Finite Group Databasebeta

← #7487732

Subgroups — #7487732

Order 512.

1993 conjugacy classes of subgroups.

The subgroup lattice has 585 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59817C211yesmaximal, characteristic
2#39618C211yesmaximal, characteristic
3#31388C214yesmaximal, complemented
4#31263C212yesmaximal, complemented
5#30507C212yesmaximal, complemented
6#30407C214yesmaximal, complemented
7#30042C212yesmaximal, complemented
8#29370C212yesmaximal, complemented
9#24437C214yesmaximal, complemented
10#23970C212yesmaximal, complemented
11#23856C212yesmaximal, complemented
12#23856C214yesmaximal, complemented
13#23355C211yesmaximal, characteristic
14C23×Q16C2212yes—
15C2×#554C2211yescharacteristic
16C2×D4×Q8C2211yescharacteristic
17C2×D4×Q8C2214yes—
18C2×#467C2214yes—
19C2×#451C2214yes—
20C2×#451C2212yes—
21C2×#451C2214yes—
22C2×#451C2212yes—
23C22×#103C2212yes—
24#2862C2211yescharacteristic
25#2692—24no—
26#2683—21no—
27#2683—21no—
28#2660—21no—
29#2566—21no—
30#2563—21no—
31#2563C2211yescharacteristic
32#2551—21no—
33#2542C2211yescharacteristic
34#2538C2214yes—
35#2452—24no—
36#2417C2214yes—
37#2409C2212yes—
38#2409C2212yes—
39#2409C2214yes—
40#2366—21no—
41#2328—21no—
42C4×C4⋊Q8C2212yes—
43C4×#127C2211yescharacteristic
44#2295C2211yescharacteristic
45#2100C2212yes—
46#2095C2212yes—
47#2076C2214yes—
48#2072C2214yes—
49#2043C2212yes—
50#2036C2214yes—
51#2036C2214yes—
52#2024C2214yes—
53#2016C2214yes—
54#2016C2212yes—
55#2016C2214yes—
56#2001C2214yes—
57#2001C2214yes—
58#2001C2214yes—
59#1928C2214yes—
60#1892C2214yes—
61#1892C2212yes—
62#1892C2212yes—
63#1878C2214yes—
64#1878C2214yes—
65#1878C2214yes—
66#1878C2214yes—
67#1872C2214yes—
68#1869C2212yes—
69#1869C2212yes—
70#1869C2214yes—
71#1869C2214yes—
72#1863C2214yes—
73#1839C2212yes—
74#1824C2212yes—
75#1787C2214yes—
76#1787C2212yes—
77C23×Q8C2312yes—
78C23×Q8C2312yes—
79C23×Q8C2312yes—
80C22×Q16—48no—
81C22×Q16—44no—
82C22×Q16C2314yes—
83C22×Q16C2314yes—
84C22×Q16C2314yes—
85C23×C8C2312yes—
86C23×C8C2312yes—
87#554—42no—
88#554—42no—
89D4×Q8—42no—
90D4×Q8—42no—
91D4×Q8—48no—
92D4×Q8—48no—
93C2×C4⋊Q8—21no—
94C2×C4⋊Q8—24no—
95C2×C4⋊Q8C2312yes—
96C2×C4⋊Q8C2311yescharacteristic
97C2×C4⋊Q8C2311yescharacteristic
98C2×#127D411yescharacteristic
99C2×#125—21no—
100C2×C22⋊Q8—24no—
101C2×C22⋊Q8—24no—
102C2×C22⋊Q8—22no—
103C2×C22⋊Q8—22no—
104C2×C22⋊Q8C2314yes—
105C2×C22⋊Q8C2312yes—
106C2×C22⋊Q8C2314yes—
107C2×C22⋊Q8C2312yes—
108C2×C22⋊Q8C2314yes—
109C2×C4×Q8C2311yescharacteristic
110C2×C4×Q8C2314yes—
111C2×C4×D4—21no—
112C2×C4×D4—21no—
113C2×C4×D4—24no—
114C2×C4×D4C2311yescharacteristic
115C2×C4×D4C2311yescharacteristic
116C2×C4×D4C2314yes—
117C22×C4⋊C4C2312yes—
118#467—48no—
119#451—44no—
120#451—48no—
121#451—48no—
122#451—44no—
