Finite Group Databasebeta

← #6650285

Subgroups — #6650285

Order 512.

2556 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35010C211yesmaximal, characteristic, complemented
2C2×#2541C211yesmaximal, characteristic
3#33098C212yesmaximal, complemented
4#32028C214yesmaximal, complemented
5#30703C218yesmaximal, complemented
6#30506C214yesmaximal, complemented
7#29625C212yesmaximal, complemented
8#24379C214yesmaximal, complemented
9#24005C212yesmaximal, complemented
10#23919C212yesmaximal, complemented
11#21738C211yesmaximal, characteristic, complemented
12C22×#127—21no—
13C22×#127C2211yescharacteristic
14C22×C22⋊Q8C2211yescharacteristic
15C22×#121C2212yes—
16C22×#121C2212yes—
17C22×C4×D4C2211yescharacteristic
18C2×#481C2214yescomplemented
19C2×#465C2218yescomplemented
20C2×#448C2214yescomplemented
21C2×#450C2214yes—
22C22×C22⋊C8C2212yes—
23#2855C2212yes—
24#2848C2212yes—
25#2627—21no—
26#2626—21no—
27#2579—22no—
28#2564—22no—
29#2541—22no—
30#2541—22no—
31#2541C2214yes—
32#2541C2214yes—
33#2487C2212yescomplemented
34#2417C2214yescomplemented
35#2409C2212yescomplemented
36C2×#392—21no—
37C2×#392—21no—
38#2315—21no—
39#2308—21no—
40C2×#390—21no—
41C2×#380C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43#2041C2218yescomplemented
44#2034C2218yescomplemented
45#2034C2218yescomplemented
46#2014C2218yescomplemented
47#2000C2218yescomplemented
48#2000C2218yescomplemented
49#1927C2218yescomplemented
50#1920C2218yescomplemented
51#1893C2218yescomplemented
52#1874C2214yescomplemented
53#1874C2214yescomplemented
54#1872C2214yes—
55#1869C2214yescomplemented
56#1869C2214yescomplemented
57#1853C2214yes—
58#1817C2212yes—
59#1786C2214yes—
60#1779C2212yes—
61C23×Q8C2311yescharacteristic
62C23×D4—22no—
63C23×D4—22no—
64C23×D4—21no—
65C23×D4—21no—
66C23×D4D411yescharacteristic
67C23×D4C2312yes—
68C24×C4C2311yescharacteristic
69C24×C4C2311yescharacteristic
70C22×SD16C2318yes—
71C23×C8C2314yes—
72C2×#127—24no—
73C2×#127—24no—
74C2×#127—22no—
75C2×#127—44no—
76C2×#127—48no—
77C2×#127C2314yes—
78C2×C22⋊Q8—21no—
79C2×C22⋊Q8—21no—
80C2×C22⋊Q8—44no—
81C2×C22⋊Q8C2312yes—
82C2×C22⋊Q8C2314yes—
83C2×C22⋊Q8C2312yes—
84C2×#121—22no—
85C2×#121—22no—
86C2×#121—24no—
87C2×#121—48no—
88C2×#121—48no—
89C2×#121C2314yes—
90C2×#121C2314yes—
91C2×#121C2318yes—
92C2×C4×D4—44no—
93C2×C4×D4D412yes—
94C2×C4×D4D412yes—
95C2×C4×D4C2312yes—
96C2×C4×D4C2312yes—
97C22×C4⋊C4—21no—
98C22×C4⋊C4D411yescharacteristic
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2311yescharacteristic
102C22×C22⋊C4—21no—
103C22×C22⋊C4—21no—
104C22×C22⋊C4—22no—
105C22×C22⋊C4—22no—
106C22×C22⋊C4D411yescharacteristic
107C22×C22⋊C4D411yescharacteristic
108C22×C42—21no—
109C22×C42C2311yescharacteristic
110#481—48no—
111#465—416no—
112#450—48no—
113#448—48no—
114C2×#106C2318yes—
115C2×#103C2318yes—
116C2×#102C2318yes—
117C2×#102C2318yes—
118C2×#102C2318yes—
119C2×C22⋊C8—44no—
120C2×C22⋊C8C2314yes—
121C2×C22⋊C8C2314yes—
