Finite Group Databasebeta

← #6506506

Subgroups — #6506506

Order 512.

2426 conjugacy classes of subgroups.

The subgroup lattice has 585 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35001C211yesmaximal, characteristic
2C2×#2440C211yesmaximal, characteristic, complemented
3#29328C212yesmaximal, complemented
4#29328C212yesmaximal, complemented
5#27973C212yesmaximal, complemented
6#27695C218yesmaximal, complemented
7#27254C218yesmaximal, complemented
8#25903C212yesmaximal, complemented
9#21758C212yesmaximal, complemented
10C2×#1874C212yesmaximal, complemented
11#20523C211yesmaximal, characteristic, complemented
12C2×#534—21no—
13C2×#534—21no—
14C22×C4⋊Q8C2211yescharacteristic, complemented
15C22×C22⋊Q8—21no—
16C22×#121C2212yes—
17C23×C4⋊C4C2211yescharacteristic
18C2×#486C2214yescomplemented
19C22×#102C2214yescomplemented
20C22×C22⋊C8C2212yes—
21#2856C2212yes—
22#2856C2212yes—
23#2656—21no—
24#2656—21no—
25#2539—24no—
26#2501C2214yes—
27#2440—24no—
28#2440C2218yescomplemented
29#2433C2214yes—
30C2×#398—21no—
31C2×#395—21no—
32C2×#389C2211yescharacteristic, complemented
33C2×#389C2211yescharacteristic, complemented
34C2×#385—21no—
35C2×#384C2211yescharacteristic, complemented
36C2×#380C2211yescharacteristic
37C2×C4×C22⋊C4C2211yescharacteristic
38#2099C2214yescomplemented
39#2099C2214yescomplemented
40#2071C2218yescomplemented
41#2041C2214yescomplemented
42#2041C2214yescomplemented
43#2023C2218yescomplemented
44#1990C2214yescomplemented
45#1957C2214yescomplemented
46#1934C2218yescomplemented
47#1934C2218yescomplemented
48#1893C2218yes—
49#1874C2218yes—
50#1874C2214yescomplemented
51#1874C2214yescomplemented
52#1874C2214yescomplemented
53#1874C2214yescomplemented
54#1834C2214yescomplemented
55#1823C2214yescomplemented
56#1786C2214yescomplemented
57C8×C22⋊C4C2218yescomplemented
58C4×C22⋊C8C2212yescomplemented
59C4×C22⋊C8C2212yescomplemented
60C2×#340C2214yescomplemented
61C23×Q8—21no—
62C23×D4—22no—
63C23×D4C2312yes—
64C24×C4—21no—
65C24×C4C2311yescharacteristic
66C23×C8C2314yes—
67#534—48no—
68#534—48no—
69C2×C4⋊Q8—22no—
70C2×C4⋊Q8—44no—
71C2×C4⋊Q8C2314yes—
72C2×#127C2312yes—
73C2×#127C2312yes—
74C2×#123—24no—
75C2×#123—24no—
76C2×C22⋊Q8—28no—
77C2×C22⋊Q8—48no—
78C2×#121—22no—
79C2×#121—22no—
80C2×#121—22no—
81C2×#121—22no—
82C2×#121—22no—
83C2×#121—22no—
84C2×#121—28no—
85C2×#121—28no—
86C2×#121C2314yes—
87C2×#121C2314yes—
88C2×#120—24no—
89C2×#120—24no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4—21no—
92C22×C4⋊C4—21no—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4—22no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—44no—
98C22×C4⋊C4C2312yes—
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2314yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2311yescharacteristic
103C22×C4⋊C4C2311yescharacteristic
104C22×C22⋊C4—21no—
105C22×C22⋊C4—22no—
106C22×C22⋊C4D411yescharacteristic
107C22×C22⋊C4D411yescharacteristic
108C22×C42C2311yescharacteristic
109C22×C42C2311yescharacteristic
110#486—48no—
111C2×#102—48no—
112C2×#102C2318yes—
113C2×#102C2318yes—
114C2×#102C2318yes—
115C2×#102C2318yes—
116C2×C22⋊C8—28no—
117C2×C22⋊C8C2314yes—
118C2×C22⋊C8C2314yes—
119C2×C4×C8C2314yes—
120C2×C4×C8C2314yes—
121#399—24no—
122#399—24no—
123#398—28no—
124#395—28no—
125#392—24no—
126#392—24no—
127#390—22no—
128#390—22no—
129#390—22no—
130#390—22no—
131#390C2314yes—
132#390C2314yes—
133#389—22no—
134#389—22no—
135#389C2314yes—
136#389C2314yes—
137#385—28no—
138#384—22no—
139#384C2314yes—
140#384C2312yes—
141#384C2312yes—
142#380—22no—
143#380C2314yes—
144C4×C22⋊C4D414yes—
145C4×C22⋊C4C2314yes—
146C2×#95—21no—
147C2×#95—21no—
148C2×#95—21no—
149C2×#95D411yescharacteristic
150C2×#95D411yescharacteristic
151#340C2318yes—
152#340C2318yes—
153#340C2318yes—
154#340C2318yes—
155#336C2314yes—
156#336C2314yes—
157C25—22no—
