Finite Group Databasebeta

← #6633301

Subgroups — #6633301

Order 512.

1720 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37641C211yesmaximal, characteristic, complemented
2#35531C211yesmaximal, characteristic
3#32916C212yesmaximal, complemented
4#31661C214yesmaximal, complemented
5#31647C214yesmaximal, complemented
6#30760C214yesmaximal, complemented
7#30741C214yesmaximal, complemented
8#29723C212yesmaximal, complemented
9#29344C212yesmaximal, complemented
10#25732C212yesmaximal
11#23935C212yesmaximal, complemented
12#23933C212yesmaximal, complemented
13#21920C211yesmaximal, characteristic, complemented
14C2×#535C2211yescharacteristic
15C2×#533C2212yes—
16C2×#520C2211yescharacteristic
17C2×#497C2212yes—
18C2×#492C2212yes—
19C2×#462C2212yescomplemented
20C2×#461C2212yescomplemented
21C2×#452C2214yes—
22C2×#450C2214yes—
23#2822—21no—
24#2822—21no—
25#2625—24no—
26#2621C2212yes—
27#2621C2212yes—
28#2582—21no—
29#2582—21no—
30#2568C2214yescomplemented
31#2546C2214yescomplemented
32#2504—21no—
33#2501—21no—
34#2500—21no—
35#2500—21no—
36#2448C2214yes—
37#2445C2214yes—
38#2418—24no—
39#2317C2211yescharacteristic
40#2309C2211yescharacteristic
41#2297C2211yescharacteristic
42C2×#390C2211yescharacteristic
43#2285C2211yescharacteristic
44#2101C2214yescomplemented
45#2073C2214yes—
46#2055C2214yes—
47#2055C2214yes—
48#2043C2214yescomplemented
49#2024C2212yescomplemented
50#2023C2214yes—
51#2023C2212yescomplemented
52#2016C2212yescomplemented
53#2015C2214yes—
54#2014C2214yes—
55#2013C2214yes—
56#2013C2214yes—
57#2013C2212yescomplemented
58#1961C2212yes—
59#1958C2212yes—
60#1935C2214yescomplemented
61#1933C2214yescomplemented
62#1917C2214yescomplemented
63#1916C2214yescomplemented
64#1880C2214yes—
65#1880C2214yes—
66#1880C2214yescomplemented
67#1878C2214yescomplemented
68#1878C2214yescomplemented
69#1877C2214yescomplemented
70#1873C2214yes—
71#1872C2214yes—
72#1817C2212yes—
73#1805C2212yescomplemented
74#1804C2212yescomplemented
75#1798C2212yes—
76#1750C2212yes—
77C22×#41C2311yescharacteristic
78C23×D4C2311yescharacteristic
79C22×Q16C2314yes—
80C22×SD16C2314yes—
81C22×SD16C2314yes—
82C22×SD16C2314yes—
83#535—42no—
84#535—42no—
85#533—48no—
86C2×C4⋊Q8C2312yes—
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2312yes—
89C2×C4⋊Q8C2312yes—
90C2×#127C2312yes—
91C2×#125—22no—
92C2×#124C2312yes—
93C2×#123—24no—
94C2×C22⋊Q8—22no—
95C2×C22⋊Q8C2314yes—
96C2×#121—21no—
97C2×#121—21no—
98C2×#120—22no—
99#520—42no—
100#520—42no—
101C2×C4×Q8—22no—
102C2×C4×Q8C2312yes—
103C2×C4×Q8C2314yes—
104C2×C4×D4—22no—
105C2×C4×D4C2312yes—
106C2×#117—22no—
107C2×#117—21no—
108C2×#117C2311yescharacteristic
109C2×#117C2311yescharacteristic
110C2×#117C2311yescharacteristic
111C22×C4⋊C4—21no—
112C22×C4⋊C4C2311yescharacteristic
113C22×C22⋊C4—22no—
114C22×C42C2311yescharacteristic
115#497—44no—
116#492—44no—
117#462—44no—
118#461—44no—
119#452—48no—
120#450—48no—
121C2×C4⋊C8C2312yes—
122C2×C4⋊C8C2312yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×C22⋊C8C2314yes—
132C2×#97C2312yes—
133C2×C4×C8C2312yes—
134#400—24no—
135#400—24no—
136#399—24no—
137#398—22no—
138#398—22no—
139#398—22no—
140#398—22no—
141#398—24no—
142#398—24no—
143#397—22no—
144#397—22no—
145#397—22no—
146#397—22no—
147#397—24no—
148#394—24no—
149#394—24no—
150#391—24no—
151#391—24no—
152#391C2314yes—
153#391C2314yes—
154#391C2312yes—
155#390—21no—
156#390—21no—
157#390—21no—
158#390—21no—
159#390C2314yes—
160#390C2314yes—
161#390C2312yes—
162#389—24no—
163#389—24no—
164#389C2314yes—
165#388—24no—
166#388—24no—
167#386—22no—
168#386—24no—
169#385—22no—
170#384—21no—
171#384D411yescharacteristic
172#384D411yescharacteristic
