Finite Group Databasebeta

← #6519031

Subgroups — #6519031

Order 512.

2166 conjugacy classes of subgroups.

The subgroup lattice has 595 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2537C211yesmaximal, characteristic
2#34186C211yesmaximal, characteristic, complemented
3#27602C214yesmaximal, complemented
4#25013C214yesmaximal, complemented
5#25011C214yesmaximal, complemented
6#24968C214yesmaximal, complemented
7#24966C214yesmaximal, complemented
8#24009C214yesmaximal, complemented
9#21982C211yesmaximal, characteristic, complemented
10#21981C211yesmaximal, characteristic, complemented
11#21786C211yesmaximal, characteristic, complemented
12#20732C211yesmaximal, characteristic, complemented
13#20732C211yesmaximal, characteristic, complemented
14C22×#121—21no—
15C22×#121C2212yes—
16C22×C4×D4C2211yescharacteristic
17C22×C4×D4C2211yescharacteristic
18C2×#482C2218yescomplemented
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21#2537—22no—
22#2537—22no—
23#2537C2214yes—
24#2537C2214yes—
25C2×#394—22no—
26#2357—22no—
27#2357—22no—
28#2354C2212yescomplemented
29#2354C2212yescomplemented
30#2354C2212yescomplemented
31#2354C2212yescomplemented
32#2312—22no—
33#2309—21no—
34#2285C2211yescharacteristic
35#2285C2211yescharacteristic
36C2×#384C2211yescharacteristic
37#2280C2214yes—
38C2×#380—21no—
39C2×#380C2212yes—
40#2274—21no—
41#2077C2218yescomplemented
42#2076C2218yescomplemented
43#2034C2218yescomplemented
44#2033C2218yescomplemented
45#1954C2214yescomplemented
46#1954C2214yescomplemented
47#1948C2214yescomplemented
48#1947C2214yescomplemented
49#1875C2218yescomplemented
50#1854C2218yescomplemented
51#1853C2214yes—
52#1852C2214yes—
53#1845C2212yescomplemented
54#1844C2212yescomplemented
55#1834C2212yescomplemented
56#1834C2212yescomplemented
57#1825C2218yescomplemented
58#1823C2212yes—
59#1823C2212yes—
60#1802C2212yescomplemented
61#1802C2212yescomplemented
62#1802C2212yescomplemented
63#1802C2212yescomplemented
64#1786C2212yes—
65#1786C2212yes—
66#1779C2214yes—
67C4×#102C2214yescomplemented
68C4×#102C2214yescomplemented
69C23×D4—22no—
70C23×D4D411yescharacteristic
71C23×D4C2311yescharacteristic
72C24×C4—21no—
73C24×C4C2312yes—
74C2×#121—24no—
75C2×#121—24no—
76C2×#121—24no—
77C2×#121—48no—
78C2×#121—48no—
79C2×#121C2318yes—
80C2×C4×D4—21no—
81C2×C4×D4—21no—
82C2×C4×D4—24no—
83C2×C4×D4—24no—
84C2×C4×D4—24no—
85C2×C4×D4—24no—
86C2×C4×D4D412yes—
87C2×C4×D4D412yes—
88C2×C4×D4C2312yes—
89C2×C4×D4C2312yes—
90C2×C4×D4C2312yes—
91C2×C4×D4C2312yes—
92C2×#117—21no—
93C2×#117—21no—
94C2×#117C2314yes—
95C2×#117C2314yes—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4D411yescharacteristic
99C22×C4⋊C4C2311yescharacteristic
100C22×C4⋊C4C2312yes—
101C22×C22⋊C4—22no—
102C22×C22⋊C4—21no—
103C22×C22⋊C4—22no—
104C22×C22⋊C4D412yes—
105C22×C42C2311yescharacteristic
106#482—416no—
107#431—24no—