122C2×C22⋊C8C2314yes—
123C2×C22⋊C8C2314yes—
124#400—22no—
125#399—22no—
126#398—24no—
127#396—24no—
128#395—22no—
129#393—22no—
130#392—24no—
131#392—24no—
132#392—24no—
133#392—24no—
134#391C2314yes—
135#390—24no—
136#390—24no—
137#389C2314yes—
138#387—24no—
139#386—24no—
140#386—22no—
141#386—22no—
142#385—24no—
143#385—22no—
144#385—22no—
145#385—22no—
146#385—22no—
147#381—22no—
148#380—21no—
149#380—21no—
150#380—24no—
151#380C2312yes—
152#380C2314yes—
153#380C2312yes—
154#379—22no—
155C4×C4⋊C4—22no—
156C4×C4⋊C4C2314yes—
157C2×#95—21no—
158C2×#95—21no—
159#340C2318yes—
160#340C2318yes—
161#336C2318yes—
162C25—22no—
163C25C2×D411yescharacteristic
164C25C2×D411yescharacteristic
165C22×Q8—21no—
166C22×Q8—42no—
167C22×Q8—44no—
168C22×Q8C2412yes—
169C22×D4—28no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4—22no—
173C22×D4—24no—
174C22×D4—24no—
175C22×D4—28no—
176C22×D4—44no—
177C22×D4—44no—
178C22×D4—42no—
179C22×D4—42no—
180C22×D4—48no—
181C22×D4—48no—
182C22×D4—44no—
183C22×D4—44no—
184C22×D4—44no—
185C22×D4—48no—
186C22×D4—48no—
187C22×D4—44no—
188C22×D4—84no—
189C22×D4—84no—
190C22×D4—84no—
191C22×D4SD1618yescomplemented
192C22×D4C2×D412yes—
193C22×D4C2×D412yes—
194C22×D4C2414yes—
195C23×C4—22no—
196C23×C4—24no—
197C23×C4—24no—
198C23×C4—21no—
199C23×C4—21no—
200C23×C4—22no—
201C23×C4—44no—
202C23×C4—44no—
203C23×C4C2×D411yescharacteristic
204C23×C4C2×D411yescharacteristic
205C23×C4C2×D411yescharacteristic
206C23×C4C2×D411yescharacteristic
207C23×C4C2×D411yescharacteristic
208C23×C4C2×D411yescharacteristic
209C23×C4C2×D412yes—
210C23×C4C2411yescharacteristic
211C23×C4C2411yescharacteristic
212C23×C4C2412yes—
213C23×C4C2411yescharacteristic
214C2×SD16—48no—
215C2×SD16—48no—
216C2×SD16—48no—
217C22×C8—44no—
218C22×C8—48no—
219C22×C8C2418yes—
220#127—44no—
221#127—44no—
222#127—48no—
223#127—416no—
224#127—416no—
225C22⋊Q8—28no—
226C22⋊Q8—44no—
227C22⋊Q8—44no—
228C22⋊Q8—44no—
229C22⋊Q8—48no—
230#121—28no—
231#121—216no—
232#121—28no—
233#121—44no—
234#121—44no—
235#121—48no—
236#121—416no—
237#121—48no—
238#121—48no—
239C4×D4—44no—
240C4×D4—44no—
241C4×D4—42no—
242C4×D4—42no—
243C4×D4—42no—
244C4×D4—42no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—28no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—22no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—24no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—44no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4C2×D412yes—
259C2×C4⋊C4C2412yes—
260C2×C4⋊C4C2414yes—
261C2×C4⋊C4C2414yes—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—22no—
266C2×C22⋊C4—28no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—24no—
269C2×C22⋊C4—24no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—42no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—48no—
274C2×C22⋊C4—48no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4C2×D412yes—
278C2×C22⋊C4C2×D412yes—
