Finite Group Databasebeta

← #7230540

Subgroups — #7230540

Order 512.

1845 conjugacy classes of subgroups.

The subgroup lattice has 595 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#45996C211yesmaximal, characteristic
2#36562C211yesmaximal, characteristic, complemented
3#33025C212yesmaximal, complemented
4#32807C212yesmaximal, complemented
5#32629C212yesmaximal, complemented
6#30909C214yesmaximal, complemented
7#30500C212yesmaximal, complemented
8#30456C214yesmaximal, complemented
9#26724C214yesmaximal, complemented
10#25630C212yesmaximal, complemented
11#25080C214yesmaximal, complemented
12#23969C212yesmaximal, complemented
13#23346C211yesmaximal, characteristic, complemented
14C2×H(3,𝔽4)—21no—
15C2×#557C2211yescharacteristic, complemented
16C2×#554C2211yescharacteristic
17C2×#492C2212yes—
18C2×#475C2212yescomplemented
19C2×#465C2214yes—
20C2×#460C2212yes—
21C2×#460C2212yes—
22C2×#450C2214yescomplemented
23C2×#450C2212yescomplemented
24C22×#102C2212yescomplemented
25#2862C2211yescharacteristic, complemented
26#2859—21no—
27#2858C2212yes—
28#2819—21no—
29#2733—21no—
30#2679—21no—
31#2670—24no—
32#2567C2211yescharacteristic, complemented
33#2566—21no—
34#2565—21no—
35#2556—24no—
36#2548C2212yes—
37#2542C2211yescharacteristic
38#2491C2214yes—
39#2433C2214yes—
40#2429C2214yescomplemented
41#2408C2212yes—
42#2360C2214yescomplemented
43C4×#124—21no—
44C4×#124C2211yescharacteristic
45C42×Q8C2211yescharacteristic, complemented
46#2065C2214yescomplemented
47#2055C2214yescomplemented
48#2055C2214yescomplemented
49#2041C2214yes—
50#2035C2214yescomplemented
51#2024C2214yescomplemented
52#2023C2214yescomplemented
53#2014C2214yes—
54#2013C2214yescomplemented
55#2013C2214yescomplemented
56#2000C2214yes—
57#2000C2214yes—
58#1988C2212yescomplemented
59#1984C2212yescomplemented
60#1982C2212yes—
61#1981C2212yescomplemented
62#1934C2214yescomplemented
63#1926C2214yescomplemented
64#1926C2214yescomplemented
65#1923C2214yescomplemented
66#1920C2214yescomplemented
67#1892C2212yescomplemented
68#1880C2214yescomplemented
69#1880C2214yescomplemented
70#1880C2214yescomplemented
71#1874C2214yes—
72#1869C2212yes—
73#1850C2214yescomplemented
74#1831C2212yescomplemented
75#1824C2212yescomplemented
76C4×#103C2212yescomplemented
77C23×D4C2312yes—
78C22×SD16C2314yes—
79C22×SD16C2314yes—
80C22×SD16C2314yes—
81C23×C8C2312yes—
82H(3,𝔽4)—48no—
83#557—42no—
84#557—42no—
85#554—42no—
86#554—42no—
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2311yescharacteristic
89C2×C4⋊Q8C2312yes—
90C2×C4⋊Q8C2311yescharacteristic
91C2×#127C2312yes—
92C2×#127C2312yes—
93C2×#125—21no—
94C2×#125—21no—
95C2×#125—22no—
96C2×#124—22no—
97C2×#124—24no—
98C2×#124—22no—
99C2×#124—22no—
100C2×#124D411yescharacteristic
101C2×C22⋊Q8—22no—
102C2×C22⋊Q8C2314yes—
103C2×C22⋊Q8C2312yes—
104C2×#121C2314yes—
105C2×#120—22no—
106C2×#120—24no—
107C2×#120—24no—
108C2×C4×Q8—21no—
109C2×C4×Q8C2311yescharacteristic
110C2×C4×Q8C2312yes—
111C2×C4×Q8C2312yes—
