Finite Group Databasebeta

← #6488767

Subgroups — #6488767

Order 512.

1698 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35323C211yesmaximal, characteristic, complemented
2#34207C211yesmaximal, characteristic
3#33017C214yesmaximal, complemented
4#30131C212yesmaximal, complemented
5#30128C212yesmaximal, complemented
6#29370C212yesmaximal, complemented
7#29367C212yesmaximal, complemented
8#29200C212yesmaximal, complemented
9#28629C212yesmaximal, complemented
10#28131C214yesmaximal, complemented
11#27966C211yesmaximal, characteristic, complemented
12#27910C211yesmaximal, characteristic, complemented
13#25858C211yesmaximal, characteristic
14#25728C211yesmaximal, characteristic
15#22278C212yesmaximal, complemented
16#22228C212yesmaximal, complemented
17#20784C211yesmaximal, characteristic, complemented
18C2×#532C2211yescharacteristic
19C2×#520C2212yes—
20C2×#497C2212yes—
21C2×#488C2212yes—
22C2×#484C2214yes—
23C2×#452C2214yes—
24#2863C2212yes—
25#2857C2211yescharacteristic
26#2851C2211yescharacteristic
27#2647C2211yescharacteristic
28#2642C2211yescharacteristic
29#2625C2212yes—
30#2590—22no—
31#2577—22no—
32#2533—22no—
33#2530—22no—
34#2361—22no—
35#2361—22no—
36#2354—22no—
37D4×C4⋊C4—22no—
38#2330C2211yescharacteristic
39#2327C2211yescharacteristic
40#2326C2212yes—
41#2326C2212yes—
42#2323C2212yes—
43#2311C2212yes—
44C4×C4⋊Q8C2211yescharacteristic
45C4×C4⋊Q8C2211yescharacteristic
46C4×#127C2211yescharacteristic
47C4×#127C2211yescharacteristic
48#2295C2212yes—
49C2×#384C2211yescharacteristic
50C4×#117C2211yescharacteristic
51#2100C2212yes—
52#2100C2212yes—
53#2095C2212yes—
54#2095C2212yes—
55#2068C2214yes—
56#2067C2214yes—
57#2056C2214yes—
58#2055C2214yes—
59#2043C2212yes—
60#2041C2212yes—
61#2016C2212yes—
62#2014C2212yes—
63#1992C2212yes—
64#1959C2212yes—
65#1958C2212yes—
66#1956C2212yes—
67#1928C2214yes—
68#1925C2214yes—
69#1880C2214yes—
70#1878C2214yes—
71#1878C2214yes—
72#1878C2214yes—
73#1877C2214yes—
74#1877C2214yes—
75#1863C2214yes—
76#1863C2214yes—
77#1862C2214yes—
78#1862C2214yes—
79#1845C2212yes—
80#1839C2212yes—
81#1836C2212yes—
82#1831C2212yes—
83#1823C2214yes—
84#1818C2212yes—
85#1798C2212yes—
86#1788C2214yes—
87#1768C2212yes—
88#1749C2211yescharacteristic
89C4×C22⋊C8C2211yescharacteristic
90#1746C2212yes—
91C22×#41C2312yes—
92C22×Q16C2314yes—
93C22×SD16C2314yes—
94#532—28no—
95C2×C4⋊Q8C2311yescharacteristic
96C2×C4⋊Q8C2311yescharacteristic
97C2×#127C2311yescharacteristic
98C2×#125C2311yescharacteristic
99C2×#124C2312yes—
100C2×C22⋊Q8C2312yes—
101C2×#121—22no—
102#520—48no—
103C2×C4×Q8C2312yes—
104C2×C4×Q8C2312yes—
105C2×C4×Q8C2312yes—
106C2×C4×D4—22no—
107C2×C4×D4—22no—
108C2×C4×D4—22no—
109C2×C4×D4—22no—
110C2×C4×D4—22no—
111C2×C4×D4C2312yes—
112C2×C4×D4C2312yes—
113C2×#117—22no—
114C2×#117C2311yescharacteristic
115C2×#117C2311yescharacteristic
116C22×C4⋊C4—22no—
117C22×C4⋊C4—22no—
118C22×C4⋊C4C2312yes—
119C22×C42C2311yescharacteristic
120#497—28no—
121#488—28no—
122#484—48no—
123#452—48no—
124C2×#107C2314yes—
125C2×#107C2314yes—
126C2×C4⋊C8C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#102C2314yes—
133C2×#102C2314yes—
134C2×#102C2314yes—
135C2×C22⋊C8C2312yes—
136C2×C22⋊C8C2312yes—
137C2×#97C2312yes—
138C2×C4×C8C2312yes—
139#400—22no—
140#400—22no—
141#400—22no—
142#400—22no—
143#398—22no—
144#398—22no—
145#397—22no—
146#397—22no—
147#391C2312yes—
148#391C2312yes—
149#391C2312yes—
150#391C2312yes—
151#391C2312yes—
152#391C2312yes—
153#390—22no—
154#390—22no—
155#390C2312yes—
156#390C2312yes—
157#389—22no—
158#389—22no—
