Finite Group Databasebeta

← #6482004

Subgroups — #6482004

Order 512.

2138 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35172C211yesmaximal, characteristic
2C2×D4×C4⋊C4C211yesmaximal, characteristic
3#32982C214yesmaximal, complemented
4#29893C214yesmaximal, complemented
5#26300C214yesmaximal, complemented
6#26234C214yesmaximal, complemented
7#25014C214yesmaximal, complemented
8#24969C214yesmaximal, complemented
9#21981C211yesmaximal, characteristic, complemented
10#21860C211yesmaximal, characteristic, complemented
11#21786C211yesmaximal, characteristic
12#21385C211yesmaximal, characteristic, complemented
13#21359C211yesmaximal, characteristic, complemented
14C2×#533C2211yescharacteristic
15C22×C4×D4—21no—
16C22×C4×D4C2211yescharacteristic
17C23×C4⋊C4C2212yes—
18C2×#483C2218yescomplemented
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21#2848C2214yes—
22#2626—22no—
23#2580—22no—
24#2566—22no—
25#2563C2212yes—
26#2542C2212yes—
27#2538C2212yes—
28#2538C2212yes—
29D4×C4⋊C4—22no—
30D4×C4⋊C4—22no—
31D4×C4⋊C4C2214yes—
32D4×C4⋊C4C2214yes—
33#2317—21no—
34#2309—21no—
35C2×#389—21no—
36C2×#385—22no—
37#2285C2211yescharacteristic
38C2×#384C2211yescharacteristic
39C2×#380C2212yes—
40C2×C4×C4⋊C4C2211yescharacteristic
41#2077C2218yescomplemented
42#2077C2218yescomplemented
43#2035C2218yescomplemented
44#2035C2218yescomplemented
45#1990C2212yescomplemented
46#1987C2212yescomplemented
47#1981C2212yescomplemented
48#1956C2212yescomplemented
49#1954C2214yes—
50#1954C2214yes—
51#1948C2212yescomplemented
52#1948C2212yescomplemented
53#1948C2214yescomplemented
54#1872C2214yes—
55#1872C2212yes—
56#1872C2212yes—
57#1872C2218yescomplemented
58#1853C2212yes—
59#1853C2212yes—
60#1853C2214yes—
61#1853C2218yescomplemented
62#1844C2212yescomplemented
63#1834C2212yescomplemented
64#1823C2218yescomplemented
65#1803C2214yescomplemented
66#1779C2214yes—
67C4×#106C2214yescomplemented
68C4×#103C2214yescomplemented
69C23×D4D411yescharacteristic
70C24×C4—22no—
71C24×C4C2312yes—
72#533—24no—
73#533—24no—
74C2×C4⋊Q8C2311yescharacteristic
75C2×C4⋊Q8C2311yescharacteristic
76C2×#125—21no—
77C2×C22⋊Q8—21no—
78C2×C22⋊Q8C2312yes—
79C2×C22⋊Q8C2314yes—
80C2×C22⋊Q8C2314yes—
81C2×C4×Q8—21no—
82C2×C4×Q8C2312yes—
83C2×C4×D4—24no—
84C2×C4×D4—24no—
85C2×C4×D4—24no—
86C2×C4×D4—24no—
87C2×C4×D4—48no—
88C2×C4×D4D412yes—
89C2×C4×D4D412yes—
90C2×C4×D4C2312yes—
91C2×C4×D4C2312yes—
92C2×#117—21no—
93C2×#117—21no—
94C22×C4⋊C4—24no—
95C22×C4⋊C4—22no—
96C22×C4⋊C4—21no—
97C22×C4⋊C4—21no—
98C22×C4⋊C4—48no—
99C22×C4⋊C4D411yescharacteristic
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2318yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C22⋊C4—22no—
105C22×C22⋊C4D412yes—
106C22×C42—21no—
107C22×C42C2311yescharacteristic
108#483—416no—