173#382—21no—
174#382D411yescharacteristic
175#382D411yescharacteristic
176#382C2312yes—
177#382C2312yes—
178#380D414yes—
179#379—22no—
180C4×C4⋊C4C2314yes—
181C4×C22⋊C4—21no—
182C4×C22⋊C4—21no—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#336C2314yes—
188#336C2314yes—
189#336C2312yes—
190#336C2312yes—
191#336C2314yes—
192C25—21no—
193C2×#41—22no—
194C2×#41—22no—
195C2×#41—24no—
196C2×#41—42no—
197C22×Q8—22no—
198C22×Q8C2414yes—
199C22×Q8C2411yescharacteristic
200C22×D4—21no—
201C22×D4—21no—
202C22×D4—22no—
203C22×D4—22no—
204C22×D4—24no—
205C22×D4—41no—
206C22×D4—41no—
207C22×D4—42no—
208C22×D4C2412yes—
209C22×D4C2411yescharacteristic
210C23×C4—21no—
211C23×C4—21no—
212C23×C4—22no—
213C23×C4C2×D412yes—
214C23×C4C2411yescharacteristic
215C2×Q16—28no—
216C2×Q16—44no—
217C2×Q16—44no—
218C2×SD16—28no—
219C2×SD16—44no—
220C2×SD16—44no—
221C2×SD16—44no—
222C2×SD16—44no—
223C2×SD16—44no—
224C2×SD16—44no—
225C2×SD16—44no—
226C2×SD16—44no—
227C22×C8C2414yes—
228C22×C8C2414yes—
229C4⋊Q8—44no—
230C4⋊Q8—44no—
231C4⋊Q8—42no—
232C4⋊Q8—42no—
233C4⋊Q8—42no—
234C4⋊Q8—42no—
235#127—24no—
236#127—42no—
237#125—48no—
238#124—24no—
239#124—42no—
240#123—88no—
241C22⋊Q8—28no—
242C22⋊Q8—28no—
243C22⋊Q8—48no—
244#121—44no—
245#121—44no—
246#120—48no—
247C4×Q8—24no—
248C4×Q8—42no—
249C4×Q8—44no—
250C4×Q8—44no—
251C4×Q8—48no—
252C4×D4—24no—
253C4×D4—42no—
254C4×D4—48no—
255#117—41no—
256#117—41no—
257#117—41no—
258#117—41no—
259#117—42no—
260#117—44no—
261#117—48no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—24no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—24no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—41no—
275C2×C4⋊C4—41no—
276C2×C4⋊C4—44no—
277C2×C4⋊C4C2×D414yes—
278C2×C4⋊C4C2×D411yescharacteristic
279C2×C4⋊C4C2×D411yescharacteristic
280C2×C4⋊C4C2×D411yescharacteristic
281C2×C4⋊C4C2×D411yescharacteristic
282C2×C4⋊C4C2×D414yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2414yes—
286C2×C22⋊C4—24no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—84no—
298C2×C22⋊C4C2×D412yes—
299C2×C22⋊C4C2×D412yes—
300C2×C42—24no—
301C2×C42—22no—
302C2×C42—24no—
303C2×C42—21no—
304C2×C42C2×D411yescharacteristic
305C2×C42C2×D411yescharacteristic
306C2×C42C2412yes—
307C2×C42C2412yes—
308C2×C42C2412yes—
309C4⋊C8—42no—
310C4⋊C8—42no—
311#103—44no—
312#103—44no—
313#103—44no—
314#103—44no—
315#103—44no—
316#102—44no—
317#102—44no—
318#102—44no—
319C22⋊C8—28no—
320#97—42no—
321C4×C8—42no—
322#95—22no—
323#95—22no—
324#95—24no—
325#95—22no—
326#95—22no—
327#95—22no—
328#95—22no—
329#95—24no—
330#95—24no—
331#95—24no—
332#95—24no—
333#95C2×D414yes—
334#95C2×D414yes—
335C24—21no—
336C24—22no—
337C24—42no—
338C24—42no—
339C24—42no—
340C24—82no—
341C24—82no—
342C24C22×D411yescharacteristic
343#41—48no—
344#41—48no—
345C2×Q8—21no—
346C2×Q8—22no—
347C2×Q8—24no—
348C2×Q8—24no—
349C2×Q8—24no—
350C2×Q8—24no—
351C2×Q8—22no—
352C2×Q8—24no—
353C2×Q8—24no—
354C2×Q8—44no—
355C2×Q8—44no—
356C2×Q8—44no—
357C2×Q8H(5,𝔽2)12yes—
358C2×D4—24no—
359C2×D4—22no—
360C2×D4—21no—
361C2×D4—24no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—24no—
366C2×D4—42no—
367C2×D4—42no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4—42no—
371C2×D4—42no—
372C2×D4—42no—
373C2×D4—84no—
374C2×D4—84no—
375C2×D4—84no—
376C2×D4—84no—
377C2×D4H(5,𝔽2)12yes—
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—24no—
383C22×C4—21no—
384C22×C4—24no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4—44no—
392C22×C4—42no—
393C22×C4—42no—
394C22×C4—84no—
395C22×C4—82no—
396C22×C4C22×D412yes—
397C22×C4C22×D412yes—
398C22×C4C22×D412yes—
399C22×C4C22×D412yes—
400C22×C4C22×D412yes—
401C22×C4C22×D411yescharacteristic
