Finite Group Databasebeta

← #6633299

Subgroups — #6633299

Order 512.

1884 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37420C211yesmaximal, characteristic, complemented
2#35554C211yesmaximal, characteristic, complemented
3#32917C212yesmaximal, complemented
4#31662C214yesmaximal, complemented
5#31658C214yesmaximal, complemented
6#30763C214yesmaximal, complemented
7#30755C214yesmaximal, complemented
8#29727C212yesmaximal, complemented
9#29339C212yesmaximal, complemented
10#25725C212yesmaximal, complemented
11#23942C211yesmaximal, characteristic, complemented
12#23942C211yesmaximal, characteristic, complemented
13#23939C211yesmaximal, characteristic, complemented
14#23939C211yesmaximal, characteristic, complemented
15#21917C211yesmaximal, characteristic, complemented
16C2×#541C2211yescharacteristic
17C2×#540C2211yescharacteristic
18C22×#127C2211yescharacteristic
19C2×#518C2211yescharacteristic
20C2×#496C2214yes—
21C2×#463C2214yescomplemented
22C2×#450C2212yes—
23C2×#450C2212yes—
24C2×#447C2212yes—
25C2×#447C2212yes—
26#2850C2212yes—
27#2707—24no—
28#2691—24no—
29#2620C2212yes—
30#2566C2212yescomplemented
31#2565C2212yescomplemented
32#2548C2212yescomplemented
33#2547C2212yescomplemented
34#2445—24no—
35#2445C2214yes—
36#2445C2214yes—
37#2405—24no—
38#2309C2211yescharacteristic
39#2309C2211yescharacteristic
40#2309C2211yescharacteristic
41#2296C2211yescharacteristic
42#2296C2211yescharacteristic
43#2101C2214yescomplemented
44#2071C2214yescomplemented
45#2055C2214yescomplemented
46#2054C2214yescomplemented
47#2042C2214yescomplemented
48#2025C2212yescomplemented
49#2025C2212yescomplemented
50#2025C2212yescomplemented
51#2025C2212yescomplemented
52#2014C2214yes—
53#2013C2212yescomplemented
54#2013C2212yescomplemented
55#2012C2214yescomplemented
56#2011C2214yes—
57#2011C2212yescomplemented
58#2011C2212yescomplemented
59#1961C2214yes—
60#1936C2214yescomplemented
61#1935C2214yescomplemented
62#1917C2214yescomplemented
63#1917C2214yescomplemented
64#1880C2214yescomplemented
65#1880C2214yescomplemented
66#1879C2214yescomplemented
67#1879C2214yescomplemented
68#1879C2214yescomplemented
69#1877C2214yescomplemented
70#1875C2212yes—
71#1875C2212yes—
72#1873C2212yes—
73#1873C2212yes—
74#1818C2211yescharacteristic
75#1818C2211yescharacteristic
76#1802C2214yescomplemented
77#1798C2212yes—
78#1749C2212yes—
79C23×D4—22no—
80C23×D4C2311yescharacteristic
81C23×D4C2311yescharacteristic
82C22×SD16C2314yes—
83C22×SD16C2314yes—
84C22×D8C2314yes—
85C22×D8C2314yes—
86#541—44no—
87#540—44no—
88C2×C4⋊Q8C2311yescharacteristic
89C2×#127—22no—
90C2×#127—44no—
91C2×#127C2314yes—
92C2×#127C2311yescharacteristic
93C2×#126—22no—
94C2×#125—22no—
95C2×#125C2311yescharacteristic
96C2×#125C2311yescharacteristic
97C2×#124C2312yes—
98C2×#124C2312yes—
99C2×#123—22no—
100C2×#123—24no—
101C2×#123—24no—
102C2×C22⋊Q8C2312yes—
103C2×#121—22no—
104C2×#121C2312yes—
105#518—44no—
106C2×C4×Q8C2312yes—
107C2×C4×D4—22no—
108C2×C4×D4C2314yes—
109C2×C4×D4C2312yes—
110C2×#117C2311yescharacteristic
111C2×#117C2311yescharacteristic
112C2×#117C2311yescharacteristic
113C2×#117C2311yescharacteristic
114C22×C22⋊C4—22no—
115C22×C42C2311yescharacteristic
116#496—48no—
117#463—48no—
118#450—44no—
119#450—44no—
120#447—44no—
121#447—44no—
122C2×C4⋊C8C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×#102C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×C22⋊C8C2312yes—
132C2×C22⋊C8C2312yes—
133C2×#97C2314yes—
134#398—24no—
135#397—24no—
136#396—24no—
137#396—24no—
138#394—24no—
139#394—24no—
140#394—24no—
141#394—24no—
142#394—24no—
143#394—24no—
144#391C2312yes—
145#391C2312yes—
146#390—22no—
147#390—22no—
148#390—22no—