123C2×#107C2314yes—
124C2×#103—44no—
125C2×#103C2314yes—
126C2×#103C2314yes—
127C2×#103C2314yes—
128C2×#103C2314yes—
129C2×#103C2314yes—
130C2×#103C2314yes—
131C2×#103C2314yes—
132C2×#103C2314yes—
133C2×C22⋊C8C2314yes—
134C2×C22⋊C8C2312yes—
135C2×C22⋊C8C2314yes—
136C2×C22⋊C8C2312yes—
137#399—22no—
138#398—24no—
139#396—24no—
140#396—24no—
141#396—24no—
142#395—24no—
143#395—22no—
144#395—22no—
145#393—24no—
146#393—22no—
147#393—22no—
148#393—22no—
149#391—22no—
150#391C2314yes—
151#391C2314yes—
152#391C2312yes—
153#390—21no—
154#390—21no—
155#390—24no—
156#390D411yescharacteristic
157#390D411yescharacteristic
158#390D414yes—
159#389—24no—
160#389—22no—
161#389—22no—
162#389C2312yes—
163#387—22no—
164#386—24no—
165#386—24no—
166#386—22no—
167#386—22no—
168#385—24no—
169#385—22no—
170#385—22no—
171#384—21no—
172#384D411yescharacteristic
173#384C2312yes—
174#382—21no—
175#382—21no—
176#382—21no—
177#380—24no—
178#380C2314yes—
179#380C2312yes—
180#380C2312yes—
181C4×C4⋊C4—22no—
182C4×C4⋊C4—22no—
183C4×C4⋊C4—21no—
184C4×C4⋊C4C2311yescharacteristic
185#376—21no—
186C43C2311yescharacteristic
187#340C2314yes—
188#340C2314yes—
189#340C2314yes—
190#340C2314yes—
191#336C2314yes—
192#336C2312yes—
193#336C2312yes—
194#336C2314yes—
195C22×Q8—22no—
196C22×Q8—22no—
197C22×Q8—21no—
198C22×Q8—22no—
199C22×Q8—28no—
200C22×Q8—24no—
201C22×Q8—28no—
202C22×Q8—42no—
203C22×Q8—44no—
204C22×Q8—44no—
205C22×Q8—42no—
206C22×Q8—44no—
207C22×Q8—42no—
208C22×Q8—42no—
209C22×Q8—42no—
210C22×Q8—44no—
211C22×Q8C2412yes—
212C22×Q8C2414yes—
213C22×Q8C2411yescharacteristic
214C22×D4C2×D411yescharacteristic
215C22×D4C2×D411yescharacteristic
216C22×D4C2×D414yes—
217C23×C4—22no—
218C23×C4—22no—
219C23×C4—22no—
220C23×C4—22no—
221C23×C4C2412yes—
222C23×C4C2412yes—
223C2×Q16—28no—
224C2×Q16—28no—
225C2×Q16—28no—
226C2×Q16—28no—
227C2×Q16—44no—
228C2×Q16—44no—
229C2×Q16—44no—
230C2×Q16—44no—
231C2×Q16—44no—
232C2×Q16—432no—
233C22×C8—42no—
234C22×C8—44no—
235C22×C8—42no—
236C22×C8—44no—
237C22×C8C2414yes—
238C22×C8C2414yes—
239C4⋊Q8—28no—
240C4⋊Q8—41no—
241C4⋊Q8—41no—
242C4⋊Q8—41no—
243C4⋊Q8—41no—
244C4⋊Q8—44no—
245C4⋊Q8—416no—
246#127Q1618yescomplemented
247#125—44no—
248C22⋊Q8—28no—
249C22⋊Q8—28no—
250C22⋊Q8—24no—
251C22⋊Q8—24no—
252C22⋊Q8—42no—
253C22⋊Q8—44no—
254C22⋊Q8—44no—
255C22⋊Q8—42no—
256C22⋊Q8—48no—
257C22⋊Q8—416no—
258C22⋊Q8—416no—
259C22⋊Q8—48no—
260C22⋊Q8—44no—
261C22⋊Q8—44no—
262C4×Q8—22no—
263C4×Q8—22no—
264C4×Q8—44no—
265C4×Q8—44no—
266C4×D4—41no—
267C4×D4—41no—
268C4×D4—44no—
269C4×D4—44no—
270C4×D4—41no—
271C4×D4—41no—
272C4×D4—44no—
273C4×D4—44no—
274C4×D4—416no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—22no—
278C2×C4⋊C4—24no—
279C2×C4⋊C4—22no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—24no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—24no—