109#431—24no—
110#431—24no—
111C2×#106C2314yes—
112C2×#106C2318yes—
113C2×#106C2314yes—
114C2×C4⋊C8C2312yes—
115C2×C4⋊C8C2312yes—
116C2×#103C2314yes—
117C2×#103C2318yes—
118C2×#103C2314yes—
119C2×C22⋊C8C2312yes—
120C2×C22⋊C8C2312yes—
121C2×C22⋊C8C2318yes—
122C2×C22⋊C8C2318yes—
123C2×C4×C8C2312yes—
124C2×C4×C8C2312yes—
125#399—24no—
126#396—24no—
127#396—24no—
128#393—24no—
129#390—22no—
130#390—22no—
131#390—22no—
132#390—22no—
133#389—22no—
134#389—22no—
135#389—24no—
136#389—24no—
137#385—24no—
138#385—24no—
139#385—24no—
140#385—28no—
141#385—28no—
142#384—22no—
143#384—21no—
144#384—21no—
145#384C2312yes—
146#384C2311yescharacteristic
147#384C2312yes—
148#384C2311yescharacteristic
149#382—22no—
150#382—21no—
151#380—24no—
152#380—21no—
153#380C2314yes—
154#380C2314yes—
155#380C2318yes—
156#380C2312yes—
157#379—24no—
158C4×C4⋊C4—21no—
159C4×C4⋊C4—21no—
160C4×C4⋊C4—21no—
161C4×C4⋊C4C2312yes—
162C4×C4⋊C4C2312yes—
163C4×C4⋊C4C2312yes—
164C2×#95—21no—
165C2×#95—22no—
166#340C2318yes—
167#340C2314yes—
168#340C2314yes—
169#340C2318yes—
170#340C2314yes—
171#340C2314yes—
172C25C2×D412yes—
173C22×Q8—21no—
174C22×Q8C2412yes—
175C22×D4—24no—
176C22×D4—24no—
177C22×D4SD1618yes—
178C22×D4C2×D412yes—
179C22×D4C2×D412yes—
180C23×C4—21no—
181C23×C4—24no—
182C23×C4—24no—
183C23×C4—21no—
184C23×C4—22no—
185C23×C4—28no—
186C23×C4—44no—
187C23×C4—44no—
188C23×C4—48no—
189C23×C4—48no—
190C23×C4—84no—
191C23×C4C2×D412yes—
192C23×C4C2×D414yes—
193C23×C4C2×D411yescharacteristic
194C23×C4C2×D411yescharacteristic
195C23×C4C2414yes—
196C23×C4C2411yescharacteristic
197C22×C8C2414yes—
198C22×C8C2414yes—
199C4⋊Q8—22no—
200C4⋊Q8—22no—
201C4⋊Q8—22no—
202C4⋊Q8—22no—
203#125—44no—
204C22⋊Q8—24no—
205C22⋊Q8—24no—
206C22⋊Q8—44no—
207C22⋊Q8—48no—
208C22⋊Q8—48no—
209C4×Q8—24no—
210C4×Q8—24no—
211C4×Q8—44no—
212C4×D4—28no—
213C4×D4—28no—
214C4×D4—24no—
215C4×D4—24no—
216C4×D4—24no—
217C4×D4—24no—
218C4×D4—416no—
219C4×D4—416no—
220#117—44no—
221#117—44no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—216no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—28no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—22no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—22no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—48no—
245C2×C4⋊C4—44no—
246C2×C4⋊C4—42no—
247C2×C4⋊C4—416no—
248C2×C4⋊C4—48no—
249C2×C4⋊C4—44no—
250C2×C4⋊C4—82no—
251C2×C4⋊C4C2×D412yes—
252C2×C4⋊C4C2×D412yes—
253C2×C4⋊C4C2414yes—
254C2×C4⋊C4C2412yes—
255C2×C4⋊C4C2412yes—
256C2×C4⋊C4C2412yes—
257C2×C4⋊C4C2412yes—
258C2×C4⋊C4C2412yes—
259C2×C22⋊C4—28no—
260C2×C22⋊C4—28no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—24no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—24no—
267C2×C22⋊C4—48no—
268C2×C22⋊C4C2×D414yes—
269C2×C22⋊C4C2×D414yes—
