Finite Group Databasebeta

← #6633280

Subgroups — #6633280

Order 512.

1702 conjugacy classes of subgroups.

The subgroup lattice has 571 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37421C211yesmaximal, characteristic, complemented
2#35626C211yesmaximal, characteristic, complemented
3#32914C211yesmaximal, characteristic, complemented
4#32911C211yesmaximal, characteristic
5#31649C214yesmaximal, complemented
6#31645C214yesmaximal, complemented
7#30745C214yesmaximal, complemented
8#30737C214yesmaximal, complemented
9#29724C211yesmaximal, characteristic, complemented
10#29722C211yesmaximal, characteristic, complemented
11#29333C212yesmaximal, complemented
12#25724C212yesmaximal, complemented
13#23946C212yesmaximal, complemented
14#23945C212yesmaximal, complemented
15#21913C211yesmaximal, characteristic, complemented
16C2×#542C2212yes—
17C22×#124C2211yescharacteristic
18C2×#518C2211yescharacteristic
19C2×#495C2214yes—
20C2×#461C2212yes—
21C2×#459C2212yes—
22C2×#451C2212yes—
23C2×#448C2212yes—
24C2×#450C2212yes—
25C2×#447C2212yes—
26#2850C2212yes—
27#2712—22no—
28#2702—24no—
29#2693—22no—
30#2621C2211yescharacteristic
31#2619C2211yescharacteristic
32#2568C2212yescomplemented
33#2563C2212yescomplemented
34#2545C2212yescomplemented
35#2544C2212yescomplemented
36#2483—22no—
37#2483—22no—
38#2447—24no—
39#2447C2214yes—
40#2447C2214yes—
41#2310C2211yescharacteristic
42#2310C2211yescharacteristic
43#2309C2211yescharacteristic
44#2296C2211yescharacteristic
45#2296C2211yescharacteristic
46#2101C2214yescomplemented
47#2074C2212yescomplemented
48#2074C2212yescomplemented
49#2056C2214yescomplemented
50#2055C2214yescomplemented
51#2042C2214yescomplemented
52#2024C2214yescomplemented
53#2023C2214yes—
54#2016C2212yescomplemented
55#2015C2214yes—
56#2015C2214yescomplemented
57#2014C2212yescomplemented
58#2012C2214yes—
59#2012C2214yes—
60#1958C2214yes—
61#1934C2214yescomplemented
62#1933C2214yescomplemented
63#1916C2214yescomplemented
64#1916C2214yescomplemented
65#1880C2214yes—
66#1879C2214yes—
67#1879C2214yes—
68#1878C2214yescomplemented
69#1877C2214yes—
70#1877C2214yescomplemented
71#1875C2212yes—
72#1875C2212yes—
73#1873C2212yes—
74#1873C2212yes—
75#1818C2212yes—
76#1804C2212yes—
77#1804C2212yes—
78#1798C2211yescharacteristic
79#1798C2211yescharacteristic
80C4×C22⋊C8C2212yes—
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×#124C2314yes—
97C2×#124C2312yes—
98C2×#124C2312yes—
99C2×#123—22no—
100C2×C22⋊Q8—22no—
101C2×C22⋊Q8C2312yes—
102C2×#121C2312yes—
103#518—44no—
104C2×C4×Q8C2312yes—
105C2×C4×D4—22no—
106C2×C4×D4C2312yes—
107C2×C4×D4C2312yes—
108C2×C4×D4C2312yes—
109C2×#117C2311yescharacteristic
110C2×#117C2311yescharacteristic
111C2×#117C2311yescharacteristic
112C2×#117C2311yescharacteristic
113C22×C22⋊C4—22no—
114C22×C22⋊C4—22no—
115C22×C42C2311yescharacteristic
116#495—48no—
117#461—44no—
118#459—44no—
119#451—44no—
120#450—44no—
121#448—44no—
122#447—44no—
123C2×C4⋊C8C2312yes—
124C2×C4⋊C8C2312yes—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#102C2314yes—
130C2×#102C2314yes—
131C2×#102C2314yes—
132C2×#102C2314yes—
133C2×C22⋊C8C2312yes—
134C2×C22⋊C8C2312yes—
135C2×C4×C8C2314yes—
136#400—24no—
137#398—22no—
138#397—24no—
139#397—24no—
140#397—22no—
141#397—24no—
142#397—24no—
143#396—24no—
144#394—22no—
145#394—24no—
146#393—24no—
147#392—22no—
148#391—22no—
149#391—22no—
150#391—24no—
151#391C2312yes—
152#391C2312yes—
153#391C2314yes—
154#391C2312yes—
155#391C2312yes—
156#390—22no—
157#390C2312yes—
158#390C2312yes—
159#390C2314yes—
160#389D414yes—
161#389C2314yes—
162#388—24no—
163#388—24no—
164#386—24no—
165#386—24no—
166#385—22no—
167#385—24no—
168#385—24no—
169#385D412yes—
