Finite Group Databasebeta

← #6506513

Subgroups — #6506513

Order 512.

2036 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37450C211yesmaximal, characteristic
2#35224C211yesmaximal, characteristic, complemented
3#29343C212yesmaximal, complemented
4#29340C212yesmaximal, complemented
5#27986C212yesmaximal, complemented
6#27711C214yesmaximal, complemented
7#27701C214yesmaximal, complemented
8#27264C214yesmaximal, complemented
9#27254C214yesmaximal, complemented
10#25915C212yesmaximal, complemented
11#21761C212yesmaximal, complemented
12C2×#1874C212yesmaximal, complemented
13#20528C211yesmaximal, characteristic, complemented
14C2×#544C2211yescharacteristic, complemented
15C22×#121C2212yes—
16C23×C4⋊C4C2211yescharacteristic
17C2×#490C2212yescomplemented
18C2×#486C2212yescomplemented
19C22×#102C2214yescomplemented
20C22×C22⋊C8C2212yes—
21#2848—22no—
22#2815—22no—
23#2627—22no—
24#2619C2212yes—
25#2619C2212yes—
26#2617—22no—
27#2617—22no—
28#2504C2214yes—
29#2442—22no—
30#2442C2214yescomplemented
31#2440—22no—
32#2440C2214yescomplemented
33#2433C2214yes—
34#2403—22no—
35#2317C2211yescharacteristic, complemented
36#2317C2211yescharacteristic, complemented
37#2316C2211yescharacteristic, complemented
38#2280C2211yescharacteristic
39#2280C2211yescharacteristic
40#2101C2214yescomplemented
41#2101C2214yescomplemented
42#2073C2214yescomplemented
43#2071C2214yescomplemented
44#2041C2214yescomplemented
45#2041C2214yescomplemented
46#2025C2214yescomplemented
47#2023C2214yescomplemented
48#1993C2212yescomplemented
49#1991C2212yescomplemented
50#1961C2212yescomplemented
51#1957C2212yescomplemented
52#1936C2214yescomplemented
53#1936C2214yescomplemented
54#1934C2214yescomplemented
55#1934C2214yescomplemented
56#1893C2214yes—
57#1893C2214yes—
58#1875C2214yescomplemented
59#1875C2214yescomplemented
60#1874C2214yes—
61#1874C2214yes—
62#1874C2214yescomplemented
63#1874C2214yescomplemented
64#1836C2212yescomplemented
65#1835C2212yescomplemented
66#1824C2212yescomplemented
67#1824C2212yescomplemented
68#1786C2212yescomplemented
69#1786C2212yescomplemented
70#1752C2214yescomplemented
71C8×C22⋊C4C2214yescomplemented
72#1750C2212yescomplemented
73#1750C2212yescomplemented
74C2×#340C2214yescomplemented
75C23×D4—22no—
76C23×D4C2312yes—
77C24×C4—21no—
78C24×C4C2311yescharacteristic
79C23×C8C2314yes—
80#544—44no—
81C2×C4⋊Q8—21no—
82C2×C4⋊Q8C2312yes—
83C2×#127C2312yes—
84C2×#127C2312yes—
85C2×#125—21no—
86C2×#125C2312yes—
87C2×C22⋊Q8—22no—
88C2×C22⋊Q8—24no—
89C2×#121—22no—
90C2×#121—22no—
91C2×#121—28no—
92C2×#121—28no—
93C2×#121C2314yes—
94C2×#121C2314yes—
95C2×#120—24no—
96C2×#120—24no—
97C2×#117C2311yescharacteristic
98C2×#117C2311yescharacteristic
99C2×#117C2311yescharacteristic
100C2×#117C2311yescharacteristic
101C22×C4⋊C4—21no—
102C22×C4⋊C4—22no—
