Finite Group Databasebeta

← #7487464

Subgroups — #7487464

Order 512.

1597 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#46152C211yesmaximal, characteristic, complemented
2#38957C211yesmaximal, characteristic
3#32647C212yesmaximal, complemented
4#32029C212yesmaximal, complemented
5#31302C214yesmaximal, complemented
6#31134C214yesmaximal, complemented
7#30227C212yesmaximal, complemented
8#29578C212yesmaximal, complemented
9#29050C214yesmaximal, complemented
10#27262C214yesmaximal, complemented
11#25662C212yesmaximal, complemented
12#24033C212yesmaximal
13#23273C211yesmaximal, characteristic, complemented
14C2×#556C2211yescharacteristic
15C2×D42C2211yescharacteristic
16C2×#489C2212yes—
17C2×#481C2212yes—
18C2×#475C2212yescomplemented
19C22×#102C2212yes—
20#2859C2212yescomplemented
21#2858C2211yescharacteristic
22#2844—21no—
23#2809—21no—
24#2720—24no—
25#2685—24no—
26#2683—21no—
27#2620—21no—
28#2567C2212yescomplemented
29#2548C2211yescharacteristic
30#2542C2211yescharacteristic
31#2500—21no—
32#2497C2214yescomplemented
33#2487C2212yes—
34#2440C2214yescomplemented
35#2435—21no—
36#2435C2214yes—
37#2426C2214yes—
38#2379—21no—
39#2309—21no—
40C4×#124C2211yescharacteristic
41#2293C2211yescharacteristic
42#2099C2212yescomplemented
43#2093C2212yes—
44#2084C2212yescomplemented
45#2071C2212yes—
46#2067C2214yescomplemented
47#2043C2214yescomplemented
48#2042C2214yes—
49#2042C2214yescomplemented
50#2042C2214yescomplemented
51#2041C2214yes—
52#2036C2214yescomplemented
53#2036C2214yescomplemented
54#2034C2214yes—
55#2023C2214yescomplemented
56#1985C2212yescomplemented
57#1983C2212yes—
58#1959C2214yescomplemented
59#1959C2214yescomplemented
60#1944C2214yescomplemented
61#1936C2214yes—
62#1936C2214yescomplemented
63#1925C2214yescomplemented
64#1924C2214yescomplemented
65#1921C2214yescomplemented
66#1893C2212yes—
67#1892C2214yescomplemented
68#1872C2214yescomplemented
69#1869C2212yes—
70#1865C2214yescomplemented
71#1864C2214yes—
72#1786C2212yes—
73#1786C2214yes—
74#1763C2212yescomplemented
75#1752C2214yescomplemented
76#1745C2212yescomplemented
77C23×D4—21no—
78C23×D4C2312yes—
79C23×C8C2312yes—
80#556—42no—
81#556—42no—
82D42—42no—
83D42—42no—
84C2×C4⋊Q8C2311yescharacteristic
85C2×C4⋊Q8C2312yes—
86C2×#127C2311yescharacteristic
87C2×#126—22no—
88C2×#125—22no—
89C2×#124—21no—
90C2×#124D411yescharacteristic
91C2×#124C2312yes—
92C2×#124C2314yes—
93C2×C22⋊Q8—21no—
94C2×C22⋊Q8—24no—
95C2×C22⋊Q8C2314yes—
96C2×#121—21no—
97C2×#121C2312yes—
98C2×#120—22no—
99C2×C4×Q8C2312yes—
100C2×C4×Q8C2311yescharacteristic
101C2×C4×D4C2311yescharacteristic
102C2×C4×D4C2311yescharacteristic
103C2×#117—21no—
104C2×#117—21no—
105C22×C4⋊C4C2312yes—
106#489—44no—
107#481—44no—
108#475—44no—
109C2×#107C2314yes—
110C2×#106C2314yes—
111C2×C4⋊C8C2312yes—
112C2×#103C2314yes—
113C2×#103C2314yes—
114C2×#103C2314yes—
115C2×#103C2314yes—
116C2×#102—44no—
117C2×#102C2314yes—
118C2×#102C2314yes—
119C2×#102C2314yes—
120C2×#102C2314yes—
121C2×#102C2314yes—
122C2×#102C2314yes—
123C2×C22⋊C8C2312yes—
124C2×C22⋊C8C2314yes—
125C2×#97C2312yes—
126C2×#97C2314yes—
127#400—22no—
128#400—24no—
129#399—22no—
130#399—22no—
131#398—24no—
132#396—24no—
133#396—24no—
134#395—22no—
135#395—22no—
136#395—24no—
137#394—22no—
138#394—24no—
139#394—24no—
140#394—24no—
141#391—22no—
142#391D411yescharacteristic
143#390—21no—
144#390—21no—
145#390—24no—
146#390—21no—
147#390—22no—
148#390—21no—
149#390D411yescharacteristic
150#390C2314yes—
151#390C2312yes—
152#389—22no—
153#389—22no—
154#389C2314yes—
155#389C2314yes—
156#389C2312yes—
157#389C2314yes—
158#388—24no—
159#388—22no—