287C2×C4⋊C4—24no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—24no—
290C2×C4⋊C4—22no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—42no—
293C2×C4⋊C4C2×D411yescharacteristic
294C2×C4⋊C4C2×D411yescharacteristic
295C2×C4⋊C4C2×D414yes—
296C2×C4⋊C4C2412yes—
297C2×C4⋊C4C2414yes—
298C2×C4⋊C4C2412yes—
299C2×C4⋊C4C2411yescharacteristic
300C2×C22⋊C4—22no—
301C2×C22⋊C4—22no—
302C2×C22⋊C4—24no—
303C2×C22⋊C4—24no—
304C2×C22⋊C4—24no—
305C2×C22⋊C4—24no—
306C2×C22⋊C4—22no—
307C2×C22⋊C4—22no—
308C2×C22⋊C4C2×D414yes—
309C2×C22⋊C4C2×D412yes—
310C2×C22⋊C4C2×D412yes—
311C2×C22⋊C4C2×D414yes—
312C2×C42—21no—
313C2×C42—24no—
314C2×C42—21no—
315C2×C42C2×D411yescharacteristic
316C2×C42C2×D412yes—
317C2×C42C2×D411yescharacteristic
318C2×C42C2412yes—
319C2×C42C2411yescharacteristic
320#107—44no—
321#103—28no—
322#103—28no—
323#103—44no—
324#103—44no—
325#103—44no—
326#103—44no—
327#103—44no—
328#103—44no—
329C22⋊C8—42no—
330C22⋊C8—44no—
331C22⋊C8—44no—
332C22⋊C8—42no—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—24no—
337#95—24no—
338#95—21no—
339#95—24no—
340#95—24no—
341#95—22no—
342#95—22no—
343#95—24no—
344C24C22×D412yes—
345C24C22×D412yes—
346C2×Q8—21no—
347C2×Q8—24no—
348C2×Q8—24no—
349C2×Q8—24no—
350C2×Q8—24no—
351C2×Q8—24no—
352C2×Q8—24no—
353C2×Q8—24no—
354C2×Q8—22no—
355C2×Q8—22no—
356C2×Q8—24no—
357C2×Q8—44no—
358C2×Q8—48no—
359C2×Q8—48no—
360C2×Q8—42no—
361C2×Q8—48no—
362C2×Q8—48no—
363C2×Q8—48no—
364C2×Q8—82no—
365C2×Q8—82no—
366C2×Q8—84no—
367C2×Q8—82no—
368C2×Q8—88no—
369C2×Q8—84no—
370C2×Q8—88no—
371C2×Q8—84no—
372C2×Q8—84no—
373C2×Q8—88no—
374C2×Q8H(5,𝔽2)18yes—
375C2×Q8H(5,𝔽2)12yes—
376C2×D4—41no—
377C2×D4—41no—
378C2×D4—44no—
379C2×D4—41no—
380C2×D4—41no—
381C2×D4—44no—
382C2×D4C2×Q1614yes—
383C2×D4C2×Q1614yescomplemented
384C2×D4C2×Q16116yescomplemented
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—22no—
391C22×C4—24no—
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—22no—
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—44no—
401C22×C4—48no—
402C22×C4—48no—
403C22×C4—44no—
404C22×C4—82no—
405C22×C4—82no—
406C22×C4—84no—
407C22×C4—84no—
408C22×C4C22×D411yescharacteristic
409C22×C4C22×D412yes—
410C22×C4C22×D412yes—
411C22×C4C22×D412yes—
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D412yes—
414C22×C4C2511yescharacteristic
415Q16—416no—
416Q16—416no—
417Q16—88no—
418C2×C8—24no—
419C2×C8—24no—
420C2×C8—24no—
421C2×C8—24no—
422C2×C8—24no—
423C2×C8—24no—
424C2×C8—48no—
425C2×C8—84no—
426C4⋊C4—28no—
427C4⋊C4—24no—
428C4⋊C4—28no—
429C4⋊C4—21no—
430C4⋊C4—24no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—24no—
434C4⋊C4—24no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—44no—
