Finite Group Databasebeta

← #6506762

Subgroups — #6506762

Order 512.

1642 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×#2441C211yesmaximal, characteristic
3#29298C212yesmaximal, complemented
4#29297C211yesmaximal, characteristic, complemented
5#29296C211yesmaximal, characteristic, complemented
6#28774C214yesmaximal, complemented
7#28773C214yesmaximal, complemented
8#27950C211yesmaximal, characteristic, complemented
9#27948C211yesmaximal, characteristic, complemented
10#26066C214yesmaximal, complemented
11#26063C214yesmaximal, complemented
12#25815C212yesmaximal, complemented
13#21941C212yesmaximal
14#20843C212yesmaximal
15#20523C211yesmaximal, characteristic
16C22×#125C2211yescharacteristic
17C22×#117C2211yescharacteristic
18C2×#499C2214yescomplemented
19C2×#427C2214yes—
20C22×C22⋊C8C2212yes—
21#2857C2211yescharacteristic
22#2857C2211yescharacteristic
23#2851C2212yes—
24#2626—21no—
25#2626—21no—
26#2608—22no—
27#2607—22no—
28#2441—22no—
29#2441—22no—
30#2441C2214yes—
31#2441C2214yes—
32#2441C2212yes—
33#2441C2212yes—
34#2439C2212yes—
35#2439C2212yes—
36C2×#399—21no—
37C2×#396—21no—
38#2329—21no—
39#2329—21no—
40C2×#389—21no—
41C2×#389C2211yescharacteristic
42C2×#389C2211yescharacteristic
43C2×#387—21no—
44C2×#384C2211yescharacteristic
45#2280C2212yes—
46C2×#380C2211yescharacteristic
47C2×C4×C22⋊C4C2211yescharacteristic
48#2085C2214yescomplemented
49#2084C2214yescomplemented
50#2084C2212yescomplemented
51#2083C2214yescomplemented
52#2083C2212yescomplemented
53#2059C2214yescomplemented
54#2059C2214yescomplemented
55#2059C2214yescomplemented
56#2058C2212yescomplemented
57#2057C2212yescomplemented
58#1991C2214yescomplemented
59#1990C2212yescomplemented
60#1990C2212yescomplemented
61#1942C2214yescomplemented
62#1942C2214yescomplemented
63#1941C2214yescomplemented
64#1941C2214yescomplemented
65#1854C2214yes—
66#1854C2214yes—
67#1854C2214yes—
68#1854C2214yes—
69#1853C2212yes—
70#1853C2212yes—
71#1853C2212yes—
72#1853C2212yes—
73#1852C2212yes—
74#1852C2212yes—
75#1852C2212yes—
76#1852C2212yes—
77#1845C2212yescomplemented
78#1844C2212yescomplemented
79#1808C2214yes—
80#1807C2214yes—
81C8×C22⋊C4C2214yes—
82C8×C22⋊C4C2214yes—
83C4×C22⋊C8C2211yescharacteristic
84C4×C22⋊C8C2211yescharacteristic
85C4×C22⋊C8C2212yes—
86#1735C2214yes—
87C24×C4C2311yescharacteristic
88C23×C8C2314yes—
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2311yescharacteristic
91C2×#125—22no—
92C2×#125—44no—
93C2×#125C2314yes—
94C2×#125C2312yes—
95C2×C22⋊Q8—21no—
96C2×C22⋊Q8—21no—
97C2×#117—22no—
98C2×#117—44no—
99C2×#117C2314yes—
100C2×#117C2312yes—
101C2×#117C2312yes—
102C22×C4⋊C4—21no—
103C22×C4⋊C4—21no—
104C22×C4⋊C4—21no—
105C22×C4⋊C4—21no—
106C22×C4⋊C4—21no—
107C22×C4⋊C4—21no—
108C22×C4⋊C4C2311yescharacteristic
109C22×C4⋊C4C2311yescharacteristic
110C22×C4⋊C4C2311yescharacteristic
111C22×C4⋊C4C2311yescharacteristic
112C22×C22⋊C4—21no—
113C22×C22⋊C4—21no—
114C22×C22⋊C4D411yescharacteristic
115C22×C22⋊C4D411yescharacteristic
116C22×C42—21no—
117C22×C42C2311yescharacteristic
118C22×C42C2311yescharacteristic
119#499—48no—
120#427—48no—
121C2×#107C2314yes—
122C2×#107C2314yes—
123C2×#107C2314yes—
124C2×#107C2314yes—
125C2×#106C2314yes—
126C2×#106C2314yes—
127C2×#106C2314yes—
128C2×#106C2314yes—
129C2×C22⋊C8Q8116yescomplemented
130C2×C22⋊C8C2312yes—
131C2×C22⋊C8C2312yes—
132C2×C22⋊C8C2314yes—
133C2×C4×C8C2312yes—
