Finite Group Databasebeta

← #6506770

Subgroups — #6506770

Order 512.

1662 conjugacy classes of subgroups.

The subgroup lattice has 627 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35033C211yesmaximal, characteristic
2C2×#2442C211yesmaximal, characteristic
3#29302C211yesmaximal, characteristic, complemented
4#29301C211yesmaximal, characteristic, complemented
5#29298C212yesmaximal, complemented
6#28787C214yesmaximal, complemented
7#28785C214yesmaximal, complemented
8#27956C211yesmaximal, characteristic, complemented
9#27956C211yesmaximal, characteristic, complemented
10#26075C214yesmaximal, complemented
11#26073C214yesmaximal, complemented
12#25815C212yesmaximal, complemented
13#21944C212yesmaximal
14#20844C212yesmaximal
15#20524C211yesmaximal, characteristic
16C22×#125C2211yescharacteristic
17C23×C4⋊C4C2211yescharacteristic
18C2×#499C2214yescomplemented
19C2×#428C2214yes—
20C2×#407C2212yes—
21#2857C2211yescharacteristic
22#2857C2211yescharacteristic
23#2851C2212yes—
24#2626—21no—
25#2626—21no—
26#2608—22no—
27#2607—22no—
28#2442—22no—
29#2442—22no—
30#2442C2214yes—
31#2442C2214yes—
32#2441C2212yes—
33#2441C2212yes—
34#2439C2212yes—
35#2439C2212yes—
36C2×#400—21no—
37C2×#398—21no—
38#2329—21no—
39#2329—21no—
40C2×#389C2211yescharacteristic
41C2×#389C2211yescharacteristic
42C2×#385—21no—
43C2×#382C2211yescharacteristic
44#2280C2212yes—
45C2×#380—21no—
46C2×#380C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48#2085C2214yescomplemented
49#2085C2214yescomplemented
50#2085C2214yescomplemented
51#2084C2212yescomplemented
52#2083C2212yescomplemented
53#2059C2214yescomplemented
54#2059C2214yescomplemented
55#2059C2214yescomplemented
56#2058C2212yescomplemented
57#2057C2212yescomplemented
58#1993C2212yescomplemented
59#1993C2212yescomplemented
60#1991C2214yescomplemented
61#1944C2214yescomplemented
62#1943C2214yescomplemented
63#1942C2214yescomplemented
64#1941C2214yescomplemented
65#1854C2214yes—
66#1854C2214yes—
67#1853C2212yes—
68#1853C2212yes—
69#1853C2214yes—
70#1853C2212yes—
71#1853C2212yes—
72#1852C2212yes—
73#1852C2212yes—
74#1852C2214yes—
75#1852C2212yes—
76#1852C2212yes—
77#1847C2212yescomplemented
78#1847C2212yescomplemented
79#1809C2214yes—
80#1809C2214yes—
81#1753C2214yes—
82#1753C2214yes—
83#1749C2211yescharacteristic
84#1749C2211yescharacteristic
85C4×C22⋊C8C2212yes—
86#1736C2214yes—
87C24×C4—21no—
88C24×C4C2311yescharacteristic
89C22×M16C2314yes—
90C2×C4⋊Q8C2311yescharacteristic
91C2×C4⋊Q8C2311yescharacteristic
92C2×#125—22no—
93C2×#125—44no—
94C2×#125C2314yes—
95C2×#125C2312yes—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—21no—
98C2×#117C2312yes—
99C2×#117C2312yes—
100C22×C4⋊C4—21no—
101C22×C4⋊C4—21no—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—22no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—44no—
107C22×C4⋊C4C2311yescharacteristic
108C22×C4⋊C4C2314yes—
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4—21no—
113C22×C22⋊C4D411yescharacteristic
114C22×C22⋊C4D411yescharacteristic
115C22×C42C2311yescharacteristic
116C22×C42C2311yescharacteristic
117#499—48no—
118#428—48no—
119C2×#107C2314yes—
120C2×#107C2314yes—
121C2×#107C2314yes—
122C2×#107C2314yes—
123C2×#106C2314yes—
124C2×#106C2314yes—
125C2×#106C2314yes—
126C2×#106C2314yes—
127#407Q8116yescomplemented
128C2×C22⋊C8C2312yes—
129C2×C22⋊C8C2312yes—
130C2×C22⋊C8C2314yes—
131C2×#97C2312yes—
132C2×#97C2312yes—
133C2×C4×C8C2314yes—
134#400—24no—
135#400—24no—
136#399—22no—
137#399—22no—
138#398—24no—
139#398—24no—
140#396—22no—
141#396—22no—
142#396—22no—
143#396—22no—
144#393—22no—
145#393—22no—
146#391—21no—
147#391—21no—
148#391—22no—
149#391—22no—
150#389—21no—
151#389—21no—
152#389—22no—
153#389—21no—
154#389—21no—
155#389—22no—
156#389—22no—
157#389C2312yes—
158#389C2312yes—
159#389C2312yes—
160#389C2312yes—
161#389C2314yes—