170#385D412yes—
171#382C2311yescharacteristic
172#382C2312yes—
173#382C2311yescharacteristic
174#381—21no—
175#381—21no—
176#381—22no—
177#381—24no—
178#381—24no—
179#380—24no—
180#379—21no—
181#379—21no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4C2312yes—
184C4×C4⋊C4C2312yes—
185#340C2314yes—
186#340C2314yes—
187#340C2314yes—
188#340C2314yes—
189#336C2312yes—
190#336C2312yes—
191#336C2314yes—
192#336C2314yes—
193#336C2314yes—
194C25—21no—
195C22×Q8—21no—
196C22×Q8—22no—
197C22×Q8—22no—
198C22×Q8—22no—
199C22×Q8—42no—
200C22×Q8—42no—
201C22×Q8C2412yes—
202C22×Q8C2412yes—
203C22×D4—24no—
204C22×D4—22no—
205C22×D4—21no—
206C22×D4—22no—
207C22×D4—22no—
208C22×D4—22no—
209C22×D4—42no—
210C22×D4C2412yes—
211C22×D4C2412yes—
212C23×C4—21no—
213C23×C4—22no—
214C23×C4—22no—
215C23×C4—22no—
216C23×C4C2×D412yes—
217C23×C4C2411yescharacteristic
218C2×Q16—44no—
219C2×Q16—44no—
220C2×Q16—44no—
221C2×SD16—28no—
222C2×SD16—44no—
223C2×SD16—44no—
224C2×SD16—44no—
225C2×SD16—44no—
226C2×SD16—44no—
227C2×D8—28no—
228C2×D8—44no—
229C2×D8—44no—
230C22×C8C2414yes—
231C22×C8C2414yes—
232C4⋊Q8—42no—
233C4⋊Q8—42no—
234C4⋊Q8—42no—
235#127—42no—
236#126—48no—
237#124—28no—
238#124—44no—
239#124—42no—
240#124—42no—
241#124—44no—
242#124—48no—
243#124—48no—
244#123—48no—
245C22⋊Q8—24no—
246C22⋊Q8—24no—
247C22⋊Q8—48no—
248#121—24no—
249#121—24no—
250C4×Q8—42no—
251C4×Q8—42no—
252C4×D4—24no—
253C4×D4—24no—
254C4×D4—42no—
255C4×D4—42no—
256C4×D4—42no—
257C4×D4—42no—
258C4×D4—48no—
259#117—22no—
260#117—22no—
261#117—42no—
262#117—42no—
263#117—42no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—24no—
269C2×C4⋊C4C2×D412yes—
270C2×C4⋊C4C2×D412yes—
271C2×C4⋊C4C2×D414yes—
272C2×C4⋊C4C2×D414yes—
273C2×C4⋊C4C2412yes—
274C2×C4⋊C4C2412yes—
275C2×C4⋊C4C2412yes—
276C2×C4⋊C4C2412yes—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—22no—
282C2×C22⋊C4—24no—
283C2×C22⋊C4—24no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4—22no—
286C2×C22⋊C4—22no—
287C2×C22⋊C4—22no—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—24no—
290C2×C22⋊C4—24no—
291C2×C22⋊C4—24no—
292C2×C22⋊C4—24no—
293C2×C22⋊C4—48no—
294C2×C22⋊C4—48no—
295C2×C22⋊C4C2×D411yescharacteristic
296C2×C22⋊C4C2×D411yescharacteristic
297C2×C22⋊C4C2×D412yes—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—22no—
301C2×C42—42no—
302C2×C42C2×D412yes—
303C2×C42C2411yescharacteristic
304C2×C42C2411yescharacteristic
305C2×C42C2412yes—
306C2×C42C2412yes—
307C4⋊C8—42no—
308C4⋊C8—42no—
309#103—44no—
310#103—44no—
311#103—44no—
312#103—44no—
313#102—44no—
314#102—44no—
315#102—44no—
316#102—44no—
317C22⋊C8—24no—
318C22⋊C8—24no—
319C4×C8—44no—
320#95—24no—
321#95—22no—
322#95—24no—
323#95—24no—
324#95—24no—
325#95—24no—
326#95—24no—
327#95—22no—
328#95—24no—
329#95—24no—
330#95C2×D414yes—
331#95C2×D414yes—
332C24—24no—
333C24—22no—
334C24—22no—
335C24—42no—
336C24—42no—
337C24—84no—
338C24C22×D411yescharacteristic
339C2×Q8—22no—
340C2×Q8—22no—
341C2×Q8—24no—
342C2×Q8—22no—
343C2×Q8—22no—
344C2×Q8—24no—
345C2×Q8—24no—
346C2×Q8—24no—
347C2×Q8—42no—
348C2×Q8—42no—
349C2×Q8—44no—
350C2×Q8—42no—
351C2×Q8—84no—
352C2×Q8—84no—
353C2×Q8—84no—
354C2×D4—24no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—22no—
358C2×D4—24no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—24no—
364C2×D4—48no—
365C2×D4—48no—
366C2×D4—42no—
367C2×D4—44no—
368C2×D4—42no—
369C2×D4—82no—
370C2×D4—84no—
371C2×D4—84no—
372C2×D4H(5,𝔽2)14yes—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—42no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—44no—
385C22×C4—44no—
386C22×C4—42no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—84no—
390C22×C4—84no—
391C22×C4—84no—
392C22×C4C22×D412yes—
