Finite Group Databasebeta

← #6635557

Subgroups — #6635557

Order 512.

1936 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37450C211yesmaximal, characteristic
2#35672C211yesmaximal, characteristic, complemented
3#32912C212yesmaximal, complemented
4#31947C214yesmaximal, complemented
5#31943C214yesmaximal, complemented
6#31004C214yesmaximal, complemented
7#30988C214yesmaximal, complemented
8#29765C212yesmaximal, complemented
9#28323C212yesmaximal, complemented
10#27986C212yesmaximal, complemented
11#24009C212yesmaximal, complemented
12#23919C212yesmaximal, complemented
13#21908C211yesmaximal, characteristic, complemented
14C22×C22⋊Q8C2211yescharacteristic
15C22×#121C2212yes—
16C2×#519C2211yescharacteristic, complemented
17C2×#482C2214yescomplemented
18C2×#460C2212yescomplemented
19C2×#460C2212yescomplemented
20C2×#448C2214yescomplemented
21C2×#450C2214yescomplemented
22#2815—22no—
23#2619C2212yes—
24#2619C2212yes—
25#2617—22no—
26#2617—22no—
27#2504C2214yes—
28#2484—22no—
29#2482—22no—
30#2433C2214yes—
31#2423—22no—
32#2423C2214yescomplemented
33#2421—22no—
34#2421C2214yescomplemented
35#2403—22no—
36#2317C2211yescharacteristic, complemented
37#2317C2211yescharacteristic, complemented
38#2310C2211yescharacteristic, complemented
39#2280C2211yescharacteristic
40#2280C2211yescharacteristic
41#2097C2214yescomplemented
42#2085C2214yescomplemented
43#2073C2214yescomplemented
44#2071C2214yescomplemented
45#2067C2214yescomplemented
46#2065C2214yescomplemented
47#2041C2214yescomplemented
48#2034C2214yes—
49#2033C2214yes—
50#2014C2214yescomplemented
51#2000C2214yes—
52#2000C2214yes—
53#1973C2214yescomplemented
54#1968C2214yescomplemented
55#1961C2212yescomplemented
56#1957C2212yescomplemented
57#1924C2214yescomplemented
58#1923C2214yescomplemented
59#1880C2214yescomplemented
60#1880C2214yescomplemented
61#1875C2214yescomplemented
62#1874C2214yescomplemented
63#1872C2214yescomplemented
64#1853C2212yescomplemented
65#1852C2212yescomplemented
66#1836C2212yescomplemented
67#1835C2212yescomplemented
68#1805C2212yescomplemented
69#1805C2212yescomplemented
70#1798C2212yescomplemented
71#1786C2212yescomplemented
72#1786C2212yescomplemented
73#1779C2212yes—
74#1750C2212yescomplemented
75C23×Q8C2311yescharacteristic
76C23×D4—22no—
77C23×D4C2312yes—
78C24×C4C2311yescharacteristic
79C22×SD16C2314yes—
80C22×SD16C2314yes—
81C2×#127C2312yes—
82C2×#127C2312yes—
83C2×C22⋊Q8—22no—
84C2×C22⋊Q8—44no—
85C2×C22⋊Q8C2314yes—
86C2×#121—22no—
87C2×#121—22no—
88C2×#121—48no—
89C2×#121C2314yes—
90C2×#121C2314yes—
91C2×#120—24no—
92C2×#120—24no—
93#519—44no—
94C2×C4×Q8—21no—
95C2×C4×Q8—21no—
96C2×C4×Q8C2312yes—
97C2×C4×Q8C2312yes—
98C2×#117—22no—
99C2×#117C2311yescharacteristic
100C2×#117C2311yescharacteristic
101C2×#117C2311yescharacteristic
102C2×#117C2311yescharacteristic
103C22×C4⋊C4—21no—
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C22⋊C4—21no—
107C22×C22⋊C4—22no—
108#482—48no—
109#460—44no—
110#460—44no—
111#450—48no—
112#448—48no—
113C2×#107C2314yes—
114C2×#106C2314yes—
115C2×C4⋊C8C2312yes—
116C2×C4⋊C8C2312yes—
117C2×#103C2314yes—
118C2×#103C2314yes—
119C2×#102C2314yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C2×#102C2314yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×C22⋊C8C2314yes—
126C2×C22⋊C8C2312yes—
127C2×C22⋊C8C2312yes—
128C2×C22⋊C8C2314yes—
129C2×#97C2312yes—
130C2×C4×C8C2312yes—
131#400—24no—
132#399—24no—
133#398—24no—
134#397—24no—
135#397—24no—
136#397—24no—
137#394—22no—
138#394—22no—
139#394—24no—
140#392—22no—
141#392—22no—
142#392—24no—
143#392—24no—
144#392—24no—
145#391—22no—
146#391C2314yes—
147#390C2314yes—
148#390C2314yes—
149#389—22no—
150#389C2314yes—
151#388—24no—
152#388—24no—
153#387—24no—
154#387—24no—
155#386—24no—
156#386—24no—
157#385—22no—
158#385—22no—
159#384—21no—
160#384C2312yes—
161#382—21no—
162#382C2312yes—
163#382C2312yes—