162#389C2312yes—
163#389C2312yes—
164#387—22no—
165#387—22no—
166#387—22no—
167#387—22no—
168#387—22no—
169#387—22no—
170#385—22no—
171#385—22no—
172#385—24no—
173#385—24no—
174#384C2311yescharacteristic
175#384C2311yescharacteristic
176#382—21no—
177#382—21no—
178#382C2312yes—
179#382C2312yes—
180#382C2312yes—
181#381—22no—
182#380—21no—
183#380—21no—
184#380—22no—
185#380—24no—
186#380—24no—
187#380C2312yes—
188#380C2312yes—
189#380C2314yes—
190#380C2312yes—
191#380C2312yes—
192#379—22no—
193C4×C22⋊C4—22no—
194C4×C22⋊C4—22no—
195C4×C22⋊C4D412yes—
196C4×C22⋊C4D412yes—
197C4×C22⋊C4C2312yes—
198C4×C22⋊C4C2312yes—
199C2×#95—21no—
200C2×#95—21no—
201C2×#95—21no—
202C2×#95—21no—
203C2×#95—21no—
204C2×#95D411yescharacteristic
205C2×#95D411yescharacteristic
206#340C2314yes—
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#340C2314yes—
213#340C2314yes—
214#336C2312yes—
215#336C2312yes—
216#336C2314yes—
217C25C2×D411yescharacteristic
218C22×Q8—21no—
219C22×Q8—21no—
220C23×C4—21no—
221C23×C4—21no—
222C23×C4—21no—
223C23×C4—21no—
224C23×C4—24no—
225C23×C4—42no—
226C23×C4—42no—
227C23×C4—44no—
228C23×C4—44no—
229C23×C4—82no—
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×D411yescharacteristic
233C23×C4C2×D411yescharacteristic
234C23×C4C2×D411yescharacteristic
235C23×C4C2×D412yes—
236C23×C4C2×Q818yes—
237C23×C4C2411yescharacteristic
238C23×C4C2412yes—
239C2×M16—44no—
240C2×M16—44no—
241C2×M16C2×Q8116yes—
242C22×C8C2414yes—
243C22×C8C2414yes—
244C4⋊Q8—42no—
245C4⋊Q8—42no—
246#125—28no—
247#125—44no—
248#125—44no—
249#125—48no—
250C22⋊Q8—44no—
251C22⋊Q8—44no—
252#117—44no—
253#117—44no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—28no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—24no—
259C2×C4⋊C4—24no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—44no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—48no—
283C2×C4⋊C4—44no—
284C2×C4⋊C4—44no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—82no—
287C2×C4⋊C4—82no—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C4⋊C4C2412yes—
291C2×C4⋊C4C2412yes—
292C2×C4⋊C4C2412yes—
293C2×C4⋊C4C2412yes—
294C2×C4⋊C4C2412yes—
295C2×C4⋊C4C2412yes—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—22no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—42no—
304C2×C22⋊C4—42no—
305C2×C22⋊C4—44no—
306C2×C22⋊C4C2×D412yes—
307C2×C22⋊C4C2×D412yes—
308C2×C22⋊C4C2×D412yes—
309C2×C22⋊C4C2×D412yes—
310C2×C42—22no—
311C2×C42—22no—
312C2×C42—42no—
313C2×C42—42no—
314C2×C42C2×D412yes—
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D411yescharacteristic
317C2×C42C2412yes—
318C2×C42C2411yescharacteristic
319C2×C42C2411yescharacteristic
320#107—28no—
321#107—44no—
322#107—44no—
323#107—44no—
324#106—28no—
325#106—44no—
326#106—44no—
327#106—44no—
328C22⋊C8C2×Q818yes—
329C22⋊C8C2×Q818yes—
330C22⋊C8C2×Q8116yes—
331#97—42no—
332#97—42no—
333C4×C8—44no—
334#95—22no—
335#95—22no—
336#95—24no—
337#95—24no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—22no—
349#95—22no—
350#95—22no—
351#95—22no—
352#95C2×D412yes—
353#95C2×D412yes—
354#95C2×D412yes—
355#95C2×D412yes—
356C24—22no—
357C24—22no—
358C24—41no—
359C24—41no—
360C24—41no—
361C24—41no—
362C24C4⋊Q814yes—
363C24C22×D412yes—
364C24C22×D411yescharacteristic
365C2×Q8—42no—
366C2×Q8—42no—
367C2×Q8—81no—
368C2×Q8—81no—
369C2×Q8—81no—
370C2×Q8—81no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
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—22no—
388C22×C4—416no—
389C22×C4—42no—
390C22×C4—41no—
391C22×C4—42no—
392C22×C4—41no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—41no—
400C22×C4—41no—
401C22×C4—48no—
402C22×C4—42no—
403C22×C4—42no—
404C22×C4—42no—
405C22×C4—42no—
406C22×C4—42no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—48no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—84no—
415C22×C4—816no—
