Finite Group Databasebeta

← #6482006

Subgroups — #6482006

Order 512.

1878 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35172C211yesmaximal, characteristic
2C2×#2354C211yesmaximal, characteristic
3#32985C214yesmaximal, complemented
4#29894C214yesmaximal, complemented
5#26297C214yesmaximal, complemented
6#26231C214yesmaximal, complemented
7#25015C214yesmaximal, complemented
8#24970C214yesmaximal, complemented
9#21981C211yesmaximal, characteristic, complemented
10#21860C211yesmaximal, characteristic, complemented
11#21786C211yesmaximal, characteristic
12#21385C211yesmaximal, characteristic, complemented
13#21358C211yesmaximal, characteristic, complemented
14C2×#533C2211yescharacteristic
15C22×C4×D4C2211yescharacteristic
16C22×#117C2212yes—
17C2×#485C2218yescomplemented
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20#2848C2214yes—
21#2626—22no—
22#2580—22no—
23#2566—22no—
24#2563C2212yes—
25#2542C2212yes—
26#2538C2212yes—
27#2538C2212yes—
28#2354—22no—
29#2354—22no—
30#2354C2214yes—
31#2354C2214yes—
32#2317—21no—
33#2309—21no—
34C2×#390—21no—
35C2×#388—22no—
36#2285C2211yescharacteristic
37C2×#384C2211yescharacteristic
38C2×#380C2212yes—
39C2×C4×C4⋊C4—21no—
40C2×C4×C4⋊C4C2211yescharacteristic
41#2077C2218yescomplemented
42#2075C2218yescomplemented
43#2036C2218yescomplemented
44#2035C2218yescomplemented
45#1990C2212yescomplemented
46#1987C2212yescomplemented
47#1981C2212yescomplemented
48#1956C2212yescomplemented
49#1954C2214yes—
50#1954C2214yes—
51#1948C2214yescomplemented
52#1947C2212yescomplemented
53#1947C2212yescomplemented
54#1873C2218yescomplemented
55#1872C2214yes—
56#1872C2212yes—
57#1872C2212yes—
58#1854C2218yescomplemented
59#1853C2214yes—
60#1852C2212yes—
61#1852C2212yes—
62#1844C2212yescomplemented
63#1834C2212yescomplemented
64#1825C2218yescomplemented
65#1803C2214yescomplemented
66#1779C2214yes—
67C4×#107C2214yescomplemented
68C4×#103C2214yescomplemented
69C23×D4D411yescharacteristic
70C24×C4C2312yes—
71#533—24no—
72#533—24no—
73C2×C4⋊Q8C2311yescharacteristic
74C2×C4⋊Q8C2311yescharacteristic
75C2×#125—21no—
76C2×C22⋊Q8—21no—
77C2×C22⋊Q8C2312yes—
78C2×C22⋊Q8C2314yes—
79C2×C22⋊Q8C2314yes—
80C2×C4×Q8—21no—
81C2×C4×Q8C2312yes—
82C2×C4×D4—24no—
83C2×C4×D4—24no—
84C2×C4×D4D412yes—
85C2×C4×D4D412yes—
86C2×C4×D4C2312yes—
87C2×C4×D4C2312yes—
88C2×#117—21no—
89C2×#117—21no—
90C2×#117—24no—
91C2×#117—48no—
92C2×#117C2318yes—
93C22×C4⋊C4—21no—
94C22×C4⋊C4—21no—
95C22×C4⋊C4D411yescharacteristic
96C22×C4⋊C4C2311yescharacteristic
97C22×C4⋊C4C2311yescharacteristic
98C22×C4⋊C4C2312yes—
99C22×C22⋊C4—22no—
100C22×C22⋊C4—22no—
101C22×C22⋊C4D412yes—
102C22×C42—22no—
103C22×C42—21no—
104C22×C42C2311yescharacteristic
105#485—416no—
106#431—24no—
107#431—24no—
108C2×#107C2314yes—
109C2×#107C2314yes—
110C2×#106C2318yes—
111C2×C4⋊C8C2312yes—
112C2×C4⋊C8C2312yes—
113C2×#103C2314yes—
114C2×#103C2318yes—
115C2×#103C2314yes—
116C2×C22⋊C8C2312yes—
117C2×C22⋊C8C2312yes—
118C2×C22⋊C8C2318yes—
119C2×C22⋊C8C2318yes—
120C2×C4×C8C2312yes—
121C2×C4×C8C2312yes—
122#399—24no—
123#396—24no—
124#396—24no—
125#393—24no—
126#390—22no—
127#390—22no—
128#390—22no—
129#390—22no—
130#390—24no—
131#390—24no—
132#389—22no—
133#389—22no—
134#388—28no—
135#388—28no—
136#385—24no—
137#385—24no—
138#385—24no—
139#384—22no—
140#384—21no—
141#384—21no—
142#384C2312yes—
143#384C2311yescharacteristic
144#384C2312yes—
145#384C2311yescharacteristic
146#382—22no—
147#382—21no—
148#380—24no—
149#380—21no—
150#380C2314yes—
151#380C2314yes—