393C22×C4C22×D414yes—
394C22×C4C22×D412yes—
395C22×C4C22×D412yes—
396C22×C4C22×D412yes—
397C22×C4C22×D411yescharacteristic
398C22×C4H(5,𝔽2)12yes—
399C22×C4H(5,𝔽2)12yes—
400C22×C4C2511yescharacteristic
401Q16—88no—
402SD16—88no—
403SD16—88no—
404D8—88no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C4⋊C4—24no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—24no—
419C4⋊C4—44no—
420C4⋊C4—44no—
421C4⋊C4—42no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—84no—
426C4⋊C4—82no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—42no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—48no—
433C22⋊C4—48no—
434C22⋊C4—44no—
435C22⋊C4—44no—
436C22⋊C4—44no—
437C22⋊C4—42no—
438C22⋊C4—42no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—84no—
442C22⋊C4—84no—
443C22⋊C4—84no—
444C22⋊C4—84no—
445C22⋊C4—84no—
446C42—24no—
447C42—22no—
448C42—22no—
449C42—22no—
450C42—22no—
451C42—42no—
452C42—42no—
453C42—42no—
454C42—44no—
455C23—24no—
456C23—21no—
457C23—21no—
458C23—24no—
459C23—48no—
460C23—44no—
461C23—44no—
462C23—42no—
463C23—42no—
464C23—44no—
465C23—44no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—88no—
470C23—164no—
471C23—164no—
472C23#53512yes—
473C23C23×D411yescharacteristic
474Q8—44no—
475Q8—44no—
476Q8—44no—
477Q8—44no—
478Q8—84no—
479Q8—164no—
480D4—24no—
481D4—24no—
482D4—48no—
483D4—42no—
484D4—42no—
485D4—44no—
486D4—44no—
487D4—164no—
488D4—164no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—24no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—24no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—44no—
505C2×C4—44no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—42no—
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—42no—
518C2×C4—84no—
519C2×C4—84no—
520C2×C4—164no—
521C2×C4—164no—
522C2×C4—164no—
523C2×C4#53512yes—
524C2×C4C2×H(5,𝔽2)11yescharacteristic
525C2×C4C2×H(5,𝔽2)11yescharacteristic
526C2×C4C2×H(5,𝔽2)12yes—
527C2×C4#58512yes—
528C8—44no—
529C8—44no—
530C22—22no—
531C22—24no—
532C22—42no—
533C22—42no—
534C22—44no—
535C22—44no—
536C22—44no—
537C22—84no—
538C22—84no—
539C22—82no—
540C22—82no—
541C22—82no—
542C22—84no—
543C22—84no—
544C22—168no—
545C22C2×#53511yescharacteristic
546C22#348411yescharacteristic
547C22#348711yescharacteristic
548C22#358414yes—
549C4—22no—
550C4—42no—
551C4—42no—
552C4—44no—
553C4—42no—
554C4—42no—
555C4—82no—
556C4—82no—
557C4—82no—
558C4—82no—
559C4—82no—
560C4—82no—
561C4#353412yes—
562C4#353412yes—
563C2—42no—
564C2—84no—
565C2—162no—
566C2#3671011yescharacteristic
567C2#6081912yes—
568C2#6163612yes—
569C2#6193012yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#32911C2
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. #37421C2semidirect
2. #35626C2semidirect
3. #32914C2semidirect
5. #31649C2semidirect
6. #31645C2semidirect
7. #30745C2semidirect
8. #30737C2semidirect
9. #29724C2semidirect
10. #29722C2semidirect
11. #29333C2semidirect
12. #25724C2semidirect
13. #23946C2semidirect
14. #23945C2semidirect
15. #21913C2semidirect
32. #2568C22semidirect
33. #2563C22semidirect
34. #2545C22semidirect
35. #2544C22semidirect
46. #2101C22semidirect
47. #2074C22semidirect
48. #2074C22semidirect
49. #2056C22semidirect
50. #2055C22semidirect
51. #2042C22semidirect
52. #2024C22semidirect
54. #2016C22semidirect
56. #2015C22semidirect
57. #2014C22semidirect
61. #1934C22semidirect
62. #1933C22semidirect
63. #1916C22semidirect
64. #1916C22semidirect
68. #1878C22semidirect
70. #1877C22semidirect

Maximal subgroups

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

subgroupindex
#374212
#356262
#329142
#329112
#316492
#316452
#307452
#307372
#297242
#297222
#293332
#257242
#239462
#239452
#219132