Finite Group Databasebeta

← #6485310

Subgroups — #6485310

Order 512.

1605 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35277C211yesmaximal, characteristic
2#34281C211yesmaximal, characteristic, complemented
3#33035C214yesmaximal, complemented
4#30115C212yesmaximal
5#30109C212yesmaximal, complemented
6#29370C212yesmaximal, complemented
7#29367C212yesmaximal, complemented
8#29225C211yesmaximal, characteristic, complemented
9#28751C211yesmaximal, characteristic, complemented
10#28511C211yesmaximal, characteristic, complemented
11#28447C211yesmaximal, characteristic, complemented
12#27899C212yesmaximal, complemented
13#25612C212yesmaximal, complemented
14#24168C214yesmaximal, complemented
15#22278C212yesmaximal, complemented
16#22228C212yesmaximal, complemented
17#20781C211yesmaximal, characteristic, complemented
18C2×#542C2212yes—
19C2×#541C2211yescharacteristic
20C2×#497C2212yes—
21C2×#488C2212yes—
22C2×#477C2214yes—
23C2×#464C2214yes—
24#2868C2211yescharacteristic
25#2858C2211yescharacteristic
26#2857C2211yescharacteristic
27#2856C2211yescharacteristic
28#2640C2212yes—
29#2620C2212yes—
30#2512—22no—
31#2491—22no—
32#2452—22no—
33#2422—22no—
34#2373—22no—
35#2373—22no—
36#2371—22no—
37#2370—22no—
38#2330C2211yescharacteristic
39#2327C2211yescharacteristic
40#2326C2212yes—
41#2323C2212yes—
42#2314C2211yescharacteristic
43C4×C4⋊Q8C2211yescharacteristic
44C4×C4⋊Q8C2211yescharacteristic
45C4×#127C2211yescharacteristic
46C4×#125C2211yescharacteristic
47C4×C22⋊Q8C2212yes—
48#2295C2212yes—
49#2276C2212yes—
50C4×#117C2211yescharacteristic
51#2100C2212yes—
52#2100C2212yes—
53#2095C2212yes—
54#2095C2212yes—
55#2043C2212yes—
56#2041C2212yes—
57#2036C2214yes—
58#2033C2214yes—
59#2016C2212yes—
60#2014C2212yes—
61#2001C2214yes—
62#2000C2214yes—
63#1992C2212yes—
64#1988C2214yes—
65#1983C2214yes—
66#1959C2212yes—
67#1958C2212yes—
68#1956C2212yes—
69#1928C2214yes—
70#1928C2214yes—
71#1925C2214yes—
72#1925C2214yes—
73#1880C2214yes—
74#1880C2214yes—
75#1878C2214yes—
76#1878C2214yes—
77#1878C2214yes—
78#1877C2214yes—
79#1863C2214yes—
80#1862C2214yes—
81#1845C2212yes—
82#1839C2212yes—
83#1836C2212yes—
84#1832C2212yes—
85#1817C2212yes—
86#1798C2212yes—
87#1767C2211yescharacteristic
88C4×C4⋊C8C2211yescharacteristic
89#1750C2212yes—
90#1746C2212yes—
91C22×Q16C2314yes—
92C22×SD16C2314yes—
93#542—48no—
94#541—24no—
95#541—24no—
96C2×C4⋊Q8—22no—
97C2×C4⋊Q8C2311yescharacteristic
98C2×C4⋊Q8C2311yescharacteristic
99C2×C4⋊Q8C2312yes—
100C2×#127C2311yescharacteristic
101C2×#126—22no—
102C2×#126—22no—
103C2×#125—22no—
104C2×#125C2311yescharacteristic
105C2×#124—22no—
106C2×#124C2312yes—
107C2×#124C2312yes—
108C2×#123—22no—
109C2×C22⋊Q8—22no—
110C2×C22⋊Q8C2312yes—
111C2×C22⋊Q8C2312yes—
112C2×#121—22no—
113C2×#121C2312yes—
114C2×C4×Q8C2312yes—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2312yes—
117C2×C4×Q8C2312yes—
118C2×C4×D4—22no—
119C2×C4×D4C2312yes—
120C2×#117C2311yescharacteristic
121C2×#117C2311yescharacteristic
122C22×C42C2311yescharacteristic
123#497—28no—
124#488—28no—
125#477—48no—
126#464—48no—
127C2×#107C2314yes—
128C2×#107C2314yes—
129C2×C4⋊C8C2312yes—
130C2×C4⋊C8C2312yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×#103C2314yes—
134C2×#103C2314yes—
135C2×#103C2314yes—
136C2×#102C2314yes—
137C2×#102C2314yes—
138C2×#102C2314yes—
139C2×C22⋊C8C2314yes—
140C2×#97C2312yes—
141C2×C4×C8C2312yes—
142#397—24no—
143#396—24no—
144#394—24no—
145#393—24no—
146#391C2312yes—
147#391C2312yes—
148#391C2312yes—
149#391C2312yes—
150#390—22no—
151#390—22no—
152#390—22no—
153#390—22no—
154#390C2312yes—
155#390C2312yes—