416C22×C4C4⋊Q818yes—
417C22×C4C4⋊Q814yes—
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D412yes—
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×D412yes—
423C22×C4C22×D411yescharacteristic
424C22×C4C22×D412yes—
425C22×C4C22×D411yescharacteristic
426C22×C4C22×Q814yes—
427C22×C4C22×Q814yes—
428C22×C4C22×Q814yes—
429C22×C4C2511yescharacteristic
430M16—416no—
431C2×C8—24no—
432C2×C8—24no—
433C2×C8—24no—
434C2×C8—24no—
435C2×C8C22×Q818yes—
436C2×C8C22×Q818yes—
437C4⋊C4—24no—
438C4⋊C4—22no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—24no—
443C4⋊C4—24no—
444C4⋊C4—24no—
445C4⋊C4—24no—
446C4⋊C4—48no—
447C4⋊C4—48no—
448C4⋊C4—48no—
449C4⋊C4—42no—
450C4⋊C4—42no—
451C4⋊C4—42no—
452C4⋊C4—44no—
453C4⋊C4—44no—
454C4⋊C4—42no—
455C4⋊C4—42no—
456C4⋊C4—81no—
457C4⋊C4—82no—
458C4⋊C4—81no—
459C4⋊C4—82no—
460C4⋊C4—82no—
461C4⋊C4—84no—
462C4⋊C4—82no—
463C4⋊C4—82no—
464C4⋊C4—82no—
465C4⋊C4—81no—
466C4⋊C4—81no—
467C4⋊C4H(5,𝔽2)14yes—
468C22⋊C4—42no—
469C22⋊C4—42no—
470C22⋊C4—42no—
471C22⋊C4—42no—
472C22⋊C4—48no—
473C22⋊C4—44no—
474C22⋊C4—44no—
475C22⋊C4—82no—
476C22⋊C4—82no—
477C22⋊C4—82no—
478C22⋊C4—82no—
479C42—24no—
480C42—22no—
481C42—22no—
482C42—41no—
483C42—41no—
484C42—42no—
485C42—82no—
486C42—82no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—22no—
492C23—22no—
493C23—28no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—44no—
499C23—44no—
500C23—44no—
501C23—44no—
502C23—44no—
503C23—44no—
504C23C2×C4⋊Q814yes—
505C23C2×C4⋊Q812yes—
506C23#57512yes—
507C23#57512yes—
508C23C23×D411yescharacteristic
509Q8—162no—
510Q8—162no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—22no—
515C2×C4—28no—
516C2×C4—28no—
517C2×C4—28no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—44no—
538C2×C4—48no—
539C2×C4—48no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—816no—
568C2×C4—84no—
569C2×C4—84no—
570C2×C4—84no—
571C2×C4C2×C4⋊Q814yes—
572C2×C4C2×C4⋊Q812yes—
573C2×C4C23×Q812yes—
574C2×C4C2×H(5,𝔽2)12yes—
575C2×C4#58512yes—
576C8—28no—
577C8—28no—
578C22—22no—
579C22—22no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—22no—
584C22—24no—
585C22—24no—
586C22—24no—
587C22—416no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22#316614yes—
593C22C22×C4⋊Q811yescharacteristic
594C22#348411yescharacteristic
595C22#348511yescharacteristic
596C22C2×#57511yescharacteristic
597C22C2×#57511yescharacteristic
598C22#358511yescharacteristic
599C22#358611yescharacteristic
600C4—22no—
601C4—24no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—42no—
607C4—44no—
608C4—42no—
609C4—82no—
610C4—82no—
611C4—82no—
612C4—82no—
613C4—82no—
614C4—82no—
615C4—82no—
616C4#350114yes—
617C2—24no—
618C2—44no—
619C2#3539111yescharacteristic
620C2C2×#316611yescharacteristic
621C2#6066411yescharacteristic
622C2#6157611yescharacteristic
623C2#6157811yescharacteristic
624C2#6171611yescharacteristic
625C2#6171811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #29302C2semidirect
4. #29301C2semidirect
5. #29298C2semidirect
6. #28787C2semidirect
7. #28785C2semidirect
8. #27956C2semidirect
9. #27956C2semidirect
10. #26075C2semidirect
11. #26073C2semidirect
12. #25815C2semidirect
18. C2×#499C22semidirect
48. #2085C22semidirect
49. #2085C22semidirect
50. #2085C22semidirect
51. #2084C22semidirect
52. #2083C22semidirect
53. #2059C22semidirect
54. #2059C22semidirect
55. #2059C22semidirect
56. #2058C22semidirect
57. #2057C22semidirect
58. #1993C22semidirect
59. #1993C22semidirect
60. #1991C22semidirect
61. #1944C22semidirect
62. #1943C22semidirect
63. #1942C22semidirect
64. #1941C22semidirect
77. #1847C22semidirect
78. #1847C22semidirect
127. #407Q8semidirect

Maximal subgroups

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

subgroupindex
#350332
C2×#24422
#293022
#293012
#292982
#287872
#287852
#279562
#260752
#260732
#258152
#219442
#208442
#205242