152#380C2318yes—
153#380C2312yes—
154#379—24no—
155C4×C4⋊C4—24no—
156C4×C4⋊C4—24no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4—21no—
160C4×C4⋊C4C2312yes—
161C4×C4⋊C4C2312yes—
162C4×C4⋊C4C2312yes—
163C2×#95—22no—
164C2×#95—21no—
165#340C2318yes—
166#340C2314yes—
167#340C2318yes—
168#340C2314yes—
169#340C2314yes—
170#340C2314yes—
171C25C2×D412yes—
172C22×Q8—21no—
173C22×Q8C2412yes—
174C22×D4—24no—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4C2×D412yes—
178C22×D4C2×D412yes—
179C23×C4—21no—
180C23×C4—24no—
181C23×C4—24no—
182C23×C4—22no—
183C23×C4—21no—
184C23×C4—44no—
185C23×C4—44no—
186C23×C4C2×D412yes—
187C23×C4C2×D414yes—
188C23×C4C2×D411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4C2414yes—
191C23×C4C2411yescharacteristic
192C22×C8C2414yes—
193C22×C8C2414yes—
194C4⋊Q8—22no—
195C4⋊Q8—22no—
196C4⋊Q8—22no—
197C4⋊Q8—22no—
198#125—44no—
199C22⋊Q8—24no—
200C22⋊Q8—24no—
201C22⋊Q8—44no—
202C22⋊Q8—48no—
203C22⋊Q8—48no—
204C4×Q8—24no—
205C4×Q8—24no—
206C4×Q8—44no—
207C4×D4—28no—
208C4×D4—28no—
209C4×D4—24no—
210C4×D4—24no—
211C4×D4—24no—
212C4×D4—24no—
213#117—216no—
214#117—48no—
215#117—416no—
216#117—44no—
217#117—44no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—82no—
242C2×C4⋊C4C2×D412yes—
243C2×C4⋊C4C2×D412yes—
244C2×C4⋊C4C2414yes—
245C2×C4⋊C4C2412yes—
246C2×C4⋊C4C2412yes—
247C2×C4⋊C4C2412yes—
248C2×C4⋊C4C2412yes—
249C2×C4⋊C4C2412yes—
250C2×C22⋊C4—28no—
251C2×C22⋊C4—28no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—28no—
255C2×C22⋊C4—24no—
256C2×C22⋊C4—24no—
257C2×C22⋊C4—24no—
258C2×C22⋊C4—24no—
259C2×C22⋊C4—48no—
260C2×C22⋊C4—84no—
261C2×C22⋊C4C2×D414yes—
262C2×C22⋊C4C2×D414yes—
263C2×C42—28no—
264C2×C42—22no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—48no—
269C2×C42—44no—
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42#4114yes—
273C2×C42#4114yes—
274C2×C42C2411yescharacteristic
275C2×C42C2411yescharacteristic
276#107—28no—
277#107—44no—
278#106—48no—
279C4⋊C8—24no—
280C4⋊C8—24no—
281#103—28no—
282#103—44no—
283#103—48no—
284C22⋊C8—24no—
285C22⋊C8—24no—
286C22⋊C8—48no—
287C22⋊C8—48no—
288C4×C8—24no—
289C4×C8—24no—
290#95—28no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—24no—
296#95—24no—
297C24—24no—
298C24—24no—
299C24—24no—
300C24—42no—
301C24—42no—
302C24—42no—
303C24—42no—
304C24C22×D414yes—
305C24C22×D411yescharacteristic
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—24no—
309C2×Q8—24no—
310C2×Q8—42no—
311C2×Q8—81no—
312C2×Q8—81no—
313C2×D4—24no—
314C2×D4—24no—
315C2×D4—24no—
316C2×D4—24no—
317C2×D4—24no—
318C2×D4—24no—
319C2×D4—416no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—24no—
324C22×C4—22no—
325C22×C4—28no—
326C22×C4—24no—
327C22×C4—24no—
328C22×C4—28no—
329C22×C4—28no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—22no—
336C22×C4—42no—
337C22×C4—41no—
338C22×C4—44no—
339C22×C4—44no—
340C22×C4—42no—
341C22×C4—41no—
342C22×C4—416no—
343C22×C4—416no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—44no—
347C22×C4—42no—
348C22×C4—44no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—44no—
352C22×C4#12314yes—
353C22×C4#12314yes—
354C22×C4#13514yes—
355C22×C4#13714yes—
356C22×C4C22×D414yes—
357C22×C4C22×D411yescharacteristic
358C22×C4C22×D411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C2×#4112yes—
363C22×C4C2×#4112yes—
364C22×C4C2511yescharacteristic
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—24no—
370C2×C8—24no—
371C4⋊C4—24no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—28no—
