Finite Group Databasebeta

← #6633296

Subgroups — #6633296

Order 512.

1716 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37299C211yesmaximal, characteristic, complemented
2#35679C211yesmaximal, characteristic, complemented
3#32918C212yesmaximal, complemented
4#31656C214yesmaximal, complemented
5#31652C214yesmaximal, complemented
6#30758C214yesmaximal, complemented
7#30750C214yesmaximal, complemented
8#29727C212yesmaximal, complemented
9#29338C211yesmaximal, characteristic, complemented
10#29337C211yesmaximal, characteristic, complemented
11#25731C211yesmaximal, characteristic
12#25730C211yesmaximal, characteristic, complemented
13#23946C212yesmaximal, complemented
14#23945C212yesmaximal, complemented
15#21917C211yesmaximal, characteristic, complemented
16C2×#542C2212yes—
17C22×#124C2211yescharacteristic
18C2×#519C2211yescharacteristic
19C2×#497C2212yes—
20C2×#496C2212yes—
21C2×#464C2214yescomplemented
22C2×#451C2212yes—
23C2×#448C2212yes—
24C2×#450C2212yes—
25C2×#447C2212yes—
26#2857C2211yescharacteristic
27#2856C2211yescharacteristic
28#2727—22no—
29#2683—22no—
30#2672—24no—
31#2620C2212yes—
32#2567C2212yescomplemented
33#2545C2212yescomplemented
34#2544C2212yescomplemented
35#2542C2212yescomplemented
36#2482—24no—
37#2448—22no—
38#2448—22no—
39#2448C2214yes—
40#2448C2214yes—
41#2310C2211yescharacteristic
42#2309C2211yescharacteristic
43#2309C2211yescharacteristic
44#2296C2211yescharacteristic
45#2296C2211yescharacteristic
46#2100C2212yescomplemented
47#2099C2212yescomplemented
48#2071C2214yes—
49#2055C2214yes—
50#2054C2214yes—
51#2043C2212yescomplemented
52#2041C2212yescomplemented
53#2024C2214yescomplemented
54#2023C2214yes—
55#2015C2214yes—
56#2015C2214yescomplemented
57#2012C2214yes—
58#2012C2214yes—
59#2012C2214yes—
60#1959C2212yes—
61#1959C2212yes—
62#1936C2214yescomplemented
63#1935C2214yescomplemented
64#1917C2214yescomplemented
65#1917C2214yescomplemented
66#1880C2214yes—
67#1880C2214yescomplemented
68#1879C2214yes—
69#1878C2214yescomplemented
70#1878C2214yescomplemented
71#1877C2214yescomplemented
72#1875C2212yes—
73#1875C2212yes—
74#1873C2212yes—
75#1873C2212yes—
76#1818C2212yes—
77#1803C2214yescomplemented
78#1798C2212yes—
79#1749C2211yescharacteristic
80#1749C2211yescharacteristic
81C23×Q8C2311yescharacteristic
82C23×D4C2311yescharacteristic
83C22×Q16C2314yes—
84C22×SD16C2314yes—
85C22×SD16C2314yes—
86C22×D8C2314yes—
87#542—44no—
88#542—44no—
89C2×C4⋊Q8C2311yescharacteristic
90C2×C4⋊Q8C2312yes—
91C2×#127C2311yescharacteristic
92C2×#126—22no—
93C2×#124—22no—
94C2×#124—22no—
95C2×#124—44no—
96C2×#124C2312yes—
97C2×#124C2312yes—
98C2×#124C2312yes—
99C2×#124C2312yes—
100C2×#123—22no—
101C2×C22⋊Q8—22no—
102C2×C22⋊Q8—24no—
103C2×C22⋊Q8C2312yes—
104C2×#121—22no—
105C2×#121C2312yes—
106#519—44no—
107C2×C4×Q8—22no—
108C2×C4×Q8C2314yes—
109C2×C4×Q8C2312yes—
110C2×C4×D4C2312yes—
111C2×#117—22no—
112C2×#117C2311yescharacteristic
113C2×#117C2311yescharacteristic
114C2×#117C2311yescharacteristic
115C2×#117C2311yescharacteristic
116C22×C22⋊C4—22no—
117C22×C42C2311yescharacteristic
118#497—44no—
119#496—44no—
120#464—48no—
121#451—44no—
122#450—44no—
123#448—44no—
124#447—44no—
125C2×C4⋊C8C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2312yes—
136C2×#97C2312yes—
137C2×#97C2312yes—
138#399—24no—
139#399—24no—
140#399—24no—
141#397—24no—
142#397—24no—
143#396—24no—
144#396—24no—
145#395—24no—
146#394—24no—
147#394—24no—
148#391—21no—
149#391—21no—
150#391C2314yes—
151#391C2312yes—
152#391C2312yes—
153#390—21no—
154#390—21no—
155#390—22no—
156#390—22no—
157#390C2314yes—
158#390C2314yes—