134C2×C4×C8C2312yes—
135C2×C4×C8C2314yes—
136#399—22no—
137#399—22no—
138#399—24no—
139#399—24no—
140#396—22no—
141#396—22no—
142#396—22no—
143#396—22no—
144#396—24no—
145#396—24no—
146#393—22no—
147#393—22no—
148#391—21no—
149#391—21no—
150#391—22no—
151#391—22no—
152#389—21no—
153#389—21no—
154#389—22no—
155#389—21no—
156#389—21no—
157#389—22no—
158#389—22no—
159#389—24no—
160#389—24no—
161#389C2312yes—
162#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#389C2314yes—
166#389C2312yes—
167#389C2312yes—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—24no—
175#387—24no—
176#385—22no—
177#385—22no—
178#384—21no—
179#384—21no—
180#384C2311yescharacteristic
181#384C2312yes—
182#384C2311yescharacteristic
183#384C2312yes—
184#382C2312yes—
185#381—22no—
186#380—21no—
187#380—21no—
188#380—22no—
189#380C2312yes—
190#380C2314yes—
191#380C2312yes—
192#380C2312yes—
193#380C2312yes—
194#379—22no—
195C4×C22⋊C4—22no—
196C4×C22⋊C4—22no—
197C4×C22⋊C4D412yes—
198C4×C22⋊C4D412yes—
199C4×C22⋊C4C2312yes—
200C4×C22⋊C4C2312yes—
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—42no—
225C23×C4—42no—
226C23×C4C2×D411yescharacteristic
227C23×C4C2×D411yescharacteristic
228C23×C4C2×D412yes—
229C23×C4C2×D411yescharacteristic
230C23×C4C2×D411yescharacteristic
231C23×C4C2×D411yescharacteristic
232C23×C4C2×Q818yes—
233C23×C4C2411yescharacteristic
234C23×C4C2412yes—
235C22×C8—44no—
236C22×C8—44no—
237C22×C8C2×Q8116yes—
238C22×C8C2414yes—
239C22×C8C2414yes—
240C4⋊Q8—42no—
241C4⋊Q8—42no—
242#125—28no—
243#125—44no—
244#125—44no—
245#125—48no—
246C22⋊Q8—44no—
247C22⋊Q8—44no—
248#117—28no—
249#117—44no—
250#117—48no—
251#117—44no—
252#117—44no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—21no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—42no—
280C2×C4⋊C4—42no—
281C2×C4⋊C4—42no—
282C2×C4⋊C4—44no—
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—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—22no—
300C2×C22⋊C4—22no—
301C2×C22⋊C4—24no—
302C2×C22⋊C4—22no—
303C2×C22⋊C4—22no—
304C2×C22⋊C4—22no—
305C2×C22⋊C4—22no—
306C2×C22⋊C4—44no—
307C2×C22⋊C4—82no—
308C2×C22⋊C4C2×D412yes—
309C2×C22⋊C4C2×D412yes—
310C2×C22⋊C4C2×D412yes—
311C2×C22⋊C4C2×D412yes—
312C2×C42—24no—
313C2×C42—22no—
314C2×C42—22no—
315C2×C42—42no—
316C2×C42—42no—
317C2×C42—44no—
318C2×C42C2×D412yes—
319C2×C42C2×D411yescharacteristic
320C2×C42C2×D411yescharacteristic
321C2×C42C2412yes—
322C2×C42C2411yescharacteristic
323C2×C42C2411yescharacteristic
324#107—28no—
325#107—44no—
326#107—44no—
327#107—44no—
328#106—28no—
329#106—44no—
330#106—44no—
331#106—44no—
332C22⋊C8C2×Q818yes—
333C22⋊C8C2×Q818yes—
334C22⋊C8C2×Q8116yes—
335C4×C8—42no—
336C4×C8—42no—
337C4×C8—44no—
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—
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—416no—
387C22×C4—41no—
388C22×C4—41no—
389C22×C4—41no—
390C22×C4—41no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—41no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
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×C4C4⋊Q818yes—
411C22×C4C4⋊Q814yes—
412C22×C4#13514yes—
413C22×C4#13514yes—
414C22×C4C22×D412yes—
415C22×C4C22×D412yes—
416C22×C4C22×D412yes—
417C22×C4C22×D411yescharacteristic
418C22×C4C22×D411yescharacteristic
419C22×C4C22×D411yescharacteristic
420C22×C4C22×D411yescharacteristic
421C22×C4C22×D411yescharacteristic
422C22×C4C22×Q814yes—