149#390—24no—
150#390C2314yes—
151#390C2312yes—
152#390C2312yes—
153#390C2312yes—
154#390C2312yes—
155#390C2312yes—
156#390C2312yes—
157#389D414yes—
158#389C2314yes—
159#388—24no—
160#387—24no—
161#386—24no—
162#386—24no—
163#386—24no—
164#385—22no—
165#385—24no—
166#385D414yes—
167#384C2312yes—
168#382C2312yes—
169#380—24no—
170#379—22no—
171#379—22no—
172#379—22no—
173C4×C4⋊C4—22no—
174C4×C4⋊C4C2312yes—
175C4×C4⋊C4C2312yes—
176#340C2314yes—
177#340C2314yes—
178#340C2314yes—
179#340C2314yes—
180#336C2314yes—
181#336C2312yes—
182#336C2312yes—
183#336C2312yes—
184#336C2312yes—
185#336C2314yes—
186C25—21no—
187C25—21no—
188C22×Q8C2412yes—
189C22×D4—24no—
190C22×D4—22no—
191C22×D4—21no—
192C22×D4—21no—
193C22×D4—24no—
194C22×D4—24no—
195C22×D4—22no—
196C22×D4—24no—
197C22×D4—24no—
198C22×D4—42no—
199C22×D4—42no—
200C22×D4—42no—
201C22×D4—48no—
202C22×D4—88no—
203C22×D4C2412yes—
204C22×D4C2412yes—
205C22×D4C2412yes—
206C23×C4—24no—
207C23×C4—22no—
208C23×C4C2×D412yes—
209C23×C4C2411yescharacteristic
210C2×SD16—28no—
211C2×SD16—44no—
212C2×SD16—44no—
213C2×SD16—44no—
214C2×SD16—44no—
215C2×SD16—44no—
216C2×D8—28no—
217C2×D8—44no—
218C2×D8—44no—
219C2×D8—44no—
220C2×D8—44no—
221C2×D8—44no—
222C22×C8C2414yes—
223C22×C8C2414yes—
224C4⋊Q8—42no—
225#127—28no—
226#127—44no—
227#127—42no—
228#127—48no—
229#126—48no—
230#125—42no—
231#125—42no—
232#125—48no—
233#124—44no—
234#124—44no—
235#123—48no—
236#123—88no—
237#123—88no—
238C22⋊Q8—24no—
239C22⋊Q8—24no—
240#121—24no—
241#121—24no—
242#121—48no—
243C4×Q8—42no—
244C4×Q8—42no—
245C4×D4—28no—
246C4×D4—44no—
247C4×D4—42no—
248C4×D4—42no—
249C4×D4—48no—
250#117—24no—
251#117—42no—
252#117—42no—
253#117—42no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—24no—
261C2×C4⋊C4C2×D412yes—
262C2×C4⋊C4C2×D412yes—
263C2×C4⋊C4C2×D414yes—
264C2×C4⋊C4C2×D414yes—
265C2×C4⋊C4C2412yes—
266C2×C4⋊C4C2412yes—
267C2×C4⋊C4C2412yes—
268C2×C4⋊C4C2412yes—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—48no—
286C2×C22⋊C4C2×D412yes—
287C2×C22⋊C4C2×D412yes—
288C2×C42—21no—
289C2×C42—21no—
290C2×C42—22no—
291C2×C42—42no—
292C2×C42C2×D412yes—
293C2×C42C2412yes—
294C2×C42C2411yescharacteristic
295C2×C42C2411yescharacteristic
296C2×C42C2412yes—
297C4⋊C8—44no—
298#103—44no—
299#103—44no—
300#102—44no—
301#102—44no—
302#102—44no—
303#102—44no—
304#102—44no—
305#102—44no—
306C22⋊C8—24no—
307C22⋊C8—24no—
308#97—44no—
309#95—24no—
310#95—24no—
311#95—24no—
312#95—24no—
313#95C2×D414yes—
314#95C2×D414yes—
315C24—24no—
316C24—22no—
317C24—22no—
318C24—22no—
319C24—42no—
320C24—42no—
321C24—42no—
322C24—42no—
323C24—42no—
324C24—84no—
325C24—84no—
326C24C22×D411yescharacteristic
327C2×Q8—24no—
328C2×Q8—22no—
329C2×Q8—22no—
330C2×Q8—24no—
331C2×D4—24no—
332C2×D4—24no—
333C2×D4—22no—
334C2×D4—24no—
335C2×D4—24no—
336C2×D4—22no—
337C2×D4—22no—
338C2×D4—24no—
339C2×D4—24no—
340C2×D4—24no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—416no—
344C2×D4—42no—
345C2×D4—48no—
346C2×D4—48no—
347C2×D4—48no—
348C2×D4—48no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—44no—
352C2×D4—44no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—88no—
356C2×D4—84no—
357C2×D4—84no—
358C2×D4—84no—
359C2×D4—832no—
360C2×D4—84no—
361C2×D4H(5,𝔽2)14yes—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—42no—
370C22×C4—44no—
371C22×C4—48no—
372C22×C4—44no—
373C22×C4—84no—
374C22×C4—84no—
375C22×C4—84no—
376C22×C4C22×D414yes—
377C22×C4C22×D414yes—
378C22×C4C22×D412yes—
379C22×C4C22×D412yes—
380C22×C4C22×D411yescharacteristic
381C22×C4H(5,𝔽2)14yes—
382C22×C4C2511yescharacteristic
383SD16—88no—
384SD16—88no—
385D8—88no—
386D8—88no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—24no—
