Finite Group Databasebeta

← #6519081

Subgroups — #6519081

Order 512.

1762 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35661C211yesmaximal, characteristic
2#34360C211yesmaximal, characteristic
3#27608C212yesmaximal, complemented
4#27602C212yesmaximal, complemented
5#25024C214yesmaximal, complemented
6#25016C214yesmaximal, complemented
7#24984C214yesmaximal, complemented
8#24976C214yesmaximal, complemented
9#24014C212yesmaximal, complemented
10#24006C212yesmaximal, complemented
11#21991C212yesmaximal, complemented
12#21792C211yesmaximal, characteristic
13#20745C212yesmaximal, complemented
14C2×#536C2211yescharacteristic
15C2×#525C2211yescharacteristic
16C22×C22⋊Q8C2211yescharacteristic
17C2×#519C2211yescharacteristic
18C2×#484C2214yes—
19C2×#484C2212yescomplemented
20C2×#483C2212yescomplemented
21C2×#433C2212yes—
22#2538C2214yes—
23#2536C2214yes—
24#2429—22no—
25#2426—22no—
26#2413—22no—
27#2411—22no—
28#2376—22no—
29#2358—22no—
30#2352C2214yes—
31D4×C4⋊C4C2214yes—
32#2329—22no—
33#2325—22no—
34#2317C2211yescharacteristic
35#2314C2212yes—
36#2311C2212yes—
37#2311C2211yescharacteristic
38#2280C2211yescharacteristic
39C2×#380C2212yes—
40#2077C2214yes—
41#2076C2214yes—
42#2076C2214yescomplemented
43#2075C2214yescomplemented
44#2036C2214yes—
45#2036C2214yes—
46#2036C2214yescomplemented
47#2035C2214yescomplemented
48#1955C2214yes—
49#1954C2214yes—
50#1949C2214yescomplemented
51#1947C2214yescomplemented
52#1873C2212yescomplemented
53#1873C2212yescomplemented
54#1872C2214yes—
55#1854C2214yes—
56#1853C2212yes—
57#1852C2212yes—
58#1852C2214yes—
59#1852C2214yescomplemented
60#1847C2212yescomplemented
61#1845C2212yescomplemented
62#1836C2212yescomplemented
63#1835C2212yescomplemented
64#1825C2214yes—
65#1825C2214yescomplemented
66#1823C2212yes—
67#1823C2212yes—
68#1805C2214yescomplemented
69#1803C2214yescomplemented
70#1787C2214yes—
71#1780C2212yes—
72#1779C2212yes—
73#1763C2214yescomplemented
74C4×#103C2214yescomplemented
75C22×#41C2311yescharacteristic
76C23×Q8C2311yescharacteristic
77C24×C4C2311yescharacteristic
78#536—44no—
79C2×C4⋊Q8—21no—
80C2×#124—21no—
81#525—28no—
82C2×#123—22no—
83C2×#123D412yes—
84C2×C22⋊Q8—22no—
85C2×C22⋊Q8—44no—
86C2×C22⋊Q8D412yes—
87C2×C22⋊Q8C2312yes—
88C2×C22⋊Q8C2314yes—
89C2×C22⋊Q8C2312yes—
90C2×#121D412yes—
91C2×#120D412yes—
92#519—28no—
93C2×C4×Q8—22no—
94C2×C4×Q8C2314yes—
95C2×C4×D4—24no—
96C2×C4×D4C2314yes—
97C2×#117—22no—
98C2×#117C2311yescharacteristic
99C2×#117C2312yes—
100C2×#117C2311yescharacteristic
101C2×#117C2311yescharacteristic
102C22×C4⋊C4—21no—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C22⋊C4—21no—
108C22×C22⋊C4—22no—
109#484—48no—
110#484—44no—
111#483—44no—
112#433—28no—
113C2×#107C2314yes—
114C2×#107C2314yes—
115C2×#107C2314yes—
116C2×#106C2314yes—
117C2×C4⋊C8C2314yes—
118C2×#103C2314yes—
119C2×#103C2314yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×C22⋊C8C2314yes—
123C2×C22⋊C8C2312yes—
124C2×C22⋊C8C2314yes—
125C2×C22⋊C8C2312yes—
126C2×C22⋊C8C2312yes—
127C2×C22⋊C8C2312yes—
128C2×C22⋊C8C2314yes—
129C2×#97C2312yes—
130C2×C4×C8C2312yes—
131#398—24no—
132#396—24no—
133#396—24no—
134#394—24no—
135#391—22no—
136#391—21no—
137#390—22no—
138#390—21no—
139#390—24no—
140#389—22no—
141#389—24no—
142#389—24no—
143#388—24no—
144#388—24no—
145#388—24no—
146#388—24no—
147#387—24no—
148#387—24no—
149#386—22no—
150#386—22no—
151#386—22no—
152#386—24no—
153#385—22no—
154#385—24no—
