Finite Group Databasebeta

← #7228768

Subgroups — #7228768

Order 512.

1731 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#46147C211yesmaximal, characteristic, complemented
2#36289C211yesmaximal, characteristic
3#33072C212yesmaximal, complemented
4#32819C212yesmaximal, complemented
5#32646C212yesmaximal, complemented
6#32259C212yesmaximal, complemented
7#32207C214yesmaximal, complemented
8#30797C214yesmaximal, complemented
9#27539C214yesmaximal, complemented
10#25958C214yesmaximal, complemented
11#25851C212yesmaximal, complemented
12#24032C212yesmaximal
13#23352C211yesmaximal, characteristic, complemented
14C2×#563—21no—
15C2×#554C2211yescharacteristic
16C2×D42C2211yescharacteristic
17C2×#501C2212yescomplemented
18C2×#480C2212yes—
19C2×#476C2212yes—
20C22×#102C2212yes—
21#2859C2212yescomplemented
22#2858C2211yescharacteristic
23#2850—21no—
24#2822—21no—
25#2734—21no—
26#2724—21no—
27#2698—24no—
28#2563C2211yescharacteristic
29#2548C2211yescharacteristic
30#2546C2212yescomplemented
31#2504—21no—
32#2496C2214yescomplemented
33#2490C2214yes—
34#2486C2212yes—
35#2472—24no—
36#2440C2214yescomplemented
37#2435—21no—
38D4×C22⋊C4C2214yes—
39C4×#124C2211yescharacteristic
40C2×#390—21no—
41C42×D4C2211yescharacteristic
42#2094C2214yescomplemented
43#2077C2214yescomplemented
44#2077C2214yescomplemented
45#2071C2214yescomplemented
46#2065C2214yescomplemented
47#2065C2214yescomplemented
48#2065C2214yescomplemented
49#2059C2214yescomplemented
50#2058C2214yescomplemented
51#2041C2214yes—
52#2033C2214yes—
53#2023C2214yescomplemented
54#1989C2212yescomplemented
55#1985C2212yescomplemented
56#1948C2212yescomplemented
57#1947C2212yescomplemented
58#1936C2214yescomplemented
59#1926C2214yescomplemented
60#1924C2214yescomplemented
61#1921C2214yescomplemented
62#1919C2214yescomplemented
63#1893C2212yes—
64#1869C2212yes—
65#1864C2214yes—
66#1864C2214yes—
67#1862C2214yes—
68#1853C2214yescomplemented
69#1850C2214yescomplemented
70#1831C2212yescomplemented
71#1807C2214yescomplemented
72#1802C2212yes—
73#1802C2212yes—
74#1786C2214yes—
75#1786C2212yes—
76#1762C2212yes—
77C23×D4—21no—
78C23×D4C2312yes—
79C23×C8C2312yes—
80#563—48no—
81#554—42no—
82#554—42no—
83D42—42no—
84D42—42no—
85C2×C4⋊Q8—22no—
86C2×C4⋊Q8C2311yescharacteristic
87C2×C4⋊Q8C2312yes—
88C2×#127C2311yescharacteristic
89C2×#126—24no—
90C2×#125—22no—
91C2×#125C2312yes—
92C2×#124—21no—
93C2×#124—22no—
94C2×#124D411yescharacteristic
95C2×#123—22no—
96C2×C22⋊Q8—21no—
97C2×C22⋊Q8—22no—
98C2×C22⋊Q8—24no—
99C2×#121—21no—
100C2×#121C2312yes—
101C2×#120—22no—
102C2×C4×Q8C2311yescharacteristic
103C2×C4×D4C2311yescharacteristic
104C2×C4×D4C2312yes—
105C2×C4×D4C2312yes—
106C2×C4×D4C2311yescharacteristic
107C22×C4⋊C4—21no—
108C22×C4⋊C4C2312yes—
109C22×C22⋊C4—24no—
110C22×C42—21no—
111#501—44no—
112#480—44no—
113#476—44no—
114C2×#107C2314yes—
115C2×#106C2314yes—