270C2×C42—22no—
271C2×C42—22no—
272C2×C42—22no—
273C2×C42—22no—
274C2×C42—44no—
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42#4114yes—
278C2×C42#4114yes—
279C2×C42C2411yescharacteristic
280C2×C42C2411yescharacteristic
281#106—28no—
282#106—44no—
283#106—48no—
284C4⋊C8—24no—
285C4⋊C8—24no—
286#103—28no—
287#103—44no—
288#103—48no—
289C22⋊C8—24no—
290C22⋊C8—24no—
291C22⋊C8—48no—
292C22⋊C8—48no—
293C4×C8—24no—
294C4×C8—24no—
295#95—22no—
296#95—22no—
297#95—28no—
298#95—22no—
299#95—22no—
300#95—24no—
301#95—24no—
302C24—24no—
303C24—24no—
304C24—42no—
305C24—42no—
306C24—42no—
307C24—42no—
308C24C2×SD1618yes—
309C24C22×D414yes—
310C24C22×D411yescharacteristic
311C2×Q8—22no—
312C2×Q8—22no—
313C2×Q8—24no—
314C2×Q8—24no—
315C2×Q8—42no—
316C2×Q8—81no—
317C2×Q8—81no—
318C2×D4—24no—
319C2×D4—24no—
320C2×D4—24no—
321C2×D4—24no—
322C2×D4—232no—
323C2×D4C2×SD1618yes—
324C2×D4C2×SD1618yes—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—24no—
328C22×C4—24no—
329C22×C4—22no—
330C22×C4—28no—
331C22×C4—24no—
332C22×C4—24no—
333C22×C4—28no—
334C22×C4—28no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—44no—
342C22×C4—42no—
343C22×C4—41no—
344C22×C4—44no—
345C22×C4—42no—
346C22×C4—41no—
347C22×C4—416no—
348C22×C4—416no—
349C22×C4—416no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—44no—
353C22×C4—42no—
354C22×C4—44no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—44no—
358C22×C4—416no—
359C22×C4—88no—
360C22×C4—832no—
361C22×C4#12314yes—
362C22×C4#12314yes—
363C22×C4C2×SD1614yes—
364C22×C4#13714yes—
365C22×C4C22×D414yes—
366C22×C4C22×D411yescharacteristic
367C22×C4C22×D411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C22×D411yescharacteristic
370C22×C4C22×D411yescharacteristic
371C22×C4C2×#4112yes—
372C22×C4C2×#4112yes—
373C22×C4C2511yescharacteristic
374C2×C8—24no—
375C2×C8—24no—
376C2×C8—24no—
377C2×C8—24no—
378C2×C8—24no—
379C2×C8—24no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—28no—
387C4⋊C4—24no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—24no—
391C4⋊C4—416no—
392C4⋊C4—42no—
393C4⋊C4—44no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—44no—
397C4⋊C4—44no—
398C4⋊C4—42no—
399C4⋊C4—44no—
400C4⋊C4—44no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—81no—
404C4⋊C4—81no—
405C4⋊C4—82no—
406C4⋊C4—82no—
407C4⋊C4—84no—
408C4⋊C4—84no—
409C4⋊C4H(5,𝔽2)14yes—
410C22⋊C4—28no—
411C22⋊C4—28no—
412C22⋊C4—44no—
413C22⋊C4—416no—
414C22⋊C4—44no—
415C22⋊C4—84no—
416C22⋊C4—84no—
417C22⋊C4—84no—
418C22⋊C4—84no—
419C42—44no—
420C42—44no—
421C42—44no—
422C42—44no—
423C42#12314yes—
424C42#12314yes—
425C42C2×#4112yes—
426C42C2×#4112yes—
427C42C2×#4112yes—
428C42C2×#4112yes—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—24no—
434C23—24no—
435C23—216no—