158C25C2×D411yescharacteristic
159C22×Q8—24no—
160C22×Q8—44no—
161C22×Q8—44no—
162C22×Q8—82no—
163C22×D4—24no—
164C22×D4—24no—
165C22×D4—22no—
166C22×D4—22no—
167C22×D4—22no—
168C22×D4—22no—
169C22×D4—22no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—44no—
173C22×D4—44no—
174C22×D4—48no—
175C22×D4—48no—
176C22×D4—48no—
177C22×D4C2414yes—
178C23×C4—21no—
179C23×C4—21no—
180C23×C4—21no—
181C23×C4—22no—
182C23×C4—21no—
183C23×C4—24no—
184C23×C4—42no—
185C23×C4—42no—
186C23×C4—44no—
187C23×C4—44no—
188C23×C4—82no—
189C23×C4C2×D412yes—
190C23×C4C2×D411yescharacteristic
191C23×C4C2×D411yescharacteristic
192C23×C4C2×D411yescharacteristic
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4#4118yes—
196C23×C4C2411yescharacteristic
197C23×C4C2412yes—
198C22×C8—28no—
199C22×C8—44no—
200C22×C8—44no—
201C22×C8C2418yes—
202C4⋊Q8—28no—
203C4⋊Q8—44no—
204C4⋊Q8—48no—
205#127—44no—
206#127—44no—
207#123—416no—
208#123—416no—
209C22⋊Q8—432no—
210#121—28no—
211#121—28no—
212#121—28no—
213#121—28no—
214#121—48no—
215#121—48no—
216#121—48no—
217#121—48no—
218#121—48no—
219#121—48no—
220#120—416no—
221#120—416no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—28no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—42no—
243C2×C4⋊C4—42no—
244C2×C4⋊C4—42no—
245C2×C4⋊C4—48no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—44no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—44no—
251C2×C4⋊C4—82no—
252C2×C4⋊C4C2412yes—
253C2×C4⋊C4C2414yes—
254C2×C4⋊C4C2412yes—
255C2×C4⋊C4C2412yes—
256C2×C4⋊C4C2412yes—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—24no—
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—24no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—48no—
266C2×C22⋊C4D818yes—
267C2×C22⋊C4SD1618yes—
268C2×C22⋊C4C2×D414yes—
269C2×C22⋊C4C2×D414yes—
270C2×C42—42no—
271C2×C42—42no—
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C2×C42C2412yes—
275C2×C42C2412yes—
276#102—216no—
277#102—48no—
278#102—48no—
279#102—48no—
280C22⋊C8—28no—
281C22⋊C8—28no—
282C4×C8—44no—
283C4×C8—44no—
284#95—24no—
285#95—24no—
286#95—24no—
287#95C2×D414yes—
288#95C2×D414yes—
289C24—24no—
290C24—41no—
291C24—41no—
292C24—41no—
293C24—41no—
294C24—44no—
295C24—44no—
296C24—44no—
297C24—82no—
298C24—82no—
299C24—82no—
300C24—82no—
301C24#12414yes—
302C24C2×D814yes—
303C24C2×SD1614yes—
304C24C22×D412yes—
305C24C22×D411yescharacteristic
306C2×Q8—48no—
307C2×Q8—48no—
308C2×Q8—84no—
309C2×Q8—816no—
310C2×D4—24no—
311C2×D4—24no—
312C2×D4—28no—
313C2×D4—28no—
314C2×D4—416no—
315C2×D4—48no—
316C2×D4—48no—
317C2×D4—48no—
318C2×D4—48no—
319C2×D4—48no—
320C2×D4—48no—
321C2×D4—44no—
322C2×D4—44no—
323C2×D4—44no—
324C2×D4—44no—
325C2×D4—44no—
326C2×D4—88no—
327C2×D4—82no—
328C2×D4—82no—
329C2×D4—82no—
330C2×D4—82no—
331C2×D4—82no—
332C2×D4—82no—
333C2×D4—82no—
334C2×D4—82no—
335C2×D4—82no—
336C2×D4—82no—
337C2×D4—832no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—24no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—24no—
348C22×C4—416no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—48no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—44no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—48no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—84no—
378C22×C4—816no—
379C22×C4#12418yes—
380C22×C4#12414yes—
381C22×C4C2×D814yes—
382C22×C4C2×D814yes—
383C22×C4C2×SD1614yes—
384C22×C4C2×SD1614yes—
385C22×C4C22×D412yes—
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D412yes—
388C22×C4C22×D412yes—
389C22×C4C22×D412yes—
390C22×C4C22×D412yes—
391C22×C4C2×#4114yes—
392C22×C4C2×#4114yes—
393C22×C4C2×#4114yes—
394C22×C4C2511yescharacteristic
395C2×C8—28no—
396C2×C8—28no—
397C2×C8—28no—
398C2×C8—88no—
399C4⋊C4—24no—
400C4⋊C4—22no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—24no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—48no—