391C2×C8—24no—
392C2×C8—24no—
393C4⋊C4—24no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—22no—
397C4⋊C4—22no—
398C4⋊C4—24no—
399C4⋊C4—24no—
400C4⋊C4—42no—
401C4⋊C4—44no—
402C4⋊C4—44no—
403C4⋊C4—44no—
404C4⋊C4—44no—
405C4⋊C4—48no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—44no—
409C4⋊C4—84no—
410C22⋊C4—24no—
411C22⋊C4—42no—
412C22⋊C4—48no—
413C22⋊C4—44no—
414C22⋊C4—42no—
415C22⋊C4—42no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—84no—
419C22⋊C4—84no—
420C22⋊C4—84no—
421C22⋊C4—84no—
422C22⋊C4—84no—
423C22⋊C4—84no—
424C22⋊C4—84no—
425C22⋊C4—84no—
426C22⋊C4—84no—
427C22⋊C4—84no—
428C42—24no—
429C42—24no—
430C42—22no—
431C42—22no—
432C42—42no—
433C42—42no—
434C42—42no—
435C42—44no—
436C23—24no—
437C23—21no—
438C23—21no—
439C23—24no—
440C23—48no—
441C23—44no—
442C23—44no—
443C23—42no—
444C23—42no—
445C23—44no—
446C23—44no—
447C23—44no—
448C23—44no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23—42no—
454C23—42no—
455C23—84no—
456C23—88no—
457C23—88no—
458C23—164no—
459C23—164no—
460C23—164no—
461C23—164no—
462C23#53512yes—
463C23C23×D411yescharacteristic
464Q8—44no—
465Q8—44no—
466D4—28no—
467D4—48no—
468D4—44no—
469D4—44no—
470D4—44no—
471D4—44no—
472D4—44no—
473D4—84no—
474D4—816no—
475D4—816no—
476D4—164no—
477D4—164no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—28no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—24no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—24no—
490C2×C4—24no—
491C2×C4—48no—
492C2×C4—44no—
493C2×C4—44no—
494C2×C4—44no—
495C2×C4—44no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—168no—
504C2×C4—164no—
505C2×C4#53512yes—
506C2×C4C2×H(5,𝔽2)12yes—
507C2×C4C2×H(5,𝔽2)12yes—
508C2×C4#58512yes—
509C8—44no—
510C8—44no—
511C22—22no—
512C22—24no—
513C22—42no—
514C22—42no—
515C22—44no—
516C22—48no—
517C22—84no—
518C22—82no—
519C22—82no—
520C22—82no—
521C22—82no—
522C22—82no—
523C22—82no—
524C22—84no—
525C22—82no—
526C22—82no—
527C22—84no—
528C22—84no—
529C22—84no—
530C22—168no—
531C22—168no—
532C22C2×#53511yescharacteristic
533C22#348411yescharacteristic
534C22#348711yescharacteristic
535C22#358514yes—
536C4—22no—
537C4—42no—
538C4—42no—
539C4—44no—
540C4—44no—
541C4—82no—
542C4—82no—
543C4—82no—
544C4—82no—
545C4—82no—
546C4#353414yes—
547C2—42no—
548C2—84no—
549C2—162no—
550C2—162no—
551C2#3671011yescharacteristic
552C2#6082512yes—
553C2#6164312yes—
554C2#6194512yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37420C2semidirect
2. #35554C2semidirect
3. #32917C2semidirect
4. #31662C2semidirect
5. #31658C2semidirect
6. #30763C2semidirect
7. #30755C2semidirect
8. #29727C2semidirect
9. #29339C2semidirect
10. #25725C2semidirect
11. #23942C2semidirect
12. #23942C2semidirect
13. #23939C2semidirect
14. #23939C2semidirect
15. #21917C2semidirect
21. C2×#463C22semidirect
30. #2566C22semidirect
31. #2565C22semidirect
32. #2548C22semidirect
33. #2547C22semidirect
43. #2101C22semidirect
44. #2071C22semidirect
45. #2055C22semidirect
46. #2054C22semidirect
47. #2042C22semidirect
48. #2025C22semidirect
49. #2025C22semidirect
50. #2025C22semidirect
51. #2025C22semidirect
53. #2013C22semidirect
54. #2013C22semidirect
55. #2012C22semidirect
57. #2011C22semidirect
58. #2011C22semidirect
60. #1936C22semidirect
61. #1935C22semidirect
62. #1917C22semidirect
63. #1917C22semidirect
64. #1880C22semidirect
65. #1880C22semidirect
66. #1879C22semidirect
67. #1879C22semidirect
68. #1879C22semidirect
69. #1877C22semidirect
76. #1802C22semidirect

Maximal subgroups

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

subgroupindex
#374202
#355542
#329172
#316622
#316582
#307632
#307552
#297272
#293392
#257252
#239422
#239392
#219172