155#385—24no—
156#384C2312yes—
157#384C2312yes—
158#382C2312yes—
159#382C2312yes—
160#381—21no—
161#380—22no—
162#380—22no—
163#380C2314yes—
164#380C2314yes—
165#380C2314yes—
166#380C2312yes—
167#380C2314yes—
168#380C2312yes—
169#379—21no—
170C4×C4⋊C4C2314yes—
171C4×C22⋊C4—22no—
172C4×C22⋊C4—24no—
173C2×#95—22no—
174#340C2314yes—
175#340C2314yes—
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#340C2314yes—
180#340C2314yes—
181#340C2314yes—
182C25C2×D411yescharacteristic
183C2×#41—24no—
184C2×#41—42no—
185C2×#41—42no—
186C22×Q8—24no—
187C22×Q8—21no—
188C22×Q8—24no—
189C22×Q8—42no—
190C22×Q8C2×D411yescharacteristic
191C22×Q8C2412yes—
192C22×D4C2×D414yes—
193C22×D4C2×D412yes—
194C22×D4C2×D411yescharacteristic
195C23×C4—21no—
196C23×C4—24no—
197C23×C4—21no—
198C23×C4—22no—
199C23×C4—22no—
200C23×C4—22no—
201C23×C4—42no—
202C23×C4—42no—
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D412yes—
205C23×C4C2412yes—
206C23×C4C2412yes—
207C23×C4C2411yescharacteristic
208C22×C8C2414yes—
209C22×C8C2414yes—
210C4⋊Q8—44no—
211#124—44no—
212#123—28no—
213#123—48no—
214C22⋊Q8—28no—
215C22⋊Q8—28no—
216C22⋊Q8—24no—
217C22⋊Q8—24no—
218C22⋊Q8—44no—
219C22⋊Q8—42no—
220C22⋊Q8—42no—
221C22⋊Q8—48no—
222#121—28no—
223#120—28no—
224C4×Q8—28no—
225C4×Q8—28no—
226C4×Q8—48no—
227C4×D4—28no—
228C4×D4—28no—
229C4×D4—88no—
230#117—24no—
231#117—42no—
232#117—48no—
233#117—44no—
234#117#4118yes—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—42no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—42no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—44no—
252C2×C4⋊C4C2×D414yes—
253C2×C4⋊C4C2×D412yes—
254C2×C4⋊C4C2414yes—
255C2×C4⋊C4C2414yes—
256C2×C4⋊C4C2412yes—
257C2×C4⋊C4C2412yes—
258C2×C4⋊C4C2411yescharacteristic
259C2×C4⋊C4C2411yescharacteristic
260C2×C22⋊C4—24no—
261C2×C22⋊C4—24no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—22no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—24no—
272C2×C22⋊C4—44no—
273C2×C22⋊C4—84no—
274C2×C22⋊C4C2×D414yes—
275C2×C22⋊C4C2×D414yes—
276C2×C22⋊C4C2×D412yes—
277C2×C42—24no—
278C2×C42—21no—
279C2×C42—21no—
280C2×C42—22no—
281C2×C42—24no—
282C2×C42C2412yes—
283#107—44no—
284#107—44no—
285#107—44no—
286#106—44no—
287C4⋊C8—28no—
288#103—28no—
289#103—28no—
290#103—44no—
291#103—44no—
292C22⋊C8—28no—
293C22⋊C8—42no—
294C22⋊C8—42no—
295C22⋊C8—44no—
296C22⋊C8—42no—
297C22⋊C8—44no—
298C22⋊C8—42no—
299#97—24no—
300C4×C8—24no—
301#95—24no—
302#95—24no—
303#95—22no—
304#95—22no—
305#95—24no—
306#95—24no—
307#95—24no—
308#95—24no—
309#95—24no—
310C24—22no—
311C24—24no—
312C24—41no—
313C24—42no—
314C24—41no—
315C24C22×D412yes—
316C24C22×D411yescharacteristic
317C24C22×D412yes—
318#41—88no—
319C2×Q8—24no—
320C2×Q8—22no—
321C2×Q8—22no—
322C2×Q8—24no—
323C2×Q8—24no—
324C2×Q8—41no—
325C2×Q8—41no—
326C2×Q8—416no—
327C2×Q8—42no—
328C2×Q8—42no—
329C2×Q8H(5,𝔽2)14yes—
330C2×D4—28no—
331C2×D4—24no—
332C2×D4—28no—
333C2×D4—22no—
334C2×D4—44no—
335C2×D4—42no—
336C2×D4—41no—
337C2×D4—41no—
338C2×D4—42no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—21no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—22no—
358C22×C4—21no—
359C22×C4—41no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—44no—
367C22×C4—44no—
368C22×C4—88no—
369C22×C4—82no—
370C22×C4—82no—
371C22×C4—82no—
372C22×C4—82no—
373C22×C4—82no—
374C22×C4—82no—
375C22×C4—84no—
376C22×C4C22×D412yes—
377C22×C4C22×D412yes—
378C22×C4C22×D411yescharacteristic