159#390C2312yes—
160#390C2312yes—
161#389—24no—
162#389—22no—
163#389D412yes—
164#389D412yes—
165#389C2312yes—
166#389C2312yes—
167#388—24no—
168#388—24no—
169#388—24no—
170#388—24no—
171#388—24no—
172#386—24no—
173#385—22no—
174#385D414yes—
175#384C2311yescharacteristic
176#384C2312yes—
177#384C2311yescharacteristic
178#381—22no—
179#380—22no—
180#379—22no—
181#379—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4C2312yes—
184C4×C4⋊C4C2312yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#336C2314yes—
190#336C2314yes—
191#336C2314yes—
192#336C2312yes—
193#336C2312yes—
194C25—21no—
195C22×Q8—24no—
196C22×Q8—21no—
197C22×Q8—22no—
198C22×Q8—24no—
199C22×Q8—42no—
200C22×Q8C2412yes—
201C22×Q8C2412yes—
202C22×D4—22no—
203C22×D4—21no—
204C22×D4—22no—
205C22×D4—24no—
206C22×D4—42no—
207C22×D4—42no—
208C22×D4C2412yes—
209C22×D4C2412yes—
210C23×C4—21no—
211C23×C4—22no—
212C23×C4C2×D412yes—
213C23×C4C2411yescharacteristic
214C2×Q16—28no—
215C2×Q16—44no—
216C2×Q16—44no—
217C2×SD16—28no—
218C2×SD16—44no—
219C2×SD16—44no—
220C2×SD16—44no—
221C2×SD16—44no—
222C2×SD16—44no—
223C2×D8—44no—
224C2×D8—44no—
225C2×D8—44no—
226C22×C8C2414yes—
227C22×C8C2414yes—
228C4⋊Q8—42no—
229C4⋊Q8—42no—
230C4⋊Q8—42no—
231#127—42no—
232#126—48no—
233#124—24no—
234#124—24no—
235#124—42no—
236#124—42no—
237#124—42no—
238#124—42no—
239#124—44no—
240#124—48no—
241#124—48no—
242#123—48no—
243C22⋊Q8—24no—
244C22⋊Q8—24no—
245C22⋊Q8—48no—
246C22⋊Q8—88no—
247#121—24no—
248#121—24no—
249#121—84no—
250C4×Q8—28no—
251C4×Q8—44no—
252C4×Q8—42no—
253C4×Q8—42no—
254C4×Q8—48no—
255C4×D4—42no—
256C4×D4—42no—
257#117—24no—
258#117—42no—
259#117—42no—
260#117—42no—
261#117—48no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4C2×D411yescharacteristic
273C2×C4⋊C4C2×D411yescharacteristic
274C2×C4⋊C4C2×D412yes—
275C2×C4⋊C4C2×D414yes—
276C2×C4⋊C4C2×D414yes—
277C2×C4⋊C4C2412yes—
278C2×C4⋊C4C2412yes—
279C2×C4⋊C4C2412yes—
280C2×C4⋊C4C2412yes—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—24no—
285C2×C22⋊C4—24no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—24no—
288C2×C22⋊C4—22no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—24no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—48no—
298C2×C22⋊C4C2×D412yes—
299C2×C22⋊C4C2×D412yes—
300C2×C42—24no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—22no—
304C2×C42—42no—
305C2×C42C2×D412yes—
306C2×C42C2412yes—
307C2×C42C2412yes—
308C2×C42C2411yescharacteristic
309C2×C42C2411yescharacteristic
310C4⋊C8—44no—
311#103—44no—
312#103—44no—
313#103—44no—
314#103—44no—
315#102—44no—
316#102—44no—
317#102—44no—
318#102—44no—
319C22⋊C8—24no—
320C22⋊C8—24no—
321#97—42no—
322#97—42no—
323#95—22no—
324#95—22no—
325#95—24no—
326#95—24no—
327#95—24no—
328#95—24no—
329#95C2×D414yes—
330#95C2×D414yes—
331C24—22no—
332C24—22no—
333C24—42no—
334C24—42no—
335C24—42no—
336C24—82no—
337C24—82no—
338C24C22×D411yescharacteristic
339C2×Q8—24no—
340C2×Q8—22no—
341C2×Q8—24no—
342C2×Q8—24no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—24no—
346C2×Q8—24no—
347C2×Q8—416no—
348C2×Q8—42no—
349C2×Q8—44no—
350C2×Q8—42no—
351C2×Q8—84no—
352C2×Q8H(5,𝔽2)14yes—
353C2×D4—24no—
354C2×D4—24no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—22no—
359C2×D4—24no—
360C2×D4—24no—
361C2×D4—42no—
362C2×D4—42no—
363C2×D4—44no—
364C2×D4—44no—
365C2×D4—42no—
366C2×D4—88no—
367C2×D4—82no—
368C2×D4—82no—
369C2×D4—84no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—44no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—84no—
386C22×C4C22×D412yes—
387C22×C4C22×D412yes—
388C22×C4C22×D412yes—
389C22×C4C22×D414yes—