159#389—22no—
160#389—22no—
161#389—22no—
162#389C2312yes—
163#389C2312yes—
164#389C2312yes—
165#389C2312yes—
166#388—22no—
167#388—22no—
168#388—24no—
169#388—24no—
170#387—22no—
171#387—22no—
172#387—22no—
173#387—22no—
174#387—22no—
175#387—22no—
176#387—22no—
177#387—22no—
178#387—22no—
179#387—22no—
180#386—24no—
181#385—22no—
182#385—24no—
183#385—24no—
184#384—22no—
185#384C2311yescharacteristic
186#384C2311yescharacteristic
187#384C2312yes—
188#384C2311yescharacteristic
189#383—22no—
190#382C2311yescharacteristic
191#380—22no—
192#380—24no—
193#380C2312yes—
194C4×C4⋊C4—22no—
195C4×C4⋊C4—22no—
196C4×C4⋊C4—22no—
197C4×C4⋊C4D412yes—
198C4×C4⋊C4D412yes—
199C4×C4⋊C4C2312yes—
200C4×C22⋊C4D412yes—
201C4×C22⋊C4D412yes—
202#376—22no—
203#376C2311yescharacteristic
204#376C2311yescharacteristic
205C43C2311yescharacteristic
206C43C2311yescharacteristic
207#340C2314yes—
208#340C2314yes—
209#340C2314yes—
210#340C2314yes—
211#336C2314yes—
212#336C2314yes—
213#336C2314yes—
214#336C2312yes—
215#336C2312yes—
216C2×#41—24no—
217C2×#41—24no—
218C2×#41—28no—
219C2×#41—44no—
220C22×Q8C2412yes—
221C22×Q8C2412yes—
222C22×D4—22no—
223C22×D4—22no—
224C22×D4C2412yes—
225C23×C4—22no—
226C23×C4—22no—
227C23×C4—22no—
228C23×C4—22no—
229C23×C4C2×D412yes—
230C23×C4C2411yescharacteristic
231C2×Q16—28no—
232C2×Q16—44no—
233C2×Q16—44no—
234C2×SD16—28no—
235C2×SD16—44no—
236C2×SD16—44no—
237C22×C8C2414yes—
238C22×C8C2414yes—
239C4⋊Q8—24no—
240C4⋊Q8—24no—
241#127—24no—
242#125—24no—
243#124—24no—
244#124#4118yes—
245C22⋊Q8—28no—
246#121—216no—
247C4×Q8—42no—
248C4×Q8—42no—
249C4×Q8—44no—
250C4×Q8—44no—
251C4×D4—216no—
252C4×D4—44no—
253C4×D4—44no—
254C4×D4—48no—
255C4×D4—48no—
256C4×D4—84no—
257C4×D4—84no—
258#117—24no—
259#117—42no—
260#117—48no—
261C2×C4⋊C4—22no—
262C2×C4⋊C4—28no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4—22no—
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—24no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—22no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—48no—
279C2×C4⋊C4—84no—
280C2×C4⋊C4C2×D411yescharacteristic
281C2×C4⋊C4C2×D411yescharacteristic
282C2×C4⋊C4C2×D412yes—
283C2×C4⋊C4C2412yes—
284C2×C4⋊C4C2412yes—
285C2×C4⋊C4C2412yes—
286C2×C4⋊C4C2412yes—
287C2×C4⋊C4C2412yes—
288C2×C22⋊C4—24no—
289C2×C22⋊C4—22no—
290C2×C22⋊C4—22no—
291C2×C22⋊C4—22no—
292C2×C22⋊C4—22no—
293C2×C22⋊C4—24no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4C2×D411yescharacteristic
296C2×C22⋊C4C2×D411yescharacteristic
297C2×C22⋊C4C2×D412yes—
298C2×C42—24no—
299C2×C42—22no—
300C2×C42—22no—
301C2×C42—22no—
302C2×C42—22no—
303C2×C42C2×D412yes—
304C2×C42C2×D412yes—
305C2×C42C2×D412yes—
306C2×C42C2×D412yes—
307C2×C42C2×D412yes—
308C2×C42C2411yescharacteristic
309C2×C42C2411yescharacteristic
310C2×C42C2411yescharacteristic
311C2×C42C2411yescharacteristic
312C2×C42C2411yescharacteristic
313C2×C42C2411yescharacteristic
314#107—44no—
315#107—44no—
316C4⋊C8—44no—
317#103—28no—
318#103—44no—
319#103—44no—
320#103—44no—
321#103—44no—
322#102—28no—
323#102—44no—
324#102—44no—
325C22⋊C8—24no—
326C22⋊C8—24no—
327#97—24no—
328C4×C8—24no—
329#95—22no—
330#95—22no—
331#95—22no—
332#95—22no—
333#95—22no—
334#95—24no—
335#95—24no—
336#95—22no—
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#95C2×D412yes—
347#95C2×D412yes—
348#95C2×D412yes—
349#95C2×D412yes—
350C24—22no—
351C24C22×D411yescharacteristic
352#41—416no—
353#41—416no—
354C2×Q8—22no—
355C2×Q8—24no—
356C2×Q8—24no—
357C2×Q8—24no—
358C2×Q8—24no—
359C2×Q8—22no—
360C2×Q8—22no—
361C2×Q8C2×#4114yes—
362C2×D4—28no—