423C22×C4C22×Q814yes—
424C22×C4C22×Q814yes—
425C22×C4C2511yescharacteristic
426C2×C8—24no—
427C2×C8—24no—
428C2×C8—24no—
429C2×C8—24no—
430C2×C8—416no—
431C2×C8C22×Q818yes—
432C2×C8C22×Q818yes—
433C4⋊C4—24no—
434C4⋊C4—22no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—24no—
440C4⋊C4—24no—
441C4⋊C4—24no—
442C4⋊C4—48no—
443C4⋊C4—48no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—44no—
448C4⋊C4—44no—
449C4⋊C4—44no—
450C4⋊C4—44no—
451C4⋊C4—42no—
452C4⋊C4—42no—
453C4⋊C4—44no—
454C4⋊C4—44no—
455C4⋊C4—81no—
456C4⋊C4—81no—
457C4⋊C4—82no—
458C4⋊C4—82no—
459C4⋊C4—82no—
460C4⋊C4—82no—
461C4⋊C4—84no—
462C4⋊C4—82no—
463C4⋊C4—82no—
464C4⋊C4—81no—
465C4⋊C4—81no—
466C4⋊C4H(5,𝔽2)14yes—
467C22⋊C4—24no—
468C22⋊C4—24no—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—48no—
472C22⋊C4—44no—
473C22⋊C4—44no—
474C22⋊C4—82no—
475C22⋊C4—82no—
476C22⋊C4—84no—
477C22⋊C4—82no—
478C22⋊C4—82no—
479C42—24no—
480C42—22no—
481C42—22no—
482C42—41no—
483C42—41no—
484C42—42no—
485C42—48no—
486C42—82no—
487C42—82no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—28no—
499C23—44no—
500C23—44no—
501C23—82no—
502C23—82no—
503C23—82no—
504C23—82no—
505C23C2×C4⋊Q814yes—
506C23C2×C4⋊Q812yes—
507C23C2×#13512yes—
508C23C2×#13512yes—
509C23C23×D411yescharacteristic
510Q8—162no—
511Q8—162no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—28no—
525C2×C4—28no—
526C2×C4—28no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—44no—
535C2×C4—48no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—48no—
539C2×C4—42no—
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—82no—
563C2×C4—82no—
564C2×C4—84no—
565C2×C4—84no—
566C2×C4—84no—
567C2×C4C2×C4⋊Q814yes—
568C2×C4C2×C4⋊Q812yes—
569C2×C4C2×#13512yes—
570C2×C4C2×#13512yes—
571C2×C4C2×#13512yes—
572C2×C4C2×#13512yes—
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—42no—
588C22—42no—
589C22—42no—
590C22—42no—
591C22—88no—
592C22#316014yes—
593C22C22×C4⋊Q811yescharacteristic
594C22#348411yescharacteristic
595C22#348511yescharacteristic
596C22C22×#13511yescharacteristic
597C22C22×#13511yescharacteristic
598C22#358511yescharacteristic
599C22#358611yescharacteristic
600C4—22no—
601C4—22no—
602C4—22no—
603C4—22no—
604C4—24no—
605C4—22no—
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×#316011yescharacteristic
621C2#6066411yescharacteristic
622C2#6156511yescharacteristic
623C2#6156611yescharacteristic
624C2#6169611yescharacteristic
625C2#6169911yescharacteristic
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. #29298C2semidirect
4. #29297C2semidirect
5. #29296C2semidirect
6. #28774C2semidirect
7. #28773C2semidirect
8. #27950C2semidirect
9. #27948C2semidirect
10. #26066C2semidirect
11. #26063C2semidirect
12. #25815C2semidirect
18. C2×#499C22semidirect
48. #2085C22semidirect
49. #2084C22semidirect
50. #2084C22semidirect
51. #2083C22semidirect
52. #2083C22semidirect
53. #2059C22semidirect
54. #2059C22semidirect
55. #2059C22semidirect
56. #2058C22semidirect
57. #2057C22semidirect
58. #1991C22semidirect
59. #1990C22semidirect
60. #1990C22semidirect
61. #1942C22semidirect
62. #1942C22semidirect
63. #1941C22semidirect
64. #1941C22semidirect
77. #1845C22semidirect
78. #1844C22semidirect
129. C2×C22⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#350332
C2×#24412
#292982
#292972
#292962
#287742
#287732
#279502
#279482
#260662
#260632
#258152
#219412
#208432
#205232