379C22×C4C22×D414yes—
380C22×C4C2×#4114yes—
381C22×C4C2511yescharacteristic
382C2×C8—24no—
383C2×C8—24no—
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—24no—
388C4⋊C4—28no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—22no—
393C4⋊C4—24no—
394C4⋊C4—22no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—48no—
398C4⋊C4—44no—
399C4⋊C4—44no—
400C4⋊C4—42no—
401C4⋊C4—48no—
402C4⋊C4—84no—
403C4⋊C4—82no—
404C4⋊C4—82no—
405C4⋊C4—82no—
406C4⋊C4—84no—
407C4⋊C4#12318yes—
408C22⋊C4—28no—
409C22⋊C4—28no—
410C22⋊C4—48no—
411C22⋊C4—44no—
412C22⋊C4—44no—
413C22⋊C4—48no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—44no—
417C22⋊C4—82no—
418C22⋊C4—84no—
419C22⋊C4—84no—
420C22⋊C4—82no—
421C22⋊C4—84no—
422C22⋊C4#12318yes—
423C42—42no—
424C42—42no—
425C42—48no—
426C42—82no—
427C42—84no—
428C42C2×#4114yes—
429C42C2×#4114yes—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—21no—
436C23—21no—
437C23—22no—
438C23—22no—
439C23—22no—
440C23—24no—
441C23—22no—
442C23—22no—
443C23—22no—
444C23—24no—
445C23—48no—
446C23—44no—
447C23—82no—
448C23—84no—
449C23—82no—
450C23C2×#12312yes—
451C23#57814yes—
452C23C23×D411yescharacteristic
453Q8—28no—
454Q8—48no—
455Q8—44no—
456Q8—82no—
457D4—416no—
458D4—82no—
459D4—84no—
460C2×C4—24no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—24no—
464C2×C4—28no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—48no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—42no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—84no—
495C2×C4—82no—
496C2×C4—84no—
497C2×C4—164no—
498C2×C4—164no—
499C2×C4—164no—
500C2×C4C2×#12318yes—
501C2×C4C2×#12312yes—
502C2×C4C22×#4112yes—
503C2×C4C2×H(5,𝔽2)12yes—
504C2×C4#58512yes—
505C8—44no—
506C8—44no—
507C22—24no—
508C22—24no—
509C22—24no—
510C22—24no—
511C22—22no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—41no—
516C22—41no—
517C22—42no—
518C22—42no—
519C22—44no—
520C22—44no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—88no—
525C22C22×#12311yescharacteristic
526C22#348411yescharacteristic
527C22#348511yescharacteristic
528C22C2×#57812yes—
529C22#358512yes—
530C4—28no—
531C4—22no—
532C4—42no—
533C4—42no—
534C4—44no—
535C4—84no—
536C4—82no—
537C4—81no—
538C4—81no—
539C4—82no—
540C4—84no—
541C4#349414yes—
542C2—44no—
543C2—42no—
544C2—82no—
545C2#3554011yescharacteristic
546C2#6052612yes—
547C2#6163411yescharacteristic
548C2#6165111yescharacteristic
549C2#6175412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#525 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #27608C2semidirect
4. #27602C2semidirect
5. #25024C2semidirect
6. #25016C2semidirect
7. #24984C2semidirect
8. #24976C2semidirect
9. #24014C2semidirect
10. #24006C2semidirect
11. #21991C2semidirect
13. #20745C2semidirect
19. C2×#484C22semidirect
20. C2×#483C22semidirect
42. #2076C22semidirect
43. #2075C22semidirect
46. #2036C22semidirect
47. #2035C22semidirect
50. #1949C22semidirect
51. #1947C22semidirect
52. #1873C22semidirect
53. #1873C22semidirect
59. #1852C22semidirect
60. #1847C22semidirect
61. #1845C22semidirect
62. #1836C22semidirect
63. #1835C22semidirect
65. #1825C22semidirect
68. #1805C22semidirect
69. #1803C22semidirect
73. #1763C22semidirect
74. C4×#103C22semidirect

Maximal subgroups

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

subgroupindex
#356612
#343602
#276082
#276022
#250242
#250162
#249842
#249762
#240142
#240062
#219912
#217922
#207452