108#431—24no—
109C2×#107C2318yes—
110C2×#106C2318yes—
111C2×C4⋊C8C2312yes—
112C2×C4⋊C8C2312yes—
113C2×#102C2314yes—
114C2×#102C2314yes—
115C2×#102C2314yes—
116C2×#102C2314yes—
117C2×C22⋊C8C2312yes—
118C2×C22⋊C8C2312yes—
119C2×C22⋊C8C2318yes—
120C2×C22⋊C8C2318yes—
121C2×C4×C8C2312yes—
122C2×C4×C8C2312yes—
123#394—28no—
124#394—28no—
125#390—22no—
126#390—22no—
127#390—22no—
128#390—22no—
129#388—24no—
130#388—24no—
131#388—24no—
132#388—24no—
133#384—21no—
134#384—21no—
135#384C2311yescharacteristic
136#384C2311yescharacteristic
137#384C2311yescharacteristic
138#384C2311yescharacteristic
139#384C2312yes—
140#384C2312yes—
141#382—21no—
142#382—21no—
143#381—24no—
144#380—21no—
145#380—21no—
146#380—24no—
147#380—24no—
148#380—24no—
149#380C2314yes—
150#380C2318yes—
151#380C2314yes—
152#380C2312yes—
153#380C2312yes—
154#379—24no—
155C4×C4⋊C4—22no—
156C4×C4⋊C4—22no—
157C4×C4⋊C4—22no—
158C4×C4⋊C4—22no—
159C4×C4⋊C4—21no—
160C4×C4⋊C4—21no—
161C4×C4⋊C4C2312yes—
162C4×C4⋊C4C2312yes—
163C4×C22⋊C4—24no—
164C4×C22⋊C4—24no—
165C2×#95—22no—
166#340C2314yes—
167#340C2314yes—
168#340C2314yes—
169#340C2314yes—
170#340C2318yes—
171#340C2318yes—
172C25—21no—
173C25C2×D412yes—
174C22×D4—24no—
175C22×D4—28no—
176C22×D4—24no—
177C22×D4—24no—
178C22×D4—24no—
179C22×D4—21no—
180C22×D4—22no—
181C22×D4—22no—
182C22×D4—42no—
183C22×D4—48no—
184C22×D4—48no—
185C22×D4—84no—
186C22×D4C2×D412yes—
187C22×D4C2×D412yes—
188C22×D4C2412yes—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—21no—
192C23×C4—22no—
193C23×C4—22no—
194C23×C4—22no—
195C23×C4—44no—
196C23×C4—44no—
197C23×C4—44no—
198C23×C4—44no—
199C23×C4—44no—
200C23×C4C2×D412yes—
201C23×C4C2×D414yes—
202C23×C4C2×D411yescharacteristic
203C23×C4C2×D411yescharacteristic
204C23×C4C2414yes—
205C23×C4C2411yescharacteristic
206C22×C8C2414yes—
207C22×C8C2414yes—
208#121—216no—
209#121—48no—
210#121—416no—
211#121—416no—
212#121—416no—
213C4×D4—28no—
214C4×D4—28no—
215C4×D4—24no—
216C4×D4—24no—
217C4×D4—24no—
218C4×D4—24no—
219C4×D4—24no—
220C4×D4—24no—
221C4×D4—24no—
222C4×D4—24no—
223C4×D4—44no—
224C4×D4—44no—
225#117—44no—
226#117—44no—
227#117—48no—
228#117—48no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—44no—
247C2×C4⋊C4—44no—
248C2×C4⋊C4—82no—
249C2×C4⋊C4C2×D412yes—
250C2×C4⋊C4C2×D412yes—
251C2×C4⋊C4C2412yes—
252C2×C4⋊C4C2414yes—
253C2×C4⋊C4C2412yes—
254C2×C4⋊C4C2412yes—
255C2×C4⋊C4C2412yes—
256C2×C4⋊C4C2412yes—
257C2×C22⋊C4—28no—
258C2×C22⋊C4—28no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—28no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—48no—
269C2×C22⋊C4—48no—
270C2×C22⋊C4—44no—
271C2×C22⋊C4C2×D414yes—
272C2×C22⋊C4C2×D414yes—
273C2×C42—22no—
274C2×C42—22no—
275C2×C42—24no—
276C2×C42—24no—
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42#4114yes—
280C2×C42#4114yes—
281C2×C42C2411yescharacteristic
282C2×C42C2411yescharacteristic