436C23—24no—
437C23—24no—
438C23—24no—
439C23—24no—
440C23—48no—
441C23—48no—
442C23—48no—
443C23—48no—
444C23—48no—
445C23—48no—
446C23C2×#12312yes—
447C23C22×SD1618yes—
448C23C22×SD1612yes—
449C23C2×#13712yes—
450C23C23×D411yescharacteristic
451Q8—44no—
452Q8—44no—
453Q8—84no—
454D4—216no—
455D4—216no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—44no—
469C2×C4—416no—
470C2×C4—48no—
471C2×C4—48no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—48no—
475C2×C4—48no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—832no—
495C2×C4—84no—
496C2×C4—88no—
497C2×C4#483116yescomplemented
498C2×C4C2×#12312yes—
499C2×C4C2×#12312yes—
500C2×C4C2×#12312yes—
501C2×C4C2×#12312yes—
502C2×C4C2×#12312yes—
503C2×C4C22×SD1612yes—
504C2×C4C22×SD1612yes—
505C2×C4C2×#13712yes—
506C2×C4C2×#13712yes—
507C2×C4C22×#4111yescharacteristic
508C2×C4C22×#4111yescharacteristic
509C2×C4C2×H(5,𝔽2)12yes—
510C2×C4#58512yes—
511C8—44no—
512C8—44no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—28no—
520C22—28no—
521C22—28no—
522C22—432no—
523C22—44no—
524C22—44no—
525C22—44no—
526C22—44no—
527C22C2×#48314yes—
528C22C22×#12311yescharacteristic
529C22#348411yescharacteristic
530C22#348511yescharacteristic
531C22C23×SD1611yescharacteristic
532C22C22×#13711yescharacteristic
533C22#358511yescharacteristic
534C22#358611yescharacteristic
535C4—22no—
536C4—22no—
537C4—22no—
538C4—44no—
539C4—44no—
540C4—82no—
541C4—82no—
542C4—82no—
543C4—84no—
544C4—82no—
545C4—82no—
546C4C2×#48314yes—
547C4C2×#48314yes—
548C4#346012yes—
549C4#350612yes—
550C2—24no—
551C2—48no—
552C2#3515211yescharacteristic
553C2C22×#48311yescharacteristic
554C2#6051711yescharacteristic
555C2#6156011yescharacteristic
556C2#6158411yescharacteristic
557C2#6169511yescharacteristic
558C2#6171911yescharacteristic
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. #32982C2semidirect
4. #29893C2semidirect
5. #26300C2semidirect
6. #26234C2semidirect
7. #25014C2semidirect
8. #24969C2semidirect
9. #21981C2semidirect
10. #21860C2semidirect
12. #21385C2semidirect
13. #21359C2semidirect
18. C2×#483C22semidirect
41. #2077C22semidirect
42. #2077C22semidirect
43. #2035C22semidirect
44. #2035C22semidirect
45. #1990C22semidirect
46. #1987C22semidirect
47. #1981C22semidirect
48. #1956C22semidirect
51. #1948C22semidirect
52. #1948C22semidirect
53. #1948C22semidirect
57. #1872C22semidirect
61. #1853C22semidirect
62. #1844C22semidirect
63. #1834C22semidirect
64. #1823C22semidirect
65. #1803C22semidirect
67. C4×#106C22semidirect
68. C4×#103C22semidirect
497. C2×C4#483semidirect

Maximal subgroups

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

subgroupindex
#351722
C2×D4×C4⋊C42
#329822
#298932
#263002
#262342
#250142
#249692
#219812
#218602
#217862
#213852
#213592