112C2×C4×Q8C2311yescharacteristic
113C2×C4×D4—21no—
114C2×C4×D4C2311yescharacteristic
115C22×C4⋊C4C2312yes—
116#492—44no—
117#475—44no—
118#465—48no—
119#460—44no—
120#460—44no—
121#450—48no—
122#450—44no—
123C2×#106C2314yes—
124C2×C4⋊C8C2312yes—
125C2×C4⋊C8C2312yes—
126C2×C4⋊C8C2312yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#102—44no—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×#102C2314yes—
136C2×C22⋊C8C2312yes—
137C2×C22⋊C8C2314yes—
138C2×C4×C8C2312yes—
139#399—22no—
140#398—24no—
141#396—24no—
142#395—22no—
143#394—22no—
144#394—22no—
145#394—22no—
146#394—24no—
147#393—24no—
148#392—22no—
149#392—24no—
150#392—24no—
151#391—21no—
152#391—22no—
153#391D411yescharacteristic
154#391C2314yes—
155#390—21no—
156#390—22no—
157#390—22no—
158#390—22no—
159#390D411yescharacteristic
160#390C2314yes—
161#390C2314yes—
162#390C2314yes—
163#389—24no—
164#389—24no—
165#389C2314yes—
166#388—24no—
167#388—24no—
168#388D414yes—
169#387—24no—
170#385—24no—
171#384—21no—
172#384—21no—
173#382—21no—
174#382—21no—
175#382D411yescharacteristic
176#380C2312yes—
177#379—22no—
178C4×C4⋊C4—21no—
179C4×C4⋊C4—22no—
180C4×C4⋊C4C2311yescharacteristic
181C4×C4⋊C4C2312yes—
182C4×C4⋊C4C2312yes—
183C4×C22⋊C4—24no—
184C4×C22⋊C4C2314yes—
185C43—21no—
186C43C2311yescharacteristic
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#336C2314yes—
192#336C2314yes—
193#336C2314yes—
194C25—22no—
195C22×Q8—21no—
196C22×Q8—22no—
197C22×Q8C2×D411yescharacteristic
198C22×Q8C2412yes—
199C22×Q8C2411yescharacteristic
200C22×D4—24no—
201C22×D4—22no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—22no—
205C22×D4—22no—
206C22×D4—44no—
207C22×D4—42no—
208C22×D4—42no—
209C22×D4—44no—
210C22×D4C2×D411yescharacteristic
211C22×D4C2414yes—
212C23×C4—22no—
213C23×C4C2412yes—
214C2×SD16—28no—
215C2×SD16—44no—
216C2×SD16—44no—
217C2×SD16—44no—
218C2×SD16—44no—
219C2×SD16—44no—
220C2×SD16—44no—
221C2×SD16—44no—
222C2×SD16—44no—
223C22×C8—42no—
224C22×C8—44no—
225C22×C8C2414yes—
226C22×C8C2414yes—
227C4⋊Q8—24no—
228C4⋊Q8—41no—
229C4⋊Q8—41no—
230C4⋊Q8—42no—
231C4⋊Q8—42no—
232C4⋊Q8—42no—
233C4⋊Q8—41no—
234C4⋊Q8—41no—
235#127—42no—
236#127—42no—
237#127—42no—
238#127—42no—
239#125—44no—
240#125—44no—
241#125—48no—
242#124—48no—
243#124—416no—
244#124—48no—
245#124—84no—
246#124SD1618yescomplemented
247C22⋊Q8—24no—
248C22⋊Q8—28no—
249C22⋊Q8—44no—
250C22⋊Q8—42no—
251C22⋊Q8—48no—
252#121—44no—
253#121—44no—
254#120—48no—
255#120—416no—
256#120—88no—
257C4×Q8—22no—
258C4×Q8—22no—
259C4×Q8—24no—
260C4×Q8—24no—
261C4×Q8—41no—
262C4×Q8—41no—
263C4×Q8—42no—
264C4×Q8—42no—
265C4×Q8—44no—
266C4×D4—41no—
267C4×D4—41no—
268C4×D4—44no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—24no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—42no—
284C2×C4⋊C4C2×D411yescharacteristic
285C2×C4⋊C4C2×D411yescharacteristic
286C2×C4⋊C4C2×D412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2414yes—
289C2×C4⋊C4C2411yescharacteristic