160#388—24no—
161#388D414yes—
162#387—24no—
163#386—22no—
164#385—24no—
165#385—22no—
166#384—22no—
167#384—21no—
168#384C2312yes—
169#384C2312yes—
170#382—22no—
171#382D411yescharacteristic
172#380C2314yes—
173#380C2312yes—
174#379—24no—
175#379—24no—
176C4×C4⋊C4C2312yes—
177C4×C4⋊C4C2311yescharacteristic
178C4×C22⋊C4—21no—
179C4×C22⋊C4C2314yes—
180C43C2311yescharacteristic
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#336C2312yes—
186#336C2312yes—
187#336C2314yes—
188#336C2314yes—
189C25—22no—
190C22×Q8—22no—
191C22×Q8C2×D411yescharacteristic
192C22×Q8C2412yes—
193C22×D4—24no—
194C22×D4—22no—
195C22×D4—21no—
196C22×D4—21no—
197C22×D4—22no—
198C22×D4—44no—
199C22×D4—42no—
200C22×D4—44no—
201C22×D4—44no—
202C22×D4—44no—
203C22×D4—82no—
204C22×D4C2×D411yescharacteristic
205C22×D4C2412yes—
206C22×D4C2411yescharacteristic
207C23×C4—21no—
208C23×C4—22no—
209C23×C4C2412yes—
210C22×C8—44no—
211C22×C8—42no—
212C22×C8C2414yes—
213C22×C8C2414yes—
214C4⋊Q8—42no—
215C4⋊Q8—42no—
216C4⋊Q8—41no—
217C4⋊Q8—41no—
218#127—41no—
219#127—41no—
220#126—48no—
221#125—84no—
222#124—24no—
223#124—24no—
224#124—42no—
225#124—44no—
226#124—44no—
227#124—82no—
228C22⋊Q8—44no—
229C22⋊Q8—44no—
230C22⋊Q8—44no—
231C22⋊Q8—88no—
232#121—24no—
233#121—42no—
234#121—44no—
235#120—48no—
236C4×Q8—24no—
237C4×Q8—41no—
238C4×Q8—41no—
239C4×Q8—42no—
240C4×D4—22no—
241C4×D4—22no—
242C4×D4—41no—
243C4×D4—41no—
244#117—44no—
245#117—82no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—24no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4C2×D411yescharacteristic
263C2×C4⋊C4C2×D411yescharacteristic
264C2×C4⋊C4C2×D412yes—
265C2×C4⋊C4C2414yes—
266C2×C4⋊C4C2412yes—
267C2×C4⋊C4C2414yes—
268C2×C4⋊C4C2411yescharacteristic
269C2×C22⋊C4—21no—
270C2×C22⋊C4—22no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—24no—
275C2×C22⋊C4—22no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—24no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4C2×D414yes—
283C2×C22⋊C4C2×D412yes—
284C2×C22⋊C4C2×D414yes—
285C2×C42—22no—
286C2×C42—21no—
287C2×C42—22no—
288C2×C42C2×D411yescharacteristic
289C2×C42C2×D411yescharacteristic
290C2×C42C2×D412yes—
291C2×C42C2411yescharacteristic
292C2×C42C2412yes—
293C2×C42C2412yes—
294#107—44no—
295#106—28no—
296C4⋊C8—42no—
297#103—44no—
298#103—44no—
299#103—44no—
300#103—44no—
301#102—28no—
302#102—44no—
303#102—44no—
304#102—44no—
305#102—44no—
306#102—44no—
307C22⋊C8—44no—
308C22⋊C8—42no—
309#97—44no—
310#97—42no—
311#95—22no—
312#95—21no—
313#95—22no—
314#95—24no—
315#95—22no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95C2×D414yes—
320#95C2×D414yes—
321C24—21no—
322C24—22no—
323C24—44no—
324C24—44no—
325C24—84no—
326C24—82no—
327C24—84no—
328C24C22×D412yes—
329C2×Q8—22no—
330C2×Q8—22no—
331C2×Q8—22no—
332C2×Q8—24no—
333C2×Q8—24no—
334C2×Q8—41no—
335C2×Q8—41no—
336C2×Q8—84no—
337C2×Q8—82no—
338C2×D4—21no—
339C2×D4—22no—
340C2×D4—22no—
341C2×D4—24no—
342C2×D4—24no—
343C2×D4—24no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4—44no—
347C2×D4—44no—
348C2×D4—42no—
349C2×D4—42no—
350C2×D4—42no—
351C2×D4—48no—
352C2×D4—44no—
353C2×D4—42no—
354C2×D4—84no—
355C2×D4—82no—
356C2×D4—84no—
357C2×D4—88no—
358C2×D4—816no—
359C2×D4—82no—
360C2×D4#13714yes—
361C2×D4H(5,𝔽2)12yes—
362C22×C4—24no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—24no—
367C22×C4—24no—
368C22×C4—24no—
369C22×C4—24no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—82no—
375C22×C4—84no—
376C22×C4C22×D412yes—
377C22×C4C22×D412yes—
378C22×C4C22×D412yes—
379C22×C4C22×D412yes—
380C22×C4C22×D411yescharacteristic
381C22×C4C22×D412yes—