164#382C2312yes—
165#381—21no—
166#381—22no—
167#381D412yes—
168#380D412yes—
169#380D412yes—
170#380C2312yes—
171#380C2312yes—
172#379—21no—
173#379D412yes—
174C4×C22⋊C4—22no—
175C4×C22⋊C4C2314yes—
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#340C2314yes—
180#336C2314yes—
181#336C2314yes—
182C25—22no—
183C25C2×D411yescharacteristic
184C22×Q8—24no—
185C22×Q8—21no—
186C22×Q8—24no—
187C22×Q8—41no—
188C22×Q8—41no—
189C22×Q8C2412yes—
190C22×D4—24no—
191C22×D4—24no—
192C22×D4—22no—
193C22×D4—24no—
194C22×D4—22no—
195C22×D4—22no—
196C22×D4—44no—
197C22×D4—42no—
198C22×D4—42no—
199C22×D4—44no—
200C22×D4—48no—
201C22×D4—84no—
202C22×D4—84no—
203C22×D4C2414yes—
204C23×C4—21no—
205C23×C4—24no—
206C23×C4—21no—
207C23×C4—22no—
208C23×C4—42no—
209C23×C4—41no—
210C23×C4—41no—
211C23×C4C2×D412yes—
212C23×C4C2×D411yescharacteristic
213C23×C4C2412yes—
214C23×C4C2411yescharacteristic
215C2×SD16—28no—
216C2×SD16—28no—
217C2×SD16—44no—
218C2×SD16—44no—
219C2×SD16—44no—
220C2×SD16—44no—
221C22×C8C2414yes—
222C22×C8C2414yes—
223#127—42no—
224#127—42no—
225#127—42no—
226#127—42no—
227C22⋊Q8—28no—
228C22⋊Q8—44no—
229C22⋊Q8—48no—
230#121—28no—
231#121—28no—
232#121—44no—
233#121—44no—
234#121—48no—
235#121—48no—
236#120—88no—
237#120—88no—
238C4×Q8—24no—
239C4×Q8—24no—
240C4×Q8—42no—
241C4×Q8—42no—
242C4×Q8—44no—
243C4×Q8—44no—
244#117—24no—
245#117—41no—
246#117—41no—
247#117—42no—
248#117—42no—
249#117—48no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—22no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—24no—
257C2×C4⋊C4—44no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4—44no—
260C2×C4⋊C4C2×D411yescharacteristic
261C2×C4⋊C4C2×D411yescharacteristic
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2414yes—
264C2×C4⋊C4C2412yes—
265C2×C4⋊C4C2411yescharacteristic
266C2×C4⋊C4C2411yescharacteristic
267C2×C4⋊C4C2412yes—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—48no—
280C2×C22⋊C4—44no—
281C2×C22⋊C4C2×D414yes—
282C2×C22⋊C4C2×D412yes—
283C2×C22⋊C4C2×D412yes—
284C2×C22⋊C4C2×D414yes—
285C2×C42—22no—
286C2×C42—22no—
287C2×C42—21no—
288C2×C42—21no—
289C2×C42C2412yes—
290C2×C42C2412yes—
291#107—44no—
292#106—44no—
293C4⋊C8—42no—
294C4⋊C8—42no—
295#103—44no—
296#103—44no—
297#102—44no—
298#102—44no—
299#102—44no—
300#102—44no—
301#102—44no—
302#102—44no—
303C22⋊C8—28no—
304C22⋊C8—42no—
305C22⋊C8—42no—
306C22⋊C8—44no—
307#97—42no—
308C4×C8—42no—
309#95—22no—
310#95—22no—
311#95—24no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—22no—
317#95—22no—
318#95C2×D414yes—
319#95C2×D414yes—
320C24—22no—
321C24—24no—
322C24—24no—
323C24—41no—
324C24—42no—
325C24—41no—
326C24—44no—
327C24—44no—
328C24—44no—
329C24—82no—
330C24—82no—
331C24—82no—
332C24—82no—
333C24C22×D411yescharacteristic
334C24C22×D412yes—
335C2×Q8—24no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—48no—
342C2×Q8—48no—
343C2×Q8—42no—
344C2×Q8—42no—
345C2×Q8H(5,𝔽2)14yes—
346C2×D4—24no—
347C2×D4—24no—
348C2×D4—24no—
349C2×D4—24no—
350C2×D4—24no—
351C2×D4—24no—
352C2×D4—44no—
353C2×D4—48no—
354C2×D4—48no—
355C2×D4—48no—
356C2×D4—48no—
357C2×D4—82no—
358C2×D4—82no—
359C2×D4—84no—
360C2×D4—84no—
361C2×D4—84no—
362C2×D4—84no—
363C2×D4—82no—
364C2×D4—82no—
365C2×D4—84no—
366C2×D4—88no—
367C2×D4—88no—
368C2×D4—84no—
369C2×D4—84no—
370C2×D4—832no—
371C22×C4—22no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—21no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—41no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—41no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—44no—
395C22×C4—84no—
396C22×C4—84no—
397C22×C4—82no—
398C22×C4—82no—
399C22×C4—82no—
400C22×C4C22×D414yes—
401C22×C4C22×D412yes—
402C22×C4C22×D412yes—
403C22×C4C22×D412yes—
404C22×C4C22×D411yescharacteristic
405C22×C4H(5,𝔽2)14yes—
406C22×C4C2511yescharacteristic
407SD16—88no—
