no—
40C2×#387—22no—
41C2×#384C2211yescharacteristic
42C2×#382C2211yescharacteristic
43#2280C2211yescharacteristic
44C2×#380C2212yes—
45C4×#117C2211yescharacteristic
46C2×C4×C4⋊C4C2212yes—
47#2100C2214yescomplemented
48#2099C2214yescomplemented
49#2096C2212yescomplemented
50#2095C2212yescomplemented
51#2094C2212yescomplemented
52#2093C2212yescomplemented
53#2085C2214yescomplemented
54#2085C2214yescomplemented
55#2077C2214yes—
56#2077C2214yes—
57#2076C2214yes—
58#2075C2214yes—
59#2074C2212yescomplemented
60#2074C2212yescomplemented
61#2073C2212yescomplemented
62#2073C2212yescomplemented
63#2036C2214yes—
64#2035C2214yes—
65#2034C2214yes—
66#2033C2214yes—
67#1776C2214yescomplemented
68#1776C2214yescomplemented
69C4×#107C2214yescomplemented
70C4×#106C2214yescomplemented
71#1763C2214yescomplemented
72#1762C2214yescomplemented
73C4×#103C2214yescomplemented
74C4×#102C2214yescomplemented
75#1750C2214yes—
76#1750C2214yes—
77#1736C2212yes—
78#1736C2212yes—
79#1736C2212yes—
80#1736C2212yes—
81#1728C2211yescharacteristic
82#1728C2211yescharacteristic
83C22×#41C2311yescharacteristic
84C24×C4C2311yescharacteristic
85C22×M16C2312yes—
86C22×M16C2312yes—
87#525—24no—
88#525—24no—
89C2×#123—21no—
90C2×C22⋊Q8C2312yes—
91C2×#121C2312yes—
92C2×#120—21no—
93C4×#41—28no—
94C2×C4×Q8C2312yes—
95C2×C4×D4—22no—
96C2×C4×D4—22no—
97C2×C4×D4C2312yes—
98C2×#117—21no—
99C2×#117—28no—
100C2×#117D412yes—
101C2×#117D412yes—
102C2×#117C2311yescharacteristic
103C2×#117C2311yescharacteristic
104C2×#117C2311yescharacteristic
105C2×#117C2312yes—
106C2×#117C2312yes—
107C22×C4⋊C4—22no—
108C22×C4⋊C4D412yes—
109C22×C4⋊C4C2312yes—
110C22×C4⋊C4C2312yes—
111C22×C22⋊C4—22no—
112C22×C22⋊C4D412yes—
113C22×C42—22no—
114C22×C42—21no—
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117#428—28no—
118#428—28no—
119C2×#107C2314yes—
120C2×#107C2314yes—
121C2×#106C2314yes—
122C2×#106C2314yes—
123#417—28no—
124#417—28no—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129#407—28no—
130C2×C22⋊C8C2314yes—
131C2×C22⋊C8C2314yes—
132C4×M16—28no—
133C2×#97C2314yes—
134C2×C4×C8C2314yes—
135#391C2311yescharacteristic
136#391C2311yescharacteristic
137#390—22no—
138#390—22no—
139#390C2311yescharacteristic
140#390C2311yescharacteristic
141#389—22no—
142#389—22no—
143#389C2312yes—
144#389C2312yes—
145#388—22no—
146#388—22no—
147#388—24no—
148#388—24no—
149#388—24no—
150#388—24no—
151#388—22no—
152#388—22no—
153#388—22no—
154#388—22no—
155#387—24no—
156#387—24no—
157#387—24no—
158#387—24no—
159#387—22no—
160#387—22no—
161#386—22no—
162#386—22no—
163#386—22no—
164#385—22no—
165#385—22no—
166#385—22no—
167#384—22no—
168#384C2312yes—
169#384C2312yes—
170#382—22no—
171#382C2312yes—
172#382C2312yes—
173#381—21no—
174#380—22no—
175#380—22no—
176#380C2314yes—
177#380C2314yes—
178#380C2312yes—
179#380C2312yes—
180#379—21no—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—22no—
184C4×C4⋊C4C2314yes—
185C4×C4⋊C4C2314yes—
186C4×C4⋊C4C2312yes—
187C4×C4⋊C4C2312yes—
188C4×C22⋊C4—22no—
189C4×C22⋊C4—22no—
190C4×C22⋊C4—22no—
191#376—21no—
192C2×#95—22no—
193C2×#95—22no—
194C2×#95—22no—
195C43—21no—
196#340C2314yes—
197#340C2314yes—
198#340C2314yes—
199#340C2314yes—
200#340C2314yes—
201#340C2314yes—
202#340C2314yes—
203#340C2314yes—
204#336C2312yes—
205#336C2312yes—
206#336C2312yes—
207#336C2312yes—
208C25C2×D411yescharacteristic
209C2×#41—22no—
210C2×#41—24no—
211C2×#41—24no—
212C2×#41#4114yes—
213C22×Q8C2411yescharacteristic
214C22×D4—21no—