438C4⋊C4—44no—
439C4⋊C4—44no—
440C4⋊C4—44no—
441C4⋊C4—48no—
442C4⋊C4—48no—
443C4⋊C4—44no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—48no—
447C4⋊C4—42no—
448C4⋊C4—82no—
449C4⋊C4—82no—
450C4⋊C4—84no—
451C4⋊C4H(5,𝔽2)12yes—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—42no—
455C22⋊C4—42no—
456C22⋊C4—44no—
457C22⋊C4—44no—
458C22⋊C4—48no—
459C22⋊C4—48no—
460C22⋊C4—82no—
461C22⋊C4—82no—
462C22⋊C4—84no—
463C22⋊C4—84no—
464C42—24no—
465C42—21no—
466C42—21no—
467C42—24no—
468C42—41no—
469C42—42no—
470C42—42no—
471C42—48no—
472C42C2×Q1614yescomplemented
473C23—22no—
474C23—22no—
475C23—24no—
476C23—24no—
477C23—24no—
478C23—24no—
479C23—22no—
480C23—22no—
481C23C22×Q1614yes—
482C23C22×Q1614yes—
483C23C23×D411yescharacteristic
484Q8—24no—
485Q8—28no—
486Q8—28no—
487Q8—44no—
488Q8—42no—
489Q8—44no—
490Q8—42no—
491Q8—42no—
492Q8—44no—
493Q8—88no—
494Q8—88no—
495Q8—88no—
496Q8—162no—
497D4—44no—
498D4—44no—
499D4—416no—
500C2×C4—21no—
501C2×C4—22no—
502C2×C4—21no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—28no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—42no—
517C2×C4—44no—
518C2×C4—42no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—44no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—44no—
528C2×C4—42no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—88no—
537C2×C4—88no—
538C2×C4—88no—
539C2×C4—164no—
540C2×C4C22×Q1612yes—
541C2×C4C22×Q1614yes—
542C2×C4C2×H(5,𝔽2)11yescharacteristic
543C2×C4C2×H(5,𝔽2)14yes—
544C2×C4C2×H(5,𝔽2)11yescharacteristic
545C8—44no—
546C8—44no—
547C22—24no—
548C22—28no—
549C22—24no—
550C22—28no—
551C22—42no—
552C22—42no—
553C22—44no—
554C22—44no—
555C22—42no—
556C22—42no—
557C22#348412yes—
558C22#348411yescharacteristic
559C22C23×Q1611yescharacteristic
560C22#358611yescharacteristic
561C22#358612yes—
562C4—22no—
563C4—24no—
564C4—42no—
565C4—42no—
566C4—42no—
567C4—48no—
568C4—42no—
569C4—41no—
570C4—42no—
571C4—82no—
572C4—82no—
573C4—84no—
574C4—82no—
575C4#353212yes—
576C4#353712yes—
577C2—44no—
578C2—44no—
579C2#5200411yescharacteristic
580C2#6156311yescharacteristic
581C2#6156312yes—
582C2#6167212yes—
583C2#6167411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#127 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #31388C2semidirect
4. #31263C2semidirect
5. #30507C2semidirect
6. #30407C2semidirect
7. #30042C2semidirect
8. #29370C2semidirect
9. #24437C2semidirect
10. #23970C2semidirect
11. #23856C2semidirect
12. #23856C2semidirect
246. #127Q16semidirect
383. C2×D4C2×Q16semidirect
384. C2×D4C2×Q16semidirect
472. C42C2×Q16semidirect

Maximal subgroups

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

subgroupindex
#598172
#396182
#313882
#312632
#305072
#304072
#300422
#293702
#244372
#239702
#238562
#233552