407C4⋊C4—48no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—44no—
412C4⋊C4—48no—
413C4⋊C4—48no—
414C4⋊C4—48no—
415C4⋊C4—44no—
416C4⋊C4—44no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4—84no—
420C4⋊C4H(5,𝔽2)14yes—
421C22⋊C4—48no—
422C22⋊C4—48no—
423C22⋊C4—48no—
424C22⋊C4—48no—
425C22⋊C4—48no—
426C22⋊C4—48no—
427C22⋊C4—48no—
428C22⋊C4C2×D8116yes—
429C22⋊C4C2×SD16116yes—
430C42—24no—
431C42—24no—
432C42—42no—
433C42—42no—
434C23—21no—
435C23—22no—
436C23—21no—
437C23—22no—
438C23—21no—
439C23—22no—
440C23—21no—
441C23—22no—
442C23—44no—
443C23—44no—
444C23—44no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—44no—
450C23—44no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—88no—
456C23—88no—
457C23—88no—
458C23—164no—
459C23—164no—
460C23—164no—
461C23—164no—
462C23#486116yescomplemented
463C23C2×#12414yes—
464C23C2×#12412yes—
465C23C22×D814yes—
466C23C22×D812yes—
467C23C22×SD1614yes—
468C23C22×SD1612yes—
469C23C23×D411yescharacteristic
470Q8—816no—
471D4—48no—
472D4—48no—
473D4—88no—
474D4—816no—
475D4—816no—
476D4—164no—
477D4—164no—
478D4—164no—
479D4—164no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—21no—
491C2×C4—22no—
492C2×C4—28no—
493C2×C4—28no—
494C2×C4—28no—
495C2×C4—24no—
496C2×C4—48no—
497C2×C4—44no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—48no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—816no—
518C2×C4—84no—
519C2×C4—84no—
520C2×C4—84no—
521C2×C4—88no—
522C2×C4C2×#12412yes—
523C2×C4C2×#12414yes—
524C2×C4C22×D814yes—
525C2×C4C22×D814yes—
526C2×C4C22×SD1614yes—
527C2×C4C22×SD1614yes—
528C2×C4C22×#4112yes—
529C2×C4C2×H(5,𝔽2)12yes—
530C2×C4#58512yes—
531C8—48no—
532C22—22no—
533C22—24no—
534C22—24no—
535C22—22no—
536C22—22no—
537C22—24no—
538C22—22no—
539C22—24no—
540C22—24no—
541C22—22no—
542C22—22no—
543C22—24no—
544C22—416no—
545C22—84no—
546C22—84no—
547C22—84no—
548C22—84no—
549C22—84no—
550C22—84no—
551C22—84no—
552C22—1616no—
553C22C2×#48618yes—
554C22C2×#48614yes—
555C22C22×#12411yescharacteristic
556C22#348411yescharacteristic
557C22#348511yescharacteristic
558C22C23×D811yescharacteristic
559C22C23×SD1611yescharacteristic
560C22#358511yescharacteristic
561C22#358611yescharacteristic
562C4—24no—
563C4—22no—
564C4—24no—
565C4—24no—
566C4—24no—
567C4—24no—
568C4—44no—
569C4—82no—
570C4—82no—
571C4—84no—
572C4—82no—
573C4#350514yes—
574C2—28no—
575C2—24no—
576C2—164no—
577C2#3537711yescharacteristic
578C2C22×#48611yescharacteristic
579C2#6062111yescharacteristic
580C2#6155911yescharacteristic
581C2#6156211yescharacteristic
582C2#6169411yescharacteristic
583C2#6169711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35001C2
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
2. C2×#2440C2semidirect
3. #29328C2semidirect
4. #29328C2semidirect
5. #27973C2semidirect
6. #27695C2semidirect
7. #27254C2semidirect
8. #25903C2semidirect
9. #21758C2semidirect
10. C2×#1874C2semidirect
11. #20523C2semidirect
14. C22×C4⋊Q8C22semidirect
18. C2×#486C22semidirect
19. C22×#102C22semidirect
28. #2440C22semidirect
32. C2×#389C22semidirect
33. C2×#389C22semidirect
35. C2×#384C22semidirect
38. #2099C22semidirect
39. #2099C22semidirect
40. #2071C22semidirect
41. #2041C22semidirect
42. #2041C22semidirect
43. #2023C22semidirect
44. #1990C22semidirect
45. #1957C22semidirect
46. #1934C22semidirect
47. #1934C22semidirect
50. #1874C22semidirect
51. #1874C22semidirect
52. #1874C22semidirect
53. #1874C22semidirect
54. #1834C22semidirect
55. #1823C22semidirect
56. #1786C22semidirect
57. C8×C22⋊C4C22semidirect
58. C4×C22⋊C8C22semidirect
59. C4×C22⋊C8C22semidirect
60. C2×#340C22semidirect
462. C23#486semidirect

Maximal subgroups

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

subgroupindex
#350012
C2×#24402
#293282
#279732
#276952
#272542
#259032
#217582
C2×#18742
#205232