382C22×C4C22×D411yescharacteristic
383C22×C4C2511yescharacteristic
384C2×C8—24no—
385C2×C8—24no—
386C2×C8—24no—
387C2×C8—24no—
388C2×C8—24no—
389C2×C8—24no—
390C2×C8—84no—
391C4⋊C4—21no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—42no—
400C4⋊C4—44no—
401C4⋊C4—42no—
402C4⋊C4—44no—
403C4⋊C4—48no—
404C4⋊C4—42no—
405C4⋊C4—82no—
406C4⋊C4—82no—
407C4⋊C4—82no—
408C4⋊C4—84no—
409C4⋊C4—84no—
410C4⋊C4—82no—
411C4⋊C4—82no—
412C4⋊C4—84no—
413C4⋊C4—82no—
414C4⋊C4H(5,𝔽2)12yes—
415C22⋊C4—24no—
416C22⋊C4—28no—
417C22⋊C4—42no—
418C22⋊C4—44no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C22⋊C4—44no—
422C22⋊C4—82no—
423C22⋊C4—84no—
424C22⋊C4—82no—
425C22⋊C4—84no—
426C22⋊C4—84no—
427C22⋊C4—84no—
428C22⋊C4—82no—
429C22⋊C4—84no—
430C22⋊C4—84no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—21no—
435C42—21no—
436C42—24no—
437C42—41no—
438C42—42no—
439C42—44no—
440C42—42no—
441C42—82no—
442C23—24no—
443C23—22no—
444C23—22no—
445C23—24no—
446C23—22no—
447C23—41no—
448C23—42no—
449C23—42no—
450C23—41no—
451C23—42no—
452C23—44no—
453C23—42no—
454C23—44no—
455C23—42no—
456C23—88no—
457C23—88no—
458C23—168no—
459C23—164no—
460C23—168no—
461C23C2×#13714yes—
462C23C23×D411yescharacteristic
463Q8—44no—
464Q8—42no—
465Q8—42no—
466Q8—82no—
467Q8—164no—
468D4—24no—
469D4—44no—
470D4—42no—
471D4—42no—
472D4—42no—
473D4—44no—
474D4—84no—
475D4—88no—
476D4—88no—
477D4—162no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—22no—
487C2×C4—21no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—48no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—164no—
514C2×C4C2×#13712yes—
515C2×C4C2×H(5,𝔽2)11yescharacteristic
516C2×C4C2×H(5,𝔽2)11yescharacteristic
517C2×C4#58514yes—
518C8—44no—
519C8—44no—
520C22—28no—
521C22—24no—
522C22—42no—
523C22—44no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—82no—
529C22—82no—
530C22—82no—
531C22—84no—
532C22—84no—
533C22—82no—
534C22—82no—
535C22—1616no—
536C22#348411yescharacteristic
537C22#350012yes—
538C22C22×#13711yescharacteristic
539C22#358511yescharacteristic
540C22#358512yes—
541C4—22no—
542C4—42no—
543C4—42no—
544C4—42no—
545C4—42no—
546C4—42no—
547C4—41no—
548C4—42no—
549C4—82no—
550C4—82no—
551C4—84no—
552C4—84no—
553C4#353312yes—
554C4#355112yes—
555C2—44no—
556C2—82no—
557C2—162no—
558C2#5197811yescharacteristic
559C2#6157011yescharacteristic
560C2#6167311yescharacteristic
561C2#6257312yes—
562C2#6258212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×D42C22
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
1. #46152C2semidirect
3. #32647C2semidirect
4. #32029C2semidirect
5. #31302C2semidirect
6. #31134C2semidirect
7. #30227C2semidirect
8. #29578C2semidirect
9. #29050C2semidirect
10. #27262C2semidirect
11. #25662C2semidirect
13. #23273C2semidirect
18. C2×#475C22semidirect
20. #2859C22semidirect
28. #2567C22semidirect
32. #2497C22semidirect
34. #2440C22semidirect
42. #2099C22semidirect
44. #2084C22semidirect
46. #2067C22semidirect
47. #2043C22semidirect
49. #2042C22semidirect
50. #2042C22semidirect
52. #2036C22semidirect
53. #2036C22semidirect
55. #2023C22semidirect
56. #1985C22semidirect
58. #1959C22semidirect
59. #1959C22semidirect
60. #1944C22semidirect
62. #1936C22semidirect
63. #1925C22semidirect
64. #1924C22semidirect
65. #1921C22semidirect
67. #1892C22semidirect
68. #1872C22semidirect
70. #1865C22semidirect
74. #1763C22semidirect
75. #1752C22semidirect
76. #1745C22semidirect

Maximal subgroups

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

subgroupindex
#461522
#389572
#326472
#320292
#313022
#311342
#302272
#295782
#290502
#272622
#256622
#240332
#232732