116C2×#106C2314yes—
117C2×#106C2314yes—
118C2×C4⋊C8C2312yes—
119C2×C4⋊C8C2312yes—
120C2×C4⋊C8C2312yes—
121C2×#102—44no—
122C2×#102C2314yes—
123C2×#102C2314yes—
124C2×#102C2314yes—
125C2×#102C2314yes—
126C2×#102C2314yes—
127C2×#102C2314yes—
128C2×C22⋊C8C2312yes—
129C2×C22⋊C8C2314yes—
130C2×#97C2312yes—
131#400—24no—
132#398—22no—
133#398—22no—
134#398—22no—
135#398—24no—
136#397—22no—
137#397—24no—
138#397—24no—
139#395—24no—
140#394—22no—
141#394—24no—
142#393—22no—
143#391—22no—
144#391D411yescharacteristic
145#390—21no—
146#390—21no—
147#390—24no—
148#390—21no—
149#390—21no—
150#390D411yescharacteristic
151#390C2314yes—
152#390C2314yes—
153#389—22no—
154#389—22no—
155#389C2314yes—
156#389C2314yes—
157#389C2314yes—
158#388—24no—
159#388D414yes—
160#387—24no—
161#386—24no—
162#386—24no—
163#385—24no—
164#384C2312yes—
165#384C2312yes—
166#382—21no—
167#382—22no—
168#382D411yescharacteristic
169#381—22no—
170#380C2314yes—
171#380C2314yes—
172#380C2312yes—
173#379—24no—
174C4×C4⋊C4C2311yescharacteristic
175C4×C4⋊C4C2312yes—
176C4×C4⋊C4C2312yes—
177C4×C22⋊C4—21no—
178C4×C22⋊C4—22no—
179C4×C22⋊C4C2314yes—
180C43C2311yescharacteristic
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#336C2314yes—
188#336C2314yes—
189#336C2314yes—
190C25—22no—
191C22×Q8—21no—
192C22×Q8—22no—
193C22×Q8C2×D411yescharacteristic
194C22×D4—24no—
195C22×D4—28no—
196C22×D4—22no—
197C22×D4—22no—
198C22×D4—21no—
199C22×D4—21no—
200C22×D4—42no—
201C22×D4—44no—
202C22×D4—44no—
203C22×D4—44no—
204C22×D4—82no—
205C22×D4C2×D411yescharacteristic
206C22×D4C2412yes—
207C22×D4C2411yescharacteristic
208C23×C4—21no—
209C23×C4—22no—
210C23×C4—22no—
211C23×C4—22no—
212C23×C4C2412yes—
213C22×C8—44no—
214C22×C8—42no—
215C22×C8C2414yes—
216C22×C8C2414yes—
217C4⋊Q8—41no—
218C4⋊Q8—41no—
219C4⋊Q8—42no—
220C4⋊Q8—42no—
221C4⋊Q8—48no—
222#127—41no—
223#127—41no—
224#126—416no—
225#125—24no—
226#125—42no—
227#125—48no—
228#124—24no—
229#124—84no—
230#124—82no—
231#123—48no—
232C22⋊Q8—48no—
233C22⋊Q8—416no—
234C22⋊Q8—44no—
235#121—24no—
236#121—42no—
237#121—44no—
238#120—48no—
239C4×Q8—41no—
240C4×Q8—41no—
241C4×D4—22no—
242C4×D4—22no—
243C4×D4—24no—
244C4×D4—24no—
245C4×D4—42no—
246C4×D4—42no—
247C4×D4—41no—
248C4×D4—41no—
249C2×C4⋊C4—22no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—24no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—24no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—22no—
260C2×C4⋊C4—22no—
261C2×C4⋊C4—42no—
262C2×C4⋊C4—42no—
263C2×C4⋊C4—44no—
264C2×C4⋊C4C2×D411yescharacteristic
265C2×C4⋊C4C2×D411yescharacteristic
266C2×C4⋊C4C2×D412yes—
267C2×C4⋊C4C2414yes—
268C2×C4⋊C4C2412yes—
269C2×C4⋊C4C2412yes—
270C2×C4⋊C4C2414yes—
271C2×C4⋊C4C2411yescharacteristic
272C2×C22⋊C4—21no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—22no—
275C2×C22⋊C4—24no—
276C2×C22⋊C4—22no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—22no—
