Finite Group Databasebeta

← #6506457

Subgroups — #6506457

Order 512.

2892 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35033C211yesmaximal, characteristic, complemented
2C2×#2435C211yesmaximal, characteristic
3#29109C212yesmaximal, complemented
4#29104C212yesmaximal, complemented
5#27281C218yesmaximal, complemented
6#27271C218yesmaximal, complemented
7#25864C212yesmaximal, complemented
8#25859C212yesmaximal, complemented
9#21768C212yesmaximal, complemented
10#20598C212yesmaximal
11#20524C211yesmaximal, characteristic, complemented
12C24×D4C2211yescharacteristic
13C22×#124C2211yescharacteristic
14C22×#124C2211yescharacteristic
15C22×C22⋊Q8C2211yescharacteristic
16C2×#489C2214yescomplemented
17C2×#488C2214yescomplemented
18C2×#418C2214yes—
19C2×#418C2214yes—
20C2×#407C2212yes—
21#2857C2212yescomplemented
22#2851C2212yescomplemented
23#2626—22no—
24#2608—22no—
25#2607—22no—
26#2441C2214yescomplemented
27#2439C2214yescomplemented
28#2435—22no—
29#2435—22no—
30#2435C2218yes—
31C2×#394—22no—
32#2329—22no—
33C2×#390C2211yescharacteristic
34C2×#390C2211yescharacteristic
35#2280C2212yes—
36C2×#379—22no—
37C2×C4×C22⋊C4C2211yescharacteristic
38#2073C2214yescomplemented
39#2071C2214yescomplemented
40#2042C2218yescomplemented
41#2042C2218yescomplemented
42#2025C2214yescomplemented
43#2023C2214yescomplemented
44#1959C2214yescomplemented
45#1958C2214yescomplemented
46#1936C2218yescomplemented
47#1934C2218yescomplemented
48#1894C2218yescomplemented
49#1873C2214yescomplemented
50#1873C2214yescomplemented
51#1824C2214yescomplemented
52#1787C2214yescomplemented
53#1786C2214yes—
54#1786C2214yes—
55#1786C2218yes—
56#1753C2218yescomplemented
57#1749C2212yescomplemented
58C4×C22⋊C8C2212yescomplemented
59C26—21no—
60C23×Q8C2311yescharacteristic
61C23×D4—21no—
62C23×D4—22no—
63C23×D4—28no—
64C23×D4—44no—
65C23×D4—44no—
66C23×D4C2314yes—
67C23×D4C2311yescharacteristic
68C23×D4C2311yescharacteristic
69C24×C4C2311yescharacteristic
70C22×M16C2314yes—
71C2×C4⋊Q8C2312yescomplemented
72C2×#125C2312yescomplemented
73C2×#124—22no—
74C2×#124—21no—
75C2×#124—21no—
76C2×#124—44no—
77C2×#124—44no—
78C2×#124C2314yes—
79C2×#124C2314yes—
80C2×C22⋊Q8—22no—
81C2×C22⋊Q8—22no—
82C2×C22⋊Q8—28no—
83C2×C22⋊Q8C2314yes—
84C2×#117C2312yes—
85C2×#117C2312yes—
86C22×C4⋊C4—21no—
87C22×C4⋊C4C2311yescharacteristic
88C22×C22⋊C4—21no—
89C22×C22⋊C4—22no—
90C22×C22⋊C4—22no—
91C22×C22⋊C4—22no—
92C22×C22⋊C4—22no—
93C22×C22⋊C4D412yes—
94C22×C42C2311yescharacteristic
95C22×C42C2311yescharacteristic
96#489—48no—
97#488—48no—
98#418—48no—
99#418—48no—
100C2×#103C2318yescomplemented
101C2×#103C2318yescomplemented
102C2×#102C2318yes—
103C2×#102C2318yes—
104C2×#102C2318yes—
105C2×#102C2318yes—
106#407—28no—
107C2×C22⋊C8C2314yes—
108C2×C22⋊C8C2314yes—