390C22×C4C22×D412yes—
391C22×C4C22×D411yescharacteristic
392C22×C4H(5,𝔽2)14yes—
393C22×C4C2511yescharacteristic
394Q16—88no—
395SD16—88no—
396SD16—88no—
397D8—88no—
398C2×C8—24no—
399C2×C8—24no—
400C2×C8—24no—
401C2×C8—24no—
402C2×C8—24no—
403C2×C8—24no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—44no—
414C4⋊C4—42no—
415C4⋊C4—44no—
416C4⋊C4—48no—
417C4⋊C4—44no—
418C4⋊C4—44no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—82no—
422C4⋊C4—82no—
423C4⋊C4—82no—
424C4⋊C4—84no—
425C4⋊C4—82no—
426C22⋊C4—24no—
427C22⋊C4—42no—
428C22⋊C4—48no—
429C22⋊C4—48no—
430C22⋊C4—44no—
431C22⋊C4—48no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—42no—
435C22⋊C4—42no—
436C22⋊C4—44no—
437C22⋊C4—44no—
438C22⋊C4—84no—
439C22⋊C4—84no—
440C22⋊C4—84no—
441C22⋊C4—84no—
442C22⋊C4—84no—
443C22⋊C4—84no—
444C42—22no—
445C42—22no—
446C42—24no—
447C42—22no—
448C42—22no—
449C42—42no—
450C42—48no—
451C42—42no—
452C42—42no—
453C42—44no—
454C23—22no—
455C23—22no—
456C23—21no—
457C23—21no—
458C23—44no—
459C23—44no—
460C23—44no—
461C23—44no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23—84no—
469C23—88no—
470C23—162no—
471C23—162no—
472C23—164no—
473C23#53512yes—
474C23C23×D411yescharacteristic
475Q8—28no—
476Q8—48no—
477Q8—44no—
478Q8—44no—
479Q8—44no—
480Q8—164no—
481D4—44no—
482D4—44no—
483D4—44no—
484D4—44no—
485D4—84no—
486D4—164no—
487D4—164no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—28no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—24no—
505C2×C4—48no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—44no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—84no—
522C2×C4—164no—
523C2×C4#53512yes—
524C2×C4C2×H(5,𝔽2)12yes—
525C2×C4C2×H(5,𝔽2)12yes—
526C2×C4#58512yes—
527C8—44no—
528C8—44no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—42no—
533C22—42no—
534C22—82no—
535C22—82no—
536C22—82no—
537C22—82no—
538C22—82no—
539C22—82no—
540C22—84no—
541C22—84no—
542C22—84no—
543C22—84no—
544C22—168no—
545C22C2×#53511yescharacteristic
546C22#348411yescharacteristic
547C22#348711yescharacteristic
548C22#358512yes—
549C22#358612yes—
550C4—22no—
551C4—44no—
552C4—42no—
553C4—42no—
554C4—44no—
555C4—42no—
556C4—42no—
557C4—82no—
558C4—82no—
559C4—82no—
560C4—82no—
561C4—82no—
562C4#353414yes—
563C2—42no—
564C2—162no—
565C2—162no—
566C2#3671011yescharacteristic
567C2#6082711yescharacteristic
568C2#6082811yescharacteristic
569C2#6165012yes—
570C2#6194412yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#25731C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #2296 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #37299C2semidirect
2. #35679C2semidirect
3. #32918C2semidirect
4. #31656C2semidirect
5. #31652C2semidirect
6. #30758C2semidirect
7. #30750C2semidirect
8. #29727C2semidirect
9. #29338C2semidirect
10. #29337C2semidirect
12. #25730C2semidirect
13. #23946C2semidirect
14. #23945C2semidirect
15. #21917C2semidirect
21. C2×#464C22semidirect
32. #2567C22semidirect
33. #2545C22semidirect
34. #2544C22semidirect
35. #2542C22semidirect
46. #2100C22semidirect
47. #2099C22semidirect
51. #2043C22semidirect
52. #2041C22semidirect
53. #2024C22semidirect
56. #2015C22semidirect
62. #1936C22semidirect
63. #1935C22semidirect
64. #1917C22semidirect
65. #1917C22semidirect
67. #1880C22semidirect
69. #1878C22semidirect
70. #1878C22semidirect
71. #1877C22semidirect
77. #1803C22semidirect

Maximal subgroups

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

subgroupindex
#372992
#356792
#329182
#316562
#316522
#307582
#307502
#297272
#293382
#293372
#257312
#257302
#239462
#239452
#219172