103C22×C4⋊C4—44no—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2314yes—
107C22×C4⋊C4C2311yescharacteristic
108C22×C22⋊C4—22no—
109#490—44no—
110#486—44no—
111C2×#102—48no—
112C2×#102C2314yes—
113C2×#102C2314yes—
114C2×#102C2314yes—
115C2×#102C2314yes—
116C2×#102C2314yes—
117C2×#102C2314yes—
118C2×#102C2314yes—
119C2×#102C2314yes—
120C2×C22⋊C8—28no—
121C2×C22⋊C8C2312yes—
122C2×C22⋊C8C2312yes—
123C2×C22⋊C8C2314yes—
124C2×#97C2312yes—
125C2×#97C2312yes—
126C2×C4×C8C2312yes—
127C2×C4×C8C2312yes—
128#400—24no—
129#398—24no—
130#398—24no—
131#397—24no—
132#397—24no—
133#395—24no—
134#394—22no—
135#394—22no—
136#394—24no—
137#392—22no—
138#392—22no—
139#392—24no—
140#392—24no—
141#392—24no—
142#390C2314yes—
143#390C2314yes—
144#389—22no—
145#389—22no—
146#389C2314yes—
147#389C2314yes—
148#386—24no—
149#386—24no—
150#385—22no—
151#385—22no—
152#385—24no—
153#385—24no—
154#384—21no—
155#384C2312yes—
156#382—21no—
157#382C2312yes—
158#382C2312yes—
159#382C2312yes—
160#381—21no—
161#381D412yes—
162#380—22no—
163#380—24no—
164#380D412yes—
165#380D412yes—
166#380C2312yes—
167#380C2312yes—
168#379—21no—
169#379D412yes—
170C4×C22⋊C4—22no—
171C4×C22⋊C4C2314yes—
172#340C2314yes—
173#340C2314yes—
174#340C2314yes—
175#340C2314yes—
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#340C2314yes—
180#336C2314yes—
181#336C2314yes—
182C25—22no—
183C25C2×D411yescharacteristic
184C22×Q8—22no—
185C22×D4—24no—
186C22×D4—24no—
187C22×D4—22no—
188C22×D4—22no—
189C22×D4—24no—
190C22×D4—22no—
191C22×D4—24no—
192C22×D4—24no—
193C22×D4—44no—
194C22×D4—44no—
195C22×D4—48no—
196C22×D4—84no—
197C22×D4—84no—
198C22×D4C2414yes—
199C23×C4—21no—
200C23×C4—24no—
201C23×C4—22no—
202C23×C4—21no—
203C23×C4—42no—
204C23×C4—42no—
205C23×C4—44no—
206C23×C4—84no—
207C23×C4C2×D412yes—
208C23×C4C2×D411yescharacteristic
209C23×C4#4118yes—
210C23×C4C2412yes—
211C23×C4C2411yescharacteristic
212C22×C8—28no—
213C22×C8—44no—
214C22×C8—44no—
215C22×C8C2414yes—
216C22×C8C2414yes—
217C4⋊Q8—24no—
218C4⋊Q8—42no—
219C4⋊Q8—44no—
220#127—42no—
221#127—42no—
222#127—42no—
223#127—42no—
224#125—24no—
225#125—42no—
226#125—44no—
227C22⋊Q8—48no—
228C22⋊Q8—88no—
229#121—28no—
230#121—28no—
231#121—28no—
232#121—28no—
233#121—48no—
234#121—48no—
235#120—88no—
236#120—88no—
237#117—42no—
238#117—42no—
239#117—42no—
240#117—42no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—44no—
256C2×C4⋊C4—42no—
257C2×C4⋊C4—48no—
258C2×C4⋊C4—44no—
259C2×C4⋊C4C2×D411yescharacteristic
260C2×C4⋊C4C2×D411yescharacteristic
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2412yes—
263C2×C4⋊C4C2414yes—
264C2×C4⋊C4C2411yescharacteristic
265C2×C4⋊C4C2411yescharacteristic
266C2×C4⋊C4C2412yes—
267C2×C4⋊C4C2412yes—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—48no—
279C2×C22⋊C4C2×D414yes—
280C2×C22⋊C4C2×D414yes—
281C2×C22⋊C4C2×D412yes—
282C2×C22⋊C4C2×D412yes—