109C2×#97C2314yescomplemented
110C2×C4×C8C2314yescomplemented
111#399—24no—
112#396—24no—
113#396—24no—
114#394—28no—
115#394—28no—
116#393—24no—
117#391—22no—
118#391—24no—
119#390—22no—
120#390—21no—
121#390—21no—
122#390C2314yes—
123#390C2314yes—
124#389—22no—
125#389—24no—
126#389C2314yescomplemented
127#389C2314yescomplemented
128#387—24no—
129#387—24no—
130#387—24no—
131#385—24no—
132#384C2312yescomplemented
133#382C2312yescomplemented
134#381—22no—
135#380—22no—
136#380C2314yes—
137#380C2314yes—
138#379—22no—
139#379—28no—
140#379—28no—
141C4×C22⋊C4—22no—
142C4×C22⋊C4—22no—
143C4×C22⋊C4D412yes—
144C4×C22⋊C4D412yes—
145C4×C22⋊C4C2314yes—
146C2×#95D412yes—
147#340C2318yescomplemented
148#340C2318yescomplemented
149#336C2314yescomplemented
150#336C2314yescomplemented
151C25—21no—
152C25—21no—
153C25—24no—
154C25—44no—
155C25—44no—
156C25—48no—
157C25—44no—
158C25—84no—
159C25C2×D411yescharacteristic
160C22×Q8—21no—
161C22×Q8—22no—
162C22×Q8—24no—
163C22×Q8—42no—
164C22×Q8—42no—
165C22×Q8C2412yes—
166C22×D4—28no—
167C22×D4—28no—
168C22×D4—24no—
169C22×D4—24no—
170C22×D4—21no—
171C22×D4—21no—
172C22×D4—24no—
173C22×D4—28no—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—28no—
177C22×D4—42no—
178C22×D4—42no—
179C22×D4—44no—
180C22×D4—44no—
181C22×D4—464no—
182C22×D4—48no—
183C22×D4—48no—
184C22×D4—44no—
185C22×D4—44no—
186C22×D4—44no—
187C22×D4—48no—
188C22×D4C2412yes—
189C22×D4C2412yes—
190C23×C4—21no—
191C23×C4—21no—
192C23×C4—42no—
193C23×C4—42no—
194C23×C4C2×D412yes—
195C23×C4C2×D412yes—
196C23×C4C2×D411yescharacteristic
197C23×C4C2×D412yes—
198C23×C4#4118yes—
199C23×C4C2411yescharacteristic
200C23×C4C2412yes—
201C2×M16—28no—
202C2×M16—44no—
203C2×M16—44no—
204C22×C8C2418yes—
205C4⋊Q8—44no—
206#125—44no—
207#124—28no—
208#124—28no—
209#124—44no—
210#124—44no—
211#124—48no—
212#124—44no—
213#124—44no—
214C22⋊Q8—28no—
215C22⋊Q8—28no—
216C22⋊Q8—48no—
217C22⋊Q8—48no—
218#117—44no—
219#117—44no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—24no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—24no—
229C2×C4⋊C4—44no—
230C2×C4⋊C4C2412yes—
231C2×C4⋊C4C2414yes—
232C2×C4⋊C4C2414yes—
233C2×C22⋊C4—24no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—28no—
236C2×C22⋊C4—24no—
237C2×C22⋊C4—24no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—28no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—24no—
244C2×C22⋊C4—44no—
245C2×C22⋊C4—48no—
246C2×C22⋊C4—48no—
247C2×C22⋊C4—44no—
248C2×C22⋊C4—84no—
249C2×C22⋊C4—84no—
250C2×C22⋊C4C2×D414yes—
251C2×C22⋊C4C2×D414yes—
252C2×C42—22no—
253C2×C42—22no—
254C2×C42—42no—
255C2×C42—42no—
256C2×C42C2×D412yes—
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D411yescharacteristic
259C2×C42C2412yes—
260C2×C42C2412yes—