156#389—22no—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389C2312yes—
162#389C2312yes—
163#388—22no—
164#388—22no—
165#388—22no—
166#388—24no—
167#387—22no—
168#387—22no—
169#387—22no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#385—22no—
177#385—24no—
178#384—22no—
179#384C2311yescharacteristic
180#384C2311yescharacteristic
181#384C2311yescharacteristic
182#384C2312yes—
183#382C2311yescharacteristic
184#380—22no—
185#380C2312yes—
186C4×C4⋊C4—22no—
187C4×C4⋊C4—22no—
188C4×C4⋊C4—22no—
189C4×C4⋊C4—22no—
190C4×C4⋊C4—22no—
191C4×C4⋊C4—22no—
192C4×C4⋊C4—22no—
193C4×C4⋊C4—22no—
194C4×C4⋊C4D412yes—
195C4×C4⋊C4D412yes—
196C4×C4⋊C4C2312yes—
197C4×C4⋊C4C2312yes—
198C4×C4⋊C4C2312yes—
199C4×C22⋊C4—22no—
200C4×C22⋊C4D412yes—
201C4×C22⋊C4D412yes—
202#376—22no—
203#376—22no—
204#376C2311yescharacteristic
205#376C2311yescharacteristic
206C43C2311yescharacteristic
207C43C2311yescharacteristic
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#340C2314yes—
212#336C2314yes—
213#336C2314yes—
214#336C2312yes—
215#336C2312yes—
216#336C2314yes—
217C22×Q8—22no—
218C22×Q8C2412yes—
219C22×Q8C2412yes—
220C22×D4—22no—
221C22×D4—22no—
222C22×D4C2412yes—
223C23×C4—22no—
224C23×C4C2×D412yes—
225C23×C4C2411yescharacteristic
226C2×Q16—28no—
227C2×Q16—44no—
228C2×Q16—44no—
229C2×SD16—28no—
230C2×SD16—44no—
231C2×SD16—44no—
232C22×C8C2414yes—
233C22×C8C2414yes—
234C4⋊Q8—22no—
235C4⋊Q8—28no—
236C4⋊Q8—42no—
237C4⋊Q8—84no—
238C4⋊Q8#4114yes—
239#127—42no—
240#126—216no—
241#126—48no—
242#125—216no—
243#125—22no—
244#125#4114yes—
245#124—24no—
246#124—24no—
247#124—28no—
248#124—48no—
249#123—48no—
250C22⋊Q8—24no—
251C22⋊Q8—24no—
252C22⋊Q8—44no—
253C22⋊Q8—48no—
254#121—44no—
255#121—84no—
256C4×Q8—24no—
257C4×Q8—24no—
258C4×Q8—28no—
259C4×Q8—44no—
260C4×Q8—44no—
261C4×D4—44no—
262C4×D4—84no—
263#117—22no—
264#117—22no—
265#117—24no—
266C2×C4⋊C4—22no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—22no—
279C2×C4⋊C4—24no—
280C2×C4⋊C4—22no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4C2×D411yescharacteristic
283C2×C4⋊C4C2×D412yes—
284C2×C4⋊C4C2×D411yescharacteristic
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C4⋊C4C2412yes—
289C2×C4⋊C4C2412yes—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—22no—
294C2×C22⋊C4—22no—
295C2×C22⋊C4—22no—
296C2×C22⋊C4—24no—
297C2×C22⋊C4—22no—
298C2×C22⋊C4—24no—
299C2×C22⋊C4—24no—
300C2×C22⋊C4C2×D411yescharacteristic
301C2×C22⋊C4C2×D411yescharacteristic
302C2×C22⋊C4C2×D412yes—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42—22no—
306C2×C42—22no—
307C2×C42—22no—
308C2×C42—22no—
309C2×C42—22no—
310C2×C42—42no—
311C2×C42C2×D412yes—
312C2×C42C2×D412yes—
313C2×C42C2×D412yes—
314C2×C42C2×D412yes—
315C2×C42C2×D412yes—
316C2×C42C2411yescharacteristic
317C2×C42C2411yescharacteristic
318C2×C42C2411yescharacteristic
319C2×C42C2411yescharacteristic
320C2×C42C2411yescharacteristic
321C2×C42C2411yescharacteristic
322#107—44no—
323#107—44no—
324C4⋊C8—24no—
325C4⋊C8—24no—
326#103—28no—
327#103—44no—
328#103—44no—
329#103—44no—
330#103—44no—
331#102—28no—
332#102—44no—
333#102—44no—
334C22⋊C8—44no—
335#97—24no—
336C4×C8—24no—
337#95—22no—
338#95—22no—
339#95—22no—
340#95—22no—
341#95—22no—
342#95—22no—
343#95—22no—
344#95—22no—
345#95—22no—
346#95—22no—
347#95—22no—
348#95—24no—
349#95—24no—
350#95C2×D412yes—
351#95C2×D412yes—
352#95C2×D412yes—
353#95C2×D412yes—
354C24—22no—
355C24C22×D411yescharacteristic
356C2×Q8—22no—
357C2×Q8—22no—
358C2×Q8—22no—
359C2×Q8—24no—
360C2×Q8—24no—
361C2×Q8—24no—
362C2×Q8—24no—
363C2×Q8—44no—
364C2×Q8—84no—
365C2×Q8C2×#4114yes—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—24no—