378C4⋊C4—24no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—24no—
382C4⋊C4—42no—
383C4⋊C4—44no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—81no—
394C4⋊C4—81no—
395C4⋊C4—82no—
396C4⋊C4—82no—
397C4⋊C4—84no—
398C4⋊C4—84no—
399C4⋊C4H(5,𝔽2)14yes—
400C22⋊C4—28no—
401C22⋊C4—28no—
402C22⋊C4—416no—
403C22⋊C4—44no—
404C22⋊C4—44no—
405C22⋊C4—84no—
406C22⋊C4—84no—
407C22⋊C4—88no—
408C22⋊C4—84no—
409C22⋊C4—84no—
410C42—416no—
411C42—44no—
412C42—44no—
413C42—44no—
414C42—44no—
415C42#12314yes—
416C42#12314yes—
417C42C2×#4112yes—
418C42C2×#4112yes—
419C42C2×#4112yes—
420C42C2×#4112yes—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—24no—
426C23—24no—
427C23—24no—
428C23—24no—
429C23—24no—
430C23—24no—
431C23—24no—
432C23—48no—
433C23—48no—
434C23—48no—
435C23—84no—
436C23—84no—
437C23—84no—
438C23—84no—
439C23C2×#12312yes—
440C23C2×#13512yes—
441C23C2×#13712yes—
442C23C23×D411yescharacteristic
443Q8—44no—
444Q8—44no—
445Q8—84no—
446D4—48no—
447D4—48no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—44no—
461C2×C4—416no—
462C2×C4—48no—
463C2×C4—48no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—48no—
467C2×C4—48no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—44no—
486C2×C4—84no—
487C2×C4—88no—
488C2×C4#485116yescomplemented
489C2×C4C2×#12312yes—
490C2×C4C2×#12312yes—
491C2×C4C2×#12312yes—
492C2×C4C2×#12312yes—
493C2×C4C2×#12312yes—
494C2×C4C2×#13512yes—
495C2×C4C2×#13512yes—
496C2×C4C2×#13712yes—
497C2×C4C2×#13712yes—
498C2×C4C22×#4111yescharacteristic
499C2×C4C22×#4111yescharacteristic
500C2×C4C2×H(5,𝔽2)12yes—
501C2×C4#58512yes—
502C8—44no—
503C8—44no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—22no—
510C22—44no—
511C22—44no—
512C22—44no—
513C22—44no—
514C22—44no—
515C22—44no—
516C22—44no—
517C22—816no—
518C22C2×#48514yes—
519C22C22×#12311yescharacteristic
520C22#348411yescharacteristic
521C22#348511yescharacteristic
522C22C22×#13511yescharacteristic
523C22C22×#13711yescharacteristic
524C22#358511yescharacteristic
525C22#358611yescharacteristic
526C4—22no—
527C4—22no—
528C4—22no—
529C4—44no—
530C4—44no—
531C4—82no—
532C4—82no—
533C4—82no—
534C4—82no—
535C4—82no—
536C4—84no—
537C4C2×#48514yes—
538C4C2×#48514yes—
539C4#346012yes—
540C4#350612yes—
541C2—24no—
542C2—84no—
543C2#3515211yescharacteristic
544C2C22×#48511yescharacteristic
545C2#6051711yescharacteristic
546C2#6156511yescharacteristic
547C2#6158811yescharacteristic
548C2#6170011yescharacteristic
549C2#6172311yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4D4
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
3. #32985C2semidirect
4. #29894C2semidirect
5. #26297C2semidirect
6. #26231C2semidirect
7. #25015C2semidirect
8. #24970C2semidirect
9. #21981C2semidirect
10. #21860C2semidirect
12. #21385C2semidirect
13. #21358C2semidirect
17. C2×#485C22semidirect
41. #2077C22semidirect
42. #2075C22semidirect
43. #2036C22semidirect
44. #2035C22semidirect
45. #1990C22semidirect
46. #1987C22semidirect
47. #1981C22semidirect
48. #1956C22semidirect
51. #1948C22semidirect
52. #1947C22semidirect
53. #1947C22semidirect
54. #1873C22semidirect
58. #1854C22semidirect
62. #1844C22semidirect
63. #1834C22semidirect
64. #1825C22semidirect
65. #1803C22semidirect
67. C4×#107C22semidirect
68. C4×#103C22semidirect
488. C2×C4#485semidirect

Maximal subgroups

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

subgroupindex
#351722
C2×#23542
#329852
#298942
#262972
#262312
#250152
#249702
#219812
#218602
#217862
#213852
#213582