290C2×C4⋊C4C2412yes—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—24no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4—24no—
301C2×C22⋊C4C2×D412yes—
302C2×C22⋊C4C2×D414yes—
303C2×C22⋊C4C2×D414yes—
304C2×C42—21no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—21no—
309C2×C42—22no—
310C2×C42C2×D412yes—
311C2×C42C2×D411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42C2411yescharacteristic
314C2×C42C2412yes—
315C2×C42C2412yes—
316#106—28no—
317C4⋊C8—42no—
318C4⋊C8—42no—
319C4⋊C8—42no—
320#103—44no—
321#103—44no—
322#103—44no—
323#102—28no—
324#102—28no—
325#102—28no—
326#102—44no—
327#102—44no—
328C22⋊C8—42no—
329C22⋊C8—44no—
330C4×C8—42no—
331#95—24no—
332#95—24no—
333#95—22no—
334#95—21no—
335#95—22no—
336#95—22no—
337#95—24no—
338#95C2×D414yes—
339#95C2×D414yes—
340C24—24no—
341C24—44no—
342C24—82no—
343C24—84no—
344C24—84no—
345C24—82no—
346C24C22×D412yes—
347C2×Q8—21no—
348C2×Q8—22no—
349C2×Q8—24no—
350C2×Q8—22no—
351C2×Q8—22no—
352C2×Q8—22no—
353C2×Q8—24no—
354C2×Q8—41no—
355C2×Q8—41no—
356C2×Q8—42no—
357C2×Q8—44no—
358C2×Q8—82no—
359C2×Q8—82no—
360C2×Q8—82no—
361C2×Q8C2×SD1614yes—
362C2×Q8H(5,𝔽2)12yes—
363C2×D4—24no—
364C2×D4—24no—
365C2×D4—24no—
366C2×D4—24no—
367C2×D4—24no—
368C2×D4—24no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—44no—
372C2×D4—48no—
373C2×D4—48no—
374C2×D4—44no—
375C2×D4—44no—
376C2×D4—82no—
377C2×D4—84no—
378C2×D4—84no—
379C2×D4—82no—
380C2×D4—82no—
381C2×D4—82no—
382C2×D4—82no—
383C2×D4—82no—
384C2×D4—82no—
385C2×D4—82no—
386C2×D4—84no—
387C2×D4—88no—
388C2×D4—84no—
389C2×D4—84no—
390C2×D4C2×SD1614yes—
391C22×C4—22no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—44no—
402C22×C4—84no—
403C22×C4C22×D412yes—
404C22×C4C22×D412yes—
405C22×C4C22×D412yes—
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D412yes—
408C22×C4C22×D412yes—
409C22×C4C22×D411yescharacteristic
410C22×C4C2511yescharacteristic
411SD16—88no—
412SD16—88no—
413SD16—88no—
414C2×C8—24no—
415C2×C8—24no—
416C2×C8—24no—
417C2×C8—24no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—84no—
421C4⋊C4—24no—
422C4⋊C4—21no—
423C4⋊C4—22no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—42no—
431C4⋊C4—44no—
432C4⋊C4—44no—
433C4⋊C4—44no—
434C4⋊C4—44no—
435C4⋊C4—48no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4—42no—
439C4⋊C4—42no—
440C4⋊C4—42no—
441C4⋊C4—84no—
442C4⋊C4—82no—
443C4⋊C4H(5,𝔽2)14yes—
444C4⋊C4H(5,𝔽2)12yes—
445C22⋊C4—42no—
446C22⋊C4—44no—
447C22⋊C4—44no—
448C22⋊C4—48no—
449C22⋊C4—48no—
450C22⋊C4—44no—
451C22⋊C4—48no—
452C22⋊C4—48no—
453C22⋊C4—48no—
454C22⋊C4—84no—
455C22⋊C4—84no—
456C22⋊C4—84no—
457C22⋊C4C2×SD16116yes—
458C42—24no—
459C42—22no—
460C42—22no—
461C42—22no—
462C42—21no—
463C42—21no—
464C42—22no—
465C42—41no—
466C42—42no—
467C42—44no—
468C42—44no—
469C42—42no—
470C42—44no—
471C42—82no—
472C42C2×SD1614yes—
473C23—24no—
474C23—22no—
475C23—22no—
476C23—24no—
477C23—44no—
478C23—44no—
479C23—44no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—88no—
485C23—164no—
486C23—168no—