363C2×D4—28no—
364C2×D4—22no—
365C2×D4—24no—
366C2×D4—24no—
367C2×D4—44no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4—44no—
371C2×D4C2×#4114yes—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—28no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—42no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—44no—
389C22×C4—44no—
390C22×C4—84no—
391C22×C4—84no—
392C22×C4—84no—
393C22×C4C22×D412yes—
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D412yes—
397C22×C4C22×D411yescharacteristic
398C22×C4C22×D411yescharacteristic
399C22×C4C22×D412yes—
400C22×C4C22×D411yescharacteristic
401C22×C4C22×D412yes—
402C22×C4H(5,𝔽2)14yes—
403C22×C4C2511yescharacteristic
404Q16—88no—
405SD16—88no—
406C2×C8—24no—
407C2×C8—24no—
408C2×C8—24no—
409C2×C8—24no—
410C2×C8—24no—
411C2×C8—24no—
412C4⋊C4—28no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—24no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—42no—
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—44no—
431C4⋊C4—44no—
432C4⋊C4—82no—
433C4⋊C4—82no—
434C4⋊C4—82no—
435C4⋊C4—82no—
436C4⋊C4—84no—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C22⋊C4—216no—
440C22⋊C4—42no—
441C22⋊C4—44no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—84no—
445C22⋊C4—84no—
446C22⋊C4—82no—
447C42—24no—
448C42—24no—
449C42—24no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—42no—
456C42—42no—
457C42—44no—
458C42—44no—
459C42—44no—
460C42—82no—
461C42C2×#4114yes—
462C23—22no—
463C23—21no—
464C23—21no—
465C23—22no—
466C23—24no—
467C23—44no—
468C23—42no—
469C23—42no—
470C23—44no—
471C23#53512yes—
472C23C23×D411yescharacteristic
473Q8—28no—
474Q8—44no—
475Q8—44no—
476Q8—44no—
477D4—28no—
478D4—416no—
479D4—416no—
480D4—44no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—22no—
487C2×C4—28no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—24no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—48no—
504C2×C4—416no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—42no—
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—42no—
522C2×C4—42no—
523C2×C4—164no—
524C2×C4—164no—
525C2×C4—164no—
526C2×C4#52514yes—
527C2×C4#53512yes—
528C2×C4#57814yes—
529C2×C4C22×#4112yes—
530C2×C4C2×H(5,𝔽2)12yes—
531C2×C4#58512yes—
532C8—44no—
533C8—44no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—42no—
538C22—42no—
539C22—44no—
540C22—48no—
541C22—84no—
542C22—82no—
543C22—82no—
544C22C2×#52511yescharacteristic
545C22C2×#53511yescharacteristic
546C22#344911yescharacteristic
547C22C2×#57812yes—
548C22#358512yes—
549C4—22no—
550C4—24no—
551C4—24no—
552C4—42no—
553C4—42no—
554C4—41no—
555C4—41no—
556C4—41no—
557C4—41no—
558C4—44no—
559C4—82no—
560C4—82no—
561C4—82no—
562C4—82no—
563C4—82no—
564C4—82no—
565C4#351314yes—
566C2—42no—
567C2—84no—
568C2#3523311yescharacteristic
569C2#6058912yes—
570C2#6082111yescharacteristic
571C2#6083011yescharacteristic
572C2#6099312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
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
1. #35323C2semidirect
3. #33017C2semidirect
4. #30131C2semidirect
5. #30128C2semidirect
6. #29370C2semidirect
7. #29367C2semidirect
8. #29200C2semidirect
9. #28629C2semidirect
10. #28131C2semidirect
11. #27966C2semidirect
12. #27910C2semidirect
15. #22278C2semidirect
16. #22228C2semidirect
17. #20784C2semidirect

Maximal subgroups

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

subgroupindex
#353232
#342072
#330172
#301312
#301282
#293702
#293672
#292002
#286292
#281312
#279662
#279102
#258582
#257282
#222782
#222282
#207842