283C2×C42—21no—
284C2×C42—21no—
285C2×C42C2412yes—
286C2×C42C2412yes—
287#102—28no—
288#102—28no—
289#102—44no—
290#102—44no—
291#102—44no—
292#102—44no—
293#102—44no—
294#102—44no—
295C22⋊C8—24no—
296C22⋊C8—24no—
297C22⋊C8—28no—
298#97—42no—
299#97—42no—
300C4×C8—42no—
301C4×C8—42no—
302#95—22no—
303#95—22no—
304#95—24no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—22no—
309#95—22no—
310#95C2×D414yes—
311#95C2×D414yes—
312C24—24no—
313C24—24no—
314C24—24no—
315C24—41no—
316C24—41no—
317C24—44no—
318C24—44no—
319C24—44no—
320C24—82no—
321C24—82no—
322C24—82no—
323C24—82no—
324C24#12414yes—
325C24C22×D411yescharacteristic
326C24C22×D412yes—
327C2×Q8—44no—
328C2×Q8—84no—
329C2×D4—24no—
330C2×D4—24no—
331C2×D4—24no—
332C2×D4—24no—
333C2×D4—24no—
334C2×D4—24no—
335C2×D4—416no—
336C2×D4—48no—
337C2×D4—48no—
338C2×D4—48no—
339C2×D4—48no—
340C2×D4—44no—
341C2×D4—88no—
342C2×D4—82no—
343C2×D4—82no—
344C2×D4—84no—
345C2×D4—84no—
346C2×D4—84no—
347C2×D4—84no—
348C2×D4—82no—
349C2×D4—82no—
350C2×D4—84no—
351C2×D4—84no—
352C2×D4—84no—
353C2×D4—832no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—24no—
358C22×C4—22no—
359C22×C4—24no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—416no—
366C22×C4—42no—
367C22×C4—41no—
368C22×C4—42no—
369C22×C4—41no—
370C22×C4—48no—
371C22×C4—42no—
372C22×C4—44no—
373C22×C4—48no—
374C22×C4—82no—
375C22×C4—84no—
376C22×C4—82no—
377C22×C4—82no—
378C22×C4—82no—
379C22×C4—816no—
380C22×C4#12418yes—
381C22×C4#12414yes—
382C22×C4C22×D412yes—
383C22×C4C22×D414yes—
384C22×C4C22×D412yes—
385C22×C4C22×D412yes—
386C22×C4C22×D411yescharacteristic
387C22×C4C2×#4114yes—
388C22×C4C2×#4114yes—
389C22×C4C2×#4114yes—
390C22×C4C2511yescharacteristic
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—88no—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—24no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—42no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—48no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—42no—
414C4⋊C4—44no—
415C4⋊C4—84no—
416C4⋊C4—84no—
417C4⋊C4—84no—
418C4⋊C4H(5,𝔽2)14yes—
419C22⋊C4—28no—
420C22⋊C4—28no—
421C22⋊C4—42no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C22⋊C4—42no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—48no—
428C22⋊C4—48no—
429C22⋊C4—84no—
430C22⋊C4—84no—
431C22⋊C4—84no—
432C22⋊C4—84no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—42no—
438C42—42no—
439C23—22no—
440C23—22no—
441C23—22no—
442C23—22no—
443C23—216no—
444C23—21no—
445C23—21no—
446C23—22no—
447C23—22no—
448C23—44no—
449C23—44no—
450C23—48no—
451C23—44no—
452C23—44no—
453C23—44no—
454C23—44no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—88no—
460C23—88no—
461C23—88no—
462C23—164no—
463C23—164no—
464C23—164no—
465C23—164no—
466C23C2×#12412yes—
467C23C2×#12414yes—
468C23#57614yes—
469C23#57714yes—