279C2×C22⋊C4C2×D412yes—
280C2×C22⋊C4C2×D412yes—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—44no—
284C2×C42C2×D412yes—
285C2×C42C2412yes—
286#106—216no—
287#103—216no—
288#102—48no—
289#102—48no—
290#102—48no—
291C22⋊C8—28no—
292C22⋊C8—44no—
293C22⋊C8—44no—
294C22⋊C8—44no—
295#95—24no—
296#95—22no—
297#95—22no—
298#95—24no—
299#95—24no—
300#95—24no—
301#95—24no—
302C24—22no—
303C24—22no—
304C24—22no—
305C24—22no—
306C24—24no—
307C24—42no—
308C24—42no—
309C24—42no—
310C24—42no—
311C24—44no—
312C24—44no—
313C24—44no—
314C24—44no—
315C24—44no—
316C24—44no—
317C24—44no—
318C24C2×SD1614yes—
319C24C2×SD1614yes—
320C24C22×D412yes—
321C24C22×D411yescharacteristic
322C24C22×D411yescharacteristic
323C24C22×D411yescharacteristic
324C2×Q8—28no—
325C2×Q8—24no—
326C2×Q8—42no—
327C2×Q8—82no—
328C2×Q8—88no—
329C2×D4—24no—
330C2×D4—28no—
331C2×D4—28no—
332C2×D4—24no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—416no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—416no—
339C2×D4—48no—
340C2×D4—48no—
341C2×D4—44no—
342C2×D4—44no—
343C2×D4—48no—
344C2×D4—48no—
345C2×D4—48no—
346C2×D4—416no—
347C2×D4—48no—
348C2×D4—48no—
349C2×D4—44no—
350C2×D4—48no—
351C2×D4—816no—
352C2×D4—88no—
353C2×D4—816no—
354C2×D4—88no—
355C2×D4—84no—
356C2×D4—84no—
357C2×D4—832no—
358C2×D4—832no—
359C2×D4—816no—
360C2×D4C2×SD1618yes—
361C2×D4C2×SD1618yes—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—28no—
366C22×C4—22no—
367C22×C4—24no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—28no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—24no—
380C22×C4—24no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—44no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—44no—
390C22×C4—44no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—44no—
394C22×C4—88no—
395C22×C4—88no—
396C22×C4—82no—
397C22×C4—82no—
398C22×C4—82no—
399C22×C4—82no—
400C22×C4C2×SD1614yes—
401C22×C4C22×D412yes—
402C22×C4C22×D412yes—
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×D411yescharacteristic
406C22×C4C22×D412yes—
407C22×C4H(5,𝔽2)14yes—
408C22×C4C2511yescharacteristic
409SD16—816no—
410C2×C8—28no—
411C2×C8—28no—
412C2×C8—28no—
413C2×C8—88no—
414C4⋊C4—28no—
415C4⋊C4—24no—
416C4⋊C4—28no—
417C4⋊C4—24no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—44no—
421C4⋊C4—42no—
422C4⋊C4—44no—
423C4⋊C4—84no—
424C4⋊C4—82no—
425C4⋊C4—84no—
426C4⋊C4H(5,𝔽2)18yes—
427C22⋊C4—42no—
428C22⋊C4—42no—
429C22⋊C4—42no—
430C22⋊C4—42no—
431C22⋊C4—416no—
432C22⋊C4—48no—
433C22⋊C4—48no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—84no—
439C22⋊C4—84no—
440C42—24no—
441C42—22no—
442C42—22no—
443C42—48no—
444C23—22no—
445C23—22no—
446C23—24no—
447C23—22no—
448C23—22no—
449C23—21no—
450C23—21no—
451C23—22no—
452C23—24no—
453C23—22no—
454C23—22no—
455C23—21no—
456C23—21no—
457C23—22no—
458C23—28no—
459C23—28no—
460C23—48no—
461C23—48no—
462C23—44no—
463C23—44no—
464C23—44no—