261#103—48no—
262#103—48no—
263#102—216no—
264#102—216no—
265#102—48no—
266#102—48no—
267C22⋊C8—28no—
268C22⋊C8—28no—
269#97—44no—
270C4×C8—44no—
271#95—24no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95C2×D414yes—
277#95C2×D414yes—
278C24—24no—
279C24—24no—
280C24—24no—
281C24—22no—
282C24—22no—
283C24—41no—
284C24—41no—
285C24—42no—
286C24—48no—
287C24—48no—
288C24—44no—
289C24—44no—
290C24—42no—
291C24—42no—
292C24—416no—
293C24—432no—
294C24—48no—
295C24—42no—
296C24—42no—
297C24—44no—
298C24—44no—
299C24—88no—
300C24—816no—
301C24—832no—
302C24—832no—
303C24—816no—
304C24#12414yes—
305C24C22×D412yes—
306C24C22×D411yescharacteristic
307C2×Q8—24no—
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—44no—
313C2×Q8—42no—
314C2×Q8—82no—
315C2×Q8—82no—
316C2×Q8—88no—
317C2×D4—232no—
318C2×D4—24no—
319C2×D4—24no—
320C2×D4—24no—
321C2×D4—24no—
322C2×D4—24no—
323C2×D4—24no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—416no—
327C2×D4—416no—
328C2×D4—416no—
329C2×D4—416no—
330C2×D4—48no—
331C2×D4—48no—
332C2×D4—42no—
333C2×D4—42no—
334C2×D4—416no—
335C2×D4—464no—
336C2×D4—432no—
337C2×D4—48no—
338C2×D4—42no—
339C2×D4—42no—
340C2×D4H(5,𝔽2)14yes—
341C2×D4H(5,𝔽2)14yes—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—416no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—44no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4#12414yes—
367C22×C4#12418yes—
368C22×C4C22×D414yes—
369C22×C4C22×D412yes—
370C22×C4C22×D412yes—
371C22×C4C22×D411yescharacteristic
372C22×C4C22×D412yes—
373C22×C4C2×#4114yes—
374C22×C4C2×#4114yes—
375C22×C4C2×#4114yes—
376C22×C4C2511yescharacteristic
377M16—88no—
378C2×C8—28no—
379C2×C8—28no—
380C2×C8—28no—
381C4⋊C4—24no—
382C4⋊C4—28no—
383C4⋊C4—28no—
384C4⋊C4—48no—
385C4⋊C4—44no—
386C4⋊C4—42no—
387C4⋊C4—48no—
388C4⋊C4—82no—
389C4⋊C4—84no—
390C4⋊C4—84no—
391C4⋊C4—82no—
392C22⋊C4—44no—
393C22⋊C4—44no—
394C22⋊C4—416no—
395C22⋊C4—48no—
396C22⋊C4—48no—
397C22⋊C4—48no—
398C22⋊C4—48no—
399C22⋊C4—88no—
400C22⋊C4—84no—
401C22⋊C4—84no—
402C22⋊C4—84no—
403C22⋊C4—82no—
404C22⋊C4—82no—
405C42—24no—
406C42—24no—
407C42—42no—
408C42—41no—
409C42—41no—
410C42—82no—
411C42—82no—
412C23—22no—
413C23—21no—
414C23—21no—
415C23—24no—
416C23—28no—
417C23—22no—
418C23—22no—
419C23—24no—
420C23—24no—
421C23—24no—
422C23—44no—
423C23—44no—
424C23—48no—
425C23—48no—
426C23—432no—
427C23—48no—
428C23—48no—
429C23—416no—
430C23—416no—
431C23—44no—
432C23—44no—
433C23—42no—
434C23—42no—
435C23—42no—
436C23—42no—
437C23—44no—
438C23—44no—
439C23—88no—
440C23—88no—
441C23—84no—
442C23—84no—
443C23—816no—
444C23—816no—
445C23—832no—
446C23—832no—
447C23—8128no—
448C23—84no—
449C23—84no—
450C23C2×#12414yes—
451C23C2×#12412yes—
452C23#57514yes—
453C23C23×D411yescharacteristic