470C23C23×D411yescharacteristic
471Q8—164no—
472D4—44no—
473D4—44no—
474D4—44no—
475D4—44no—
476D4—88no—
477D4—816no—
478D4—816no—
479D4—168no—
480D4—164no—
481D4—164no—
482C2×C4—21no—
483C2×C4—22no—
484C2×C4—21no—
485C2×C4—22no—
486C2×C4—28no—
487C2×C4—28no—
488C2×C4—28no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—48no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—84no—
513C2×C4—816no—
514C2×C4—88no—
515C2×C4—84no—
516C2×C4C2×#12412yes—
517C2×C4C2×#12414yes—
518C2×C4C22×#4112yes—
519C2×C4C2×H(5,𝔽2)12yes—
520C2×C4#58512yes—
521C8—44no—
522C8—44no—
523C22—24no—
524C22—24no—
525C22—24no—
526C22—24no—
527C22—24no—
528C22—24no—
529C22—22no—
530C22—24no—
531C22—416no—
532C22—42no—
533C22—41no—
534C22—41no—
535C22—84no—
536C22—84no—
537C22—84no—
538C22—84no—
539C22—84no—
540C22—84no—
541C22—84no—
542C22—1616no—
543C22#313618yes—
544C22C22×#12411yescharacteristic
545C22#348411yescharacteristic
546C22#348511yescharacteristic
547C22C2×#57612yes—
548C22C2×#57712yes—
549C4—22no—
550C4—24no—
551C4—44no—
552C4—42no—
553C4—44no—
554C4—42no—
555C4—82no—
556C4—82no—
557C4—81no—
558C4—81no—
559C4—84no—
560C4#350514yes—
561C2—28no—
562C2—42no—
563C2—164no—
564C2#3537711yescharacteristic
565C2C2×#313612yes—
566C2#6163011yescharacteristic
567C2#6163311yescharacteristic
568C2#6176612yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#37450C2
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. #35224C2semidirect
3. #29343C2semidirect
4. #29340C2semidirect
5. #27986C2semidirect
6. #27711C2semidirect
7. #27701C2semidirect
8. #27264C2semidirect
9. #27254C2semidirect
10. #25915C2semidirect
11. #21761C2semidirect
12. C2×#1874C2semidirect
13. #20528C2semidirect
14. C2×#544C22semidirect
17. C2×#490C22semidirect
18. C2×#486C22semidirect
19. C22×#102C22semidirect
30. #2442C22semidirect
32. #2440C22semidirect
35. #2317C22semidirect
36. #2317C22semidirect
37. #2316C22semidirect
40. #2101C22semidirect
41. #2101C22semidirect
42. #2073C22semidirect
43. #2071C22semidirect
44. #2041C22semidirect
45. #2041C22semidirect
46. #2025C22semidirect
47. #2023C22semidirect
48. #1993C22semidirect
49. #1991C22semidirect
50. #1961C22semidirect
51. #1957C22semidirect
52. #1936C22semidirect
53. #1936C22semidirect
54. #1934C22semidirect
55. #1934C22semidirect
58. #1875C22semidirect
59. #1875C22semidirect
62. #1874C22semidirect
63. #1874C22semidirect
64. #1836C22semidirect
65. #1835C22semidirect
66. #1824C22semidirect
67. #1824C22semidirect
68. #1786C22semidirect
69. #1786C22semidirect
70. #1752C22semidirect
71. C8×C22⋊C4C22semidirect
72. #1750C22semidirect
73. #1750C22semidirect
74. C2×#340C22semidirect

Maximal subgroups

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

subgroupindex
#374502
#352242
#293432
#293402
#279862
#277112
#277012
#272642
#272542
#259152
#217612
C2×#18742
#205282