402C22×C4C22×D412yes—
403C22×C4H(5,𝔽2)14yes—
404C22×C4C2511yescharacteristic
405Q16—88no—
406SD16—88no—
407SD16—88no—
408SD16—88no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C2×C8—24no—
413C2×C8—24no—
414C2×C8—24no—
415C4⋊C4—28no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—41no—
422C4⋊C4—44no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—44no—
427C4⋊C4—42no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—48no—
431C4⋊C4—44no—
432C4⋊C4—48no—
433C4⋊C4—84no—
434C4⋊C4—82no—
435C4⋊C4—82no—
436C4⋊C4—82no—
437C4⋊C4—82no—
438C22⋊C4—42no—
439C22⋊C4—42no—
440C22⋊C4—44no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—42no—
444C22⋊C4—48no—
445C22⋊C4—48no—
446C22⋊C4—44no—
447C22⋊C4—84no—
448C22⋊C4—82no—
449C22⋊C4—84no—
450C22⋊C4—84no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—24no—
458C42—44no—
459C42—42no—
460C42—48no—
461C23—22no—
462C23—21no—
463C23—21no—
464C23—22no—
465C23—22no—
466C23—42no—
467C23—41no—
468C23—41no—
469C23—42no—
470C23—44no—
471C23—42no—
472C23—42no—
473C23—42no—
474C23—44no—
475C23—84no—
476C23—88no—
477C23—164no—
478C23—162no—
479C23—162no—
480C23#53512yes—
481C23C23×D411yescharacteristic
482Q8—24no—
483Q8—42no—
484Q8—44no—
485Q8—44no—
486Q8—44no—
487Q8—44no—
488Q8—88no—
489D4—24no—
490D4—48no—
491D4—42no—
492D4—44no—
493D4—44no—
494D4—84no—
495D4—164no—
496D4—164no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—24no—
502C2×C4—24no—
503C2×C4—24no—
504C2×C4—22no—
505C2×C4—24no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—48no—
516C2×C4—48no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—44no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—164no—
531C2×C4—162no—
532C2×C4#53512yes—
533C2×C4#57714yes—
534C2×C4C2×H(5,𝔽2)12yes—
535C2×C4C2×H(5,𝔽2)11yescharacteristic
536C2×C4C2×H(5,𝔽2)11yescharacteristic
537C2×C4#58512yes—
538C8—44no—
539C8—44no—
540C22—22no—
541C22—24no—
542C22—42no—
543C22—41no—
544C22—41no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—82no—
549C22—82no—
550C22—82no—
551C22—84no—
552C22—84no—
553C22—82no—
554C22—82no—
555C22—168no—
556C22C2×#53511yescharacteristic
557C22#348411yescharacteristic
558C22#348711yescharacteristic
559C22C2×#57712yes—
560C22#358612yes—
561C4—22no—
562C4—24no—
563C4—42no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—42no—
568C4—42no—
569C4—84no—
570C4—84no—
571C4—82no—
572C4—82no—
573C4#353412yes—
574C4#353412yes—
575C2—42no—
576C2—82no—
577C2—162no—
578C2#3671011yescharacteristic
579C2#6081312yes—
580C2#6166211yescharacteristic
581C2#6167411yescharacteristic
582C2#6191112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#535C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2285 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37641C2semidirect
3. #32916C2semidirect
4. #31661C2semidirect
5. #31647C2semidirect
6. #30760C2semidirect
7. #30741C2semidirect
8. #29723C2semidirect
9. #29344C2semidirect
11. #23935C2semidirect
12. #23933C2semidirect
13. #21920C2semidirect
19. C2×#462C22semidirect
20. C2×#461C22semidirect
30. #2568C22semidirect
31. #2546C22semidirect
44. #2101C22semidirect
48. #2043C22semidirect
49. #2024C22semidirect
51. #2023C22semidirect
52. #2016C22semidirect
57. #2013C22semidirect
60. #1935C22semidirect
61. #1933C22semidirect
62. #1917C22semidirect
63. #1916C22semidirect
66. #1880C22semidirect
67. #1878C22semidirect
68. #1878C22semidirect
69. #1877C22semidirect
73. #1805C22semidirect
74. #1804C22semidirect

Maximal subgroups

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

subgroupindex
#376412
#355312
#329162
#316612
#316472
#307602
#307412
#297232
#293442
#257322
#239352
#239332
#219202