279C2×C22⋊C4—24no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4—24no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—88no—
284C2×C22⋊C4C2×D414yes—
285C2×C22⋊C4C2×D414yes—
286C2×C22⋊C4C2×D412yes—
287C2×C42—21no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42—82no—
291C2×C42C2×D412yes—
292C2×C42C2×D411yescharacteristic
293C2×C42C2×D411yescharacteristic
294C2×C42C2411yescharacteristic
295C2×C42C2412yes—
296C2×C42C2412yes—
297#107—28no—
298#106—44no—
299#106—44no—
300#106—44no—
301C4⋊C8—42no—
302C4⋊C8—42no—
303C4⋊C8—42no—
304#102—28no—
305#102—28no—
306#102—44no—
307#102—44no—
308#102—44no—
309#102—44no—
310C22⋊C8—44no—
311C22⋊C8—42no—
312#97—42no—
313#95—21no—
314#95—22no—
315#95—22no—
316#95—24no—
317#95—24no—
318#95—22no—
319#95—24no—
320#95—24no—
321#95—24no—
322#95C2×D414yes—
323#95C2×D414yes—
324C24—21no—
325C24—22no—
326C24—44no—
327C24—48no—
328C24—44no—
329C24—84no—
330C24—82no—
331C24C22×D412yes—
332C2×Q8—22no—
333C2×Q8—41no—
334C2×Q8—41no—
335C2×Q8—42no—
336C2×Q8—44no—
337C2×Q8—82no—
338C2×Q8—82no—
339C2×Q8—82no—
340C2×D4—21no—
341C2×D4—22no—
342C2×D4—22no—
343C2×D4—24no—
344C2×D4—22no—
345C2×D4—24no—
346C2×D4—41no—
347C2×D4—41no—
348C2×D4—44no—
349C2×D4—44no—
350C2×D4—48no—
351C2×D4—42no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—48no—
355C2×D4—44no—
356C2×D4—42no—
357C2×D4—82no—
358C2×D4—84no—
359C2×D4—88no—
360C2×D4—816no—
361C2×D4—82no—
362C2×D4AGL(1,ℤ/8)14yes—
363C2×D4H(5,𝔽2)12yes—
364C2×D42−1+414yes—
365C22×C4—24no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—24no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—22no—
373C22×C4—24no—
374C22×C4—22no—
375C22×C4—42no—
376C22×C4—44no—
377C22×C4—44no—
378C22×C4—42no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—82no—
382C22×C4—84no—
383C22×C4—82no—
384C22×C4—82no—
385C22×C4C22×D412yes—
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D412yes—
388C22×C4C22×D412yes—
389C22×C4C22×D412yes—
390C22×C4C22×D411yescharacteristic
391C22×C4C22×D412yes—
392C22×C4C2511yescharacteristic
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C2×C8—24no—
398C2×C8—24no—
399C2×C8—84no—
400C4⋊C4—21no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—24no—
406C4⋊C4—22no—
407C4⋊C4—24no—
408C4⋊C4—22no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—42no—
412C4⋊C4—44no—
413C4⋊C4—44no—
414C4⋊C4—42no—
415C4⋊C4—44no—
416C4⋊C4—48no—
417C4⋊C4—48no—
418C4⋊C4—44no—
419C4⋊C4—48no—
420C4⋊C4—44no—
421C4⋊C4—82no—
422C4⋊C4—82no—
423C4⋊C4—82no—
424C4⋊C4H(5,𝔽2)12yes—
425C22⋊C4—24no—
426C22⋊C4—28no—
427C22⋊C4—28no—
428C22⋊C4—42no—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—48no—
433C22⋊C4—44no—
434C22⋊C4—48no—
435C22⋊C4—84no—
436C22⋊C4—84no—
437C22⋊C4—84no—
438C22⋊C4—84no—
439C42—22no—
440C42—24no—
441C42—22no—
442C42—22no—
443C42—21no—
444C42—21no—
445C42—22no—