465C23—44no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—82no—
470C23—82no—
471C23—82no—
472C23—82no—
473C23—84no—
474C23—84no—
475C23—88no—
476C23—88no—
477C23—88no—
478C23—88no—
479C23—88no—
480C23—88no—
481C23—88no—
482C23#53412yes—
483C23C22×SD1614yes—
484C23C22×SD1612yes—
485C23C22×SD1612yes—
486C23C22×SD1612yes—
487C23#57512yes—
488C23C23×D411yescharacteristic
489Q8—48no—
490Q8—84no—
491D4—48no—
492D4—48no—
493D4—48no—
494D4—48no—
495D4—816no—
496D4—88no—
497D4—832no—
498D4—816no—
499D4—816no—
500D4—816no—
501C2×C4—28no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—28no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—22no—
511C2×C4—28no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—22no—
515C2×C4—44no—
516C2×C4—416no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—44no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—82no—
531C2×C4—84no—
532C2×C4—82no—
533C2×C4—82no—
534C2×C4—82no—
535C2×C4—84no—
536C2×C4—84no—
537C2×C4—164no—
538C2×C4#53412yes—
539C2×C4C22×SD1614yes—
540C2×C4C2×H(5,𝔽2)12yes—
541C2×C4C2×H(5,𝔽2)14yes—
542C8—48no—
543C22—22no—
544C22—22no—
545C22—24no—
546C22—24no—
547C22—22no—
548C22—22no—
549C22—24no—
550C22—28no—
551C22—24no—
552C22—24no—
553C22—41no—
554C22—42no—
555C22—41no—
556C22—41no—
557C22—42no—
558C22—41no—
559C22—42no—
560C22—44no—
561C22—42no—
562C22—88no—
563C22—88no—
564C22—84no—
565C22—84no—
566C22—84no—
567C22—84no—
568C22—84no—
569C22—84no—
570C22—84no—
571C22—1616no—
572C22#318714yes—
573C22C2×#53411yescharacteristic
574C22#348412yes—
575C22C23×SD1611yescharacteristic
576C22C2×#57511yescharacteristic
577C22#358512yes—
578C4—24no—
579C4—22no—
580C4—42no—
581C4—41no—
582C4—42no—
583C4—41no—
584C4—48no—
585C4—82no—
586C4—82no—
587C4—84no—
588C4#354314yes—
589C2—24no—
590C2—42no—
591C2—42no—
592C2—44no—
593C2—164no—
594C2#3723211yescharacteristic
595C2C2×#318711yescharacteristic
596C2#6075311yescharacteristic
597C2#6156012yes—
598C2#6157712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2541C2
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
1. #35010C2semidirect
3. #33098C2semidirect
4. #32028C2semidirect
5. #30703C2semidirect
6. #30506C2semidirect
7. #29625C2semidirect
8. #24379C2semidirect
9. #24005C2semidirect
10. #23919C2semidirect
11. #21738C2semidirect
18. C2×#481C22semidirect
19. C2×#465C22semidirect
20. C2×#448C22semidirect
33. #2487C22semidirect
34. #2417C22semidirect
35. #2409C22semidirect
43. #2041C22semidirect
44. #2034C22semidirect
45. #2034C22semidirect
46. #2014C22semidirect
47. #2000C22semidirect
48. #2000C22semidirect
49. #1927C22semidirect
50. #1920C22semidirect
51. #1893C22semidirect
52. #1874C22semidirect
53. #1874C22semidirect
55. #1869C22semidirect
56. #1869C22semidirect
191. C22×D4SD16semidirect

Maximal subgroups

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

subgroupindex
#350102
C2×#25412
#330982
#320282
#307032
#305062
#296252
#243792
#240052
#239192
#217382