487C23—168no—
488C23—164no—
489C23C22×SD1614yes—
490C23C23×D411yescharacteristic
491Q8—24no—
492Q8—44no—
493Q8—41no—
494Q8—41no—
495Q8—42no—
496Q8—42no—
497Q8—44no—
498Q8—164no—
499Q8—162no—
500D4—44no—
501D4—44no—
502D4—44no—
503D4—44no—
504D4—44no—
505D4—88no—
506D4—164no—
507D4—164no—
508D4—164no—
509D4—164no—
510D4—168no—
511C2×C4—22no—
512C2×C4—21no—
513C2×C4—22no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—24no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—24no—
528C2×C4—24no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—44no—
541C2×C4—44no—
542C2×C4—44no—
543C2×C4—164no—
544C2×C4C22×SD1614yes—
545C2×C4C22×SD1612yes—
546C2×C4C22×SD1614yes—
547C2×C4C2×H(5,𝔽2)11yescharacteristic
548C2×C4C2×H(5,𝔽2)12yes—
549C2×C4C2×H(5,𝔽2)11yescharacteristic
550C2×C4#58512yes—
551C8—44no—
552C8—44no—
553C22—24no—
554C22—28no—
555C22—44no—
556C22—42no—
557C22—42no—
558C22—84no—
559C22—84no—
560C22—84no—
561C22—84no—
562C22—84no—
563C22—84no—
564C22—84no—
565C22—1616no—
566C22#348411yescharacteristic
567C22#348812yes—
568C22C23×SD1611yescharacteristic
569C22#358512yes—
570C22#358611yescharacteristic
571C4—22no—
572C4—24no—
573C4—24no—
574C4—42no—
575C4—44no—
576C4—42no—
577C4—42no—
578C4—42no—
579C4—41no—
580C4—42no—
581C4—82no—
582C4—82no—
583C4—84no—
584C4#352911yescharacteristic
585C4#354911yescharacteristic
586C4#355812yes—
587C2—44no—
588C2—164no—
589C2#4622511yescharacteristic
590C2#6156211yescharacteristic
591C2#6167311yescharacteristic
592C2#6195112yes—
593C2#6208912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#45996C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#124 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #36562C2semidirect
3. #33025C2semidirect
4. #32807C2semidirect
5. #32629C2semidirect
6. #30909C2semidirect
7. #30500C2semidirect
8. #30456C2semidirect
9. #26724C2semidirect
10. #25630C2semidirect
11. #25080C2semidirect
12. #23969C2semidirect
13. #23346C2semidirect
15. C2×#557C22semidirect
18. C2×#475C22semidirect
22. C2×#450C22semidirect
23. C2×#450C22semidirect
24. C22×#102C22semidirect
25. #2862C22semidirect
32. #2567C22semidirect
40. #2429C22semidirect
42. #2360C22semidirect
45. C42×Q8C22semidirect
46. #2065C22semidirect
47. #2055C22semidirect
48. #2055C22semidirect
50. #2035C22semidirect
51. #2024C22semidirect
52. #2023C22semidirect
54. #2013C22semidirect
55. #2013C22semidirect
58. #1988C22semidirect
59. #1984C22semidirect
61. #1981C22semidirect
62. #1934C22semidirect
63. #1926C22semidirect
64. #1926C22semidirect
65. #1923C22semidirect
66. #1920C22semidirect
67. #1892C22semidirect
68. #1880C22semidirect
69. #1880C22semidirect
70. #1880C22semidirect
73. #1850C22semidirect
74. #1831C22semidirect
75. #1824C22semidirect
76. C4×#103C22semidirect
246. #124SD16semidirect

Maximal subgroups

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

subgroupindex
#459962
#365622
#330252
#328072
#326292
#309092
#305002
#304562
#267242
#256302
#250802
#239692
#233462