454Q8—48no—
455Q8—84no—
456Q8—164no—
457D4—28no—
458D4—28no—
459D4—432no—
460D4—48no—
461D4—48no—
462D4—44no—
463D4—44no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—28no—
469C2×C4—28no—
470C2×C4—28no—
471C2×C4—24no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—24no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—44no—
481C2×C4—48no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—44no—
493C2×C4—84no—
494C2×C4—84no—
495C2×C4—84no—
496C2×C4—84no—
497C2×C4C2×#12414yes—
498C2×C4C2×#12412yes—
499C2×C4C22×#4112yes—
500C2×C4C2×H(5,𝔽2)12yes—
501C2×C4C2×H(5,𝔽2)12yes—
502C8—48no—
503C22—22no—
504C22—24no—
505C22—24no—
506C22—22no—
507C22—24no—
508C22—24no—
509C22—24no—
510C22—416no—
511C22—44no—
512C22—44no—
513C22—44no—
514C22—44no—
515C22—48no—
516C22—48no—
517C22—816no—
518C22—816no—
519C22—864no—
520C22—84no—
521C22—84no—
522C22—82no—
523C22—82no—
524C22—82no—
525C22—82no—
526C22#314014yes—
527C22C22×#12411yescharacteristic
528C22#348411yescharacteristic
529C22#348411yescharacteristic
530C22C2×#57512yes—
531C22#358512yes—
532C4—24no—
533C4—22no—
534C4—44no—
535C4—42no—
536C4—42no—
537C4—82no—
538C4—84no—
539C4—84no—
540C4—82no—
541C4#349014yes—
542C2—24no—
543C2—44no—
544C2—84no—
545C2—84no—
546C2#3536711yescharacteristic
547C2C2×#314011yescharacteristic
548C2#6061911yescharacteristic
549C2#6157612yes—
550C2#6158112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24×D4C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35033C2semidirect
3. #29109C2semidirect
4. #29104C2semidirect
5. #27281C2semidirect
6. #27271C2semidirect
7. #25864C2semidirect
8. #25859C2semidirect
9. #21768C2semidirect
11. #20524C2semidirect
16. C2×#489C22semidirect
17. C2×#488C22semidirect
21. #2857C22semidirect
22. #2851C22semidirect
26. #2441C22semidirect
27. #2439C22semidirect
38. #2073C22semidirect
39. #2071C22semidirect
40. #2042C22semidirect
41. #2042C22semidirect
42. #2025C22semidirect
43. #2023C22semidirect
44. #1959C22semidirect
45. #1958C22semidirect
46. #1936C22semidirect
47. #1934C22semidirect
48. #1894C22semidirect
49. #1873C22semidirect
50. #1873C22semidirect
51. #1824C22semidirect
52. #1787C22semidirect
56. #1753C22semidirect
57. #1749C22semidirect
58. C4×C22⋊C8C22semidirect
71. C2×C4⋊Q8C23semidirect
72. C2×#125C23semidirect
100. C2×#103C23semidirect
101. C2×#103C23semidirect
109. C2×#97C23semidirect
110. C2×C4×C8C23semidirect
126. #389C23semidirect
127. #389C23semidirect
132. #384C23semidirect
133. #382C23semidirect
147. #340C23semidirect
148. #340C23semidirect
149. #336C23semidirect
150. #336C23semidirect

Maximal subgroups

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

subgroupindex
#350332
C2×#24352
#291092
#291042
#272812
#272712
#258642
#258592
#217682
#205982
#205242