408SD16—88no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—24no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—42no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—48no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—44no—
428C4⋊C4—42no—
429C4⋊C4—84no—
430C4⋊C4—84no—
431C22⋊C4—28no—
432C22⋊C4—42no—
433C22⋊C4—42no—
434C22⋊C4—44no—
435C22⋊C4—48no—
436C22⋊C4—48no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—44no—
440C22⋊C4—48no—
441C22⋊C4—84no—
442C22⋊C4—84no—
443C22⋊C4—84no—
444C22⋊C4—84no—
445C22⋊C4—82no—
446C42—22no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—44no—
451C42—44no—
452C42—42no—
453C42—42no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—21no—
459C23—22no—
460C23—21no—
461C23—22no—
462C23—22no—
463C23—22no—
464C23—48no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—44no—
471C23—44no—
472C23—44no—
473C23—82no—
474C23—84no—
475C23—88no—
476C23—82no—
477C23—88no—
478C23—88no—
479C23—164no—
480C23—164no—
481C23—164no—
482C23—164no—
483C23#53512yes—
484C23#57714yes—
485C23C23×D411yescharacteristic
486Q8—24no—
487Q8—24no—
488Q8—48no—
489Q8—42no—
490Q8—42no—
491D4—44no—
492D4—44no—
493D4—44no—
494D4—44no—
495D4—88no—
496D4—816no—
497D4—816no—
498D4—168no—
499D4—164no—
500D4—164no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—24no—
511C2×C4—22no—
512C2×C4—24no—
513C2×C4—22no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—48no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—44no—
533C2×C4—84no—
534C2×C4—82no—
535C2×C4—84no—
536C2×C4—88no—
537C2×C4#53512yes—
538C2×C4C2×H(5,𝔽2)12yes—
539C2×C4C2×H(5,𝔽2)12yes—
540C2×C4#58511yescharacteristic
541C2×C4#58511yescharacteristic
542C8—44no—
543C8—44no—
544C22—24no—
545C22—22no—
546C22—24no—
547C22—24no—
548C22—24no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—41no—
554C22—41no—
555C22—42no—
556C22—88no—
557C22—84no—
558C22—84no—
559C22—84no—
560C22—84no—
561C22—84no—
562C22—84no—
563C22—84no—
564C22—1616no—
565C22C2×#53511yescharacteristic
566C22#348411yescharacteristic
567C22#348511yescharacteristic
568C22C2×#57712yes—
569C22#358512yes—
570C4—22no—
571C4—44no—
572C4—42no—
573C4—44no—
574C4—42no—
575C4—42no—
576C4—82no—
577C4—84no—
578C4—81no—
579C4—81no—
580C4—82no—
581C4#353412yes—
582C4#353412yes—
583C2—42no—
584C2—44no—
585C2—164no—
586C2#3672111yescharacteristic
587C2#6081212yes—
588C2#6163311yescharacteristic
589C2#6165111yescharacteristic
590C2#6175612yes—
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. #35672C2semidirect
3. #32912C2semidirect
4. #31947C2semidirect
5. #31943C2semidirect
6. #31004C2semidirect
7. #30988C2semidirect
8. #29765C2semidirect
9. #28323C2semidirect
10. #27986C2semidirect
11. #24009C2semidirect
12. #23919C2semidirect
13. #21908C2semidirect
16. C2×#519C22semidirect
17. C2×#482C22semidirect
18. C2×#460C22semidirect
19. C2×#460C22semidirect
20. C2×#448C22semidirect
21. C2×#450C22semidirect
32. #2423C22semidirect
34. #2421C22semidirect
36. #2317C22semidirect
37. #2317C22semidirect
38. #2310C22semidirect
41. #2097C22semidirect
42. #2085C22semidirect
43. #2073C22semidirect
44. #2071C22semidirect
45. #2067C22semidirect
46. #2065C22semidirect
47. #2041C22semidirect
50. #2014C22semidirect
53. #1973C22semidirect
54. #1968C22semidirect
55. #1961C22semidirect
56. #1957C22semidirect
57. #1924C22semidirect
58. #1923C22semidirect
59. #1880C22semidirect
60. #1880C22semidirect
61. #1875C22semidirect
62. #1874C22semidirect
63. #1872C22semidirect
64. #1853C22semidirect
65. #1852C22semidirect
66. #1836C22semidirect
67. #1835C22semidirect
68. #1805C22semidirect
69. #1805C22semidirect
70. #1798C22semidirect
71. #1786C22semidirect
72. #1786C22semidirect
74. #1750C22semidirect

Maximal subgroups

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

subgroupindex
#374502
#356722
#329122
#319472
#319432
#310042
#309882
#297652
#283232
#279862
#240092
#239192
#219082