283#107—48no—
284#106—48no—
285C4⋊C8—24no—
286C4⋊C8—24no—
287#102—28no—
288#102—28no—
289#102—44no—
290#102—44no—
291C22⋊C8—24no—
292C22⋊C8—24no—
293C22⋊C8—48no—
294C22⋊C8—48no—
295C4×C8—24no—
296C4×C8—24no—
297#95—28no—
298#95—22no—
299#95—22no—
300#95—24no—
301#95—24no—
302#95—22no—
303#95—22no—
304C24—24no—
305C24—24no—
306C24—22no—
307C24—24no—
308C24—24no—
309C24—42no—
310C24—42no—
311C24—42no—
312C24—42no—
313C24—42no—
314C24—42no—
315C24—42no—
316C24—42no—
317C24—42no—
318C24—42no—
319C24C22×D414yes—
320C24C22×D411yescharacteristic
321C2×D4—24no—
322C2×D4—24no—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—24no—
327C2×D4—24no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—48no—
334C2×D4—48no—
335C2×D4—42no—
336C2×D4—416no—
337C2×D4—416no—
338C2×D4—42no—
339C2×D4—416no—
340C2×D4—88no—
341C2×D4—832no—
342C2×D4H(5,𝔽2)14yes—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—28no—
349C22×C4—28no—
350C22×C4—28no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—24no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—44no—
361C22×C4—44no—
362C22×C4—42no—
363C22×C4—44no—
364C22×C4—44no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4—88no—
376C22×C4—88no—
377C22×C4—816no—
378C22×C4#12314yes—
379C22×C4#12314yes—
380C22×C4#13514yes—
381C22×C4AGL(1,ℤ/8)14yes—
382C22×C4C22×D414yes—
383C22×C4C22×D411yescharacteristic
384C22×C4C22×D411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C22×D411yescharacteristic
387C22×C4C22×D411yescharacteristic
388C22×C4C2×#4112yes—
389C22×C4C2×#4112yes—
390C22×C4C2511yescharacteristic
391C2×C8—24no—
392C2×C8—24no—
393C2×C8—24no—
394C2×C8—24no—
395C2×C8—24no—
396C2×C8—24no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—24no—
402C4⋊C4—24no—
403C4⋊C4—28no—
404C4⋊C4—24no—
405C4⋊C4—24no—
406C4⋊C4—44no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—81no—
415C4⋊C4—81no—
416C4⋊C4—81no—
417C4⋊C4—81no—
418C4⋊C4—82no—
419C4⋊C4—82no—
420C22⋊C4—28no—
421C22⋊C4—28no—
422C22⋊C4—416no—
423C22⋊C4—416no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—84no—
429C22⋊C4—84no—
430C22⋊C4—84no—
431C22⋊C4—84no—
432C42—44no—
433C42—44no—
434C42—84no—
435C42—84no—
436C42#12314yes—
437C42#12314yes—
438C42C2×#4112yes—
439C42C2×#4112yes—
440C42C2×#4112yes—
441C42C2×#4112yes—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—24no—
447C23—24no—
448C23—24no—
449C23—24no—
450C23—24no—
451C23—24no—
452C23—24no—
453C23—24no—
454C23—48no—
455C23—48no—
456C23—48no—
457C23—48no—
458C23—48no—
459C23—42no—
460C23—42no—
461C23—48no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—84no—
467C23—84no—
468C23—84no—
469C23—84no—
470C23—84no—
471C23—84no—
472C23—84no—
473C23—84no—
474C23C2×#12312yes—
475C23C2×#13512yes—
476C23C2×AGL(1,ℤ/8)12yes—
477C23C23×D411yescharacteristic
478D4—24no—
479D4—24no—
480D4—48no—
481D4—48no—
482D4—48no—
483D4—42no—
484D4—42no—