369C2×D4—24no—
370C2×D4—44no—
371C2×D4—44no—
372C2×D4—44no—
373C2×D4—44no—
374C2×D4—84no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—21no—
378C22×C4—21no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—24no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—42no—
388C22×C4—42no—
389C22×C4—44no—
390C22×C4—84no—
391C22×C4C22×D411yescharacteristic
392C22×C4C22×D412yes—
393C22×C4C22×D412yes—
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D412yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D412yes—
400C22×C4C2511yescharacteristic
401Q16—88no—
402SD16—88no—
403C2×C8—24no—
404C2×C8—24no—
405C2×C8—24no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C4⋊C4—28no—
410C4⋊C4—28no—
411C4⋊C4—28no—
412C4⋊C4—28no—
413C4⋊C4—24no—
414C4⋊C4—22no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—24no—
420C4⋊C4—24no—
421C4⋊C4—44no—
422C4⋊C4—44no—
423C4⋊C4—44no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—44no—
427C4⋊C4—44no—
428C4⋊C4—44no—
429C4⋊C4—44no—
430C4⋊C4—82no—
431C4⋊C4—82no—
432C4⋊C4—84no—
433C4⋊C4C2×#4114yes—
434C22⋊C4—28no—
435C22⋊C4—24no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—44no—
439C22⋊C4—44no—
440C22⋊C4—44no—
441C22⋊C4—48no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—84no—
445C22⋊C4—84no—
446C22⋊C4—82no—
447C42—28no—
448C42—24no—
449C42—24no—
450C42—21no—
451C42—21no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—42no—
456C42—42no—
457C42—42no—
458C42—44no—
459C42—44no—
460C42—44no—
461C42—82no—
462C42—82no—
463C42—82no—
464C42C2×#4112yes—
465C42C2×#4112yes—
466C42H(5,𝔽2)14yes—
467C23—22no—
468C23—21no—
469C23—21no—
470C23—21no—
471C23—22no—
472C23—44no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—44no—
477C23C23×D411yescharacteristic
478Q8—28no—
479Q8—44no—
480Q8—44no—
481Q8—44no—
482Q8—164no—
483D4—44no—
484D4—44no—
485D4—88no—
486D4—164no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
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—44no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—84no—
528C2×C4—164no—
529C2×C4#52514yes—
530C2×C4#53512yes—
531C2×C4#53512yes—
532C2×C4#57814yes—
533C2×C4C22×#4111yescharacteristic
534C2×C4C22×#4111yescharacteristic
535C2×C4C2×H(5,𝔽2)12yes—
536C2×C4#58512yes—
537C8—44no—
538C8—44no—
539C22—42no—
540C22—41no—
541C22—41no—
542C22—41no—
543C22—42no—
544C22—84no—
545C22—82no—
546C22—82no—
547C22—82no—
548C22—84no—
549C22C2×#52511yescharacteristic
550C22C2×#53511yescharacteristic
551C22#344911yescharacteristic
552C22C2×#57812yes—
553C22#358512yes—
554C4—24no—
555C4—22no—
556C4—22no—
557C4—24no—
558C4—42no—
559C4—41no—
560C4—44no—
561C4—41no—
562C4—41no—
563C4—42no—
564C4—44no—
565C4—82no—
566C4—82no—
567C4—82no—
568C4—82no—
569C4—82no—
570C4#351412yes—
571C4#351712yes—
572C2—81no—
573C2—162no—
574C2#3519011yescharacteristic
575C2#6060012yes—
576C2#6082111yescharacteristic
577C2#6083011yescharacteristic
578C2#6098112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#2856C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34281C2semidirect
3. #33035C2semidirect
5. #30109C2semidirect
6. #29370C2semidirect
7. #29367C2semidirect
8. #29225C2semidirect
9. #28751C2semidirect
10. #28511C2semidirect
11. #28447C2semidirect
12. #27899C2semidirect
13. #25612C2semidirect
14. #24168C2semidirect
15. #22278C2semidirect
16. #22228C2semidirect
17. #20781C2semidirect

Maximal subgroups

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

subgroupindex
#352772
#342812
#330352
#301152
#301092
#293702
#293672
#292252
#287512
#285112
#284472
#278992
#256122
#241682
#222782
#222282
#207812