215C22×D4—22no—
216C22×D4C2411yescharacteristic
217C23×C4—21no—
218C23×C4—24no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4—24no—
222C23×C4—22no—
223C23×C4—22no—
224C23×C4—22no—
225C23×C4C2×D411yescharacteristic
226C23×C4C2×D412yes—
227C23×C4C2×D412yes—
228C23×C4C2×D412yes—
229C23×C4C2412yes—
230C23×C4C2411yescharacteristic
231C2×M16—24no—
232C2×M16—24no—
233C2×M16—24no—
234C2×M16—24no—
235C2×M16—24no—
236C2×M16#4118yes—
237C22×C8C2414yes—
238C22×C8C2414yes—
239#123—28no—
240C22⋊Q8—24no—
241C22⋊Q8—24no—
242#121—24no—
243#121—24no—
244#120—28no—
245C4×Q8—24no—
246C4×Q8—24no—
247C4×D4—216no—
248C4×D4—24no—
249C4×D4—24no—
250C4×D4—48no—
251#117—22no—
252#117—28no—
253#117—28no—
254#117—24no—
255#117—24no—
256#117—24no—
257#117—24no—
258#117—24no—
259#117—24no—
260#117—44no—
261#117#4114yes—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—24no—
272C2×C4⋊C4—24no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—84no—
276C2×C4⋊C4C2×D412yes—
277C2×C4⋊C4C2×D412yes—
278C2×C4⋊C4C2×D412yes—
279C2×C4⋊C4C2×D412yes—
280C2×C4⋊C4C2414yes—
281C2×C4⋊C4C2414yes—
282C2×C4⋊C4C2412yes—
283C2×C4⋊C4C2411yescharacteristic
284C2×C4⋊C4C2411yescharacteristic
285C2×C4⋊C4C2412yes—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—28no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—22no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—84no—
300C2×C22⋊C4C2×D412yes—
301C2×C22⋊C4C2×D412yes—
302C2×C22⋊C4C2×D412yes—
303C2×C22⋊C4C2×D412yes—
304C2×C42—21no—
305C2×C42—24no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—22no—
311C2×C42—24no—
312C2×C42—24no—
313C2×C42—44no—
314C2×C42—84no—
315C2×C42C2×D412yes—
316C2×C42C2×D411yescharacteristic
317C2×C42C2×D411yescharacteristic
318C2×C42#4118yes—
319C2×C42C2411yescharacteristic
320C2×C42C2411yescharacteristic
321C2×C42C2412yes—
322#107—28no—
323#107—28no—
324#106—28no—
325#106—28no—
326#103—28no—
327#103—28no—
328#102—28no—
329#102—28no—
330C22⋊C8—28no—
331C22⋊C8—28no—
332#97—28no—
333C4×C8—28no—
334#95—24no—
335#95—24no—
336#95—24no—
337#95—24no—
338#95—24no—
339#95—24no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348C24—22no—
349C24—21no—
350C24—22no—
351C24—22no—
352C24—42no—
353C24—42no—
354C24C22×D412yes—
355C24C22×D411yescharacteristic
356#41—216no—
357#41—48no—
358C2×Q8—21no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8C2×#4112yes—
362C2×D4—24no—
363C2×D4—28no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—21no—
367C2×D4—42no—
368C2×D4—42no—
369C2×D4—82no—
370C2×D4—82no—
371C2×D4C2×#4112yes—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—24no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—22no—
392C22×C4—24no—
393C22×C4—42no—
394C22×C4—42no—
395C22×C4—42no—
396C22×C4—42no—
397C22×C4—42no—
398C22×C4—48no—
399C22×C4—48no—
400C22×C4—42no—
401C22×C4—44no—
402C22×C4—44no—
403C22×C4—44no—
404C22×C4—44no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—44no—
408C22×C4—82no—
409C22×C4—82no—
410C22×C4#12318yes—
411C22×C4C22×D412yes—
412C22×C4C22×D412yes—
413C22×C4C22×D412yes—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D411yescharacteristic
417C22×C4C2×#4114yes—
418C22×C4C2×#4112yes—
419C22×C4C2×#4112yes—
420C22×C4C2×#4114yes—
421C22×C4C2511yescharacteristic
422M16—216no—
423M16—48no—
424C2×C8—24no—
425C2×C8—24no—
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8C2×#4118yes—
430C4⋊C4—28no—
431C4⋊C4—24no—
432C4⋊C4—24no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—22no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—44no—