446C42—22no—
447C42—41no—
448C42—42no—
449C42—44no—
450C42—82no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—24no—
455C23—24no—
456C23—24no—
457C23—41no—
458C23—41no—
459C23—42no—
460C23—42no—
461C23—42no—
462C23—44no—
463C23—42no—
464C23—44no—
465C23—88no—
466C23—88no—
467C23—816no—
468C23—168no—
469C23—164no—
470C23C2×AGL(1,ℤ/8)14yes—
471C23C23×D411yescharacteristic
472Q8—82no—
473Q8—164no—
474Q8—162no—
475D4—22no—
476D4—24no—
477D4—44no—
478D4—42no—
479D4—42no—
480D4—44no—
481D4—84no—
482D4—88no—
483D4—88no—
484D4—162no—
485D4#56314yescomplemented
486C2×C4—21no—
487C2×C4—22no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—24no—
503C2×C4—48no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—88no—
521C2×C4—164no—
522C2×C4—164no—
523C2×C4C2×AGL(1,ℤ/8)12yes—
524C2×C4#57714yes—
525C2×C4C2×H(5,𝔽2)11yescharacteristic
526C2×C4C2×H(5,𝔽2)11yescharacteristic
527C2×C4C2×2−1+412yes—
528C2×C4#58512yes—
529C8—44no—
530C8—44no—
531C22—24no—
532C22—24no—
533C22—28no—
534C22—42no—
535C22—42no—
536C22—44no—
537C22—44no—
538C22—44no—
539C22—42no—
540C22—42no—
541C22—82no—
542C22—82no—
543C22—84no—
544C22—84no—
545C22—82no—
546C22—1616no—
547C22#348411yescharacteristic
548C22#348912yes—
549C22C22×AGL(1,ℤ/8)11yescharacteristic
550C22C2×#57712yes—
551C22#358511yescharacteristic
552C4—22no—
553C4—24no—
554C4—42no—
555C4—42no—
556C4—42no—
557C4—42no—
558C4—42no—
559C4—41no—
560C4—84no—
561C4—84no—
562C4—82no—
563C4—82no—
564C4C2×#56311yescharacteristic
565C4#354511yescharacteristic
566C4#356112yes—
567C2—44no—
568C2—44no—
569C2—162no—
570C2#4613611yescharacteristic
571C2#6156911yescharacteristic
572C2#6166211yescharacteristic
573C2#6212812yes—
574C2#6216412yes—
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. #46147C2semidirect
3. #33072C2semidirect
4. #32819C2semidirect
5. #32646C2semidirect
6. #32259C2semidirect
7. #32207C2semidirect
8. #30797C2semidirect
9. #27539C2semidirect
10. #25958C2semidirect
11. #25851C2semidirect
13. #23352C2semidirect
17. C2×#501C22semidirect
21. #2859C22semidirect
30. #2546C22semidirect
32. #2496C22semidirect
36. #2440C22semidirect
42. #2094C22semidirect
43. #2077C22semidirect
44. #2077C22semidirect
45. #2071C22semidirect
46. #2065C22semidirect
47. #2065C22semidirect
48. #2065C22semidirect
49. #2059C22semidirect
50. #2058C22semidirect
53. #2023C22semidirect
54. #1989C22semidirect
55. #1985C22semidirect
56. #1948C22semidirect
57. #1947C22semidirect
58. #1936C22semidirect
59. #1926C22semidirect
60. #1924C22semidirect
61. #1921C22semidirect
62. #1919C22semidirect
68. #1853C22semidirect
69. #1850C22semidirect
70. #1831C22semidirect
71. #1807C22semidirect
485. D4#563semidirect

Maximal subgroups

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

subgroupindex
#461472
#362892
#330722
#328192
#326462
#322592
#322072
#307972
#275392
#259582
#258512
#240322
#233522