485D4—832no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—44no—
498C2×C4—48no—
499C2×C4—48no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—44no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—88no—
519C2×C4—84no—
520C2×C4—88no—
521C2×C4—84no—
522C2×C4—88no—
523C2×C4—88no—
524C2×C4#482116yescomplemented
525C2×C4C2×#12312yes—
526C2×C4C2×#12312yes—
527C2×C4C2×#12312yes—
528C2×C4C2×#12312yes—
529C2×C4C2×#12312yes—
530C2×C4C2×#13512yes—
531C2×C4C2×#13512yes—
532C2×C4C2×AGL(1,ℤ/8)12yes—
533C2×C4C2×AGL(1,ℤ/8)12yes—
534C2×C4C22×#4111yescharacteristic
535C2×C4C22×#4111yescharacteristic
536C2×C4C2×H(5,𝔽2)12yes—
537C2×C4#58512yes—
538C8—44no—
539C8—44no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—22no—
544C22—22no—
545C22—22no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—44no—
550C22—44no—
551C22—44no—
552C22—44no—
553C22—44no—
554C22—44no—
555C22—44no—
556C22—816no—
557C22—816no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22C2×#48214yes—
563C22C22×#12311yescharacteristic
564C22#348411yescharacteristic
565C22#348511yescharacteristic
566C22C22×#13511yescharacteristic
567C22C22×AGL(1,ℤ/8)11yescharacteristic
568C22#358511yescharacteristic
569C22#358611yescharacteristic
570C4—22no—
571C4—22no—
572C4—22no—
573C4—44no—
574C4—82no—
575C4—82no—
576C4—82no—
577C4—82no—
578C4—82no—
579C4—84no—
580C4C2×#48214yes—
581C4C2×#48214yes—
582C4#349412yes—
583C4#349412yes—
584C2—24no—
585C2—84no—
586C2—84no—
587C2#3554011yescharacteristic
588C2C22×#48211yescharacteristic
589C2#6052011yescharacteristic
590C2#6156511yescharacteristic
591C2#6158211yescharacteristic
592C2#6170011yescharacteristic
593C2#6171711yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2537C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34186C2semidirect
3. #27602C2semidirect
4. #25013C2semidirect
5. #25011C2semidirect
6. #24968C2semidirect
7. #24966C2semidirect
8. #24009C2semidirect
9. #21982C2semidirect
10. #21981C2semidirect
11. #21786C2semidirect
12. #20732C2semidirect
13. #20732C2semidirect
18. C2×#482C22semidirect
28. #2354C22semidirect
29. #2354C22semidirect
30. #2354C22semidirect
31. #2354C22semidirect
41. #2077C22semidirect
42. #2076C22semidirect
43. #2034C22semidirect
44. #2033C22semidirect
45. #1954C22semidirect
46. #1954C22semidirect
47. #1948C22semidirect
48. #1947C22semidirect
49. #1875C22semidirect
50. #1854C22semidirect
53. #1845C22semidirect
54. #1844C22semidirect
55. #1834C22semidirect
56. #1834C22semidirect
57. #1825C22semidirect
60. #1802C22semidirect
61. #1802C22semidirect
62. #1802C22semidirect
63. #1802C22semidirect
67. C4×#102C22semidirect
68. C4×#102C22semidirect
524. C2×C4#482semidirect

Maximal subgroups

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

subgroupindex
C2×#25372
#341862
#276022
#250132
#250112
#249682
#249662
#240092
#219822
#219812
#217862
#207322