445C4⋊C4—44no—
446C4⋊C4—44no—
447C4⋊C4—82no—
448C4⋊C4—82no—
449C4⋊C4—84no—
450C4⋊C4—84no—
451C4⋊C4C2×#4114yes—
452C22⋊C4—28no—
453C22⋊C4—28no—
454C22⋊C4—24no—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—24no—
458C22⋊C4—24no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—44no—
462C22⋊C4—82no—
463C22⋊C4—82no—
464C22⋊C4—82no—
465C22⋊C4—82no—
466C22⋊C4—84no—
467C22⋊C4—84no—
468C42—24no—
469C42—24no—
470C42—22no—
471C42—22no—
472C42—42no—
473C42—44no—
474C42—42no—
475C42—82no—
476C42—82no—
477C42—84no—
478C42—84no—
479C42#12314yes—
480C42#12314yes—
481C42C2×#4114yes—
482C42C2×#4114yes—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—22no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—48no—
494C23—42no—
495C23—42no—
496C23—44no—
497C23—41no—
498C23—41no—
499C23—82no—
500C23—82no—
501C23—84no—
502C23C2×#12312yes—
503C23#52512yes—
504C23#53212yes—
505C23C23×D411yescharacteristic
506Q8—24no—
507Q8—42no—
508D4—24no—
509D4—48no—
510D4—42no—
511D4—88no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—48no—
528C2×C4—48no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—44no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—48no—
535C2×C4—48no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—44no—
548C2×C4—42no—
549C2×C4—44no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—44no—
553C2×C4—88no—
554C2×C4—88no—
555C2×C4—88no—
556C2×C4C2×#12314yes—
557C2×C4C2×#12314yes—
558C2×C4C2×#12312yes—
559C2×C4#52512yes—
560C2×C4#52918yes—
561C2×C4#52918yes—
562C2×C4#53212yes—
563C2×C4C22×#4112yes—
564C2×C4C22×#4111yescharacteristic
565C2×C4C22×#4111yescharacteristic
566C2×C4C22×#4112yes—
567C8—28no—
568C8—44no—
569C22—22no—
570C22—22no—
571C22—24no—
572C22—24no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—44no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—88no—
583C22—81no—
584C22—81no—
585C22—82no—
586C22#239514yes—
587C22C22×#12311yescharacteristic
588C22C2×#52511yescharacteristic
589C22C2×#53211yescharacteristic
590C22#358512yes—
591C22#358612yes—
592C4—24no—
593C4—24no—
594C4—42no—
595C4—42no—
596C4—44no—
597C4—42no—
598C4—42no—
599C4—81no—
600C4—84no—
601C4—81no—
602C4—82no—
603C4—84no—
604C4#239514yes—
605C4C2×#52912yes—
606C4C2×#52912yes—
607C2—24no—
608C2—82no—
609C2—82no—
610C2C2×#239511yescharacteristic
611C2#6051712yes—
612C2#6056712yes—
613C2#6070411yescharacteristic
614C2#6070511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2314C2semidirect
3. #23772C2semidirect
4. #23770C2semidirect
5. #23768C2semidirect
6. #23766C2semidirect
7. #23745C2semidirect
8. #23734C2semidirect
9. #23659C2semidirect
10. #23642C2semidirect
11. #23631C2semidirect
13. C4×#428C2semidirect
14. C4×#417C2semidirect
15. #20377C2semidirect
47. #2100C22semidirect
48. #2099C22semidirect
49. #2096C22semidirect
50. #2095C22semidirect
51. #2094C22semidirect
52. #2093C22semidirect
53. #2085C22semidirect
54. #2085C22semidirect
59. #2074C22semidirect
60. #2074C22semidirect
61. #2073C22semidirect
62. #2073C22semidirect
67. #1776C22semidirect
68. #1776C22semidirect
69. C4×#107C22semidirect
70. C4×#106C22semidirect
71. #1763C22semidirect
72. #1762C22semidirect
73. C4×#103C22semidirect
74. C4×#102C22semidirect

Maximal subgroups

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

subgroupindex
#340652
C2×#23142
#237722
#237702
#237682
#237662
#237452
#237342
#236592
#236422
#236312
#234822
C4×#4282
C4×#4172
#203772